Content-Type: multipart/mixed; boundary="-------------9811040152360" This is a multi-part message in MIME format. ---------------9811040152360 Content-Type: text/plain; name="98-697.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-697.keywords" random Schr{\"o}dinger operator, density of states, Poisson potential ---------------9811040152360 Content-Type: application/postscript; name="dpp.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dpp.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: dpp.dvi %%Pages: 46 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips dpp.dvi -o dpp.ps %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.10.30:1400 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{ rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 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Fs(\))35 b(of)f(a)g(compact)g(supp)s(ort.)50 b(In)35 b(this)f(case)h Fo(E)i Fs(=)31 b(0)k(is)f(a)g(\015uctuational)f (edge)118 2220 y(and)g(according)f(to)g(I.Lifshitz)1134 2392 y Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b Fl(')g Fs(exp)q(\()p Fl(\000)p Fo(const)23 b Fl(\001)f Fo(E)2152 2351 y Fj(\000)p Fn(d=)p Fi(2)2318 2392 y Fs(\))p Fo(;)114 b(E)34 b Fl(!)27 b Fs(+0)p Fo(:)-2764 b Fs(\(0.2\))118 2575 y(The)35 b(righ)m(t)f(hand)g (side)g(of)g(this)g(form)m(ula)e(is)i(just)g(the)h(probabilit)m(y)d(to) i(ha)m(v)m(e)h(a)f(w)m(ell)f(\(a)h(region)g(of)f Fk(R)3851 2539 y Fn(d)118 2694 y Fs(free)38 b(of)f Fo(x)481 2658 y Fj(0)481 2719 y Fn(j)518 2694 y Fs('s\))h(of)f(width)h Fo(L)e Fl(')h Fo(E)1351 2658 y Fj(\000)p Fi(1)p Fn(=)p Fi(2)1516 2694 y Fs(;)j(the)e(latter)f(relation)f(is)h(due)h(to)f(the)h (uncertain)m(t)m(y)h(principle.)118 2810 y(In)32 b(other)g(w)m(ords,)h (the)f(asymptotics)g(of)f(the)h(IDS)f(in)g(this)h(case)g(is)f (determined)h(b)m(y)h(an)e(optimization)118 2926 y(pro)s(cedure,)39 b(balancing)c(the)i(comp)s(etition)d(of)i(the)h(quan)m(tum)g(and)g(the) g(probabilistic)d(comp)s(onen)m(ts)118 3043 y(of)48 b(the)g(problem.)88 b(This)48 b(is)g(wh)m(y)h(form)m(ula)d(\(0.2\))h(is)h(often)g(called)e (the)j(quan)m(tum)f(Lifshitz)f(tail.)118 3159 y(Rigorous)f(deriv)-5 b(ations)46 b(of)h(v)-5 b(arious)46 b(v)m(ersions)i(of)54 b(\(0.2\))46 b(\(e.g.)87 b(its)47 b(logarithmic)c(or)k(ev)m(en)i(its)d (the)118 3275 y(double)41 b(logarithmic)c(forms\))j(ha)m(v)m(e)j (required)e(a)g(n)m(um)m(b)s(er)g(of)g(rather)g(sophisticated)f (probabilistic)118 3391 y(and)c(sp)s(ectral)g(tec)m(hniques)j(\(see)e (e.g.)54 b([30,)36 b(31,)g(3,)g(21,)g(27,)g(12)o(])h(for)e(results)i (and)f(references\).)57 b(Here)118 3508 y(and)24 b(b)s(elo)m(w)g(w)m(e) h(use)g(the)f(sym)m(b)s(ol)f(")p Fl(')p Fs(")h(to)g(denote)h(the)f (asymptotic)f(equiv)-5 b(alence)24 b(without)g(indicating)118 3624 y(explicitly)31 b(the)i(order)g(of)f(the)h(remainder)e(and)i(resp) s(ectiv)m(e)h(constan)m(ts.)331 3740 y(Other)h(asymptotic)g(regimes)f (of)h(the)h(IDS)f(for)g(the)g(p)s(oten)m(tial)f(\(0.1\))h(corresp)s (ond)h(to)f(the)h(case)118 3856 y(when)h(the)g(one-site)e(p)s(oten)m (tial)g(has)h(a)g(non)g(p)s(ositiv)m(e)g(part,)g(i.e.)54 b(inf)22 b Fo(V)g Fs(\()p Fo(x)p Fs(\))34 b Fo(<)f Fs(0,)k(so)f(that)g (the)g(lo)m(w)m(er)118 3973 y(edge)d(of)e(the)h(sp)s(ectrum)h(is)e Fo(E)j Fs(=)28 b Fl(\0001)p Fs(.)43 b(In)32 b(this)g(case)h(one)f(has)g (to)g(distinguish)e(the)j(t)m(w)m(o)f(asymptotic)118 4089 y(regimes,)45 b(called)d(usually)g(the)h(quan)m(tum)h(and)f(the)g (classical)f(one.)74 b(W)-8 b(e)44 b(will)c(presen)m(t)45 b(resp)s(ectiv)m(e)118 4205 y(asymptotic)32 b(form)m(ulae)f(b)m(y)i (using)g(a)f(v)m(ersion)h(of)f(Lifshitz's)g(argumen)m(ts)h(adapted)g (to)f(this)g(case.)331 4321 y(Recall)c(the)j(de\014nition)e(of)g(the)i (IDS)e(\(implicit)d(also)j(in)h([17)o(,)g(16]\):)42 b(it)29 b(is)h(the)g(limit)d(as)j Fo(L)e Fl(!)f Fs(+)p Fl(1)118 4437 y Fs(of)34 b(the)h(exp)s(ectation)f(of)g(the)h(normalized)d(coun)m (ting)i(function)g(of)g(eigen)m(v)-5 b(alues)34 b(of)g(the)g(Sc)m (hr\177)-49 b(odinger)118 4554 y(op)s(erator)30 b Fo(H)590 4569 y Fi(\003)639 4580 y Fh(L)689 4554 y Fs(,)h(where)h Fo(H)1108 4569 y Fi(\003)1157 4580 y Fh(L)1237 4554 y Fs(is)e(the)h(restriction)f(of)g Fo(H)2154 4569 y Fn(!)2232 4554 y Fs(=)d Fl(\000)p Fs(\001)19 b(+)e Fo(V)2662 4569 y Fn(!)2743 4554 y Fs(to)30 b Fo(L)2926 4518 y Fi(2)2966 4554 y Fs(\(\003)3072 4569 y Fn(L)3124 4554 y Fs(\))g(\(see)i(e.g.)43 b([21)o(,)31 b(7]\).)118 4670 y(Here)36 b(\003)419 4685 y Fn(L)506 4670 y Fs(is)f(the)h(cub)s(e)g(of)f(the)h Fk(R)1362 4634 y Fn(d)1444 4670 y Fs(of)f(cen)m(ter)h(zero)g(and)g(of)f (side)g(length)g Fo(L)p Fs(.)52 b(The)37 b(de\014nition)d(sho)m(ws)118 4787 y(that)d(the)h(IDS)f(can)h(b)s(e)f(regarded)h(as)g(the)f (probabilit)m(y)f(to)h(\014nd)g(an)h(eigen)m(v)-5 b(alue)31 b(of)f Fo(H)3283 4802 y Fi(\003)3332 4813 y Fh(L)3414 4787 y Fs(lying)g(b)s(elo)m(w)118 4903 y(a)36 b(giv)m(en)g(energy)i Fo(E)6 b Fs(.)54 b(F)-8 b(or)36 b Fo(E)j Fl(!)34 b(\0001)i Fs(these)i(eigen)m(v)-5 b(alues)36 b(are)g(pro)s(duced)h(b)m(y)g(v)m (ery)h(deep)f(p)s(oten)m(tial)118 5019 y(w)m(ells,)j(created)g(b)m(y)g (large)e(clusters)h(of)g Fo(k)i Fs(P)m(oisson)f(p)s(oin)m(ts)e Fo(x)2406 5034 y Fn(j)2443 5019 y Fs('s,)j(con\014ned)f(to)f(a)f (su\016cien)m(tly)i(small)118 5135 y(regions)32 b(of)g(the)h(space,)h (sa)m(y)g(a)e(cub)s(e)h(of)f(the)h(side)g(length)f Fo(l)e(<<)d(L)p Fs(.)44 b(The)34 b(resp)s(ectiv)m(e)g(probabilit)m(y)c(is)1716 5316 y Fo(e)1761 5274 y Fj(\000)p Fn(\026l)1880 5251 y Fh(d)1921 5316 y Fs(\()p Fo(\026l)2049 5274 y Fn(d)2089 5316 y Fs(\))2127 5274 y Fn(k)2170 5316 y Fo(=k)s Fs(!)-2182 b(\(0.3\))1989 5454 y Fp(2)p eop %%Page: 3 3 3 2 bop 118 241 a Fs(Assume)31 b(\014rst)f(that)g Fo(l)i Fs(is)d(indep)s(enden)m(t)i(of)e Fo(k)s Fs(.)43 b(This)30 b(corresp)s(onds)h(to)e(a)h(single)f(site)g(p)s(oten)m(tial)f(that)i (is)118 357 y(v)-5 b(arying)36 b(relativ)m(ely)f(slo)m(wly)h(with)g (resp)s(ect)h(to)f(the)h(exp)s(onen)m(tial)f(deca)m(y)i(of)d(the)i (single)e(site)i(ground)118 473 y(state.)42 b(Then,)30 b(the)e(IDS)f(has)g(to)g(b)s(e)h(asymptotically)d(equiv)-5 b(alen)m(t)27 b(to)g(the)g(probabilit)m(y)g(\(0.3\))g(in)f(whic)m(h)118 589 y Fo(k)37 b Fs(is)c(determined)h(b)m(y)g(the)h(equation)e Fo(E)6 b Fs(\()p Fo(k)s Fs(\))30 b(=)f Fo(E)40 b Fs(where)35 b Fo(E)6 b Fs(\()p Fo(g)t Fs(\))33 b(is)g(the)h(ground)g(state)g (energy)h(of)e(the)118 706 y(w)m(ell)25 b Fo(g)t(V)c Fs(\()p Fo(x)p Fs(\).)41 b(In)27 b(other)e(w)m(ords,)k(for)c(a)h(non)f (p)s(ositiv)m(e)h(one-site)f(p)s(oten)m(tial)f Fo(V)e Fs(\()p Fo(x)p Fs(\))k(optimal)d(\015uctuations)118 822 y(are)33 b(p)s(oten)m(tial)e(w)m(ells)h(whose)i(form)d(is)h(roughly)g Fo(k)s(V)22 b Fs(\()p Fo(x)p Fs(\).)44 b(Th)m(us,)34 b(w)m(e)g(ha)m(v)m(e)g(to)e(\014nd)h(the)h(ground)e(state)118 938 y(energy)42 b Fo(E)6 b Fs(\()p Fo(g)t Fs(\))40 b(of)g(the)h(w)m (ell)f Fo(g)t(V)21 b Fs(\()p Fo(x)p Fs(\))41 b(as)g(the)g(function)f (of)h(its)f(coupling)f(constan)m(t)j Fo(g)i Fs(and)d(then)g(the)118 1054 y(asymptotic)35 b(form)g(of)g(the)h(IDS)g(will)e(b)s(e)i(giv)m(en) g(b)m(y)i(\(0.3\))d(in)g(whic)m(h)h Fo(k)h Fs(=)c Fo(g)t Fs(\()p Fo(E)6 b Fs(\))35 b(where)i Fo(g)t Fs(\()p Fo(E)6 b Fs(\))35 b(is)g(the)118 1171 y(asymptotic)f(in)m(v)m(erse)j(of)d Fl(\000)p Fo(E)6 b Fs(\()p Fo(g)t Fs(\))35 b(for)f Fo(E)k Fl(!)31 b(\0001)p Fs(,)36 b Fo(g)f Fl(!)c Fs(+)p Fl(1)p Fs(.)50 b(The)36 b(leading)d(con)m(tribution)h(is)g(giv)m(en)118 1287 y(b)m(y)g(the)f(factor)f(1)p Fo(=k)s Fs(!.)43 b(Th)m(us)34 b(the)f(asymptotic)f(form)m(ula)f(should)h(ha)m(v)m(e)i(the)f(form)1110 1450 y Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b Fl(')g Fs(exp)q(\()p Fl(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\))p Fo(;)113 b(E)34 b Fl(!)27 b(\0001)-2788 b Fs(\(0.4\))118 1613 y(or)1151 1776 y(log)16 b Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b Fl(')h(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\))p Fo(;)114 b(E)34 b Fl(!)27 b(\0001)-2747 b Fs(\(0.5\))118 1945 y(The)32 b(simplest)e(case)j(when)f(this)f(form)m(ula)e(can)j(b)s (e)f(made)g(rigorous)f(is)h(when)i Fo(V)21 b Fs(\()p Fo(x)p Fs(\))32 b(is)e(con)m(tin)m(uous)i(in)118 2062 y(its)g(\(negativ)m(e\))h(absolute)f(minim)m(um.)40 b(In)33 b(this)f(case)1230 2225 y Fo(E)6 b Fs(\()p Fo(g)t Fs(\))26 b(=)i Fo(g)t(V)21 b Fs(\(0\)\(1)g(+)h Fo(o)p Fs(\(1\)\))p Fo(;)114 b(E)34 b Fl(!)27 b(\0001)-2668 b Fs(\(0.6\))118 2388 y(The)34 b(resp)s(ectiv)m(e)g(rigorous)d(form)m(ula)1008 2598 y(log)17 b Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b(=)1597 2531 y Fo(E)p 1534 2575 204 4 v 1534 2667 a(V)22 b Fs(\(0\))1764 2598 y(log)17 b Fl(j)p Fo(E)6 b Fl(j)p Fs(\(1)21 b(+)h Fo(o)p Fs(\(1\)\))p Fo(;)114 b(E)33 b Fl(!)28 b(\0001)-2890 b Fs(\(0.7\))118 2822 y(w)m(as)27 b(pro)m(v)m(ed)h(in)d([20])h(using)g (a)g(com)m(bination)f(of)g(v)-5 b(ariational)23 b(and)j(Wiener)h(in)m (tegral)d(tec)m(hniques.)44 b(The)118 2938 y(form)m(ula)31 b(is)i(ob)m(viously)g(classical)e(b)s(ecause)k(it)d(includes)h(only)f (the)i(c)m(haracteristics)f(of)g(the)g(p)s(oten)m(tial)118 3054 y(but)k(not)f(the)g(Laplacian.)53 b(In)36 b(addition)f(it)g(can)h (b)s(e)h(easily)e(sho)m(wn)i(that)f(the)h(r.h.s.)55 b(of)43 b(\(0.4\))35 b(is)h(the)118 3170 y(asymptotic)e(form)g(of)g(the)i (logarithm)31 b(of)k Fo(P)14 b Fs(\()p Fo(V)1871 3185 y Fn(!)1920 3170 y Fs(\()p Fo(x)p Fs(\))32 b Fo(<)g(E)6 b Fs(\).)50 b(This)35 b(pro)m(vides)h(one)f(more)f(evidence)i(of)118 3286 y(the)d(classical)e(nature)i(of)f(the)h(form)m(ula.)331 3403 y(This)24 b(pap)s(er)g(is)g(dev)m(oted)i(to)e(quan)m(tum)g(v)m (ersions)h(of)31 b(\(0.4\))o(,)26 b(i.e.)40 b(to)24 b(singular)f (one-site)h(p)s(oten)m(tials)118 3519 y Fo(V)e Fs(\()p Fo(x)p Fs(\).)45 b(Despite)33 b(a)g(considerable)g(amoun)m(t)f(of)h(ph) m(ysical)g(results)g(in)g(the)g(question)h(\(see)g(e.g.)45 b([18,)33 b(4]\))118 3635 y(rigorous)48 b(results)h(are)f(almost)f (absen)m(t,)54 b(except)c(the)f(one)g(dimensional)d(case)j(of)f(p)s (oin)m(t)g(one-site)118 3751 y(p)s(oten)m(tials)1463 3915 y Fo(V)21 b Fs(\()p Fo(x)p Fs(\))28 b(=)g Fl(\000)p Fo(g)1928 3930 y Fi(0)1967 3915 y Fo(\016)t Fs(\()p Fo(x)p Fs(\))p Fo(;)115 b(g)2334 3930 y Fi(0)2401 3915 y Fo(>)27 b Fs(0)-2435 b(\(0.8\))118 4084 y(where)30 b Fo(\016)t Fs(\()p Fo(x)p Fs(\))g(is)e(the)i(Dirac)e(delta-function.)40 b(In)30 b(this)f(case)h(b)m(y)g(using)e(the)i(sp)s(ecial)e(Mark)m(o)m (v)j(pro)s(cesses)118 4200 y(tec)m(hnique)j(one)f(can)g(sho)m(w)g (\(see)h([18]\))e(an)h(asymptotic)f(of)g(the)h(form)892 4439 y(log)17 b Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b(=)f Fl(\000)p Fs(2)1534 4288 y Fm(r)p 1634 4288 132 4 v 1661 4371 a Fo(E)p 1644 4416 112 4 v 1644 4507 a(E)1716 4522 y Fi(0)1782 4439 y Fs(log)1925 4288 y Fm(r)p 2025 4288 132 4 v 2051 4371 a Fo(E)p 2035 4416 112 4 v 2035 4507 a(E)2107 4522 y Fi(0)2156 4439 y Fs(\(1)22 b(+)g Fo(o)p Fs(\(1\)\))p Fo(;)114 b(E)33 b Fl(!)28 b(\0001)118 4657 y Fs(where)34 b Fo(E)472 4672 y Fi(0)539 4657 y Fs(=)28 b Fl(\000)p Fo(g)771 4621 y Fi(2)767 4682 y(0)810 4657 y Fo(=)p Fs(4.)43 b(W)-8 b(e)33 b(see)h(that)e(form)m(ula)f(\(0.5\))h (is)g(only)g(v)-5 b(alid)31 b(up)i(to)f(a)g(factor.)43 b(2.)331 4773 y(This)33 b(di\013erence)g(is)g(the)g(result)g(of)f(a)h (tunneling)f(phenomenon)h(related)g(to)f(the)i(question)f(ho)m(w)118 4890 y(close)f(to)g(one)g(another)g(should)g Fo(k)i Fs(p)s(oten)m (tials)d(\(0.8\))h(b)s(e)g(in)f(order)h(to)f(b)s(e)h(regarded)h(as)f (the)g(p)s(oten)m(tial)118 5006 y Fo(k)172 4970 y Fj(\003)212 5006 y Fs(\()p Fo(k)s Fs(\))p Fo(\016)t Fs(\()p Fo(x)p Fs(\))26 b(i.e.)40 b(the)27 b(p)s(oten)m(tial)d(of)h(same)h(shap)s(e)h (as)f(\(0.8\))f(and)h(of)f(amplitude)f Fo(k)3022 4970 y Fj(\003)3062 5006 y Fs(\()p Fo(k)s Fs(\).)41 b(In)26 b(other)g(w)m(ords,)118 5122 y(in)i(this)f(case)i(con)m(trary)g(to)f (the)h(classical)d(case,)k(the)f(radius)f(of)g(the)g(exp)s(onen)m(tial) g(deca)m(y)h(of)f(the)h(single)118 5238 y(site)42 b(ground)g(state)h (is)e(m)m(uc)m(h)i(larger)e(than)h(the)g(width)g(of)g(the)g(single)f (site)h(p)s(oten)m(tial.)71 b(Th)m(us)43 b(the)118 5355 y(optimal)31 b(cluster)j(should)g(b)s(e)g(m)m(uc)m(h)g(smaller)e (\(i.e.)46 b(its)34 b(radius)f(should)h(tend)g(to)g(0)f(su\016cien)m (tly)i(fast\))1989 5454 y Fp(3)p eop %%Page: 4 4 4 3 bop 118 241 a Fs(in)36 b(order)h(to)f(b)s(e)h(mo)s(deled)f(b)m(y)h (a)g(single)e(site)i(p)s(oten)m(tial)e(of)h(some)g(e\013ectiv)m(e)i (amplitude)d Fo(k)3510 205 y Fj(\003)3549 241 y Fs(\()p Fo(k)s Fs(\).)56 b(W)-8 b(e)118 357 y(shall)29 b(see)j(b)s(elo)m(w)f (that)f(in)g(man)m(y)g(in)m(teresting)h(cases)h Fo(l)e Fl(')e Fo(k)2299 321 y Fj(\000)p Fn(\013)2399 297 y Fg(\003)2470 357 y Fs(for)i(some)g Fo(\013)2922 321 y Fj(\003)2989 357 y Fo(>)e Fs(0.)42 b(Because)33 b(of)d(that,)118 473 y(the)k(factor)g Fo(k)622 437 y Fj(\000)p Fn(\013)722 414 y Fg(\003)758 437 y Fn(dk)871 473 y Fs(in)f(\(0.3\))g(will)e(also)i (con)m(tribute)h(to)f(the)i(asymptotic)e(form)m(ula)e(of)j(the)g(IDS.)g (The)118 589 y(study)g(of)e(this)g(phenomenon)h(is)f(one)h(of)f(the)h (topics)g(dealt)f(with)g(in)g(the)h(presen)m(t)h(pap)s(er.)331 706 y(In)27 b(this)h(pap)s(er)f(w)m(e)i(study)f(the)g(case)g(of)f (singular)f(one-site)h(p)s(oten)m(tial)f(\(mainly)f(with)j(p)s(o)m(w)m (er-la)m(w)118 822 y(singularities\))g(follo)m(wing)h(in)h(essence)j (the)f(sc)m(heme)g(outlined)e(ab)s(o)m(v)m(e.)43 b(W)-8 b(e)32 b(\014nd)f(the)h(precise)f(form)f(of)118 938 y Fo(k)172 902 y Fj(\003)212 938 y Fs(\()p Fo(k)s Fs(\))35 b(\(see)i(Theorem)f(1.4)f(and)g(1.6)g(b)s(elo)m(w\).)53 b(This)35 b(b)s(ecame)h(p)s(ossible)f(o)m(wing)g(to)g(an)g(impro)m (ving)f(of)118 1054 y(one)d(of)g(the)h(tec)m(hnique)g(in)e(the)i (\014eld,)f(based)h(on)f(appro)m(ximations)e(of)i(the)h(Sc)m(hr\177)-49 b(odinger)31 b(op)s(erator)f(in)118 1171 y(the)f(whole)f(space)h(b)m(y) g(the)g(op)s(erator)e(with)h(the)h(same)f(p)s(oten)m(tial)e(but)j (de\014ned)g(in)f(a)g(\014nite)g(b)s(o)m(x)g(whose)118 1287 y(size)36 b(is)g(prop)s(erly)g(c)m(hosen)h(as)g(a)e(function)h(of) g(energy)-8 b(.)55 b(Previous)36 b(v)m(ersions)h(of)f(this)g(tec)m (hnique)h(w)m(ere)118 1403 y(based)31 b(on)f(the)h(so)g(called)e (Neumann-Diric)m(hlet)e(brac)m(k)m(eting)k(where)h(the)e(b)s(o)m(xes)i (with)e(the)h(Neumann)118 1519 y(and)c(Diric)m(hlet)d(b)s(oundary)j (conditions)e(w)m(ere)j(used)g(to)e(construct)h(the)g(upp)s(er)g(and)f (the)h(lo)m(w)m(er)g(b)s(ounds)118 1636 y(for)34 b(the)h(IDS.)g(The)h (error)e(in)g(these)i(b)s(ounds)g(is)e(of)g(the)h(order)g Fo(O)s Fs(\()p Fo(L)2640 1599 y Fj(\000)p Fi(1)2734 1636 y Fs(\))g(where)h Fo(L)f Fs(is)f(the)h(size)g(of)f(the)118 1752 y(b)s(o)m(x.)44 b(This)33 b(precision)f(is)g(not)g(su\016cien)m(t) i(to)e(treat)h(the)g(quan)m(tum)g(case.)331 1868 y(In)38 b(the)h(presen)m(t)h(pap)s(er)f(w)m(e)g(appro)m(ximate)f(the)h(IDS)f (of)g(the)g(random)g(mo)s(del)f(b)m(y)i(the)g(IDS)f(of)118 1984 y(some)g(w)m(ell)g(c)m(hosen)h(p)s(erio)s(dic)e(Sc)m(hr\177)-49 b(odinger)38 b(op)s(erators)g(and)h(obtain)e(m)m(uc)m(h)h(more)g (precise)h(b)s(ounds)118 2100 y(\(see)g(section)e(2)g(and)h(more)f (precisely)h(Lemmas)e(2.1)h(and)h(2.3\)\).)57 b(This)38 b(metho)s(d)f(w)m(as)h(prop)s(osed)g(in)118 2217 y([12])h(and)h(has)g (b)s(een)h(used)f(to)g(solv)m(e)g(sev)m(eral)g(problems)f(in)g(the)h (\014eld)f(\([11,)g(10]\).)65 b(W)-8 b(e)40 b(obtain)e(the)118 2333 y(once)f(logarithmic)32 b(v)m(ersions)37 b(of)43 b(\(0.4\))o(,)37 b(i.e.)f(\(0.5\))f(with)h(explicit)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\))35 b(and)i(constan)m(ts)g(in)e(fron)m(t)h (of)118 2449 y Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\))31 b(\(see)j(Theorems)f(1.5)f(and)h (1.6\).)331 2565 y(Let)e(us)i(no)m(w)f(giv)m(e)f(just)h(one)g(example)f (of)g(the)h(results)g(giv)m(en)g(in)f(section)g(1.2;)h(let)f Fo(V)53 b Fs(the)32 b(single)118 2746 y(site)c(p)s(oten)m(tial)e(b)s(e) i(the)h(3-dimensional)c(attractiv)m(e)i(screened)k(Coulom)m(b)c(p)s (oten)m(tial)f Fo(V)21 b Fs(\()p Fo(x)p Fs(\))28 b(=)g Fl(\000)3677 2679 y Fo(e)3722 2643 y Fj(\000j)p Fn(x)p Fj(j)p 3677 2724 184 4 v 3714 2815 a Fl(j)p Fo(x)p Fl(j)3871 2746 y Fs(.)118 2917 y(In)33 b(this)f(case)i(w)m(e)f(pro)m(v)m(e)918 3092 y(log)17 b Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b(=)f Fl(\000)p Fs(2)1560 3002 y Fm(p)p 1660 3002 134 4 v 90 x Fl(j)p Fo(E)6 b Fl(j)16 b Fs(log)h Fl(j)p Fo(E)6 b Fl(j)p Fs(\(1)21 b(+)h Fo(o)p Fs(\(1\)\))p Fo(;)114 b(E)34 b Fl(!)27 b(\0001)p Fo(;)118 3262 y Fs(\(see)34 b(Theorem)f(1.6)f(and)g (the)h(discussion)g(follo)m(wing)d(it\).)331 3379 y(The)48 b(role)e(of)g(the)i(IDS)e(in)h(the)g(sp)s(ectral)g(theory)g(and)g (theoretical)f(ph)m(ysics)j(of)d(disordered)118 3495 y(systems)30 b(is)d(w)m(ell)g(kno)m(wn)j(and)e(appreciated.)42 b(Ho)m(w)m(ev)m(er,)31 b(for)d(us)g(there)h(is)f(one)g(more)f(reason)i (to)e(study)118 3611 y(this)40 b(quan)m(tit)m(y)-8 b(.)68 b(Since)40 b(the)h(pioneering)e(pap)s(ers)i(of)f(I.Lifshitz,)i(the)f (study)g(of)f(the)h(IDS)f(has)h(b)s(een)118 3727 y(pro)m(viding)j(a)h (\014rst)h(imp)s(ortan)m(t)d(step)j(in)f(the)g(understanding)h(and)f (study)i(of)e(more)f(complicated)118 3844 y(prop)s(erties)g(and)h(quan) m(tities)f(in)g(a)g(resp)s(ectiv)m(e)h(v)m(ersion)g(of)f(the)h(strong)f (lo)s(calization)d(regime.)77 b(In)118 3960 y(particular,)38 b(the)h(IDS)g(is)f(the)h(\014rst)g(momen)m(t)e(\(see)j(form)m(ula)d (\(1.4\))o(\))i(of)f(the)h(sp)s(ectral)f(k)m(ernel)h(of)f(the)118 4076 y(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erator.)42 b(F)-8 b(rom)31 b(the)i(mathematical)c(ph)m(ysics)34 b(p)s(oin)m(t)e(of)g(view,)g(the)h(IDS)f(determines)118 4192 y(the)24 b(equilibrium)c(prop)s(erties)j(of)g(disordered)h (systems,)i(i.e.)40 b(the)24 b(ideal)d(gas)j(of)e(elemen)m(tary)i (excitation)118 4309 y(\(electrons,)j(phonons,)f(spin)f(w)m(a)m(v)m(es) h(etc.\))42 b(in)23 b(the)i(random)e(en)m(vironmen)m(t.)42 b(The)25 b(study)g(of)f(the)h(kinetic)118 4425 y(prop)s(erties)h(of)f (this)g(gas)h(and)f(the)h(in)m(teraction)f(e\013ects)i(requires)f(the)g (kno)m(wledge)h(of)e(higher)g(momen)m(ts)118 4541 y(of)k(the)g(sp)s (ectral)g(k)m(ernel,)h(the)g(second)g(momen)m(ts)e(\014rst)i(of)e(all.) 41 b(The)30 b(kno)m(wledge)g(of)e(these)i(correlators)118 4657 y(allo)m(ws)42 b(one)g(to)h(answ)m(er)h(a)e(n)m(um)m(b)s(er)h(of)f (relev)-5 b(an)m(t)42 b(questions)i(concerning)e(the)h(existence)i(and) d(the)118 4773 y(nature)31 b(of)g(the)g(lo)s(calization)c(and)k(b)s (eha)m(vior)g(of)f(related)h(quan)m(tities.)42 b(In)32 b(particular,)d(in)h(subsequen)m(t)118 4890 y(pap)s(er)e(\([13)o(]\),)h (w)m(e)f(use)h(the)f(tec)m(hnique)h(dev)m(elop)s(ed)f(in)f(this)g(pap)s (er)h(in)e(order)i(to)f(\014nd)h(the)g(large)f(de\014cit)118 5006 y(asymptotic)32 b(b)s(eha)m(vior)g(of)g(the)h(in)m(terband)g(ligh) m(t)e(absorption)h(co)s(e\016cien)m(t.)331 5122 y(The)e(pap)s(er)g(is)f (organized)f(as)i(follo)m(ws.)41 b(In)30 b(section)g(1)f(w)m(e)h (de\014ne)h(the)f(framew)m(ork)f(of)g(our)g(study)118 5238 y(and)38 b(giv)m(e)g(a)g(brief)f(accoun)m(t)i(of)e(our)h(results.) 60 b(W)-8 b(e)38 b(also)f(presen)m(t)j(sev)m(eral)f(examples)e(at)h (the)g(end)h(of)118 5355 y(the)44 b(section.)78 b(In)44 b(section)g(2)g(w)m(e)h(\014rst)f(pro)m(v)m(e)i(the)e(basic)g(relation) e(\(1.4\))h(expressing)i(the)g(IDS)e(in)1989 5454 y Fp(4)p eop %%Page: 5 5 5 4 bop 118 241 a Fs(terms)45 b(of)f(the)h(sp)s(ectral)f(family)e(of)i (the)i(random)d(Sc)m(hr\177)-49 b(odinger)45 b(op)s(erator.)79 b(Then)46 b(w)m(e)f(construct)118 357 y(our)d(main)e(tec)m(hnical)h(to) s(ol,)i(the)f(p)s(erio)s(dic)f(appro)m(ximations)f(of)h(the)h(IDS.)g (Sections)g(3)g(and)g(4)g(are)118 473 y(dev)m(oted)c(to)d(the)i(deriv) -5 b(ation)34 b(of)i(the)g(lo)m(w)m(er)g(and)g(upp)s(er)h(b)s(ounds)f (for)g(the)g(IDS)g(using)g(the)g(p)s(erio)s(dic)118 589 y(appro)m(ximations.)k(Section)26 b(5)g(con)m(tains)h(auxiliary)d (facts)j(on)f(the)h(statistics)f(of)g(the)h(P)m(oisson)g(\014eld,)h(on) 118 706 y(random)d(Sc)m(hr\177)-49 b(odinger)27 b(op)s(erators)f(and)g (on)g(the)h(structure)g(of)f(the)g(ground)g(state)h(of)f(the)g(Sc)m (hr\177)-49 b(odinger)118 822 y(op)s(erator)32 b(with)g(a)h(singular)e (single)g(site)h(p)s(oten)m(tial.)1766 1058 y Fr(Contents)218 1233 y Fs(0.)97 b(In)m(tro)s(duction:)43 b(problems)32 b(and)h(history)1950 b(1)218 1349 y(1.)97 b(The)34 b(assumptions)e(and) h(the)g(results)2068 b(5)218 1465 y(1.1.)97 b(The)34 b(in)m(tegrated)e(densit)m(y)h(of)g(states)2026 b(6)218 1581 y(1.2.)97 b(The)34 b(asymptotics)e(of)g(the)h(IDS)2198 b(6)218 1698 y(2.)97 b(P)m(erio)s(dic)32 b(appro)m(ximations)2419 b(9)218 1814 y(2.1.)97 b(A)33 b(general)f(appro)m(ximation)e(result) 2051 b(9)218 1930 y(2.2.)97 b(Application)30 b(to)j(the)g(estimation)d (of)i(the)h(in)m(tegrated)g(densit)m(y)g(of)f(states)606 b(18)218 2046 y(3.)97 b(The)34 b(lo)m(w)m(er)e(b)s(ounds)2648 b(19)218 2162 y(3.1.)97 b(The)34 b(general)e(case)2623 b(20)218 2279 y(3.2.)97 b(The)34 b(case)f(when)h Fo(V)54 b Fs(is)32 b(b)s(ounded)h(from)f(b)s(elo)m(w)1589 b(22)218 2395 y(3.3.)97 b(The)34 b(case)f(when)h Fo(V)54 b Fs(has)33 b(p)s(o)m(w)m(er)g(la)m(w)f(singularities)1413 b(23)218 2511 y(4.)97 b(The)34 b(upp)s(er)f(b)s(ounds)2623 b(24)218 2627 y(4.1.)97 b(The)34 b(general)e(case)2623 b(24)218 2744 y(4.2.)97 b(Pro)s(of)32 b(of)g(Prop)s(osition)f(4.1)2305 b(25)218 2860 y(4.3.)97 b(The)34 b(case)f(when)h Fo(V)54 b Fs(has)33 b(p)s(o)m(w)m(er)g(la)m(w)f(singularities)1413 b(30)218 2976 y(5.)97 b(App)s(endix)2995 b(35)218 3092 y(5.1.)97 b(The)34 b(structure)f(of)f(the)h(P)m(oisson)g(p)s(oten)m (tial)1703 b(35)218 3209 y(5.2.)97 b(An)33 b(a-priori)d(estimate)h(on)i (the)g(densit)m(y)g(of)g(states)1440 b(37)218 3325 y(5.3.)97 b(Exp)s(onen)m(tial)32 b(deca)m(y)i(estimates)2119 b(39)218 3441 y(5.4.)97 b(Some)32 b(useful)h(facts)g(ab)s(out)f(the)h(single)e (site)i(p)s(oten)m(tial)e(Hamiltonian)d Fo(H)3189 3456 y Fn(g)3800 3441 y Fs(40)218 3557 y(References)3135 b(45)1103 3943 y(1.)55 b Fr(The)37 b(assumptions)g(and)h(the)g(resul)-7 b(ts)331 4117 y Fs(Let)32 b Fo(V)50 b Fs(:)27 b Fk(R)732 4081 y Fn(d)807 4117 y Fl(!)g Fk(R)43 b Fs(b)s(e)33 b(a)f(function)g (suc)m(h)i(that)1747 4276 y Fo(V)49 b Fs(=)27 b Fo(V)2013 4291 y Fi(1)2075 4276 y Fs(+)22 b Fo(V)2230 4291 y Fi(2)118 4434 y Fs(where)118 4571 y Fx(H1)41 b Fs(for)32 b(some)h Fo(C)h(>)28 b Fs(0)k(and)h(an)m(y)g Fo(x)28 b Fl(2)g Fk(R)1603 4535 y Fn(d)1650 4571 y Fs(,)k Fl(j)p Fo(V)1794 4586 y Fi(1)1833 4571 y Fs(\()p Fo(x)p Fs(\))p Fl(j)c(\024)g Fo(C)7 b(e)2247 4535 y Fj(\000j)p Fn(x)p Fj(j)p Fn(=C)2476 4571 y Fs(.)118 4688 y Fx(H2)41 b Fo(V)360 4703 y Fi(2)442 4688 y Fs(is)h(a)h(compactly)f(supp)s(orted)h(function)g(satisfying)e Fo(V)2483 4703 y Fi(2)2567 4688 y Fl(2)46 b Fo(L)2745 4652 y Fn(p)2785 4688 y Fs(\()p Fk(R)2889 4652 y Fn(d)2935 4688 y Fs(\))d(where)h Fo(p)h(>)f(p)p Fs(\()p Fo(d)p Fs(\))e(and)303 4804 y Fo(p)p Fs(\()p Fo(d)p Fs(\))27 b(=)h(2)k(if)g Fo(d)27 b Fl(\024)h Fs(2)33 b(and)f Fo(p)p Fs(\()p Fo(d)p Fs(\))27 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Fs(\001)c(+)f Fo(V)2268 530 y Fn(!)2319 515 y Fo(:)-2228 b Fs(\(1.2\))118 673 y(One)33 b(has)118 848 y Fx(Theorem)k(1.1)h Fs(\([7]\))p Fx(.)49 b Fe(Under)28 b(the)h(assumptions)f(made)f(ab)-5 b(ove,)29 b Fo(H)2649 863 y Fn(!)2728 848 y Fe(is)g(essential)5 b(ly)28 b(self-adjoint)f(on)118 964 y Fl(C)176 928 y Fj(1)170 989 y Fi(0)251 964 y Fs(\()p Fk(R)355 928 y Fn(d)401 964 y Fs(\))35 b Fo(!)t Fe(-almost)f(sur)-5 b(ely.)118 1139 y Fs(Under)39 b(our)g(assumptions)f(on)h Fo(V)21 b Fs(,)40 b(w)m(e)g(kno)m(w)f(that)g(the)g(almost)e(sure)i(sp)s (ectrum)g(of)f Fo(H)3411 1154 y Fn(!)3500 1139 y Fs(is)g(\006)g(=)g Fk(R)118 1256 y Fs(\([21,)32 b(7]\).)118 1432 y(1.1.)56 b Fx(The)31 b(in)m(tegrated)e(densit)m(y)h(of)h(states.)49 b Fs(Let)27 b(\003)f(b)s(e)h(a)f(cub)s(e)h(cen)m(tered)i(at)d(0)g(in)g Fk(R)3372 1396 y Fn(d)3418 1432 y Fs(.)41 b(W)-8 b(e)27 b(de\014ne)118 1548 y Fo(H)207 1512 y Fn(D)199 1574 y(!)r(;)p Fi(\003)350 1548 y Fs(to)33 b(b)s(e)f(the)h(Diric)m(hlet)e(restriction) g(of)h Fo(H)1831 1563 y Fn(!)1914 1548 y Fs(to)g(\003.)44 b(Pic)m(k)33 b Fo(E)g Fl(2)c Fk(R)5 b Fs(.)49 b(Consider)33 b(the)g(quan)m(tit)m(y)539 1768 y Fo(N)617 1783 y Fn(!)r(;)p Fi(\003)736 1768 y Fs(\()p Fo(E)6 b Fs(\))28 b(=)1149 1701 y(1)p 1031 1745 285 4 v 1031 1836 a(V)-8 b(ol)o(\(\003\))1326 1768 y Fo(])p Fl(f)p Fs(eigen)m(v)j(alues)32 b(of)g Fo(H)2118 1727 y Fn(D)2110 1793 y(!)r(;)p Fi(\003)2262 1768 y Fs(smaller)e(than)j (or)f(equal)g(to)h Fo(E)6 b Fl(g)p Fo(:)-3359 b Fs(\(1.3\))118 1981 y(Then)34 b(one)f(has)118 2156 y Fx(Theorem)k(1.2)h Fs(\([7]\))p Fx(.)49 b Fe(Under)31 b(the)g(assumptions)g(made)f(ab)-5 b(ove,)32 b(ther)-5 b(e)31 b(exists)g(a)g(non)g(r)-5 b(andom,)31 b(non)118 2272 y(de)-5 b(cr)g(e)g(asing,)34 b(non)h(ne)-5 b(gative,)35 b(right)h(c)-5 b(ontinuous)35 b(function)g Fo(N)10 b Fs(\()p Fo(E)c Fs(\))36 b Fe(such)g(that,)g Fo(!)t Fe(-almost)e(sur)-5 b(ely,)36 b(for)118 2388 y(al)5 b(l)35 b Fo(E)e Fl(2)28 b Fk(R)5 b Fe(,)41 b Fo(E)g Fe(a)35 b(c)-5 b(ontinuity)35 b(p)-5 b(oint)35 b(of)f Fo(N)10 b Fe(,)35 b Fo(N)1842 2403 y Fn(!)r(;)p Fi(\003)1962 2388 y Fs(\()p Fo(E)6 b Fs(\))34 b Fe(c)-5 b(onver)g(ges)34 b(to)h Fo(N)10 b Fs(\()p Fo(E)c Fs(\))35 b Fe(as)g Fs(\003)f Fe(exhausts)h Fk(R)3661 2352 y Fn(d)3707 2388 y Fe(.)118 2563 y Fo(N)10 b Fs(\()p Fo(E)c Fs(\))39 b(is)g(the)g Fe(inte)-5 b(gr)g(ate)g(d)40 b(density)h(of)f(states)f Fs(\(IDS\))f(of)g Fo(H)2342 2578 y Fn(!)2393 2563 y Fs(.)62 b(As)39 b Fo(N)49 b Fs(is)39 b(non)f(decreasing,)j(one)e(can)118 2679 y(de\014ne)e(its)f(distributional)d(deriv)-5 b(ativ)m(e)35 b Fo(dN)10 b Fs(.)54 b(It)36 b(is)g(a)f(p)s(ositiv)m(e)h(measure)g(and) g(is)f(supp)s(orted)i(on)f(the)118 2795 y(almost)31 b(sure)i(sp)s (ectrum)g(of)g Fo(H)1258 2810 y Fn(!)1340 2795 y Fs(\(see)h([7,)e (21]\).)43 b(One)33 b(has)g(the)g(follo)m(wing)d(result)118 2970 y Fx(Theorem)37 b(1.3.)49 b Fe(F)-7 b(or)34 b Fo(')28 b Fl(2)g(C)1242 2934 y Fj(1)1236 2995 y Fi(0)1317 2970 y Fs(\()p Fk(R)5 b Fs(\))p Fe(,)40 b(we)35 b(have)1226 3128 y Fs(\()p Fo(';)17 b(dN)10 b Fs(\))27 b(=)h Fk(E)12 b Fs(\()p Fe(tr)7 b Fs(\()p Fx(1)1952 3144 y Fn(C)e Fi(\(0)p Fn(;)p Fi(1\))2156 3128 y Fo(')p Fs(\()p Fo(H)2339 3143 y Fn(!)2389 3128 y Fs(\))p Fx(1)2483 3144 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))2687 3128 y Fs(\)\))p Fo(:)-2672 b Fs(\(1.4\))118 3292 y Fe(wher)-5 b(e)34 b Fo(C)7 b Fs(\(0)p Fo(;)17 b Fs(1\))34 b Fe(is)h(the)g(cub)-5 b(e)34 b(of)h(c)-5 b(enter)35 b Fs(0)f Fe(and)h(side)f(length)g Fs(1)p Fe(.)118 3467 y Fs(F)-8 b(orm)m(ula)34 b(\(1.4\))i(is)g(w)m(ell)f(kno)m(wn)j (under)f(more)e(restrictiv)m(e)i(assumptions)f(on)g(the)h(p)s(oten)m (tial)d Fo(V)3686 3482 y Fn(!)3773 3467 y Fs(i.e.)118 3583 y(for)e(a)g(less)h(singular)e(single)h(site)g(p)s(oten)m(tial)f Fo(V)54 b Fs(\(see)34 b([21]\).)118 3760 y(1.2.)56 b Fx(The)35 b(asymptotics)e(of)i(the)g(IDS.)49 b Fs(T)-8 b(o)30 b(describ)s(e)h(the)g(asymptotic)e(b)s(eha)m(vior)h(of)g Fo(N)10 b Fs(\()p Fo(E)c Fs(\))30 b(near)118 3876 y Fl(\0001)p Fs(,)j(w)m(e)g(will)e(need)i(to)g(de\014ne)g(an)g(auxiliary)d(op)s (erator.)43 b(F)-8 b(or)32 b Fo(g)f Fl(2)d Fk(R)5 b Fs(,)39 b(de\014ne)1626 4034 y Fo(H)8 b Fs(\()p Fo(g)t Fs(\))26 b(=)i Fl(\000)p Fs(\001)22 b(+)g Fo(g)t(V)5 b(:)-2272 b Fs(\(1.5\))118 4192 y(Under)45 b(our)f(assumptions)g(on)g Fo(V)21 b Fs(,)47 b Fo(V)66 b Fs(is)43 b(relativ)m(ely)g(form)g(b)s (ounded)i(with)e(resp)s(ect)j(to)d Fl(\000)p Fs(\001)i(with)118 4309 y(relativ)m(e)27 b(b)s(ound)h(0.)41 b(Hence)29 b Fo(H)8 b Fs(\()p Fo(g)t Fs(\))27 b(admits)f(a)h(unique)h(self-adjoin)m (t)e(extension.)43 b(Let)28 b Fo(\033)t Fs(\()p Fo(H)8 b Fs(\()p Fo(g)t Fs(\)\))26 b(denote)118 4425 y(its)38 b(sp)s(ectrum.)62 b(It)39 b(is)f(lo)m(w)m(er)g(semi-b)s(ounded.)61 b(The)40 b(in\014m)m(um)d(of)h Fo(\033)t Fs(\()p Fo(H)8 b Fs(\()p Fo(g)t Fs(\)\))37 b(i.e.)61 b(the)39 b(ground)f(state)118 4541 y(energy)g(of)d Fo(H)8 b Fs(\()p Fo(g)t Fs(\))35 b(will)f(b)s(e)j(denoted)g(b)m(y)h Fo(E)6 b Fs(\()p Fo(g)t Fs(\).)54 b(Let)36 b Fo(')2164 4556 y Fn(g)2240 4541 y Fs(b)s(e)h(the)g(resp)s(ectiv)m(e)h(ground)e(state)h(i.e.)54 b(the)118 4657 y(unique)42 b(p)s(ositiv)m(e)g(normalized)d (eigenfunction)j(of)f Fo(H)8 b Fs(\()p Fo(g)t Fs(\))40 b(asso)s(ciated)i(to)g(energy)h Fo(E)6 b Fs(\()p Fo(g)t Fs(\))40 b(\([22,)i(26]\).)118 4773 y(In)e(the)f(sequel)h(it)e(will)f (often)i(b)s(e)g(more)g(con)m(v)m(enien)m(t)i(to)e(w)m(ork)h(with)e Fo(E)2816 4788 y Fj(\000)2876 4773 y Fs(\()p Fo(g)t Fs(\))g(=)g Fl(\000)p Fo(E)6 b Fs(\()p Fo(g)t Fs(\))39 b(instead)g(of)118 4890 y Fo(E)6 b Fs(\()p Fo(g)t Fs(\))39 b(itself.)62 b(F)-8 b(rom)38 b(assumption)h(H3,)i(one)f(easily)e(infers)i(that)f Fo(E)2654 4905 y Fj(\000)2713 4890 y Fs(\()p Fo(g)t Fs(\))f Fl(!)h Fs(+)p Fl(1)g Fs(when)i Fo(g)h Fl(!)d Fs(+)p Fl(1)118 5006 y Fs(\(see)31 b(section)f(5\).)43 b(Moreo)m(v)m(er)31 b Fo(E)1288 5021 y Fj(\000)1378 5006 y Fs(is)e(strictly)h(increasing)f (in)g(a)h(neigh)m(b)s(orho)s(o)s(d)f(of)h(+)p Fl(1)p Fs(.)42 b(Let)30 b Fo(g)j Fs(b)s(e)e(an)118 5122 y(in)m(v)m(erse)j(of)e Fo(E)624 5137 y Fj(\000)716 5122 y Fs(in)g(a)g(neigh)m(b)s(orho)s(o)s (d)f(of)i(+)p Fl(1)p Fs(.)43 b Fo(g)35 b Fs(is)d(strictly)g (increasing.)331 5238 y(In)e(the)g(regular)f(\(classical\))f(case,)k (it)d(w)m(as)i(found)f(that)f Fo(g)k Fs(is)d(go)m(v)m(erning)g(the)g (\014rst)h(term)e(asymp-)118 5355 y(totic)41 b(of)h(log)16 b Fo(N)53 b Fs(\(cf)42 b([21,)g(20)o(]\).)72 b(In)42 b(the)h(singular)d(\(quan)m(tum\))j(case,)i(the)e(singular)d(set)j(of)e Fo(V)64 b Fs(will)1989 5454 y Fp(6)p eop %%Page: 7 7 7 6 bop 118 241 a Fs(pla)m(y)35 b(a)f(sp)s(ecial)g(part)h(in)f(the)h (asymptotics.)51 b(T)-8 b(o)35 b(measure)g(this)f(role,)h(w)m(e)h(in)m (tro)s(duce)f(the)g(notion)f(of)118 357 y Fe(asymptotic)h(gr)-5 b(ound)34 b(state)f Fs(i.e.)118 533 y Fx(De\014nition)j(1.1.)49 b Fs(Let)34 b Fo(g)e Fl(2)e Fs(\(1)p Fo(;)17 b Fs(+)p Fl(1)p Fs(\))28 b Fl(7!)h Fo( )1780 548 y Fn(g)1849 533 y Fl(2)h Fo(H)2034 497 y Fi(1)2073 533 y Fs(\()p Fk(R)2177 497 y Fn(d)2223 533 y Fs(\).)46 b(W)-8 b(e)34 b(will)d(sa)m(y)j(that)g Fo( )3133 548 y Fn(g)3207 533 y Fs(is)f(an)g Fe(asymptotic)118 650 y(gr)-5 b(ound)35 b(state)d Fs(if)g(and)h(only)f(if)244 788 y Fl(\017)41 b Fs(the)33 b(v)m(ector)h Fo( )859 803 y Fn(g)932 788 y Fs(is)e(normalized.)244 904 y Fl(\017)41 b(9)p Fo(g)437 919 y Fi(0)516 904 y Fo(>)e Fs(1,)j Fo(l)778 919 y Fi(0)857 904 y Fo(>)d Fs(0)g(suc)m(h)i(that)e Fl(8)p Fo(g)44 b Fl(\025)c Fo(g)1815 919 y Fi(0)1854 904 y Fs(,)h(supp)17 b Fo( )2202 919 y Fn(g)2283 904 y Fl(\032)40 b Fo(C)7 b Fs(\(0)p Fo(;)17 b(l)2637 919 y Fi(0)2676 904 y Fs(\))39 b(\(where)i Fo(C)7 b Fs(\()p Fo(x;)17 b(l)r Fs(\))39 b(denotes)i(the)335 1020 y(cub)s(e)33 b(of)g(cen)m(ter)g Fo(x)g Fs(and)g(side)g(length)f Fo(l)r Fs(\).)244 1137 y Fl(\017)1126 1264 y(jh)p Fs(\()p Fo(H)8 b Fs(\()p Fo(g)t Fs(\))20 b Fl(\000)j Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))p Fo( )1873 1279 y Fn(g)1912 1264 y Fo(;)17 b( )2019 1279 y Fn(g)2060 1264 y Fl(ij)p 1126 1308 1001 4 v 1496 1400 a(j)p Fo(E)6 b Fs(\()p Fo(g)t Fs(\))p Fl(j)2164 1331 y(!)27 b Fs(0)32 b(as)h Fo(g)e Fl(!)c Fs(+)p Fl(1)p Fo(:)-2782 b Fs(\(1.6\))118 1575 y(In)40 b(Lemma)f(5.6,)j(w)m(e)f(pro)m(v)m(e)g (the)g(existence)g(of)f(an)g(asymptotic)f(ground)h(state.)67 b(F)-8 b(or)39 b Fo(a)h Fl(2)h Fk(R)3671 1539 y Fn(d)3718 1575 y Fs(,)h(w)m(e)118 1692 y(de\014ne)34 b(the)f(translation)e Fo(\034)1104 1707 y Fn(a)1178 1692 y Fs(b)m(y)j Fo(\034)1356 1707 y Fn(a)1398 1692 y Fo(V)21 b Fs(\()p Fo(x)p Fs(\))28 b(=)g Fo(V)21 b Fs(\()p Fo(x)i Fl(\000)f Fo(a)p Fs(\))33 b(and)g(w)m(e)g(de\014ne)760 1943 y Fl(A)840 1958 y Fn( )886 1966 y Fh(g)954 1943 y Fs(=)1057 1773 y Fm(\()1137 1943 y Fo(\013)c(>)e Fs(0;)98 b(lim)1456 2003 y Fn(g)r Fj(!)p Fi(+)p Fj(1)1761 1943 y Fs(sup)1705 2030 y Fj(j)p Fn(a)p Fj(j\024)p Fn(g)1873 2011 y Fg(\000)p Fh(\013)1981 1803 y Fm(\024)2044 1876 y Fo(g)t Fl(jh)p Fs(\()p Fo(\034)2242 1891 y Fn(a)2282 1876 y Fo(V)44 b Fl(\000)23 b Fo(V)e Fs(\))p Fo( )2662 1891 y Fn(g)2703 1876 y Fo(;)c( )2810 1891 y Fn(g)2850 1876 y Fl(ij)p 2044 1920 873 4 v 2351 2012 a Fo(E)2423 2027 y Fj(\000)2483 2012 y Fs(\()p Fo(g)t Fs(\))2926 1803 y Fm(\025)3007 1943 y Fs(=)27 b(0)3159 1773 y Fm(\))118 2195 y Fs(If)35 b Fl(A)298 2210 y Fn( )344 2218 y Fh(g)416 2195 y Fl(6)p Fs(=)c Fl(;)p Fs(,)36 b(then)f(w)m(e)h (de\014ne)g Fo(\013)1353 2159 y Fj(\003)1392 2195 y Fs(\()p Fo( )1493 2210 y Fn(g)1534 2195 y Fs(\))31 b(:=)h(inf)23 b Fl(A)1953 2210 y Fn( )1999 2218 y Fh(g)2039 2195 y Fs(.)50 b(Moreo)m(v)m(er,)37 b(w)m(e)f(de\014ne)g Fl(A)f Fs(to)f(b)s(e)h(the)h(union)e(of)118 2311 y(all)d Fl(A)334 2326 y Fn( )380 2334 y Fh(g)420 2311 y Fs(.)43 b(By)33 b(Lemma)e(5.6,)i(w)m(e)g(kno)m(w)h(that)e Fl(A)c(6)p Fs(=)f Fl(;)p Fs(.)43 b(W)-8 b(e)33 b(de\014ne)1757 2472 y Fo(\013)1820 2431 y Fj(\003)1887 2472 y Fs(:=)27 b(inf)c Fl(A)p Fo(:)-2141 b Fs(\(1.7\))118 2633 y(Roughly)27 b(sp)s(eaking,)h(the)g(dep)s(endence)i(of)d(the)h(radius)f(of)f(the)i (exp)s(onen)m(tial)f(deca)m(y)i(of)e(the)g(single)g(site)118 2749 y(p)s(oten)m(tial)36 b(ground)h(state)h(on)f(the)h(coupling)e (constan)m(t)i Fo(g)i Fs(is)d(of)g(the)h(form)e Fo(g)2985 2713 y Fj(\000)p Fn(\013)3085 2689 y Fg(\003)3124 2749 y Fs(.)58 b(This)37 b(determines)118 2865 y(the)c(c)m(haracteristic)g (size)f Fo(l)j Fs(of)d(the)h(optimal)d(cluster.)331 2981 y(Then)j(w)m(e)h(pro)m(v)m(e)118 3158 y Fx(Theorem)j(1.4.)49 b Fe(Under)35 b(the)g(assumptions)f(H1,)h(H2)g(and)f(H3,)h(for)f (su\016ciently)h(lar)-5 b(ge)34 b Fo(E)6 b Fe(,)35 b(one)f(has)470 3318 y Fl(\000)p Fs(\(1)23 b(+)f Fo(\013)818 3277 y Fj(\003)857 3318 y Fo(d)p Fs(\))p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(o)p Fs(\(1\)\))27 b Fl(\024)h Fs(log)17 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))28 b Fl(\024)g(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(o)p Fs(\(1\)\))p Fo(:)-3628 b Fs(\(1.8\))331 3495 y(One)36 b(ma)m(y)f(complain)f(that)h(Theorem)i(1.4)e(is)g (somewhat)h(imprecise)f(in)g(that)h(it)e(only)i(giv)m(es)g(a)118 3611 y(t)m(w)m(o)30 b(sided)f(estimate.)41 b(But,)30 b(as)g(w)m(e)g(will)c(see)k(b)s(elo)m(w,)g(this)e(is)h(in)f(some)h(w)m (a)m(y)h(una)m(v)m(oidable)f(as)g(the)g(true)118 3727 y(asymptotic)h(do)s(es)g(not)g(only)g(dep)s(end)h(on)g Fo(g)i Fs(but)e(also)e(on)h(the)h(singular)d(set)j(of)f(the)h(negativ)m (e)f(part)g(of)118 3844 y Fo(V)22 b Fs(.)41 b(More)28 b(precisely)-8 b(,)29 b(as)f(can)g(b)s(e)g(seen)h(from)d(Theorem)i(1.6) f(\(and)h(from)e(the)i(pro)s(of)f(of)g(Theorem)h(1.4\),)118 3960 y(the)h(asymptotics)f(of)f(the)i(IDS)f(dep)s(ends)i(on)e(the)g(w)m (a)m(y)i(the)e(eigenfunction)g(asso)s(ciated)g(to)f(the)i(lo)m(w)m(est) 118 4076 y(eigen)m(v)-5 b(alue)37 b(for)g(the)h(op)s(erator)e Fl(\000)p Fs(\001)27 b(+)e Fo(g)t(V)58 b Fs(concen)m(trates)39 b(near)f(the)g(singular)e(set)i(of)f(the)g(negativ)m(e)118 4192 y(part)c(of)h Fo(V)55 b Fs(as)33 b Fo(g)k Fs(b)s(ecomes)d(large.) 45 b(In)34 b(general)f(the)h(correction)f(also)g(dep)s(ends)i(on)f(the) g(geometry)f(of)118 4309 y(the)i(singular)f(set.)51 b(F)-8 b(or)34 b(example,)h(if)f(the)h(singular)f(set)h(is)g(a)f(segmen)m(t)i (\(e.g.)50 b(a)35 b(dislo)s(cation\),)e(using)118 4425 y(the)k(tec)m(hniques)i(dev)m(elop)s(ed)f(in)e(section)h(3,)h(one)f (can)g(see)h(that)f(neither)g(the)g(lo)m(w)m(er)g(nor)g(the)g(upp)s(er) 118 4541 y(b)s(ound)d(giv)m(en)f(b)m(y)h(Theorem)g(1.4)e(are)i(sharp.) 46 b(The)34 b(t)m(w)m(o)g(sided)f(estimate)g(\(1.8\))g(can)g(b)s(e)g (made)g(more)118 4657 y(precise)g(if)f(w)m(e)h(kno)m(w)h(more)e(on)h Fo(V)21 b Fs(.)331 4773 y(The)35 b(\014rst)h(and)f(simplest)e(example)h (w)m(e)i(giv)m(e)f(is)f(the)h(case)h(when)g Fo(V)56 b Fs(is)35 b(b)s(ounded)g(from)f(b)s(elo)m(w,)118 4890 y(reac)m(hes)39 b(its)e(minim)m(um)e(at)i(a)g(single)g(p)s(oin)m(t,)h (sa)m(y0,)i(and)d(is)g(con)m(tin)m(uous)i(near)e(0.)59 b(Then)38 b(one)g(easily)118 5006 y(pro)m(v)m(es)c(that)f Fo(\013)694 4970 y Fj(\003)761 5006 y Fs(=)27 b(0)32 b(and)h(the)g(upp)s(er)g(and)g(lo)m(w)m(er)g(b)s(ounds)g(in)f(\(1.8\))g (coalesce)h(to)f(giv)m(e)h(\(0.7\))o(.)331 5122 y(W)-8 b(e)28 b(will)e(no)m(w)j(giv)m(e)f(other)h(results)g(that,)g(w)m(e)g (think,)g(enclose)g(most)e(of)h(the)h(ph)m(ysically)f(relev)-5 b(an)m(t)118 5238 y(examples.)331 5355 y(Let)29 b Fo(v)549 5370 y Fj(\000)637 5355 y Fs(b)s(e)g(the)h(essen)m(tial)f(in\014m)m(um) e(of)i Fo(V)51 b Fs(and)29 b(assume)g(that)g Fo(V)51 b Fs(is)28 b(b)s(ounded)i(from)e(b)s(elo)m(w,)i(sa)m(y)1989 5454 y Fp(7)p eop %%Page: 8 8 8 7 bop 118 241 a Fx(H1')41 b Fl(\0001)28 b Fo(<)g(v)690 256 y Fj(\000)777 241 y Fo(<)f Fs(0.)118 389 y(It)33 b(is)f(easy)h(to)g(sho)m(w)g(that)g Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b Fl(\030)f Fo(E)6 b(=)p Fl(j)p Fo(v)1626 404 y Fj(\000)1684 389 y Fl(j)33 b Fs(when)g Fo(E)h Fl(!)27 b Fs(+)p Fl(1)32 b Fs(\(see)i(Lemma)d(5.5\).)43 b(W)-8 b(e)33 b(obtain)118 577 y Fx(Theorem)k(1.5.)49 b Fe(Under)35 b(the)g(assumptions)f(H1,)h(H2)g(and)f(H1',)h(one)g(has)974 805 y Fs(log)17 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))104 b Fl(\030)1453 867 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1721 805 y Fl(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\))103 b Fl(\030)2384 867 y Fn(E)t Fj(!)p Fi(+)p Fj(1)2675 738 y Fo(E)p 2661 782 107 4 v 2661 874 a(v)2708 889 y Fj(\000)2794 805 y Fs(log)16 b Fo(E)6 b(:)-2923 b Fs(\(1.9\))118 1043 y Fe(Her)-5 b(e)35 b(and)f(in)h(the)g(r)-5 b(est)35 b(of)g(the)f(p)-5 b(ap)g(er,)35 b Fo(a)27 b Fl(\030)i Fo(b)35 b Fe(wil)5 b(l)34 b(always)h(me)-5 b(an)34 b Fo(a)28 b Fs(=)f Fo(b)p Fs(\(1)22 b(+)g Fo(o)p Fs(\(1\)\))p Fe(.)331 1231 y Fs(This)32 b(result)h(extends)h(\(0.7\))f(remo)m(ving)e(the)i(con)m(tin)m(uit)m(y) g(assumption)f(near)h(the)g(minim)m(um.)331 1348 y(Consider)j(no)m(w)h (an)f(example)f(a)h(bit)f(more)g(general)h(that)g(could)f(b)s(e)h (regarded)h(as)f(a)g(mo)s(del)e(of)118 1464 y(line)d(defect)j(\(c)m (harged)f(dislo)s(cation\).)41 b(In)33 b(this)f(case,)i Fo(d)27 b Fs(=)h(2)k(and)1457 1646 y Fo(V)1514 1661 y Fi(2)1554 1646 y Fs(\()p Fo(x)p Fs(\))c(=)f(log)1942 1669 y Fj(\000)2018 1646 y Fl(j)p Fo(x)p Fl(j)p Fo(;)114 b(x)28 b Fl(2)g Fk(R)2513 1605 y Fi(2)118 1832 y Fs(where,)58 b(for)52 b Fo(a)61 b Fl(\025)g Fs(0,)56 b(log)1128 1856 y Fj(\000)1204 1832 y Fo(a)61 b Fs(=)g(min)n Fl(f)p Fs(log)17 b Fo(a;)g Fs(0)p Fl(g)p Fs(.)101 b(Using)51 b(the)i(inequalit)m(y)e (log)3209 1856 y Fj(\000)3284 1832 y Fl(j)p Fo(x)p Fl(j)36 b Fs(+)f(log)16 b Fo(R)62 b Fl(\024)118 1949 y Fs(log)244 1972 y Fj(\000)320 1949 y Fo(R)q Fl(j)p Fo(x)p Fl(j)49 b(\024)h Fs(log)17 b Fl(j)p Fo(x)p Fl(j)45 b Fs(for)g(0)k Fo(<)g(R)i(<)e Fs(1)c(and)h(the)f(v)-5 b(ariational)42 b(principle)i(for)h(the)h(ground)f(state)118 2065 y(energy)-8 b(,)47 b(one)c(sho)m(ws)i(that,)g(in)d(this)h(case,)k Fo(E)1837 2080 y Fj(\000)1896 2065 y Fs(\()p Fo(g)t Fs(\))93 b Fl(\030)2040 2125 y Fn(g)r Fj(!)p Fi(+)p Fj(1)2287 2065 y Fo(g)t(=)p Fs(2)17 b(log)e Fo(g)46 b Fs(hence)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\))103 b Fl(\030)3191 2127 y Fn(E)t Fj(!)p Fi(+)p Fj(1)3458 2065 y Fs(2)p Fo(E)6 b(=)17 b Fs(log)e Fo(E)6 b Fs(.)118 2225 y(One)33 b(also)e(sho)m(ws)j (that)e Fo(\013)1073 2189 y Fj(\003)1139 2225 y Fs(=)c(0)k(for)g(this)f (single)g(site)h(p)s(oten)m(tial.)42 b(Therefore,)34 b(Theorem)e(1.4)g(tells)f(us)118 2341 y(that)1054 2522 y(log)16 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))105 b Fl(\030)1532 2584 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1801 2522 y Fl(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\))103 b Fl(\030)2464 2584 y Fn(E)t Fj(!)p Fi(+)p Fj(1)2731 2522 y Fl(\000)p Fs(2)p Fo(E)6 b(:)118 2749 y Fs(Hence)34 b(asymptotic)e(\(0.5\))g(is)g(also)g (v)-5 b(alid)30 b(for)i(certain)h(mildly)c(singular)i(p)s(oten)m (tials.)331 2865 y(Another)h(case)h(where)g(one)f(can)g(\014nd)h(an)f (asymptotic)f(for)g(log)17 b Fo(N)42 b Fs(is)32 b(when)h Fo(V)53 b Fs(has)32 b(only)g(p)s(o)m(w)m(er)118 2981 y(la)m(w)f(singularities.)41 b(Let)31 b Fo(q)k Fs(b)s(e)d(a)f(p)s (ositiv)m(e)g(in)m(teger)g(and)h(pic)m(k)f Fo(q)36 b Fs(p)s(ositiv)m(e)30 b(exp)s(onen)m(ts)k(\()p Fo(\027)3393 2996 y Fn(i)3421 2981 y Fs(\))3459 2996 y Fn(i)p Fi(=1)p Fn(;:::)n(;q)3740 2981 y Fs(and)118 3100 y Fo(q)47 b Fs(functions)d(\()p Fo(h)734 3115 y Fn(i)762 3100 y Fs(\()p Fo(\022)s Fs(\)\))924 3115 y Fn(i)p Fi(=1)p Fn(;:::)n(;q)1217 3100 y Fs(con)m(tin)m(uous)g(on)g(the)f(sphere)i Fk(S)2418 3064 y Fn(d)p Fj(\000)p Fi(1)2542 3100 y Fs(.)76 b(F)-8 b(or)43 b(1)i Fl(\024)i Fo(i)f Fl(\024)h Fo(q)t Fs(,)e(consider)f(the) 118 3216 y(p)s(oten)m(tials)1276 3450 y Fo(V)1333 3465 y Fn(i)1361 3450 y Fs(\()p Fo(x)p Fs(\))28 b(=)1634 3383 y Fo(h)1690 3398 y Fn(i)1718 3383 y Fs(\()p Fo(\022)s Fs(\()p Fo(x)p Fs(\)\))p 1634 3428 340 4 v 1716 3519 a Fl(j)p Fo(x)p Fl(j)1827 3490 y Fn(\027)1862 3500 y Fh(i)2016 3450 y Fs(where)34 b Fo(\022)s Fs(\()p Fo(x)p Fs(\))28 b(=)2646 3383 y Fo(x)p 2619 3428 111 4 v 2619 3519 a Fl(j)p Fo(x)p Fl(j)118 3450 y Fs(\(1.10\))118 3686 y(Assume)34 b(that)1447 3933 y(0)28 b Fo(<)f(\027)1675 3948 y Fn(i)1731 3933 y Fo(<)1835 3763 y Fm(\()1915 3867 y Fs(2)32 b(if)g Fo(d)27 b Fs(=)h(1)p Fo(;)17 b Fs(2)p Fo(;)1915 4006 y(d=)p Fs(2)32 b(if)f Fo(d)c Fl(\025)i Fs(3)p Fo(:)118 3933 y Fs(\(1.11\))118 4200 y(Then)43 b Fo(V)439 4215 y Fn(i)509 4200 y Fs(is)e(relativ)m(ely)f(form)h(b)s (ounded)h(with)g(resp)s(ect)g(to)g Fl(\000)p Fs(\001)g(with)g(relativ)m (e)e(b)s(ound)i(0)g(and)g(w)m(e)118 4316 y(can)35 b(consider)h(the)f (op)s(erators)g Fo(H)1375 4280 y Fn(i)1435 4316 y Fs(=)d Fl(\000)p Fs(\001)24 b(+)g Fo(V)1882 4331 y Fn(i)1945 4316 y Fs(with)35 b(form)e(domain)h Fo(H)2841 4280 y Fi(1)2880 4316 y Fs(\()p Fk(R)2984 4280 y Fn(d)3030 4316 y Fs(\).)51 b(F)-8 b(or)34 b(1)e Fl(\024)g Fo(i)g Fl(\024)h Fo(q)t Fs(,)i Fo(E)3869 4331 y Fn(i)118 4432 y Fs(denotes)f(the)f (ground)g(state)g(energy)g(of)f Fo(H)1720 4396 y Fn(i)1748 4432 y Fs(.)43 b(No)m(w)34 b(w)m(e)f(assume)g(that)118 4581 y Fx(H1")117 b Fl(\017)41 b Fs(there)35 b(exists)g Fo(q)j Fs(distinct)33 b(p)s(oin)m(ts)h(\()p Fo(x)1873 4596 y Fn(i)1902 4581 y Fs(\))1940 4596 y Fn(i)p Fi(=1)p Fn(;:::)n(;q)2224 4581 y Fs(in)f Fk(R)2405 4545 y Fn(d)2486 4581 y Fs(and)h Fo(q)k Fs(con)m(tin)m(uous)d(compactly)e(sup-)528 4697 y(p)s(orted)g(function)f(\()p Fo(W)1352 4712 y Fn(i)1380 4697 y Fs(\))1418 4712 y Fn(i)p Fi(=1)p Fn(;:::)o(;q)1701 4697 y Fs(suc)m(h)i(that)e Fo(W)2224 4712 y Fn(i)2253 4697 y Fs(\(0\))27 b(=)h(1)k(suc)m(h)i(that)900 4965 y Fo(V)957 4980 y Fi(2)997 4965 y Fs(\()p Fo(x)p Fs(\))28 b(=)1314 4837 y Fn(q)1259 4871 y Fm(X)1274 5081 y Fn(i)p Fi(=1)1420 4965 y Fo(W)1512 4980 y Fn(i)1540 4965 y Fs(\()p Fo(x)23 b Fl(\000)f Fo(x)1810 4980 y Fn(i)1839 4965 y Fs(\))p Fo(V)1934 4980 y Fn(i)1962 4965 y Fs(\()p Fo(x)g Fl(\000)h Fo(x)2232 4980 y Fn(i)2261 4965 y Fs(\))k(=)2485 4837 y Fn(q)2430 4871 y Fm(X)2445 5081 y Fn(i)p Fi(=1)2590 4965 y Fo(\034)2632 4980 y Fn(x)2672 4990 y Fh(i)2703 4965 y Fs(\()p Fo(W)2833 4980 y Fn(i)2861 4965 y Fo(V)2918 4980 y Fn(i)2946 4965 y Fs(\)\()p Fo(x)p Fs(\))-2997 b(\(1.12\))528 5238 y(where)34 b(the)f(exp)s(onen)m(ts)i(\()p Fo(\027)1523 5253 y Fn(i)1551 5238 y Fs(\))1589 5253 y Fi(1)p Fj(\024)p Fn(i)p Fj(\024)p Fn(q)1829 5238 y Fs(satisfy)e(\(1.11\).)437 5355 y Fl(\017)41 b Fs(for)32 b(some)h(1)27 b Fl(\024)h Fo(i)1136 5370 y Fi(0)1204 5355 y Fl(\024)g Fo(q)t Fs(,)k(w)m(e)i(ha)m(v)m(e)g Fo(E)1856 5370 y Fn(i)1880 5379 y Fd(0)1947 5355 y Fo(<)27 b Fs(0.)1989 5454 y Fp(8)p eop %%Page: 9 9 9 8 bop 118 241 a Fs(Notice)32 b(that)h(assumption)f(H1")g(implies)e (assumptions)i(H2)h(and)f(H3.)44 b(De\014ne)1090 432 y Fo(\027)1144 391 y Fj(y)1208 432 y Fs(=)27 b(sup)18 b Fl(f)o Fo(\027)1572 447 y Fn(i)1601 432 y Fs(;)49 b(1)27 b Fl(\024)i Fo(i)f Fl(\024)g Fo(q)36 b Fs(suc)m(h)e(that)e Fo(E)2607 447 y Fn(i)2663 432 y Fo(<)c Fs(0)p Fl(g)16 b Fo(;)-2791 b Fs(\(1.13\))761 596 y Fo(E)833 611 y Fj(\000)920 596 y Fs(=)28 b(sup)1187 515 y Fm(\010)1245 596 y Fl(j)p Fo(E)1345 611 y Fn(i)1373 596 y Fl(j)p Fs(;)49 b(1)28 b Fl(\024)g Fo(i)g Fl(\024)g Fo(q)36 b Fs(suc)m(h)e(that)f Fo(E)2408 611 y Fn(i)2464 596 y Fo(<)27 b Fs(0)33 b(and)f Fo(\027)2886 611 y Fn(i)2942 596 y Fs(=)c Fo(\027)3100 555 y Fj(y)3136 515 y Fm(\011)3211 596 y Fo(;)-3120 b Fs(\(1.14\))1739 804 y Fo(\013)1802 763 y Fj(y)1865 804 y Fs(=)2085 736 y(1)p 1979 781 261 4 v 1979 872 a(2)22 b Fl(\000)g Fo(\027)2203 843 y Fj(y)2249 804 y Fo(:)-2158 b Fs(\(1.15\))118 1058 y(Then,)38 b(b)m(y)f(Lemma)d(5.7,)i(w)m(e)h(kno) m(w)g(that)e Fo(E)1773 1073 y Fj(\000)1833 1058 y Fs(\()p Fo(g)t Fs(\))17 b Fl(\030)f Fo(E)2142 1073 y Fj(\000)2201 1058 y Fo(g)2252 1022 y Fi(2)p Fn(\013)2332 998 y Fg(y)2404 1058 y Fs(when)37 b Fo(g)g Fl(!)32 b Fs(+)p Fl(1)p Fs(.)53 b(In)36 b(addition,)f(using)118 1174 y(the)f(p)s(erio)s(dic)d(appro)m (ximation)g(sc)m(heme,)j(in)e(this)h(case,)h(w)m(e)g(can)f(\014nd)h(an) f(upp)s(er)g(b)s(ound)g(ha)m(ving)g(the)118 1291 y(same)h(form)e(as)j (the)f(lo)m(w)m(er)g(b)s(ound)g(in)f(\(1.8\))g(where)i Fo(\013)2139 1254 y Fj(\003)2212 1291 y Fs(is)f(replaced)g(b)m(y)h Fo(\013)2898 1254 y Fj(y)2967 1291 y Fs(\(Prop)s(osition)d(4.4\).)47 b(On)118 1407 y(the)42 b(other)g(hand,)i(w)m(e)e(pro)m(v)m(e)h(that)e Fo(\013)1548 1371 y Fj(\003)1630 1407 y Fl(\024)i Fo(\013)1813 1371 y Fj(y)1890 1407 y Fs(\(Lemma)d(3.1\).)70 b(F)-8 b(rom)40 b(these)i(t)m(w)m(o)h(facts)e(and)h(from)118 1523 y(Theorem)33 b(1.4,)f(w)m(e)i(deduce)g Fo(\013)1246 1487 y Fj(\003)1313 1523 y Fs(=)27 b Fo(\013)1479 1487 y Fj(y)1547 1523 y Fs(and)118 1717 y Fx(Theorem)37 b(1.6.)49 b Fe(Under)35 b(the)g(assumptions)f(H1)h(and)f(H1",)h(one)f(has)784 1907 y Fs(log)17 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))104 b Fl(\030)1263 1969 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1531 1907 y Fl(\000)p Fs(\(1)23 b(+)f Fo(\013)1879 1866 y Fj(y)1914 1907 y Fo(d)p Fs(\))p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\))1350 2200 y Fl(\030)1263 2262 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1531 2200 y Fl(\000)1625 2059 y Fm(\022)1699 2200 y Fs(1)22 b(+)1878 2132 y Fo(d)f Fl(\000)i Fo(\027)2104 2096 y Fj(y)p 1878 2177 263 4 v 1985 2268 a Fs(2)2150 2059 y Fm(\023)17 b(\022)2350 2132 y Fo(E)p 2323 2177 132 4 v 2323 2268 a(E)2395 2283 y Fj(\000)2465 2059 y Fm(\023)2538 2081 y Fi(1)p Fj(\000)p Fn(\027)2667 2058 y Fg(y)2700 2081 y Fn(=)p Fi(2)2791 2200 y Fs(log)2934 2059 y Fm(\022)3044 2132 y Fo(E)p 3017 2177 V 3017 2268 a(E)3089 2283 y Fj(\000)3158 2059 y Fm(\023)118 2082 y Fs(\(1.16\))331 2455 y(W)-8 b(e)32 b(see)h(that)f(the)g(lo)m(w)m(er)g(b)s(ound)g(in)f(\(1.8\))g(is) h(rather)f(univ)m(ersal.)44 b(There)33 b(exist)f(other)g(metho)s(ds)118 2571 y(to)41 b(obtain)e(this)i(b)s(ound,)i(for)d(instance)h(a)f(v)m (ersion)i(of)e(the)h(v)-5 b(ariational)37 b(metho)s(d)k(of)f([18)o(,)h (21])g(\(also)118 2687 y([13].)k(This)33 b(metho)s(d)g(also)f(giv)m(es) h(a)g(sharp)h(lo)m(w)m(er)f(b)s(ound)g(for)g(practically)e(all)g (asymptotic)h(form)m(ulae)118 2804 y(for)37 b(log)16 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))38 b(kno)m(wn)h(so)e(far.)57 b(This)37 b(is)g(true)g(in)g(particular)e(for)i(the)g(P)m(oisson)h(p)s (oten)m(tial)d(with)i(a)118 2920 y(non-negativ)m(e)j(single)g(site)g(p) s(oten)m(tial.)65 b(Th)m(us)42 b(it)d(is)h(the)g(upp)s(er)h(b)s(ound)g (that)f(has)h(b)s(een)g(requiring)118 3036 y(di\013eren)m(t)33 b(tec)m(hniques)h(dep)s(ending)f(on)g(the)g(sp)s(eci\014c)g(p)s(oten)m (tial)e(\(see)i(e.g.)44 b([3,)33 b(27)o(,)g(30)o(,)g(12)o(]\))331 3152 y(Assumptions)h(H1)g(and)g(H1")f(include)g(most)h(ph)m(ysically)f (in)m(teresting)h(cases)h(as,)g(for)e(example,)118 3333 y(the)h(3-dimensional)c(attractiv)m(e)j(screened)i(Coulom)m(b)d(p)s (oten)m(tial)g Fo(V)21 b Fs(\()p Fo(x)p Fs(\))29 b(=)g Fl(\000)3000 3266 y Fo(e)3045 3230 y Fj(\000j)p Fn(x)p Fj(j)p 3000 3311 184 4 v 3037 3402 a Fl(j)p Fo(x)p Fl(j)3194 3333 y Fs(.)45 b(In)34 b(this)e(case)j(w)m(e)118 3517 y(ha)m(v)m(e)f Fo(\013)406 3481 y Fj(\003)473 3517 y Fs(=)27 b Fo(\013)639 3481 y Fj(y)702 3517 y Fs(=)h(1)k(and)h Fo(E)1149 3532 y Fj(\000)1236 3517 y Fs(=)27 b(1)33 b(\(see)g([15]\);)f (th)m(us)1358 3723 y(log)17 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))105 b Fl(\030)1837 3785 y Fn(E)t Fj(!)p Fi(+)p Fj(1)2106 3723 y Fl(\000)p Fs(2)2232 3634 y Fl(p)p 2315 3634 79 4 v 89 x Fo(E)23 b Fs(log)16 b Fo(E)6 b(:)118 3954 y Fs(There)41 b(is)e(another)h(ph)m(ysically)g(in)m(teresting)f (case)i(that)e(has)i(not)e(b)s(een)i(discussed)g(here:)59 b(it)39 b(is)g(the)118 4071 y(case)34 b(of)e(p)s(oin)m(t)f(p)s(oten)m (tials.)43 b(Suc)m(h)33 b(p)s(oten)m(tials)f(will)e(b)s(e)j(studied)g (in)f(a)g(sequel)h(to)g(this)f(pap)s(er)h([13)o(].)1321 4367 y(2.)55 b Fr(Periodic)38 b(appr)n(o)n(xima)-7 b(tions)118 4541 y Fs(2.1.)56 b Fx(A)38 b(general)g(appro)m(ximation)f(result.)48 b Fs(In)34 b(this)f(subsection,)h(w)m(e)g(will)d(sho)m(w)k(ho)m(w)f(to) f(adapt)118 4657 y(the)c(p)s(erio)s(dic)e(appro)m(ximation)f(sc)m(heme) j(dev)m(elop)s(ed)h(in)d([12])h(to)g(the)h(case)g(of)f(un)m(b)s(ounded) i(p)s(oten)m(tials.)118 4773 y(The)37 b(main)e(new)i(tec)m(hnical)e (problems)h(will)d(come)j(from)f(the)i(fact)f(that,)g(in)g(the)g (presen)m(t)i(case,)g(the)118 4890 y(negativ)m(e)29 b(part)g(of)g(the)h (random)e(p)s(oten)m(tial)g(gro)m(ws)h(m)m(uc)m(h)h(faster)g(at)e (in\014nit)m(y)h(than)g(in)g(the)g(case)h(dealt)118 5006 y(with)i(in)g([12])g(or)h(when)g(the)g(single)f(site)g(p)s(oten)m(tial) f(is)h(b)s(ounded.)331 5122 y(In)i(the)h(sequel)h Fo(C)7 b Fs(\()p Fo(x;)17 b(l)r Fs(\))34 b(will)e(denote)k(the)e(cub)s(e)i(of) e(cen)m(ter)h Fo(x)g Fs(and)g(side)f(length)g Fo(l)r Fs(.)49 b(F)-8 b(or)34 b Fo(y)g Fl(2)d Fk(R)3851 5086 y Fn(d)118 5238 y Fs(and)i Fo(f)43 b Fs(a)32 b(function)g(on)h Fk(R)1064 5202 y Fn(d)1110 5238 y Fs(,)g Fo(f)1218 5253 y Fn(y)1292 5238 y Fs(will)d(denote)k(the)f(translate)f(of)g Fo(f)43 b Fs(b)m(y)33 b Fo(y)j Fs(i.e.)43 b Fo(f)3003 5253 y Fn(y)3044 5238 y Fs(\()p Fo(x)p Fs(\))28 b(=)g Fo(f)11 b Fs(\()p Fo(x)22 b Fl(\000)h Fo(y)t Fs(\).)118 5355 y(Fix)41 b Fo(!)t Fs(;)k(the)d(supp)s(ort)g(of)f(the)h(P)m(oisson) g(measure)g Fo(\026)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\))38 b(is)k(a)f(set)h(of)f(discrete)h(p)s(oin)m(ts)f(that)h(w)m(e)1989 5454 y Fp(9)p eop %%Page: 10 10 10 9 bop 118 241 a Fs(denote)33 b(b)m(y)h(\()p Fo(x)661 256 y Fn(k)704 241 y Fs(\()p Fo(!)t Fs(\)\))883 256 y Fn(k)r Fj(2)p Ff(N)1020 241 y Fs(.)43 b(W)-8 b(e)33 b(then)g(de\014ne)h (the)f(follo)m(wing)d(p)s(erio)s(dic)h(p)s(oten)m(tial)1049 533 y Fo(V)1106 548 y Fn(!)r(;n)1246 533 y Fs(=)1385 439 y Fm(X)1350 658 y Fn(\014)s Fj(2)p Fn(n)p Ff(Z)1533 639 y Fh(d)1580 333 y Fm(0)1580 512 y(@)1809 439 y(X)1667 654 y Fn(k)r Fi(;)h Fn(!)1802 666 y Fh(k)1840 654 y Fj(2)p Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(\))2111 533 y Fo(V)21 b Fs(\()p Fo(x)i Fl(\000)f Fo(\014)28 b Fl(\000)23 b Fo(x)2642 548 y Fn(k)2685 533 y Fs(\()p Fo(!)t Fs(\)\))2864 333 y Fm(1)2864 512 y(A)118 820 y Fs(and)33 b(the)g(corresp)s(onding)f(p)s (erio)s(dic)f(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erator)1608 991 y Fo(H)1689 1006 y Fn(!)r(;n)1829 991 y Fs(=)27 b Fl(\000)p Fs(\001)c(+)f Fo(V)2268 1006 y Fn(!)r(;n)2381 991 y Fo(:)-2290 b Fs(\(2.1\))118 1163 y(By)43 b(our)f(regularit)m(y)f (assumptions)g(on)i Fo(V)21 b Fs(,)45 b Fo(!)t Fs(-almost)39 b(surely)-8 b(,)45 b Fo(H)2595 1178 y Fn(!)r(;n)2749 1163 y Fs(is)d(relativ)m(ely)f(form)g(b)s(ounded)118 1281 y(with)35 b(resp)s(ect)h(to)e Fl(\000)p Fs(\001)h(\(see)h([2]\).) 50 b(It)35 b(is)f(essen)m(tially)h(self-adjoin)m(t)e(on)i Fl(C)2787 1245 y Fj(1)2781 1306 y Fi(0)2862 1281 y Fs(\()p Fk(R)2965 1245 y Fn(d)3012 1281 y Fs(\).)50 b(W)-8 b(e)35 b(can)g(de\014ne)h(the)118 1398 y(in)m(tegrated)28 b(densit)m(y)h(of)f (states)i Fo(N)1368 1413 y Fn(!)r(;n)1480 1398 y Fs(\()p Fo(E)6 b Fs(\))29 b(of)e(this)h(p)s(erio)s(dic)f(op)s(erator)h(b)m(y)h (the)g(same)f(limit)d(pro)s(cedure)118 1514 y(as)41 b(in)f(the)i(case)g (of)e(the)i(random)e(op)s(erator)g(\(see)i([22,)f(14)o(]\).)69 b(W)-8 b(e)41 b(will)e(no)m(w)j(compare)e Fo(N)10 b Fs(\()p Fo(E)c Fs(\))41 b(and)118 1630 y Fk(E)13 b Fs(\()p Fo(N)295 1645 y Fn(!)r(;n)413 1630 y Fs(\()p Fo(E)6 b Fs(\)\).)44 b(W)-8 b(e)33 b(pro)m(v)m(e)118 1813 y Fx(Lemma)k(2.1.)49 b Fe(F)-7 b(or)40 b(any)h Fo(\013)e Fl(2)h Fs(\(0)p Fo(;)17 b Fs(1\))p Fe(,)41 b(ther)-5 b(e)41 b(exists)f Fo(C)46 b(>)39 b Fs(1)h Fe(and)h Fo(\032)e(>)f Fs(0)j Fe(\(dep)-5 b(ending)39 b(only)i(on)f Fo(d)p Fe(\))118 1929 y(such)35 b(that,)g(for)g(any)f Fo(')28 b Fl(2)g(C)1155 1893 y Fj(1)1149 1954 y Fi(0)1230 1929 y Fs(\()p Fk(R)5 b Fs(\))p Fe(,)41 b(for)34 b Fo(k)d Fl(2)d Fk(N)1840 1893 y Fj(\003)1920 1929 y Fe(and)35 b Fo(n)28 b Fl(2)g Fk(N)2356 1893 y Fj(\003)2401 1929 y Fe(,)35 b(we)f(have)440 2165 y Fl(j)p Fk(E)12 b Fs(\(\()p Fo(';)17 b(dN)841 2180 y Fn(!)r(;n)959 2165 y Fs(\)\))22 b Fl(\000)h Fs(\()p Fo(';)17 b(dN)10 b Fs(\))p Fl(j)27 b(\024)h Fo(C)7 b(e)1762 2124 y Fn(C)e(\026)1864 2165 y Fo(n)1922 2124 y Fj(\000)p Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))p Fn(k)2210 2165 y Fo(e)2255 2124 y Fn(C)g(k)14 b Fi(log)e Fn(k)2603 2165 y Fs(sup)2609 2246 y Fn(x)p Fj(2)p Ff(R)2523 2306 y Fi(0)p Fj(\024)p Fn(j)t Fj(\024)p Fn(k)r Fi(+)p Fn(\032)2846 2020 y Fm(\014)2846 2080 y(\014)2846 2140 y(\014)2846 2200 y(\014)2879 2165 y Fs(\()p Fl(j)p Fo(x)p Fl(j)22 b Fs(+)g Fo(C)7 b Fs(\))3263 2124 y Fn(\032)p Fi(+)p Fn(k)3407 2097 y Fo(d)3458 2061 y Fn(j)3494 2097 y Fo(')p 3407 2142 152 4 v 3411 2233 a(d)3462 2204 y Fn(j)3498 2233 y Fo(x)3568 2165 y Fs(\()p Fo(x)p Fs(\))3699 2020 y Fm(\014)3699 2080 y(\014)3699 2140 y(\014)3699 2200 y(\014)3749 2165 y Fo(:)-3658 b Fs(\(2.2\))118 2472 y Fx(Remark)37 b(2.1.)49 b Fs(The)23 b(pro)s(of)f(sho)m(ws)h(that)f (the)h(constan)m(t)g Fo(C)29 b Fs(obtained)22 b(in)f(Lemma)g(2.1)h(is)f (indep)s(enden)m(t)118 2589 y(of)32 b(the)h(concen)m(tration)g Fo(\026)f Fs(of)g(the)h(P)m(oisson)g(pro)s(cess.)118 2771 y(Before)j(starting)f(the)h(pro)s(of)f(of)g(Lemma)f(2.1,)j(let)e (us)h(recall)e(some)i(basic)f(facts)h(ab)s(out)f(the)i(densit)m(y)118 2887 y(of)d(states)i(of)f(a)f(p)s(erio)s(dic)f(Sc)m(hr\177)-49 b(odinger)36 b(op)s(erator.)49 b(Let)35 b Fk(T)2298 2851 y Fj(\003)2298 2912 y Fn(n)2380 2887 y Fs(=)d Fk(R)2554 2851 y Fn(d)2600 2887 y Fo(=)p Fs(\(2)p Fo(\031)t(n)p Fk(Z)2922 2851 y Fn(d)2960 2887 y Fs(\).)50 b(F)-8 b(or)34 b Fo(\022)h Fl(2)d Fk(T)3493 2851 y Fj(\003)3493 2912 y Fn(n)3543 2887 y Fs(,)j(w)m(e)h(can)118 3004 y(consider)h Fo(H)583 3019 y Fn(!)r(;n;\022)787 3004 y Fs(the)g(unique)h (self-adjoin)m(t)d(op)s(erator)h(de\014ned)i(b)m(y)g(the)f(quadratic)f (form)g Fl(kr)p Fo(')p Fl(k)3758 2967 y Fi(2)3822 3004 y Fs(+)118 3120 y Fl(h)p Fo(V)214 3135 y Fn(!)264 3120 y Fo(';)17 b(')p Fl(i)46 b Fs(on)h Fo(L)737 3084 y Fi(2)737 3146 y Fn(\022)r(;)p Fi(lo)r(c)932 3120 y Fs(\(i.e.)86 b(the)48 b(set)g(of)e Fo(L)1722 3084 y Fi(2)1722 3146 y(lo)r(c)1815 3120 y Fs(-functions)h(that)f(satisfy)i(the)f(b)s (oundary)h(conditions)118 3249 y Fo(')p Fs(\()p Fo(x)14 b Fs(+)g Fo(n\015)5 b Fs(\))28 b(=)f Fo(e)707 3213 y Fn(in\022)r(\015)854 3249 y Fo(')p Fs(\()p Fo(x)p Fs(\))i(that)f(is)g (endo)m(w)m(ed)i(with)f(the)g(usual)f(scalar)g(pro)s(duct)g(on)h Fo(L)3236 3213 y Fi(2)3276 3249 y Fs(\()p Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\)\)\).)41 b(W)-8 b(e)118 3366 y(kno)m(w)38 b(that)e Fo(H)673 3381 y Fn(!)r(;n;\022)876 3366 y Fs(has)h(a)f (compact)g(resolv)m(en)m(t)i(\(see)f([23]\);)h(hence)g(its)e(sp)s (ectrum)h(is)f(discrete.)55 b(Let)118 3482 y(us)33 b(denote)h(its)e (eigen)m(v)-5 b(alues)32 b(b)m(y)946 3653 y Fo(E)1018 3668 y Fi(0)1058 3653 y Fs(\()p Fo(\022)s(;)17 b(!)t(;)g(n)p Fs(\))27 b Fl(\024)h Fo(E)1597 3668 y Fi(1)1636 3653 y Fs(\()p Fo(\022)s(;)17 b(!)t(;)g(n)p Fs(\))27 b Fl(\024)h(\001)17 b(\001)g(\001)26 b(\024)i Fo(E)2424 3668 y Fn(n)2472 3653 y Fs(\()p Fo(\022)s(;)17 b(!)t(;)g(n)p Fs(\))26 b Fl(\024)i Fo(:)17 b(:)g(:)118 3824 y Fs(The)34 b(functions)e(\()p Fo(\022)f Fl(7!)c Fo(E)1052 3839 y Fn(n)1099 3824 y Fs(\()p Fo(\022)s(;)17 b(!)t(;)g(n)p Fs(\)\))1472 3839 y Fn(n)p Fj(2)p Ff(N)1646 3824 y Fs(are)32 b(Lipsc)m(hitz)h(con)m(tin)m(uous)g (in)f Fo(\022)k Fs(and)c(one)h(has)975 3996 y Fo(E)1047 4011 y Fn(n)1094 3996 y Fs(\()p Fo(\022)s(;)17 b(!)t(;)g(n)p Fs(\))26 b Fl(!)i Fs(+)p Fl(1)k Fs(as)h Fo(n)27 b Fl(!)h Fs(+)p Fl(1)k Fs(\(uniformly)e(in)i Fo(\022)s Fs(\))p Fo(:)118 4167 y Fs(One)h(pro)m(v)m(es)h(that)f(the)g(IDS)f(of)g Fo(H)1393 4182 y Fn(!)r(;n)1538 4167 y Fs(satis\014es)1115 4394 y Fo(N)1193 4409 y Fn(!)r(;n)1306 4394 y Fs(\()p Fo(E)6 b Fs(\))27 b(=)1689 4326 y(1)p 1601 4371 225 4 v 1601 4462 a(\(2)p Fo(\031)t Fs(\))1785 4433 y Fn(d)1851 4299 y Fm(X)1855 4511 y Fn(n)p Fj(2)p Ff(N)2012 4258 y Fm(Z)2067 4484 y Fj(f)p Fn(\022)r Fj(2)p Ff(T)2234 4465 y Fg(\003)2234 4500 y Fh(n)2271 4484 y Fi(;)32 b Fn(E)2375 4492 y Fh(n)2418 4484 y Fi(\()p Fn(\022)r(;!)r(;n)p Fi(\))p Fj(\024)p Fn(E)t Fj(g)2802 4394 y Fo(d\022)-2780 b Fs(\(2.3\))118 4658 y(and)1187 4829 y(\()p Fo(';)17 b(dN)1462 4844 y Fn(!)r(;n)1574 4829 y Fs(\)\))28 b(=)f(tr\()p Fx(1)1951 4844 y Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(\))2163 4829 y Fo(')p Fs(\()p Fo(H)2346 4844 y Fn(!)r(;n)2458 4829 y Fs(\))p Fx(1)2552 4844 y Fn(C)g Fi(\(0)p Fn(;n)p Fi(\))2764 4829 y Fs(\))p Fo(:)118 5006 y Fs(for)32 b(an)m(y)h Fo(')28 b Fl(2)g(C)695 4970 y Fj(1)689 5031 y Fi(0)770 5006 y Fs(\()p Fk(R)5 b Fs(\))38 b(\(see)c([24,)e(23])g(or)h([28)o (]\).)118 5122 y(The)28 b(rest)f(of)f(this)g(subsection)i(will)c(b)s(e) j(dev)m(oted)h(to)f(the)g(pro)s(of)e(of)h(Lemma)g(2.1.)41 b(T)-8 b(o)26 b(pro)m(v)m(e)i(this)f(result)118 5238 y(w)m(e)38 b(will)c(need)j(the)g(form)m(ula)e(giv)m(en)h(in)g(Theorem)h (1.3.)54 b(The)38 b(pro)s(of)d(of)h(this)g(form)m(ula)f(will)f(b)s(e)i (giv)m(en)118 5355 y(at)e(the)h(end)f(of)g(the)h(section.)48 b(Let)34 b(us)h(pro)s(ceed)g(with)f(the)h(pro)s(of)e(of)h(Lemma)e(2.1.) 48 b(Fix)34 b Fo(')c Fl(2)h(C)3648 5318 y Fj(1)3642 5379 y Fi(0)3723 5355 y Fs(\()p Fk(R)5 b Fs(\).)1969 5454 y Fp(10)p eop %%Page: 11 11 11 10 bop 118 241 a Fs(W)-8 b(e)41 b(w)m(an)m(t)g(to)e(estimate)h Fl(j)p Fk(E)12 b Fs(\(\()p Fo(';)17 b(dN)1466 256 y Fn(!)r(;n)1584 241 y Fs(\)\))27 b Fl(\000)h Fs(\()p Fo(';)17 b(dN)10 b Fs(\))p Fl(j)p Fs(.)65 b(The)41 b(computation)d(done)j(in)e(the)i (pro)s(of)e(of)118 357 y(Theorem)33 b(5.1)f(in)g([12])g(giv)m(es)1071 610 y Fk(E)12 b Fs(\(\()p Fo(';)17 b(dN)1444 625 y Fn(!)r(;n)1563 610 y Fs(\)\))27 b(=)h Fk(E)12 b Fs(\(tr)6 b(\()p Fx(1)2044 626 y Fn(C)f Fi(\(0)p Fn(;)p Fi(1\))2248 610 y Fo(')p Fs(\()p Fo(H)2431 625 y Fn(!)r(;n)2544 610 y Fs(\))p Fx(1)2638 626 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))2842 610 y Fs(\)\))p Fo(:)118 877 y Fs(Here)48 b(w)m(e)g(used)h(the)f(fact)f (that)g(the)h(P)m(oisson)f(pro)s(cess)i(is)e Fk(Z)2461 841 y Fn(d)2498 877 y Fs(-homogeneous.)88 b(Notice)47 b(that)g(our)118 993 y(regularit)m(y)35 b(assumptions)i(on)f Fo(V)1321 1008 y Fn(!)r(;n)1471 993 y Fs(are)g(w)m(eak)m(er)j(than)d (the)h(one)g(needed)h(in)e([12])g(and)h([22].)55 b(Indeed)118 1110 y Fo(!)42 b Fs(almost)c(surely)-8 b(,)41 b Fo(V)912 1125 y Fn(!)r(;n)1064 1110 y Fs(is)d(only)h(relativ)m(ely)f(form)f(b)s (ounded)j(with)f(resp)s(ect)h(to)f Fl(\000)p Fs(\001)h(with)e(relativ)m (e)118 1226 y(b)s(ound)33 b(0)f(\(see)h(Lemma)e(5.1\).)43 b(Nev)m(ertheless)35 b(the)d(pro)s(ofs)g(of)g(the)h(relev)-5 b(an)m(t)32 b(results)h(in)e(these)j(pap)s(ers)118 1342 y(extend)g(easily)e(to)g(the)h(case)h(of)e(relativ)m(ely)f(form)h(b)s (ounded)h(p)s(erturbations)f(of)g Fl(\000)p Fs(\001.)118 1458 y(No)m(w)40 b(w)m(e)h(only)f(ha)m(v)m(e)h(to)e(estimate)g Fl(j)p Fk(E)12 b Fs(\(tr)6 b(\()p Fx(1)1778 1474 y Fn(C)f Fi(\(0)p Fn(;)p Fi(1\))1982 1458 y Fs(\()p Fo(')p Fs(\()p Fo(H)2203 1473 y Fn(!)r(;n)2316 1458 y Fs(\))27 b Fl(\000)g Fo(')p Fs(\()p Fo(H)2668 1473 y Fn(!)2718 1458 y Fs(\)\)\)\))p Fl(j)p Fs(.)65 b(This)40 b(is)f(done)h(with)g(an)118 1574 y(in)m(tegral)34 b(represen)m(tation)h(of)g Fo(')p Fs(\()p Fo(H)8 b Fs(\))34 b(using)h(an)g(almost)e(analytic)h(extension) h(of)g Fo(')p Fs(.)51 b(Pic)m(k)35 b Fo(')d Fl(2)g(S)7 b Fs(\()p Fk(R)f Fs(\))118 1691 y(\(the)35 b(Sc)m(h)m(w)m(artz)i(space) f(of)f(rapidly)f(decreasing)h(functions\).)50 b(An)36 b(almost)d(analytic)h(extension)h(of)g Fo(')118 1807 y Fs(is)d(a)g(function)48 b(~)-64 b Fo(')27 b Fs(:)60 b Fk(C)54 b Fl(!)27 b Fk(C)59 b Fs(satisfying)218 1992 y(1.)41 b(for)32 b Fo(z)h Fl(2)28 b Fk(R)5 b Fs(,)54 b(~)-65 b Fo(')p Fs(\()p Fo(z)t Fs(\))28 b(=)g Fo(')p Fs(\()p Fo(z)t Fs(\),)218 2108 y(2.)41 b(supp\()16 b(~)-65 b Fo(')q Fs(\))28 b Fl(\032)g(f)p Fo(z)k Fl(2)c Fk(C)20 b Fs(;)55 b Fl(j)p Fs(Im)o(\()p Fo(z)t Fs(\))p Fl(j)28 b Fo(<)g Fs(1)p Fl(g)p Fs(,)218 2224 y(3.)57 b(~)-65 b Fo(')28 b Fl(2)g(S)7 b Fs(\()p Fl(f)p Fo(z)33 b Fl(2)28 b Fk(C)20 b Fs(;)55 b Fl(j)p Fs(Im)o(\()p Fo(z)t Fs(\))p Fl(j)28 b Fo(<)g Fs(1)p Fl(g)p Fs(\),)218 2394 y(4.)41 b(The)33 b(family)c(of)i(functions)h Fo(x)c Fl(7!)1582 2327 y Fo(@)22 b Fs(~)-66 b Fo(')p 1582 2371 121 4 v 1589 2462 a(@)p 1645 2408 50 4 v 5 w(z)1713 2394 y Fs(\()p Fo(x)20 b Fs(+)g Fo(iy)t Fs(\))g Fl(\001)g(j)p Fo(y)t Fl(j)2221 2353 y Fj(\000)p Fn(n)2353 2394 y Fs(\(for)31 b(0)c Fo(<)h Fl(j)p Fo(y)t Fl(j)e Fo(<)h Fs(1\))32 b(is)f(b)s(ounded)h (in)f Fl(S)7 b Fs(\()p Fk(R)f Fs(\))335 2540 y(for)32 b(an)m(y)h Fo(n)28 b Fl(2)g Fk(N)9 b Fs(.)118 2725 y(Suc)m(h)30 b(extensions)f(alw)m(a)m(ys)g(exist)g(for)f Fo(')g Fl(2)g(S)36 b Fs(\(see)29 b([19]\))f(and,)i(one)f(has)f(the)h(follo)m(wing)d (estimates:)41 b(for)118 2841 y Fo(n)28 b Fl(\025)g Fs(0,)33 b Fo(\013)28 b Fl(\025)g Fs(0,)k Fo(\014)i Fl(\025)28 b Fs(0,)k(there)i(exists)f Fo(C)1612 2856 y Fn(n;\013;\014)1814 2841 y Fo(>)27 b Fs(0)33 b(suc)m(h)h(that)557 3169 y(sup)502 3255 y Fi(0)p Fn(<)p Fj(j)p Fn(y)r Fj(j\024)p Fi(1)776 3169 y Fs(sup)782 3251 y Fn(x)p Fj(2)p Ff(R)939 3024 y Fm(\014)939 3084 y(\014)939 3144 y(\014)939 3204 y(\014)972 3169 y Fo(x)1027 3128 y Fn(\013)1115 3101 y Fo(@)1171 3065 y Fn(\014)p 1087 3146 160 4 v 1087 3237 a Fo(@)5 b(x)1198 3208 y Fn(\014)1273 3028 y Fm(\022)1347 3169 y Fl(j)p Fo(y)t Fl(j)1455 3128 y Fj(\000)p Fn(n)1577 3169 y Fl(\001)1637 3101 y Fo(@)21 b Fs(~)-65 b Fo(')p 1637 3146 121 4 v 1644 3237 a(@)p 1700 3182 50 4 v 5 w(z)1768 3169 y Fs(\()p Fo(x)22 b Fs(+)g Fo(iy)t Fs(\))2104 3028 y Fm(\023)2177 3024 y(\014)2177 3084 y(\014)2177 3144 y(\014)2177 3204 y(\014)2238 3169 y Fl(\024)28 b Fo(C)2413 3184 y Fn(n;\013;\014)2706 3169 y Fs(sup)2604 3256 y Fn(\014)2647 3232 y Fg(0)2669 3256 y Fj(\024)p Fn(n)p Fi(+)p Fn(\014)s Fi(+2)2695 3329 y Fn(\013)2740 3305 y Fg(0)2763 3329 y Fj(\024)p Fn(\013)2972 3169 y Fs(sup)2978 3251 y Fn(x)p Fj(2)p Ff(R)3135 3024 y Fm(\014)3135 3084 y(\014)3135 3144 y(\014)3135 3204 y(\014)3168 3169 y Fo(x)3223 3128 y Fn(\013)3268 3104 y Fg(0)3306 3101 y Fo(@)3362 3065 y Fn(\014)3405 3042 y Fg(0)3432 3101 y Fo(')p 3306 3146 191 4 v 3310 3237 a(@)5 b(x)3421 3208 y Fn(\014)3464 3189 y Fg(0)3506 3169 y Fs(\()p Fo(x)p Fs(\))3637 3024 y Fm(\014)3637 3084 y(\014)3637 3144 y(\014)3637 3204 y(\014)3687 3169 y Fo(:)-3596 b Fs(\(2.4\))118 3570 y(Let)48 b(~)-65 b Fo(')32 b Fs(b)s(e)h(an)f(almost)e(analytic)h (extension)i(of)f(\()p Fo(i)21 b Fs(+)g Fo(x)p Fs(\))2165 3534 y Fn(q)2204 3570 y Fo(')p Fs(\()p Fo(x)p Fs(\))32 b(for)g(some)g Fo(q)f(>)d(d=)p Fs(2,)j Fo(q)36 b Fs(in)m(teger.)43 b(Then,)118 3686 y(b)m(y)34 b([5)o(])f(and)g([8],)f(w)m(e)i(kno)m(w)g (that,)e(for)g(an)m(y)h Fo(n)g Fs(and)g Fo(!)e Fl(2)d Fs(\012,)33 b(the)g(follo)m(wing)d(form)m(ula)h(holds,)777 3993 y Fo(')p Fs(\()p Fo(H)960 4008 y Fn(!)r(;n)1073 3993 y Fs(\))c(=)1289 3926 y Fo(i)p 1252 3971 108 4 v 1252 4062 a Fs(2)p Fo(\031)1386 3858 y Fm(Z)1441 4083 y Ff(C)1520 3926 y Fo(@)22 b Fs(~)-66 b Fo(')p 1520 3971 121 4 v 1527 4062 a(@)p 1583 4007 50 4 v 5 w(z)1651 3993 y Fs(\()p Fo(z)t Fs(\))23 b Fl(\001)f Fs(\()p Fo(i)g Fs(+)g Fo(H)2121 4008 y Fn(!)r(;n)2234 3993 y Fs(\))2272 3952 y Fj(\000)p Fn(q)2364 3993 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)2655 4008 y Fn(!)r(;n)2768 3993 y Fs(\))2806 3952 y Fj(\000)p Fi(1)2900 3993 y Fo(dz)j Fl(^)d Fo(d)p 3162 3939 V(z)t(:)-3120 b Fs(\(2.5\))118 4308 y(F)-8 b(or)32 b Fo(q)f(>)d(d=)p Fs(2,)k Fx(1)735 4323 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))939 4308 y Fs(\()p Fo(i)22 b Fs(+)g Fo(H)1211 4323 y Fn(!)r(;n)1324 4308 y Fs(\))1362 4272 y Fj(\000)p Fn(q)1455 4308 y Fs(\()p Fo(z)27 b Fl(\000)22 b Fo(H)1745 4323 y Fn(!)r(;n)1858 4308 y Fs(\))1896 4272 y Fj(\000)p Fi(1)2023 4308 y Fs(is)32 b(trace-class)g(and)h(w)m(e)h(ha)m(v)m(e)118 4599 y(\(2.6\))83 b(tr)494 4518 y Fm(\000)540 4599 y Fx(1)596 4615 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))800 4599 y Fo(')p Fs(\()p Fo(H)983 4614 y Fn(!)r(;n)1096 4599 y Fs(\))p Fx(1)1190 4615 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))1394 4518 y Fm(\001)994 4812 y Fs(=)1144 4745 y Fo(i)p 1107 4789 108 4 v 1107 4880 a Fs(2)p Fo(\031)1241 4676 y Fm(Z)1297 4902 y Ff(C)1375 4745 y Fo(@)22 b Fs(~)-66 b Fo(')p 1375 4789 121 4 v 1382 4880 a(@)p 1438 4826 50 4 v 5 w(z)1506 4812 y Fs(\()p Fo(z)t Fs(\))23 b Fl(\001)f Fs(tr)1796 4731 y Fm(\000)1842 4812 y Fx(1)1898 4828 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))2102 4812 y Fs(\()p Fo(i)23 b Fs(+)f Fo(H)2375 4827 y Fn(!)r(;n)2487 4812 y Fs(\))2525 4771 y Fj(\000)p Fn(q)2618 4812 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)2909 4827 y Fn(!)r(;n)3021 4812 y Fs(\))3059 4771 y Fj(\000)p Fi(1)3153 4812 y Fx(1)3209 4828 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3414 4731 y Fm(\001)3476 4812 y Fo(dz)27 b Fl(^)22 b Fo(d)p 3738 4757 V(z)5 b(:)118 5122 y Fs(By)40 b(Lemma)e(5.1)g(and)i(section)f(B.12)g(in)f([26],)j(w)m (e)f(kno)m(w)h(that,)f Fo(!)t Fs(-almost)d(surely)i Fo(H)3347 5137 y Fn(!)3437 5122 y Fs(is)g(essen)m(tial)118 5238 y(self-adjoin)m(t)d(on)i Fl(C)824 5202 y Fj(1)818 5263 y Fi(0)899 5238 y Fs(\()p Fk(R)1002 5202 y Fn(d)1049 5238 y Fs(\))f(and)h(that)f Fx(1)1591 5254 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))1796 5238 y Fs(\()p Fo(i)25 b Fs(+)h Fo(H)2075 5253 y Fn(!)2125 5238 y Fs(\))2163 5202 y Fj(\000)p Fn(q)2256 5238 y Fs(\()p Fo(z)k Fl(\000)c Fo(H)2553 5253 y Fn(!)2604 5238 y Fs(\))2642 5202 y Fj(\000)p Fi(1)2773 5238 y Fs(is)38 b(trace-class.)58 b(Hence)39 b(\(2.5\))118 5355 y(and)33 b(\(2.6\))f(also)f(hold)h(for)g Fo(H)1183 5370 y Fn(!)1234 5355 y Fs(.)1969 5454 y Fp(11)p eop %%Page: 12 12 12 11 bop 118 241 a Fs(W)-8 b(e)33 b(are)g(no)m(w)g(going)e(to)h(use)i (Lemma)d(5.1.)43 b(W)-8 b(e)33 b(pic)m(k)g Fo(p)2166 205 y Fj(0)2217 241 y Fl(2)28 b Fs(\()p Fo(p)p Fs(\()p Fo(d)p Fs(\))p Fo(;)17 b(p)p Fs(\))32 b(and)g Fo(b)c Fs(=)g(1,)k(and)h(compute)565 408 y Fl(j)p Fk(E)13 b Fs(\()5 b(tr\()p Fx(1)867 423 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))1072 408 y Fs(\()p Fo(')p Fs(\()p Fo(H)1293 423 y Fn(!)r(;n)1405 408 y Fs(\))22 b Fl(\000)h Fo(')p Fs(\()p Fo(H)1748 423 y Fn(!)1798 408 y Fs(\)\)\)\))p Fl(j)725 629 y(\024)870 561 y Fs(1)p 840 606 108 4 v 840 697 a(2)p Fo(\031)975 493 y Fm(Z)1030 719 y Ff(C)1099 484 y Fm(\014)1099 544 y(\014)1099 604 y(\014)1099 664 y(\014)1142 561 y Fo(@)e Fs(~)-65 b Fo(')p 1142 606 121 4 v 1149 697 a(@)p 1205 642 50 4 v 5 w(z)1273 629 y Fs(\()p Fo(z)t Fs(\))1398 484 y Fm(\014)1398 544 y(\014)1398 604 y(\014)1398 664 y(\014)1448 629 y Fk(E)1525 548 y Fm(\000)1577 544 y(\015)1577 604 y(\015)1632 629 y Fx(1)1688 644 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))1909 548 y Fm(\000)1955 629 y Fs(\()p Fo(i)22 b Fs(+)g Fo(H)2227 644 y Fn(!)r(;n)2340 629 y Fs(\))2378 588 y Fj(\000)p Fn(q)2470 629 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)2761 644 y Fn(!)r(;n)2874 629 y Fs(\))2912 588 y Fj(\000)p Fi(1)1689 845 y Fl(\000)p Fs(\()p Fo(i)g Fs(+)f Fo(H)2039 860 y Fn(!)2089 845 y Fs(\))2127 804 y Fj(\000)p Fn(q)2220 845 y Fs(\()p Fo(z)27 b Fl(\000)22 b Fo(H)2510 860 y Fn(!)2560 845 y Fs(\))2598 804 y Fj(\000)p Fi(1)2693 765 y Fm(\001)2755 845 y Fx(1)2811 861 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3015 761 y Fm(\015)3015 820 y(\015)3071 884 y Fi(tr)3130 765 y Fm(\001)3192 845 y Fo(dxdy)725 1066 y Fl(\024)830 972 y Fm(X)838 1184 y Fn(k)r Fj(\025)p Fi(1)1030 999 y Fs(1)p 1001 1043 108 4 v 1001 1135 a(2)p Fo(\031)1135 931 y Fm(Z)1190 1156 y Ff(C)1259 922 y Fm(\014)1259 982 y(\014)1259 1041 y(\014)1259 1101 y(\014)1302 999 y Fo(@)22 b Fs(~)-66 b Fo(')p 1302 1043 121 4 v 1309 1135 a(@)p 1365 1080 50 4 v 5 w(z)1433 1066 y Fs(\()p Fo(z)t Fs(\))1558 922 y Fm(\014)1558 982 y(\014)1558 1041 y(\014)1558 1101 y(\014)1608 1066 y Fk(E)1686 986 y Fm(\000)1737 1066 y Fx(1)1793 1082 y Fj(f)p Fn(!)r Fi(;)33 b Fn(V)1968 1090 y Fh(!)2013 1082 y Fj(2)p Fi(\012)2111 1094 y Fh(k)2149 1082 y Fi(\()p Fn(\013;)p Fi(1)p Fn(;p)2332 1063 y Fg(0)2354 1082 y Fi(\))p Fj(n)p Fi(\012)2467 1094 y Fh(k)q Fg(\000)p Fd(1)2584 1082 y Fi(\()p Fn(\013;)p Fi(1)p Fn(;p)2767 1063 y Fg(0)2789 1082 y Fi(\))p Fj(g)2856 1066 y Fo(K)7 b Fs(\()p Fo(z)t(;)17 b(!)t Fs(\))3180 986 y Fm(\001)3242 1066 y Fo(dxdy)118 794 y Fs(\(2.7\))118 1342 y(where)415 1517 y Fo(K)7 b Fs(\()p Fo(z)t(;)17 b(!)t Fs(\))28 b(=)870 1432 y Fm(\015)870 1492 y(\015)926 1517 y Fx(1)982 1533 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))1202 1436 y Fm(\000)1248 1517 y Fs(\()p Fo(i)22 b Fs(+)g Fo(H)1520 1532 y Fn(!)r(;n)1633 1517 y Fs(\))1671 1476 y Fj(\000)p Fn(q)1764 1517 y Fs(\()p Fo(z)27 b Fl(\000)22 b Fo(H)2054 1532 y Fn(!)r(;n)2167 1517 y Fs(\))2205 1476 y Fj(\000)p Fi(1)2321 1517 y Fl(\000)h Fs(\()p Fo(i)f Fs(+)g Fo(H)2693 1532 y Fn(!)2744 1517 y Fs(\))2782 1476 y Fj(\000)p Fn(q)2875 1517 y Fs(\()p Fo(z)k Fl(\000)d Fo(H)3165 1532 y Fn(!)3215 1517 y Fs(\))3253 1476 y Fj(\000)p Fi(1)3348 1436 y Fm(\001)3410 1517 y Fx(1)3466 1533 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3670 1432 y Fm(\015)3670 1492 y(\015)3725 1556 y Fi(tr)118 1698 y Fs(Here)33 b Fl(k)22 b(\001)g(k)520 1713 y Fi(tr)612 1698 y Fs(denotes)33 b(the)g(trace-class)g(norm.)118 1814 y(W)-8 b(e)29 b(need)h(to)e(estimate)g Fx(1)1064 1830 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))1268 1814 y Fs(\(\()p Fo(i)14 b Fs(+)g Fo(H)1562 1829 y Fn(!)r(;n)1676 1814 y Fs(\))1714 1778 y Fj(\000)p Fn(q)1806 1814 y Fs(\()p Fo(z)19 b Fl(\000)14 b Fo(H)2080 1829 y Fn(!)r(;n)2194 1814 y Fs(\))2232 1778 y Fj(\000)p Fi(1)2340 1814 y Fl(\000)g Fs(\()p Fo(i)g Fs(+)g Fo(H)2687 1829 y Fn(!)2739 1814 y Fs(\))2777 1778 y Fj(\000)p Fn(q)2870 1814 y Fs(\()p Fo(z)19 b Fl(\000)14 b Fo(H)3144 1829 y Fn(!)3195 1814 y Fs(\))3233 1778 y Fj(\000)p Fi(1)3327 1814 y Fs(\))p Fx(1)3421 1830 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3654 1814 y Fs(under)118 1931 y(the)28 b(assumption)f Fo(V)849 1946 y Fn(!)927 1931 y Fl(2)h Fs(\012)1091 1946 y Fn(k)1135 1931 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)1422 1895 y Fj(0)1444 1931 y Fs(\).)41 b(Therefore)29 b(w)m(e)g(imitate)c (the)j(metho)s(d)f(used)i(in)e([12].)42 b(W)-8 b(e)28 b(write)473 2023 y Fm(\015)473 2083 y(\015)528 2108 y Fx(1)584 2123 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))788 2108 y Fs(\(\()p Fo(i)23 b Fs(+)f Fo(H)1099 2123 y Fn(!)r(;n)1211 2108 y Fs(\))1249 2067 y Fj(\000)p Fn(q)1342 2108 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)1633 2123 y Fn(!)r(;n)1745 2108 y Fs(\))1783 2067 y Fj(\000)p Fi(1)1900 2108 y Fl(\000)f Fs(\()p Fo(i)h Fs(+)f Fo(H)2272 2123 y Fn(!)2322 2108 y Fs(\))2360 2067 y Fj(\000)p Fn(q)2453 2108 y Fs(\()p Fo(z)27 b Fl(\000)22 b Fo(H)2743 2123 y Fn(!)2793 2108 y Fs(\))2831 2067 y Fj(\000)p Fi(1)2926 2108 y Fs(\))p Fx(1)3020 2123 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3224 2023 y Fm(\015)3224 2083 y(\015)3279 2147 y Fi(tr)3366 2108 y Fl(\024)28 b Fo(A)22 b Fs(+)g Fo(B)-3620 b Fs(\(2.8\))118 2283 y(where)513 2436 y Fo(A)28 b Fs(=)718 2351 y Fm(\015)718 2411 y(\015)773 2436 y Fx(1)829 2451 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))1050 2355 y Fm(\000)1096 2436 y Fs(\()p Fo(z)27 b Fl(\000)22 b Fo(H)1386 2451 y Fn(!)r(;n)1499 2436 y Fs(\))1537 2394 y Fj(\000)p Fi(1)1653 2436 y Fl(\000)h Fs(\()p Fo(z)k Fl(\000)22 b Fo(H)2043 2451 y Fn(!)2094 2436 y Fs(\))2132 2394 y Fj(\000)p Fi(1)2226 2355 y Fm(\001)2288 2436 y Fs(\()p Fo(i)g Fs(+)g Fo(H)2560 2451 y Fn(!)2611 2436 y Fs(\))2649 2394 y Fj(\000)p Fn(q)2742 2436 y Fx(1)2798 2451 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3002 2351 y Fm(\015)3002 2411 y(\015)3057 2475 y Fi(tr)614 2597 y Fs(=)718 2512 y Fm(\015)718 2572 y(\015)773 2597 y Fx(1)829 2612 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))1050 2516 y Fm(\000)1096 2597 y Fs(\()p Fo(z)27 b Fl(\000)22 b Fo(H)1386 2612 y Fn(!)r(;n)1499 2597 y Fs(\))1537 2556 y Fj(\000)p Fi(1)1648 2597 y Fs(\()p Fo(V)1743 2612 y Fn(!)r(;n)1877 2597 y Fl(\000)h Fo(V)2034 2612 y Fn(!)2084 2597 y Fs(\))17 b(\()p Fo(z)27 b Fl(\000)22 b Fo(H)2429 2612 y Fn(!)2480 2597 y Fs(\))2518 2556 y Fj(\000)p Fi(1)2612 2516 y Fm(\001)2674 2597 y Fs(\()p Fo(i)g Fs(+)g Fo(H)2946 2612 y Fn(!)2997 2597 y Fs(\))3035 2556 y Fj(\000)p Fn(q)3128 2597 y Fx(1)3184 2612 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3388 2512 y Fm(\015)3388 2572 y(\015)3443 2636 y Fi(tr)118 2772 y Fs(and)444 2925 y Fo(B)33 b Fs(=)655 2840 y Fm(\015)655 2900 y(\015)710 2925 y Fx(1)766 2940 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))970 2925 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)1261 2940 y Fn(!)r(;n)1373 2925 y Fs(\))1411 2883 y Fj(\000)p Fi(1)1522 2844 y Fm(\000)1568 2925 y Fs(\()p Fo(i)f Fs(+)g Fo(H)1840 2940 y Fn(!)r(;n)1953 2925 y Fs(\))1991 2883 y Fj(\000)p Fn(q)2106 2925 y Fl(\000)h Fs(\()p Fo(i)f Fs(+)g Fo(H)2478 2940 y Fn(!)2528 2925 y Fs(\))2566 2883 y Fj(\000)p Fn(q)2659 2844 y Fm(\001)2721 2925 y Fx(1)2777 2940 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))2981 2840 y Fm(\015)2981 2900 y(\015)3037 2964 y Fi(tr)551 3182 y Fs(=)655 3008 y Fm(\015)655 3068 y(\015)655 3127 y(\015)655 3187 y(\015)655 3247 y(\015)710 3182 y Fx(1)766 3198 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))970 3182 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)1261 3197 y Fn(!)r(;n)1373 3182 y Fs(\))1411 3141 y Fj(\000)p Fi(1)1532 3054 y Fn(q)r Fj(\000)p Fi(1)1522 3088 y Fm(X)1538 3300 y Fn(l)q Fi(=1)1666 3182 y Fs(\()p Fo(i)f Fs(+)g Fo(H)1938 3197 y Fn(!)r(;n)2051 3182 y Fs(\))2089 3141 y Fn(l)q Fj(\000)p Fn(q)2220 3182 y Fs(\()p Fo(V)2315 3197 y Fn(!)r(;n)2450 3182 y Fl(\000)h Fo(V)2607 3197 y Fn(!)2657 3182 y Fs(\))17 b(\()p Fo(i)22 b Fs(+)g Fo(H)2984 3197 y Fn(!)3034 3182 y Fs(\))3072 3141 y Fj(\000)p Fn(l)3153 3182 y Fx(1)3209 3198 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3413 3008 y Fm(\015)3413 3068 y(\015)3413 3127 y(\015)3413 3187 y(\015)3413 3247 y(\015)3469 3314 y Fi(tr)3544 3182 y Fo(:)118 3451 y Fs(The)34 b(estimates)f(for)f Fo(A)h Fs(and)g Fo(B)38 b Fs(b)s(eing)33 b(obtained)f(essen)m(tially)h (in)f(the)i(same)e(w)m(a)m(y)-8 b(,)35 b(w)m(e)f(will)c(write)j(the)118 3567 y(details)f(for)g Fo(A)g Fs(only)-8 b(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)201 3787 y Fo(A)28 b Fl(\024)430 3693 y Fm(X)407 3912 y Fn(\015)447 3893 y Fg(0)470 3912 y Fj(2)p Ff(Z)567 3893 y Fh(d)614 3703 y Fm(\015)614 3762 y(\015)669 3787 y Fx(1)725 3803 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))929 3787 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)1220 3802 y Fn(!)r(;n)1333 3787 y Fs(\))1371 3746 y Fj(\000)p Fi(1)1465 3787 y Fs(\()p Fo(V)1560 3802 y Fn(!)r(;n)1695 3787 y Fl(\000)f Fo(V)1851 3802 y Fn(!)1901 3787 y Fs(\))p Fx(1)1995 3803 y Fn(C)5 b Fi(\()p Fn(\015)2117 3784 y Fg(0)2141 3803 y Fn(;)p Fi(1\))2227 3703 y Fm(\015)2227 3762 y(\015)2282 3826 y Fj(L)p Fi(\()p Fn(L)2406 3807 y Fd(2)2441 3826 y Fi(\))2495 3787 y Fl(\001)2104 4057 y(\001)2154 3972 y Fm(\015)2154 4032 y(\015)2209 4057 y Fx(1)2265 4073 y Fn(C)g Fi(\()p Fn(\015)2387 4054 y Fg(0)2410 4073 y Fn(;)p Fi(1\))2497 4057 y Fs(\()p Fo(z)27 b Fl(\000)22 b Fo(H)2787 4072 y Fn(!)2838 4057 y Fs(\))2876 4016 y Fj(\000)p Fi(1)2970 4057 y Fs(\()p Fo(i)g Fs(+)g Fo(H)3242 4072 y Fn(!)3292 4057 y Fs(\))3330 4016 y Fj(\000)p Fn(q)3423 4057 y Fx(1)3479 4073 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))3684 3972 y Fm(\015)3684 4032 y(\015)3739 4096 y Fi(tr)118 4238 y Fs(By)35 b(Lemma)e(5.4)i(applied)e(to)h Fo(H)1328 4253 y Fn(!)1413 4238 y Fs(and)h(to)f Fo(H)1807 4253 y Fn(!)r(;n)1920 4238 y Fs(,)h(for)f Fo(V)2190 4253 y Fn(!)2271 4238 y Fl(2)e Fs(\012)2439 4253 y Fn(k)2482 4238 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)2769 4202 y Fj(0)2791 4238 y Fs(\),)35 b(w)m(e)h(kno)m(w)g(that,)f(for)f (some)118 4357 y Fo(\017)28 b(>)g Fs(0,)k Fo(\032)c(>)g Fs(0)k(and)h Fo(C)h(>)28 b Fs(0,)k(for)g(all)f Fo(\015)1507 4321 y Fj(0)1558 4357 y Fl(2)d Fk(Z)1721 4321 y Fn(d)1759 4357 y Fs(,)k(w)m(e)i(ha)m(v)m(e)433 4501 y Fm(\015)433 4561 y(\015)488 4586 y Fx(1)544 4601 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))748 4586 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)1039 4601 y Fn(!)r(;n)1152 4586 y Fs(\))1190 4545 y Fj(\000)p Fi(1)1284 4586 y Fs(\()p Fo(V)1379 4601 y Fn(!)r(;n)1514 4586 y Fl(\000)f Fo(V)1670 4601 y Fn(!)1721 4586 y Fs(\))p Fx(1)1815 4601 y Fn(C)5 b Fi(\()p Fn(\015)1937 4582 y Fg(0)1960 4601 y Fn(;)p Fi(1\))2046 4501 y Fm(\015)2046 4561 y(\015)2101 4625 y Fj(L)p Fi(\()p Fn(L)2225 4606 y Fd(2)2260 4625 y Fi(\))2319 4586 y Fl(\024)28 b Fo(C)2511 4518 y Fs(\(1)22 b(+)g Fl(j)p Fo(\015)2802 4482 y Fj(0)2825 4518 y Fl(j)p Fs(\))2891 4482 y Fn(\013)p 2511 4563 429 4 v 2550 4654 a Fo(\021)t Fs(\()p Fo(z)t(;)17 b(K)7 b Fs(\))2861 4625 y Fn(\032)2950 4586 y Fo(e)2995 4545 y Fj(\000)p Fn(\017)p Fj(\001)p Fn(\021)r Fi(\()p Fn(z)s(;K)e Fi(\))p Fj(j)p Fn(\015)3371 4521 y Fg(0)3393 4545 y Fj(j)3413 4521 y Fd(1)p Fg(\000)p Fh(\013)3539 4586 y Fo(;)548 4791 y Fm(\015)548 4851 y(\015)604 4875 y Fx(1)660 4891 y Fn(C)g Fi(\()p Fn(\015)782 4872 y Fg(0)805 4891 y Fn(;)p Fi(1\))891 4875 y Fs(\()p Fo(z)27 b Fl(\000)c Fo(H)1182 4890 y Fn(!)1232 4875 y Fs(\))1270 4834 y Fj(\000)p Fi(1)1364 4875 y Fs(\()p Fo(i)g Fs(+)f Fo(H)1637 4890 y Fn(!)1687 4875 y Fs(\))1725 4834 y Fj(\000)p Fn(q)1818 4875 y Fx(1)1874 4891 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))2078 4791 y Fm(\015)2078 4851 y(\015)2133 4915 y Fi(tr)2220 4875 y Fl(\024)28 b Fo(C)7 b Fs(\(1)22 b(+)g Fl(j)p Fo(\015)2693 4834 y Fj(0)2716 4875 y Fl(j)p Fs(\))2782 4834 y Fn(\013)2841 4808 y Fo(e)2886 4772 y Fj(\000)p Fn(\017)p Fj(\001)p Fn(\021)r Fi(\()p Fn(z)s(;K)5 b Fi(\))p Fj(j)p Fn(\015)3262 4748 y Fg(0)3284 4772 y Fj(j)3304 4748 y Fd(1)p Fg(\000)p Fh(\013)p 2841 4853 589 4 v 2960 4944 a Fo(\021)t Fs(\()p Fo(z)t(;)17 b(K)7 b Fs(\))3271 4915 y Fn(\032)3440 4875 y Fo(:)118 5178 y Fs(where)36 b Fo(K)j Fs(=)31 b Fo(C)7 b(k)823 5097 y Fh(p)p 772 5113 135 3 v 772 5158 a(p)p Fg(\000)p Fh(p)884 5144 y Fg(0)920 5178 y Fs(,)36 b Fo(\021)t Fs(\()p Fo(z)t(;)17 b(K)7 b Fs(\))31 b(=)g Fo(\021)t Fs(\()p Fo(z)t(;)17 b(K)r(;)g Fs(1\))31 b(=)2125 5111 y Fl(j)p Fs(Im)o Fo(z)t Fl(j)p 1979 5155 513 4 v 1979 5247 a(j)p Fo(z)t Fl(j)22 b Fs(+)g Fo(K)30 b Fs(+)22 b Fo(C)2536 5178 y Fs(and)35 b Fo(\032)g Fs(dep)s(ends)i(only)d(on)g Fo(d)h Fs(and)g Fo(q)t Fs(.)118 5355 y(On)i(the)f(other)h(hand,)h(due)f (to)f(the)h(exp)s(onen)m(tial)f(deca)m(y)h(of)f Fo(V)22 b Fs(,)37 b(for)f Fl(j)p Fo(\015)2770 5318 y Fj(0)2793 5355 y Fl(j)e(\024)h Fo(n=)p Fs(2)h(and)g(some)h Fo(C)k(>)34 b Fs(0,)1969 5454 y Fp(12)p eop %%Page: 13 13 13 12 bop 118 241 a Fs(w)m(e)34 b(ha)m(v)m(e)870 377 y Fm(\015)870 437 y(\015)925 462 y Fx(1)981 478 y Fn(C)5 b Fi(\(0)p Fn(;)p Fi(1\))1186 462 y Fs(\()p Fo(z)26 b Fl(\000)d Fo(H)1476 477 y Fn(!)r(;n)1589 462 y Fs(\))1627 421 y Fj(\000)p Fi(1)1721 462 y Fs(\()p Fo(V)1816 477 y Fn(!)r(;n)1951 462 y Fl(\000)f Fo(V)2107 477 y Fn(!)2158 462 y Fs(\))p Fx(1)2252 478 y Fn(C)5 b Fi(\()p Fn(\015)2374 459 y Fg(0)2397 478 y Fn(;)p Fi(1\))2483 377 y Fm(\015)2483 437 y(\015)2538 501 y Fj(L)p Fi(\()p Fn(L)2662 482 y Fd(2)2697 501 y Fi(\))2756 462 y Fl(\024)2871 395 y Fo(e)2916 359 y Fj(\000)p Fn(n=C)p 2871 439 238 4 v 2879 531 a Fl(j)p Fs(Im)o Fo(z)t Fl(j)3119 462 y Fo(:)118 693 y Fs(Hence,)34 b(if)e(w)m(e)h(sum)g(up)g(these)g(estimates,)g(w)m(e)g (get)g(that)1402 933 y Fo(A)28 b Fl(\024)g Fo(C)1726 866 y(n)1784 830 y Fi(\()p Fn(d)p Fi(+3\))p Fn(\013)p 1695 910 352 4 v 1695 1002 a Fo(\021)t Fs(\()p Fo(z)t(;)17 b(K)7 b Fs(\))2006 973 y Fn(\032)2056 933 y Fo(e)2101 892 y Fj(\000)p Fn(\017)p Fj(\001)p Fn(\021)r Fi(\()p Fn(z)s(;K)e Fi(\))p Fn(n)2460 869 y Fd(1)p Fg(\000)p Fh(\013)2586 933 y Fo(:)-2495 b Fs(\(2.9\))118 1169 y(F)-8 b(or)48 b Fo(B)5 b Fs(,)53 b(w)m(e)d(get)e(an)h(estimate)f(analogous)f (to)i(\(2.9\))o(;)57 b(only)48 b(the)h(constan)m(ts)i(c)m(hange.)92 b(Plugging)118 1286 y(this)36 b(in)m(to)f(\(2.8\))h(and)g(\(2.7\),)h (summing)d(o)m(v)m(er)j Fo(k)i Fs(using)d(the)h(estimate)e(\(5.1\))h (for)f(the)i(probabilit)m(y)d(of)118 1402 y(\012)188 1417 y Fn(k)231 1402 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)518 1366 y Fj(0)540 1402 y Fs(\),)33 b(w)m(e)g(get)618 1639 y Fl(j)p Fk(E)12 b Fs(\(tr)6 b(\()p Fx(1)920 1655 y Fn(C)f Fi(\(0)p Fn(;)p Fi(1\))1124 1639 y Fs(\()p Fo(')p Fs(\()p Fo(H)1345 1654 y Fn(!)r(;n)1457 1639 y Fs(\))22 b Fl(\000)h Fo(')p Fs(\()p Fo(H)1800 1654 y Fn(!)1850 1639 y Fs(\)\)\)\))p Fl(j)k(\024)h Fo(C)7 b(n)2297 1598 y Fi(\()p Fn(d)p Fi(+3\))p Fn(\013)2568 1572 y Fs(1)p 2538 1616 108 4 v 2538 1708 a(2)p Fo(\031)2673 1504 y Fm(Z)2728 1729 y Ff(C)2797 1495 y Fm(\014)2797 1555 y(\014)2797 1614 y(\014)2797 1674 y(\014)2840 1572 y Fo(@)21 b Fs(~)-65 b Fo(')p 2840 1616 121 4 v 2847 1708 a(@)p 2903 1653 50 4 v 5 w(z)2971 1639 y Fs(\()p Fo(z)t Fs(\))3096 1495 y Fm(\014)3096 1555 y(\014)3096 1614 y(\014)3096 1674 y(\014)3146 1639 y Fo(S)6 b Fs(\()p Fo(z)t(;)17 b(n)p Fs(\))p Fo(dxdy)-3526 b Fs(\(2.10\))118 1871 y(where)686 2117 y Fo(S)6 b Fs(\()p Fo(z)t(;)17 b(n)p Fs(\))28 b(:=)1137 2022 y Fm(X)1145 2234 y Fn(k)r Fj(\025)p Fi(1)1308 2049 y Fs(\()p Fo(C)7 b(\026)p Fs(\))1520 2013 y Fn(k)p 1308 2094 255 4 v 1394 2185 a Fo(k)s Fs(!)1589 1946 y Fm( )1677 2049 y Fl(j)p Fo(z)t Fl(j)23 b Fs(+)f Fo(C)7 b(k)2095 1968 y Fh(p)p 2044 1984 135 3 v 2044 2029 a(p)p Fg(\000)p Fh(p)2156 2015 y Fg(0)2214 2049 y Fs(+)22 b Fo(C)p 1677 2094 712 4 v 1922 2185 a Fl(j)p Fs(Im)p Fo(z)t Fl(j)2399 1946 y Fm(!)2478 1968 y Fn(\032)2535 2117 y Fo(e)2580 2010 y Fj(\000)p Fn(\017)2828 1977 y Fg(j)p Fd(Im)p Fh(z)r Fg(j)p 2673 1995 450 3 v 2673 2093 a(j)p Fh(z)r Fg(j)p Fd(+)p Fh(C)s(k)2932 2019 y(p)p 2881 2034 135 3 v 2881 2078 a(p)p Fg(\000)p Fh(p)2993 2064 y Fg(0)3029 2093 y Fd(+)p Fh(C)3133 2010 y Fn(n)3176 1987 y Fd(1)p Fg(\000)p Fh(\013)3303 2117 y Fo(:)118 2398 y Fs(As)33 b(supp)16 b(~)-65 b Fo(')28 b Fl(\032)g(f)p Fo(z)33 b Fl(2)28 b Fk(C)20 b Fs(;)54 b Fl(j)p Fs(Im\()p Fo(z)t Fs(\))p Fl(j)28 b Fo(<)f Fs(1)p Fl(g)p Fs(,)32 b(using)f(the)i(notation)e Fo(z)h Fs(=)c Fo(x)21 b Fs(+)g Fo(iy)t Fs(,)31 b(for)h Fo(l)e Fl(2)e Fk(N)3292 2362 y Fj(\003)3337 2398 y Fs(,)k(w)m(e)h (estimate)118 2514 y Fo(S)6 b Fs(\()p Fo(z)t(;)17 b(n)p Fs(\))33 b(for)f Fl(j)p Fo(y)t Fl(j)26 b Fo(<)i Fs(1)k(b)m(y)482 2785 y Fo(S)6 b Fs(\()p Fo(z)t(;)17 b(n)p Fs(\))28 b(=)906 2690 y Fm(X)914 2902 y Fn(k)r Fj(\025)p Fi(1)1077 2717 y Fs(\()p Fo(C)7 b(\026)p Fs(\))1289 2681 y Fn(k)p 1077 2762 255 4 v 1163 2853 a Fo(k)s Fs(!)1358 2614 y Fm( )1446 2717 y Fs(\()p Fl(j)p Fo(x)p Fl(j)22 b Fs(+)g Fo(C)7 b Fs(\))p Fo(k)1945 2636 y Fh(p)p 1894 2652 135 3 v 1894 2697 a(p)p Fg(\000)p Fh(p)2006 2683 y Fg(0)p 1446 2762 597 4 v 1691 2853 a Fl(j)p Fo(y)t Fl(j)2053 2614 y Fm(!)2132 2636 y Fn(\032)2188 2785 y Fo(e)2233 2678 y Fj(\000)p Fn(\017)2495 2645 y Fg(j)p Fh(y)r Fg(j)p 2327 2663 407 3 v 2327 2761 a Fd(\()p Fg(j)p Fh(x)p Fg(j)p Fd(+)p Fh(C)s Fd(\))p Fh(k)2637 2687 y(p)p 2586 2702 135 3 v 2586 2746 a(p)p Fg(\000)p Fh(p)2698 2732 y Fg(0)2744 2678 y Fn(n)2787 2655 y Fd(1)p Fg(\000)p Fh(\013)803 3127 y Fs(=)27 b Fo(n)964 3086 y Fj(\000)p Fn(l)q Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))1253 2986 y Fm(\022)1336 3059 y Fl(j)p Fo(x)p Fl(j)22 b Fs(+)g Fo(C)p 1336 3104 308 4 v 1437 3195 a Fl(j)p Fo(y)t Fl(j)1654 2986 y Fm(\023)1727 3009 y Fn(\032)p Fi(+)p Fn(l)1861 3032 y Fm(X)1868 3244 y Fn(k)r Fj(\025)p Fi(1)2031 3059 y Fs(\()p Fo(C)7 b(\026)p Fs(\))2243 3023 y Fn(k)p 2031 3104 255 4 v 2118 3195 a Fo(k)s Fs(!)2295 3127 y Fo(k)2410 3046 y Fh(p)p 2359 3061 135 3 v 2359 3106 a(p)p Fg(\000)p Fh(p)2471 3092 y Fg(0)2504 3077 y Fn(\032)2561 2986 y Fm(\022)2683 3059 y Fl(j)p Fo(y)t Fl(j)p Fo(n)2849 3023 y Fi(1)p Fj(\000)p Fn(\013)p 2644 3104 384 4 v 2644 3195 a Fs(\()p Fl(j)p Fo(x)p Fl(j)22 b Fs(+)g Fo(C)7 b Fs(\))3038 2986 y Fm(\023)3111 3007 y Fn(l)3153 3127 y Fo(e)3198 3078 y Fj(\000)p Fn(\017)3292 3044 y Fh(n)3330 3023 y Fd(1)p Fg(\000)p Fh(\013)3453 3044 y Fg(j)p Fh(y)r Fg(j)p 3292 3062 233 3 v 3301 3104 a Fd(\()p Fg(j)p Fh(x)p Fg(j)p Fd(+)p Fh(C)s Fd(\))3534 3078 y Fn(k)3573 3028 y Fg(\000)3682 3002 y Fh(p)p 3631 3017 135 3 v 3631 3061 a(p)p Fg(\000)p Fh(p)3743 3047 y Fg(0)803 3467 y Fs(=)27 b Fo(n)964 3426 y Fj(\000)p Fn(l)q Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))1253 3327 y Fm(\022)1336 3400 y Fl(j)p Fo(x)p Fl(j)22 b Fs(+)g Fo(C)p 1336 3444 308 4 v 1437 3535 a Fl(j)p Fo(y)t Fl(j)1654 3327 y Fm(\023)1727 3349 y Fn(\032)p 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Fo(L)p Fs(!\()p Fo(e)2907 4431 y Fn(C)e(\026e)3064 4472 y Fl(\000)23 b Fo(e)3209 4431 y Fn(C)5 b(\026)3311 4472 y Fs(\))p Fo(;)118 4299 y Fs(\(2.12\))118 4789 y(where)40 b Fo(L)e Fs(denotes)i(the)f(smallest)e(in)m(teger)h(larger)f(than)h(\()p Fo(\032)27 b Fs(+)f Fo(l)r Fs(\))2660 4722 y Fo(p)p 2563 4766 243 4 v 2563 4857 a(p)c Fl(\000)h Fo(p)2783 4829 y Fj(0)2816 4789 y Fs(.)61 b(Here)39 b(w)m(e)g(used)h(Stirling's)118 4954 y(form)m(ula)31 b(and)h(the)h(iden)m(tit)m(y)429 5121 y Fm(X)444 5333 y Fn(l)q Fj(\025)p Fi(1)604 5149 y Fs(1)p 599 5193 59 4 v 599 5284 a Fo(l)r Fs(!)684 5046 y Fm( )763 5121 y(X)770 5333 y Fn(k)r Fj(\025)p Fi(1)933 5149 y Fs(\()p Fo(C)7 b(\026)p Fs(\))1145 5112 y Fn(k)p 933 5193 255 4 v 1020 5284 a Fo(k)s Fs(!)1197 5216 y Fo(k)1251 5175 y Fn(l)1277 5046 y Fm(!)1384 5216 y Fs(=)1488 5121 y Fm(X)1495 5333 y Fn(k)r Fj(\025)p Fi(1)1658 5149 y Fs(\()p Fo(C)g(\026)p Fs(\))1870 5112 y Fn(k)p 1658 5193 V 1745 5284 a Fo(k)s Fs(!)1939 5046 y Fm( )2018 5121 y(X)2034 5333 y Fn(l)q Fj(\025)p Fi(1)2188 5149 y Fo(k)2242 5112 y Fn(l)p 2188 5193 81 4 v 2199 5284 a Fo(l)r Fs(!)2278 5046 y Fm(!)2385 5216 y Fs(=)2488 5121 y Fm(X)2496 5333 y Fn(k)r Fj(\025)p Fi(1)2659 5149 y Fs(\()p Fo(C)g(\026)p Fs(\))2871 5112 y Fn(k)p 2659 5193 255 4 v 2745 5284 a Fo(k)s Fs(!)2923 5216 y(\()p Fo(e)3006 5175 y Fn(k)3071 5216 y Fl(\000)22 b Fs(1\))28 b(=)f Fo(e)3433 5175 y Fn(C)5 b(\026e)3590 5216 y Fl(\000)23 b Fo(e)3735 5175 y Fn(C)5 b(\026)1969 5454 y Fp(13)p eop %%Page: 14 14 14 13 bop 118 241 a Fs(Hence,)34 b(for)e(some)h Fo(C)h(>)28 b Fs(0)k(\(indep)s(enden)m(t)i(of)e Fo(l)r Fs(,)h Fo(n)p Fs(,)g Fo(z)k Fs(and)32 b Fo(\026)p Fs(\),)h(w)m(e)g(ha)m(v)m(e)1061 479 y Fo(S)6 b Fs(\()p Fo(z)t(;)17 b(n)p Fs(\))28 b Fl(\024)g Fo(C)7 b(e)1609 438 y Fn(C)e(\026)1711 479 y Fo(n)1769 438 y Fj(\000)p Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))p Fn(l)2040 479 y Fo(e)2085 438 y Fn(C)g(l)13 b Fi(log)g Fn(l)2320 338 y Fm(\022)2403 412 y Fl(j)p Fo(x)p Fl(j)22 b Fs(+)g Fo(C)p 2403 456 308 4 v 2504 547 a Fl(j)p Fo(y)t Fl(j)2721 338 y Fm(\023)2794 361 y Fn(\032)p Fi(+)p Fn(l)2928 479 y Fo(:)-2837 b Fs(\(2.13\))118 697 y(Plugging)46 b(this)i(in)m(to)f(\(2.10\))h(and)g(using)g(estimate)f(\(2.4\))h(for)f (almost)f(analytic)h(extensions,)54 b(w)m(e)118 814 y(get)33 b(\(2.2\))f(and)g(end)i(the)f(pro)s(of)e(of)h(Lemma)g(2.1.)p 3827 930 4 66 v 3831 867 59 4 v 3831 930 V 3889 930 4 66 v 118 1104 a Fx(Remark)37 b(2.2.)49 b Fs(In)37 b(the)g(pro)s(of)e (of)h(Lemma)f(2.1,)i(w)m(e)h(only)e(ha)m(v)m(e)i(used)f(the)g(fact)f (that)h(the)g(space)g(of)118 1220 y(realization)30 b(\012)j(could)g(b)s (e)g(written)f(as)h(\012)c(=)f Fl([)1822 1235 y Fn(n)1869 1220 y Fs(\012)1939 1235 y Fn(n)2019 1220 y Fs(where)34 b(the)f(probabilit)m(y)e Fo(P)14 b Fs(\(\012)3153 1235 y Fn(n)3200 1220 y Fs(\))32 b(w)m(as)i(decreasing)118 1336 y(fast)28 b(enough,)h(and)f(that)f(in)g(these)i(subsets,)i(w)m(e)d (had)g(uniform)e(estimates)h(on)h(the)g(quan)m(tit)m(y)g(w)m(e)h(w)m (an)m(t)118 1452 y(to)g(compute.)42 b(Ob)m(viously)-8 b(,)30 b(to)f(get)g(suc)m(h)h(a)f(decomp)s(osition,)f(one)i(do)s(es)f (not)g(need)h(to)f(ha)m(v)m(e)h(a)f(P)m(oisson)118 1569 y(p)s(oten)m(tial)40 b(but)i(only)f(a)h(homogeneous)f(random)g(\014eld) h(with)f(suitable)g(b)s(ounds)h(at)f(in\014nit)m(y)-8 b(.)70 b(This)118 1685 y(idea)32 b(is)g(applicable)f(to)h(man)m(y)h (other)f(random)g(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erators.)118 1859 y Fx(Pro)s(of)40 b(of)h(Theorem)f(1.3.)51 b Fs(By)35 b([7],)h(w)m(e)g(kno)m(w)g(that,)g(for)e Fo(\036)e Fl(2)g(C)2627 1823 y Fj(1)2621 1884 y Fi(0)2702 1859 y Fs(\()p Fk(R)5 b Fs(\))41 b(and)35 b(for)g(almost)e(ev)m(ery)k Fo(!)t Fs(,)118 1975 y(w)m(e)d(ha)m(v)m(e)780 2156 y Fl(h)p Fo(\036;)17 b(dN)10 b Fl(i)27 b Fs(=)61 b(lim)1229 2226 y Fi(\003)p Fj(!)p Ff(R)1397 2207 y Fh(d)1577 2088 y Fs(1)p 1459 2133 285 4 v 1459 2224 a(V)-8 b(ol)o(\(\003\))1754 2156 y(tr\()p Fo(\036)p Fs(\()p Fo(H)2053 2115 y Fn(D)2045 2180 y(!)r(;)p Fi(\003)2163 2156 y Fs(\)\))1126 2418 y(=)61 b(lim)1229 2488 y Fi(\003)p Fj(!)p Ff(R)1397 2469 y Fh(d)1577 2351 y Fs(1)p 1459 2395 V 1459 2486 a(V)-8 b(ol)o(\(\003\))1754 2418 y Fk(E)12 b Fs(\(tr)6 b(\()p Fo(\036)p Fs(\()p Fo(H)2157 2377 y Fn(D)2149 2443 y(!)r(;)p Fi(\003)2268 2418 y Fs(\)\)\))1126 2680 y(=)61 b(lim)1229 2750 y Fi(\003)p Fj(!)p Ff(R)1397 2731 y Fh(d)1577 2613 y Fs(1)p 1459 2657 V 1459 2748 a(V)-8 b(ol)o(\(\003\))1830 2585 y Fm(X)1771 2805 y Fn(\015)t Fj(2)p Fi(\003)p Fj(\\)p Ff(Z)2004 2786 y Fh(d)2051 2680 y Fk(E)12 b Fs(\(tr)6 b(\()p Fx(1)2325 2696 y Fn(C)f Fi(\()p Fn(\015)t(;)p Fi(1\))2534 2680 y Fo(\036)p Fs(\()p Fo(H)2719 2639 y Fn(D)2711 2705 y(!)r(;)p Fi(\003)2830 2680 y Fs(\))p Fx(1)2924 2696 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))3133 2680 y Fs(\)\))p Fo(:)118 2451 y Fs(\(2.14\))118 2959 y(On)33 b(the)g(other)f(hand,)h(as)g Fo(H)1175 2974 y Fn(!)1258 2959 y Fs(is)f(homogeneous,)h(for)f(an)m(y)h Fo(\015)g Fl(2)28 b Fk(Z)2560 2923 y Fn(d)2598 2959 y Fs(,)k(w)m(e)i(ha)m(v)m(e)844 3116 y Fk(E)13 b Fs(\(tr)6 b(\()p Fx(1)1119 3132 y Fn(C)f Fi(\()p Fn(\015)t(;)p Fi(1\))1328 3116 y Fo(\036)p Fs(\()p Fo(H)1505 3131 y Fn(!)1555 3116 y Fs(\))p Fx(1)1649 3132 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))1858 3116 y Fs(\)\))27 b(=)h Fk(E)12 b Fs(\(tr)6 b(\()p Fx(1)2339 3132 y Fn(C)f Fi(\(0)p Fn(;)p Fi(1\))2543 3116 y Fo(\036)p Fs(\()p Fo(H)2720 3131 y Fn(!)2770 3116 y Fs(\))p Fx(1)2864 3132 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))3068 3116 y Fs(\)\))p Fo(:)118 3273 y Fs(So)33 b(that)668 3454 y Fk(E)13 b Fs(\(tr)6 b(\()p Fx(1)943 3469 y Fn(C)f Fi(\(0)p Fn(;)p Fi(1\))1147 3454 y Fo(\036)p Fs(\()p Fo(H)1324 3469 y Fn(!)1374 3454 y Fs(\))p Fx(1)1468 3469 y Fn(C)g Fi(\(0)p Fn(;)p Fi(1\))1672 3454 y Fs(\)\))27 b(=)2007 3386 y(1)p 1889 3431 V 1889 3522 a(V)-8 b(ol)o(\(\003\))2260 3359 y Fm(X)2200 3579 y Fn(\015)t Fj(2)p Fi(\003)p Fj(\\)p Ff(Z)2433 3560 y Fh(d)2480 3454 y Fk(E)13 b Fs(\(tr)5 b(\()p Fx(1)2754 3469 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))2964 3454 y Fo(\036)p Fs(\()p Fo(H)3141 3469 y Fn(!)3191 3454 y Fs(\))p Fx(1)3285 3469 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))3494 3454 y Fs(\)\))p Fo(:)118 3725 y Fs(Hence,)34 b(b)m(y)h(\(2.14\))o(,)e(to)f(get)h(equation)f(\(1.4\),) g(w)m(e)i(just)f(ha)m(v)m(e)h(to)e(pro)m(v)m(e)i(that)490 4003 y(lim)456 4074 y Fi(\003)p Fj(!)p Ff(R)624 4055 y Fh(d)804 3936 y Fs(1)p 686 3981 V 686 4072 a(V)-8 b(ol)o(\(\003\))997 3799 y Fm(\014)997 3859 y(\014)997 3919 y(\014)997 3979 y(\014)997 4038 y(\014)997 4098 y(\014)1090 3909 y(X)1030 4128 y Fn(\015)t Fj(2)p Fi(\003)p Fj(\\)p Ff(Z)1264 4109 y Fh(d)1310 4003 y Fk(E)1388 3923 y Fm(\000)1439 4003 y Fs(tr\()p Fx(1)1609 4019 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1818 4003 y Fo(\036)p Fs(\()p Fo(H)2003 3962 y Fn(D)1995 4028 y(!)r(;)p Fi(\003)2114 4003 y Fs(\))p Fx(1)2208 4019 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))2417 4003 y Fs(\))22 b Fl(\000)g Fs(tr\()p Fx(1)2746 4019 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2955 4003 y Fo(\036)p Fs(\()p Fo(H)3132 4018 y Fn(!)3182 4003 y Fs(\))p Fx(1)3276 4019 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))3486 4003 y Fs(\))3524 3923 y Fm(\001)3569 3799 y(\014)3569 3859 y(\014)3569 3919 y(\014)3569 3979 y(\014)3569 4038 y(\014)3569 4098 y(\014)3630 4003 y Fs(=)28 b(0)p Fo(:)-3692 b Fs(\(2.15\))118 4282 y(T)-8 b(o)40 b(shorten)h(the)f(notations,)h(let)f(\003)1481 4297 y Fn(L)1573 4282 y Fs(=)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(L)p Fs(\))40 b(b)s(e)g(the)g(cub)s(e)h(of)e(cen)m(ter)j(0)e(and)g (side)g(length)f Fo(L)p Fs(.)118 4398 y(Pic)m(k)33 b Fo(\017)28 b(>)g Fs(0.)43 b(T)-8 b(o)33 b(pro)m(v)m(e)h(\(2.15\))o(,)e (one)h(could)f(try)h(to)g(pro)m(v)m(e)g(that,)g(for)f(an)m(y)h Fo(\015)g Fl(2)28 b Fk(Z)3141 4362 y Fn(d)3179 4398 y Fs(,)k(one)h(has)293 4680 y(lim)266 4742 y Fn(L)p Fj(!1)472 4680 y Fo(A)p Fs(\()p Fo(\015)5 b(;)17 b Fs(\003)751 4695 y Fn(L)802 4680 y Fs(\))28 b(:=)85 b(lim)998 4750 y Fi(\003)1047 4761 y Fh(L)1094 4750 y Fj(!)p Ff(R)1213 4731 y Fh(d)1265 4680 y Fk(E)1342 4600 y Fm(\000)1394 4680 y Fs(tr\()p Fx(1)1564 4696 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1773 4680 y Fo(\036)p Fs(\()p Fo(H)1958 4639 y Fn(D)1950 4705 y(!)r(;)p Fi(\003)2065 4716 y Fh(L)2114 4680 y Fs(\))p Fx(1)2208 4696 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))2418 4680 y Fs(\))22 b Fl(\000)g Fs(tr\()p Fx(1)2747 4696 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2956 4680 y Fo(\036)p Fs(\()p Fo(H)3133 4695 y Fn(!)3183 4680 y Fs(\))p Fx(1)3277 4696 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))3486 4680 y Fs(\))3524 4600 y Fm(\001)3598 4680 y Fs(=)27 b(0)118 4546 y(\(2.16\))118 4890 y(in)f(some)g(uniform)f(w)m (a)m(y)-8 b(.)42 b(Ho)m(w)m(ev)m(er,)30 b(this)c(ma)m(y)g(b)s(e)h (di\016cult)f(as,)h(b)s(ecause)h(of)e(the)h(Diric)m(hlet)d(b)s(oundary) 118 5006 y(conditions,)30 b Fo(A)p Fs(\()p Fo(\015)5 b(;)17 b Fs(\003)889 5021 y Fn(L)941 5006 y Fs(\))31 b(ma)m(y)f(ha)m(v)m(e)i(some)f(non)f(uniform)f(b)s(eha)m(vior)i(for)f Fo(\015)36 b Fs(close)30 b(to)h(the)g(b)s(oundary)g(of)118 5122 y(\003)186 5137 y Fn(L)238 5122 y Fs(.)47 b(So)34 b(w)m(e)h(are)f(going)e(to)i(split)f(the)h(di\016cult)m(y)g(in)m(to)f (t)m(w)m(o)h(parts;)h(on)f(the)g(one)g(hand,)h(w)m(e)g(will)c(sho)m(w) 118 5238 y(that,)37 b(for)f(an)m(y)h Fo(\017)d(>)g Fs(0)i(and)g(for)g Fo(\015)j Fl(2)34 b Fs(\003)1574 5253 y Fn(L)1651 5238 y Fl(n)24 b Fs(\003)1793 5254 y Fi(\(1)p Fj(\000)p Fn(\017)p Fi(\))p Fn(L)2019 5238 y Fs(,)37 b(w)m(e)g(will)d(sho)m(w)j(that)f Fo(A)p Fs(\()p Fo(\015)5 b(;)17 b Fs(\003)3157 5253 y Fn(L)3209 5238 y Fs(\))36 b(sta)m(ys)i(b)s(ounded)118 5355 y(\(uniformly)29 b(in)h Fo(L)p Fs(,)h Fo(!)t Fs(\).)43 b(As)31 b(there)h(are)f(only)f(v)m(ery)i(few)g(suc)m(h)g(terms,)f(this) g(part)f(of)h(the)g(sum)g(tends)h(to)1969 5454 y Fp(14)p eop %%Page: 15 15 15 14 bop 118 241 a Fs(0.)53 b(On)36 b(the)g(other)g(hand,)i(for)d Fo(\015)j Fl(2)c Fs(\003)1527 256 y Fi(\(1)p Fj(\000)p Fn(\017)p Fi(\))p Fn(L)1752 241 y Fs(,)j(w)m(e)g(will)d(sho)m(w)j(that) e Fo(A)p Fs(\()p Fo(\015)5 b(;)17 b Fs(\003)2889 256 y Fn(L)2941 241 y Fs(\))36 b(tends)h(to)e(0)h(uniformly)118 357 y(in)e Fo(L)h Fs(and)f Fo(!)t Fs(.)49 b(As)35 b(in)e(the)i(pro)s (of)f(of)g(Lemma)f(2.1,)i(uniformit)m(y)d(in)i Fo(!)j Fs(cannot)e(b)s(e)g(ac)m(hiev)m(ed)h(o)m(v)m(er)f(the)118 473 y(whole)27 b(set)h(of)e(realization,)g(but)h(only)g(o)m(v)m(er)h (subsets)h(whose)f(measure)g(w)m(e)g(con)m(trol)e(\(see)i(Lemma)e (5.1\).)118 589 y(This)33 b(will)d(su\016ce.)331 706 y(T)-8 b(o)32 b(estimate)f Fo(A)p Fs(\()p Fo(\015)5 b(;)17 b Fs(\003)1143 721 y Fn(L)1195 706 y Fs(\),)32 b(w)m(e)h(will)d(use)j (\(2.5\))f(to)g(compute)g Fo(\036)p Fs(\()p Fo(H)2711 721 y Fn(!)2761 706 y Fs(\))g(and)g Fo(\036)p Fs(\()p Fo(H)3205 670 y Fn(D)3197 731 y(!)r(;)p Fi(\003)3312 742 y Fh(L)3362 706 y Fs(\).)43 b(Hence,)34 b(w)m(e)118 827 y(see)g(that)e(w)m(e)i(only)e(to)g(estimate)g(the)h(follo)m(wing)d (expression)k(\(cf)39 b(\(2.8\)\))224 1113 y(tr)o(\()p Fx(1)393 1129 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))603 1113 y Fs(\()p Fo(z)26 b Fl(\000)d Fo(H)901 1072 y Fn(D)893 1138 y(!)r(;)p Fi(\003)1008 1149 y Fh(L)1058 1113 y Fs(\))1096 1072 y Fj(\000)p Fi(1)1190 1113 y Fs(\()p Fo(i)g Fs(+)f Fo(H)1471 1072 y Fn(D)1463 1138 y(!)r(;)p Fi(\003)1578 1149 y Fh(L)1628 1113 y Fs(\))1666 1072 y Fj(\000)p Fn(q)1759 1113 y Fx(1)1815 1129 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2024 1113 y Fs(\))22 b Fl(\000)h Fs(tr)o(\()p Fx(1)2353 1129 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2563 1113 y Fs(\()p Fo(z)26 b Fl(\000)d Fo(H)2853 1128 y Fn(!)2903 1113 y Fs(\))2941 1072 y Fj(\000)p Fi(1)3036 1113 y Fs(\()p Fo(i)f Fs(+)g Fo(H)3308 1128 y Fn(!)3358 1113 y Fs(\))3396 1072 y Fj(\000)p Fn(q)3489 1113 y Fx(1)3545 1129 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))3754 1113 y Fs(\))118 1285 y(where)34 b Fo(q)d(>)d(d=)p Fs(2)k(is)g(an)g(ev)m(en)i(in)m (teger.)118 1402 y(T)-8 b(o)35 b(do)f(this)g(w)m(e)i(need)g(a)e(w)m(a)m (y)i(to)e(compare)g(the)h(resolv)m(en)m(t)h(of)e(the)h(Diric)m(hlet)e (problem)g(on)i(\003)3622 1417 y Fn(L)3708 1402 y Fs(with)118 1518 y(the)29 b(resolv)m(en)m(t)h(of)f Fo(H)877 1533 y Fn(!)956 1518 y Fs(o)m(v)m(er)h(the)f(whole)g(space.)43 b(W)-8 b(e)29 b(use)h(the)g(follo)m(wing)c(resolv)m(en)m(t)k(iden)m (tit)m(y)e(\(see)i(e.g.)118 1679 y([1]\):)43 b(let)32 b Fo(\037)c Fl(2)g(C)711 1643 y Fi(2)705 1704 y(0)751 1679 y Fs(\()831 1591 y Fj(\016)789 1679 y Fs(\003)857 1694 y Fn(L)909 1679 y Fs(\))k(then,)h(for)g(Im)p Fo(z)e Fl(6)p Fs(=)d(0,)k(w)m(e)i(ha)m(v)m(e)713 1849 y Fo(\037)p Fs(\()p Fo(z)26 b Fl(\000)d Fo(H)1064 1864 y Fn(!)1114 1849 y Fs(\))1152 1808 y Fj(\000)p Fi(1)1274 1849 y Fs(=)28 b(\()p Fo(z)f Fl(\000)22 b Fo(H)1676 1808 y Fn(D)1668 1874 y(!)r(;)p Fi(\003)1783 1885 y Fh(L)1833 1849 y Fs(\))1871 1808 y Fj(\000)p Fi(1)1966 1849 y Fo(\037)g Fs(+)g(\()p Fo(z)27 b Fl(\000)22 b Fo(H)2445 1808 y Fn(D)2437 1874 y(!)r(;)p Fi(\003)2552 1885 y Fh(L)2602 1849 y Fs(\))2640 1808 y Fj(\000)p Fi(1)2735 1849 y Fs([)p Fl(\000)p Fs(\001)p Fo(;)17 b(\037)p Fs(]\()p Fo(z)27 b Fl(\000)c Fo(H)3343 1864 y Fn(!)3393 1849 y Fs(\))3431 1808 y Fj(\000)p Fi(1)3525 1849 y Fo(:)-3434 b Fs(\(2.17\))118 2020 y(Pic)m(k)33 b Fo(\013)q Fs(,)g Fo(k)i Fs(and)e Fo(p)784 1984 y Fj(0)840 2020 y Fs(are)f(tak)m(en)i(as)e(in)g(Lemma)f(2.1.)44 b(W)-8 b(e)33 b(will)d(use)j(the)g(follo)m(wing)d(lemma)118 2200 y Fx(Lemma)37 b(2.2.)49 b Fe(Assume)f(that)f Fo(q)55 b(>)50 b(d=)p Fs(2)d Fe(is)g(an)f(even)h(inte)-5 b(ger)47 b(and)f(that)i Fo(!)j Fe(is)c(such)g(that)g Fo(V)3730 2215 y Fn(!)3831 2200 y Fl(2)118 2316 y Fs(\012)188 2331 y Fn(k)231 2316 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)518 2280 y Fj(0)540 2316 y Fs(\))p Fe(.)60 b(Then,)41 b(ther)-5 b(e)40 b(exists)g Fo(C)1549 2331 y Fn(q)1624 2316 y Fo(>)d Fs(0)j Fe(such)g(that,)h(for)f(any)g Fs(\003)2709 2280 y Fj(0)2769 2316 y Fl(\032)e Fs(\003)2952 2331 y Fn(L)3044 2316 y Fe(\()p Fs(\003)3152 2280 y Fj(0)3215 2316 y Fe(me)-5 b(asur)g(able\),)40 b(we)118 2432 y(have)1190 2609 y Fl(k)p Fs(\()p Fo(i)22 b Fs(+)g Fo(H)1520 2568 y Fn(D)1512 2634 y(!)r(;)p Fi(\003)1627 2645 y Fh(L)1677 2609 y Fs(\))1715 2568 y Fj(\000)p Fi(1)1809 2609 y Fl(k)1859 2624 y Fj(T)1898 2632 y Fh(q)1964 2609 y Fl(\024)28 b Fo(C)2139 2624 y Fn(q)2177 2609 y Fo(k)2331 2525 y Fh(pd)p 2241 2541 243 3 v 2241 2586 a Fd(2)p Fh(q)r Fd(\()p Fh(p)p Fg(\000)p Fh(p)2437 2572 y Fg(0)2460 2586 y Fd(\))2498 2609 y Fo(L)2574 2536 y Fh(d)p 2575 2548 33 3 v 2576 2589 a(q)2617 2563 y Fi(\(1+)2744 2536 y Fh(\013)p 2744 2548 41 3 v 2749 2589 a Fd(2)2794 2563 y Fi(\))118 2609 y Fs(\(2.18\))1332 2777 y Fl(k)p Fs(\(1)21 b Fl(\000)i Fs(\001)1671 2735 y Fn(D)1671 2801 y Fi(\003)1720 2812 y Fh(L)1770 2777 y Fs(\))1808 2735 y Fj(\000)p Fi(1)1903 2777 y Fx(1)1959 2792 y Fi(\003)2008 2773 y Fg(0)2034 2777 y Fl(k)2084 2792 y Fj(T)2123 2800 y Fh(q)2189 2777 y Fl(\024)28 b Fo(C)2364 2792 y Fn(q)2402 2777 y Fl(j)p Fs(\003)2498 2735 y Fj(0)2521 2777 y Fl(j)2549 2735 y Fi(1)p Fn(=q)2657 2777 y Fo(:)-2566 b Fs(\(2.19\))118 2949 y Fe(Her)-5 b(e)35 b Fl(j)p Fs(\003)447 2913 y Fj(0)470 2949 y Fl(j)f Fe(denotes)h(the)f(me)-5 b(asur)g(e)35 b(of)f Fs(\003)1605 2913 y Fj(0)1663 2949 y Fe(and)1538 3131 y Fl(k)22 b(\001)g(k)1710 3146 y Fj(T)1749 3154 y Fh(q)1815 3131 y Fs(=)27 b(\()p Fe(tr)17 b Fs(\()p Fl(j)22 b(\001)g(j)2212 3089 y Fn(q)2250 3131 y Fs(\))o(\))2325 3082 y Fi(1)p Fn(=q)2451 3131 y Fo(:)331 3311 y Fs(Let)k(us)g(p)s(ostp)s(one)g(the)g(pro)s(of)f(of)g (this)h(result)f(to)h(\014nish)g(the)g(pro)s(of)f(of)g(Theorem)h(1.3.) 41 b(Pic)m(k)26 b Fo(\017)i(>)g Fs(0)118 3427 y(to)k(b)s(e)g(c)m(hosen) h(precisely)g(later.)42 b(F)-8 b(or)31 b Fo(!)k Fs(suc)m(h)f(that)d Fo(V)2101 3442 y Fn(!)2179 3427 y Fl(2)e Fs(\012)2344 3442 y Fn(k)2387 3427 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)2674 3391 y Fj(0)2696 3427 y Fs(\).)43 b(Hence,)33 b(b)m(y)g(Lemma)e(5.1,)h(w)m(e)118 3561 y(can)h(decomp)s(ose)g Fo(V)845 3576 y Fn(!)923 3561 y Fs(=)27 b Fo(V)1105 3510 y Fi(\()p Fn(i)p Fi(\))1083 3571 y Fn(!)1210 3561 y Fs(+)22 b Fo(V)1387 3510 y Fi(\()p Fn(r)r Fi(\))1365 3571 y Fn(!)1512 3561 y Fs(where)244 3714 y Fl(\017)41 b Fo(V)414 3673 y Fi(\()p Fn(i)p Fi(\))392 3739 y Fn(!)524 3714 y Fl(2)28 b Fo(L)684 3667 y Fn(p)720 3644 y Fg(0)684 3742 y Fi(lo)r(c,unif)916 3714 y Fs(\(\003)1022 3729 y Fn(L)1074 3714 y Fs(\))33 b(and)f(sup)1481 3738 y Fn(x)p Fj(2)p Fi(\003)1617 3749 y Fh(L)1684 3714 y Fl(k)p Fo(V)1812 3663 y Fi(\()p Fn(i)p Fi(\))1791 3725 y Fn(!)1895 3714 y Fl(k)1945 3741 y Fn(L)1993 3722 y Fh(p)2025 3708 y Fg(0)2052 3741 y Fi(\()p Fn(C)5 b Fi(\()p Fn(x;)p Fi(1\)\))2343 3714 y Fl(\024)28 b Fs(1.)244 3868 y Fl(\017)41 b Fs(for)32 b Fo(x)c Fl(2)g Fs(\003)729 3883 y Fn(L)781 3868 y Fs(,)33 b Fl(j)p Fo(V)947 3817 y Fi(\()p Fn(r)r Fi(\))926 3879 y Fn(!)1040 3868 y Fs(\()p Fo(x)p Fs(\))p Fl(j)28 b(\024)g Fo(C)7 b(k)1524 3787 y Fh(p)p 1473 3803 135 3 v 1473 3848 a(p)p Fg(\000)p Fh(p)1585 3834 y Fg(0)1621 3868 y Fo(L)1687 3832 y Fn(\013)1737 3868 y Fs(.)118 4010 y(Hence,)34 b(com)m(bining)d(the)i(t)m(w)m(o)g (estimates)f(of)g(Lemma)g(2.2,)g(w)m(e)i(get)813 4212 y Fl(k)p Fs(\()p Fo(i)22 b Fs(+)g Fo(H)1143 4170 y Fn(D)1135 4236 y(!)r(;)p Fi(\003)1250 4247 y Fh(L)1300 4212 y Fs(\))1338 4170 y Fj(\000)p Fi(1)1432 4212 y Fx(1)1488 4227 y Fi(\003)1537 4238 y Fh(L)1583 4227 y Fj(n)p Fi(\003)1667 4241 y Fd(\(1)p Fg(\000)p Fd(2)p Fh(\017)p Fd(\))p Fh(L)1901 4212 y Fl(k)1951 4227 y Fj(T)1990 4235 y Fh(q)2056 4212 y Fl(\024)28 b Fo(C)7 b(\017)2287 4138 y Fd(1)p 2287 4150 31 3 v 2287 4191 a Fh(q)2332 4212 y Fo(k)2408 4125 y Fd(\()p Fh(p)p Fd(+2)p Fh(q)r Fd(\))p Fh(d)p 2396 4143 243 3 v 2396 4188 a Fd(2)p Fh(q)r Fd(\()p Fh(p)p Fg(\000)p Fh(p)2592 4174 y Fg(0)2615 4188 y Fd(\))2653 4212 y Fo(L)2719 4170 y Fn(\013)2769 4212 y Fo(L)2845 4138 y Fh(d)p 2845 4150 33 3 v 2846 4191 a(q)2887 4165 y Fi(\(1+)3014 4138 y Fh(\013)p 3015 4150 41 3 v 3020 4191 a Fd(2)3065 4165 y Fi(\)+)3157 4138 y Fh(d)p 3157 4150 33 3 v 3158 4191 a(q)118 4212 y Fs(\(2.20\))118 4378 y(as)710 4545 y(\()p Fo(i)23 b Fs(+)f Fo(H)991 4503 y Fn(D)983 4569 y(!)r(;)p Fi(\003)1098 4580 y Fh(L)1148 4545 y Fs(\))1186 4503 y Fj(\000)p Fi(1)1308 4545 y Fs(=)1411 4464 y Fm(\002)1453 4545 y Fs(1)g(+)g(\()p Fo(i)g Fs(+)g Fo(H)1902 4503 y Fn(D)1894 4569 y(!)r(;)p Fi(\003)2009 4580 y Fh(L)2059 4545 y Fs(\))2097 4503 y Fj(\000)p Fi(1)2192 4545 y Fs(\(1)f Fl(\000)i Fo(i)f Fs(+)g Fo(V)2610 4560 y Fn(!)2661 4545 y Fs(\))2699 4464 y Fm(\003)2757 4545 y Fs(\(1)g Fl(\000)g Fs(\001)3046 4503 y Fn(D)3046 4569 y Fi(\003)3095 4580 y Fh(L)3146 4545 y Fs(\))3184 4503 y Fj(\000)p Fi(1)3278 4545 y Fo(:)118 4717 y Fs(So,)33 b(b)m(y)g(form)m(ula)e(\(2.5\))o(,)i (w)m(e)h(get)e(that,)h(for)f Fo(!)k Fs(suc)m(h)e(that)e Fo(V)2313 4732 y Fn(!)2391 4717 y Fl(2)c Fs(\012)2555 4732 y Fn(k)2598 4717 y Fs(\()p Fo(\013)q(;)17 b(p)2792 4681 y Fj(0)2815 4717 y Fo(;)g Fs(1\),)32 b(w)m(e)h(ha)m(v)m(e)809 4924 y Fl(j)p Fs(tr)o(\()p Fo(\036)p Fs(\()p Fo(H)1135 4883 y Fn(D)1127 4949 y(!)r(;)p Fi(\003)1242 4960 y Fh(L)1292 4924 y Fs(\))p Fx(1)1386 4940 y Fi(\003)1435 4951 y Fh(L)1481 4940 y Fj(n)p Fi(\003)1565 4954 y Fd(\(1)p Fg(\000)p Fd(2)p Fh(\017)p Fd(\))p Fh(L)1799 4924 y Fs(\))p Fl(j)27 b(\024)h Fo(C)7 b(k)2150 4838 y Fd(\()p Fh(p)p Fd(+2)p Fh(q)r Fd(\))p Fh(d)p 2138 4856 243 3 v 2138 4901 a Fd(2)p Fh(q)r Fd(\()p Fh(p)p Fg(\000)p Fh(p)2334 4887 y Fg(0)2357 4901 y Fd(\))2395 4924 y Fo(\017)2444 4851 y Fd(1)p 2444 4863 31 3 v 2444 4904 a Fh(q)2489 4924 y Fo(L)2555 4878 y Fn(\013)p Fi(+)2665 4851 y Fh(d)p 2666 4863 33 3 v 2667 4904 a(q)2707 4878 y Fi(+)2772 4844 y Fh(d)p Fd(\()p Fh(q)r Fg(\000)p Fd(1\))p 2772 4863 189 3 v 2852 4904 a Fh(q)2971 4878 y Fi(\(1+)3098 4851 y Fh(\013)p 3098 4863 41 3 v 3103 4904 a Fd(2)3149 4878 y Fi(\))3180 4924 y Fo(:)118 5095 y Fs(Hence,)34 b(taking)e(the)h(exp)s(ectation)g(in)e Fo(!)36 b Fs(and)d(summing)e(o)m(v)m(er)i(all)e Fo(k)s Fs(,)i(w)m(e)g(get)g(that)889 5175 y Fm(\014)889 5235 y(\014)889 5295 y(\014)922 5290 y Fk(E)999 5179 y Fm(h)1052 5290 y Fs(tr\()p Fo(\036)p Fs(\()p Fo(H)1351 5249 y Fn(D)1343 5314 y(!)r(;)p Fi(\003)1458 5325 y Fh(L)1507 5290 y Fs(\))p Fx(1)1601 5305 y Fi(\003)1650 5316 y Fh(L)1697 5305 y Fj(n)p Fi(\003)1781 5319 y Fd(\(1)p Fg(\000)p Fd(2)p Fh(\017)p Fd(\))p Fh(L)2014 5290 y Fs(\))2052 5179 y Fm(i)2099 5175 y(\014)2099 5235 y(\014)2099 5295 y(\014)2160 5290 y Fl(\024)28 b Fo(C)7 b(\017)2391 5216 y Fd(1)p 2392 5228 31 3 v 2392 5269 a Fh(q)2436 5290 y Fo(L)2502 5243 y Fn(\013)p Fi(+)2612 5216 y Fh(d)p 2613 5228 33 3 v 2614 5269 a(q)2655 5243 y Fi(+)2720 5210 y Fh(d)p Fd(\()p Fh(q)r Fg(\000)p Fd(1\))p 2719 5228 189 3 v 2799 5269 a Fh(q)2918 5243 y Fi(\(1+)3045 5216 y Fh(\013)p 3046 5228 41 3 v 3051 5269 a Fd(2)3096 5243 y Fi(\))1969 5454 y Fp(15)p eop %%Page: 16 16 16 15 bop 118 241 a Fs(Finally)-8 b(,)30 b(w)m(e)k(obtain)888 538 y(1)p 744 582 337 4 v 744 674 a(V)-8 b(ol)o(\(\003)991 689 y Fn(L)1042 674 y Fs(\))1107 311 y Fm(\014)1107 371 y(\014)1107 431 y(\014)1107 491 y(\014)1107 550 y(\014)1107 610 y(\014)1107 670 y(\014)1107 730 y(\014)1107 789 y(\014)1316 511 y(X)1140 726 y Fn(\015)t Fj(2)p Fi(\003)1276 737 y Fh(L)1323 726 y Fj(n)p Fi(\003)1407 740 y Fd(\(1)p Fg(\000)p Fd(2)p Fh(\017)p Fd(\))p Fh(L)1304 820 y Fn(\015)t Fj(2)p Ff(Z)1442 797 y Fh(d)1653 605 y Fk(E)1730 524 y Fm(\002)1777 605 y Fs(tr\()p Fx(1)1947 621 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2156 605 y Fo(\036)p Fs(\()p Fo(H)2341 564 y Fn(D)2333 630 y(!)r(;)p Fi(\003)2448 641 y Fh(L)2498 605 y Fs(\))p Fx(1)2592 621 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))2801 605 y Fs(\))2839 524 y Fm(\003)2881 311 y(\014)2881 371 y(\014)2881 431 y(\014)2881 491 y(\014)2881 550 y(\014)2881 610 y(\014)2881 670 y(\014)2881 730 y(\014)2881 789 y(\014)761 1018 y Fs(=)1019 951 y(1)p 875 995 V 875 1087 a(V)-8 b(ol)o(\(\003)1122 1102 y Fn(L)1174 1087 y Fs(\))1238 904 y Fm(\014)1238 963 y(\014)1238 1023 y(\014)1271 1018 y Fk(E)1349 908 y Fm(h)1401 1018 y Fs(tr\()p Fo(\036)p Fs(\()p Fo(H)1700 977 y Fn(D)1692 1043 y(!)r(;)p Fi(\003)1807 1054 y Fh(L)1857 1018 y Fs(\))p Fx(1)1951 1034 y Fi(\003)2000 1045 y Fh(L)2046 1034 y Fj(n)p Fi(\003)2130 1048 y Fd(\(1)p Fg(\000)p Fd(2)p Fh(\017)p Fd(\))p Fh(L)2364 1018 y Fs(\))2402 908 y Fm(i)2449 904 y(\014)2449 963 y(\014)2449 1023 y(\014)2510 1018 y Fl(\024)28 b Fo(C)7 b(\017)2741 945 y Fd(1)p 2741 957 31 3 v 2741 998 a Fh(q)2786 1018 y Fo(L)2852 972 y Fn(\013)p Fi(\(1+)3024 938 y Fh(d)p Fd(\()p Fh(q)r Fg(\000)p Fd(1\))p 3025 957 189 3 v 3089 998 a(2)p Fh(q)3223 972 y Fi(\))3255 1018 y Fo(:)118 736 y Fs(\(2.21\))118 1232 y(On)30 b(the)h(other)g (hand,)g(let)f Fo(\037)g Fs(b)s(e)h(a)f Fl(C)1464 1196 y Fj(1)1458 1256 y Fi(0)1539 1232 y Fs(-function)f(supp)s(orted)i(in)f (\003)2587 1247 y Fn(L)2669 1232 y Fs(that)g(is)g(constan)m(t)h(equal)f (to)g(1)g(in)118 1348 y(\003)186 1363 y Fi(\(1)p Fj(\000)p Fn(\017)p Fi(\))p Fn(L)412 1348 y Fs(.)43 b(F)-8 b(or)32 b Fo(\015)g Fl(2)d Fs(\003)903 1363 y Fi(\(1)p Fj(\000)p Fi(2)p Fn(\017)p Fi(\))p Fn(L)1163 1348 y Fs(,)k(w)m(e)g(use)h(the)f (resolv)m(en)m(t)h(form)m(ula)c(\(2.17\))i(to)g(write)121 1634 y(\()p Fo(i)22 b Fs(+)g Fo(H)401 1593 y Fn(D)393 1659 y(!)r(;)p Fi(\003)508 1670 y Fh(L)558 1634 y Fs(\))596 1593 y Fj(\000)p Fn(q)689 1634 y Fx(1)745 1649 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))982 1634 y Fs(=)28 b(\()p Fo(i)22 b Fs(+)g Fo(H)1366 1593 y Fn(D)1358 1659 y(!)r(;)p Fi(\003)1473 1670 y Fh(L)1523 1634 y Fs(\))1561 1593 y Fj(\000)p Fn(q)1654 1634 y Fo(\037)p Fx(1)1771 1649 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))982 1801 y Fs(=)28 b(\()p Fo(i)22 b Fs(+)g Fo(H)1366 1760 y Fn(D)1358 1826 y(!)r(;)p Fi(\003)1473 1837 y Fh(L)1523 1801 y Fs(\))1561 1760 y Fj(\000)p Fi(\()p Fn(q)r Fj(\000)p Fi(1\))1799 1801 y Fo(\037)p Fs(\()p Fo(i)h Fs(+)f Fo(H)2133 1816 y Fn(!)2183 1801 y Fs(\))2221 1760 y Fj(\000)p Fi(1)2315 1801 y Fx(1)2371 1817 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1922 1969 y Fl(\000)22 b Fs(\()p Fo(i)h Fs(+)f Fo(H)2302 1928 y Fn(D)2294 1993 y(!)r(;)p Fi(\003)2409 2004 y Fh(L)2459 1969 y Fs(\))2497 1928 y Fj(\000)p Fi(\()p Fn(q)r Fj(\000)p Fi(1\))2751 1969 y Fs([\001)p Fo(;)17 b(\037)p Fs(])g(\()p Fo(i)22 b Fs(+)g Fo(H)3280 1984 y Fn(!)3331 1969 y Fs(\))3369 1928 y Fj(\000)p Fi(1)3463 1969 y Fx(1)3519 1984 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))982 2212 y Fs(=)28 b Fo(\037)p Fs(\()p Fo(i)22 b Fs(+)g Fo(H)1419 2227 y Fn(!)1469 2212 y Fs(\))1507 2171 y Fj(\000)p Fn(q)1600 2212 y Fx(1)1656 2228 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1888 2212 y Fl(\000)2042 2084 y Fn(q)1987 2118 y Fm(X)1998 2327 y Fn(j)t Fi(=1)2131 2212 y Fs(\()p Fo(i)23 b Fs(+)f Fo(H)2412 2171 y Fn(D)2404 2237 y(!)r(;)p Fi(\003)2519 2248 y Fh(L)2569 2212 y Fs(\))2607 2171 y Fj(\000)p Fn(j)2715 2212 y Fs([\001)p Fo(;)17 b(\037)p Fs(])g(\()p Fo(i)22 b Fs(+)g Fo(H)3244 2227 y Fn(!)3294 2212 y Fs(\))3332 2171 y Fj(\000)p Fn(q)r Fi(+)p Fn(j)t Fj(\000)p Fi(1)3602 2212 y Fx(1)3658 2228 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))3868 2212 y Fo(:)118 2470 y Fs(Hence)997 2629 y(\()p Fo(i)23 b Fs(+)f Fo(H)1278 2587 y Fn(D)1270 2653 y(!)r(;)p Fi(\003)1385 2664 y Fh(L)1435 2629 y Fs(\))1473 2587 y Fj(\000)p Fn(q)1566 2629 y Fx(1)1622 2644 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1859 2629 y Fs(=)27 b(\()p Fo(i)c Fs(+)f Fo(H)2235 2644 y Fn(!)2285 2629 y Fs(\))2323 2587 y Fj(\000)p Fn(q)2416 2629 y Fx(1)2472 2644 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2703 2629 y Fs(+)22 b Fo(R)q Fs(\()p Fo(L)p Fs(\))-2900 b(\(2.22\))118 2788 y(where)267 3122 y Fo(R)q Fs(\()p Fo(L)p Fs(\))28 b(=)g Fl(\000)p Fs(\(1)22 b Fl(\000)h Fo(\037)p Fs(\)\()p Fo(i)f Fs(+)g Fo(H)1273 3137 y Fn(!)1323 3122 y Fs(\))1361 3081 y Fj(\000)p Fn(q)1454 3122 y Fx(1)1510 3138 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1742 3122 y Fl(\000)1896 2994 y Fn(q)1841 3028 y Fm(X)1852 3238 y Fn(j)t Fi(=1)1985 3122 y Fs(\()p Fo(i)22 b Fs(+)g Fo(H)2265 3081 y Fn(D)2257 3147 y(!)r(;)p Fi(\003)2372 3158 y Fh(L)2423 3122 y Fs(\))2461 3081 y Fj(\000)p Fn(j)2568 3122 y Fs([)q(\001)p Fo(;)17 b(\037)p Fs(])f(\()p Fo(i)23 b Fs(+)f Fo(H)3098 3137 y Fn(!)3148 3122 y Fs(\))3186 3081 y Fj(\000)p Fn(q)r Fi(+)p Fn(j)t Fj(\000)p Fi(1)3456 3122 y Fx(1)3512 3138 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))3721 3122 y Fo(:)118 3386 y Fs(By)34 b(the)h(supp)s(ort)f(prop)s(erties)g(of)f(1)23 b Fl(\000)g Fo(\037)34 b Fs(and)g([\001)p Fo(;)17 b(\037)p Fs(],)35 b(b)m(y)g(Lemma)d(5.4)h(and)h(b)m(y)h(the)f(assumption)g(on) 118 3503 y Fo(!)t Fs(,)e(w)m(e)i(ha)m(v)m(e)g(that,)e(for)g(some)g Fo(C)j(>)28 b Fs(0,)1190 3686 y Fl(k)p Fo(R)q Fs(\()p Fo(L)p Fs(\))p Fl(k)1507 3701 y Fj(T)1546 3710 y Fd(1)1612 3686 y Fl(\024)g Fo(C)7 b(k)1909 3605 y Fh(p)p 1858 3621 135 3 v 1858 3666 a(p)p Fg(\000)p Fh(p)1970 3652 y Fg(0)2007 3686 y Fo(e)2052 3645 y Fj(\000)p Fi(\()p Fn(\017L)p Fi(\))2238 3622 y Fd(1)p Fg(\000)p Fh(\013)2361 3645 y Fn(=)p Fi(\()p Fn(C)e(k)2517 3622 y Fh(p=)p Fd(\()p Fh(p)p Fg(\000)p Fh(p)2717 3601 y Fg(0)2739 3622 y Fd(\))2767 3645 y Fi(\))2799 3686 y Fo(:)118 3851 y Fs(Plugging)31 b(this)h(in)m(to)g(\(2.5\))o(,)h(w)m(e)h(get)e(that,)h(for)f(some)g Fo(C)j(>)27 b Fs(0,)419 4040 y Fl(j)p Fs(tr)o(\()p Fx(1)616 4056 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))825 4040 y Fo(\036)p Fs(\()p Fo(H)1010 3999 y Fn(D)1002 4065 y(!)r(;)p Fi(\003)1117 4076 y Fh(L)1167 4040 y Fs(\))p Fx(1)1261 4056 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))1470 4040 y Fs(\))22 b Fl(\000)h Fs(tr\()p Fx(1)1800 4056 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2009 4040 y Fo(\036)p Fs(\()p Fo(H)2186 4055 y Fn(!)2236 4040 y Fs(\))p Fx(1)2330 4056 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))2539 4040 y Fs(\))p Fl(j)27 b(\024)i Fo(C)7 b(k)2930 3959 y Fh(p)p 2879 3975 V 2879 4020 a(p)p Fg(\000)p Fh(p)2991 4006 y Fg(0)3027 4040 y Fo(e)3072 3999 y Fj(\000)p Fi(\()p Fn(\017L)p Fi(\))3258 3976 y Fd(1)p Fg(\000)p Fh(\013)3381 3999 y Fn(=)p Fi(\()p Fn(C)e(k)3537 3976 y Fh(p=)p Fd(\()p Fh(p)p Fg(\000)p Fh(p)3737 3955 y Fg(0)3760 3976 y Fd(\))3788 3999 y Fi(\))3819 4040 y Fo(:)-3728 b Fs(\(2.23\))118 4205 y(T)-8 b(aking)47 b(the)h(exp)s(ectation)f(of)54 b(\(2.23\))o(,)d(summing)46 b(in)g Fo(k)51 b Fs(and)c(using)i(\(5.1\))d (in)h(the)h(same)f(w)m(a)m(y)h(as)118 4322 y(in)32 b(\(2.13\))o(,)h(w)m (e)g(get)g(that)618 4489 y Fk(E)696 4408 y Fm(\000)747 4489 y Fl(j)p Fs(tr\()p Fx(1)945 4504 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1154 4489 y Fo(\036)p Fs(\()p Fo(H)1339 4447 y Fn(D)1331 4513 y(!)r(;)p Fi(\003)1446 4524 y Fh(L)1496 4489 y Fs(\))p Fx(1)1590 4504 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))1799 4489 y Fs(\))22 b Fl(\000)g Fs(tr\()p Fx(1)2128 4504 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2337 4489 y Fo(\036)p Fs(\()p Fo(H)2514 4504 y Fn(!)2564 4489 y Fs(\))p Fx(1)2658 4504 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))2867 4489 y Fs(\))p Fl(j)2933 4408 y Fm(\001)3006 4489 y Fl(\024)29 b Fo(C)7 b Fs(\()p Fo(\017L)p Fs(\))3370 4447 y Fj(\000)p Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))3620 4489 y Fo(:)-3529 b Fs(\(2.24\))118 4653 y(Then,)34 b(b)m(y)h(\(2.21\)) d(and)g(\(2.24\),)g(w)m(e)i(ha)m(v)m(e)g(that)355 4823 y(1)p 211 4867 337 4 v 211 4958 a(V)-8 b(ol)o(\(\003)458 4973 y Fn(L)510 4958 y Fs(\))657 4795 y Fm(X)574 5015 y Fn(\015)t Fj(2)p Fi(\003)710 5026 y Fh(L)757 5015 y Fj(\\)p Ff(Z)854 4996 y Fh(d)901 4890 y Fk(E)978 4809 y Fm(\000)1030 4890 y Fs(tr)o(\()p Fx(1)1199 4906 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1409 4890 y Fo(\036)p Fs(\()p Fo(H)1594 4849 y Fn(D)1586 4915 y(!)r(;)p Fi(\003)1701 4926 y Fh(L)1750 4890 y Fs(\))p Fx(1)1844 4906 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))2054 4890 y Fs(\))2092 4809 y Fm(\001)2165 4890 y Fs(=)1550 5207 y(=)1808 5139 y(1)p 1664 5184 V 1664 5275 a(V)-8 b(ol)o(\(\003)1911 5290 y Fn(L)1962 5275 y Fs(\))2110 5112 y Fm(X)2027 5332 y Fn(\015)t Fj(2)p Fi(\003)2163 5343 y Fh(L)2209 5332 y Fj(\\)p Ff(Z)2307 5313 y Fh(d)2354 5207 y Fk(E)2431 5126 y Fm(\000)2482 5207 y Fs(tr\()p Fx(1)2652 5222 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))2861 5207 y Fo(\036)p Fs(\()p Fo(H)3038 5222 y Fn(!)3088 5207 y Fs(\))p Fx(1)3182 5222 y Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(1\))3391 5207 y Fs(\))3429 5126 y Fm(\001)3497 5207 y Fs(+)22 b Fo(Q)p Fs(\()p Fo(L)p Fs(\))1969 5454 y Fp(16)p eop %%Page: 17 17 17 16 bop 118 241 a Fs(where)977 528 y Fl(j)p Fo(Q)p Fs(\()p Fo(L)p Fs(\))p Fl(j)28 b(\024)g Fo(C)7 b Fs(\()p Fo(\017L)p Fs(\))1643 487 y Fj(\000)p Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))1915 528 y Fs(+)22 b Fo(C)7 b(\017)2139 455 y Fd(1)p 2140 467 31 3 v 2140 508 a Fh(q)2184 528 y Fo(L)2250 482 y Fn(\013)p Fi(\(1+)2422 448 y Fh(d)p Fd(\()p Fh(q)r Fg(\000)p Fd(1\))p 2423 467 189 3 v 2487 508 a(2)p Fh(q)2622 482 y Fi(\))2676 528 y Fs(+)22 b Fo(C)7 b(L)2917 487 y Fj(\000)p Fi(1)3011 528 y Fo(:)118 820 y Fs(If)40 b(w)m(e)h(c)m(ho)s(ose)g Fo(\014)46 b Fs(=)41 b Fo(\013)q Fs(\()p Fo(q)30 b Fs(+)d Fo(d)p Fs(\()p Fo(q)k Fl(\000)d Fs(1\))p Fo(=)p Fs(2\),)41 b(set)g Fo(\017)g Fs(=)f Fo(L)2167 784 y Fj(\000)p Fn(\014)2309 820 y Fs(and)h(pic)m(k)f (0)g Fo(<)h(\013)g(<)f Fs(1)p Fo(=)p Fs(3)f(small)f(enough)118 936 y(so)g(that)g(1)26 b Fl(\000)g Fo(\014)42 b(>)37 b Fs(1)p Fo(=)p Fs(2,)i(w)m(e)g(get)f(that)g Fo(Q)p Fs(\()p Fo(L)p Fs(\))f Fl(!)g Fs(0)g(as)i Fo(L)e Fl(!)f Fs(+)p Fl(1)p Fs(.)60 b(This)38 b(completes)g(the)g(pro)s(of)f(of)118 1052 y(Theorem)c(1.3.)p 3827 1168 4 66 v 3831 1106 59 4 v 3831 1168 V 3889 1168 4 66 v 118 1412 a Fx(Remark)k(2.3.)49 b Fs(Remark)32 b(2.2)g(applies)g(also)g(for)g(the)h(pro)s(of)f(of)g (Theorem)h(1.3.)118 1655 y Fx(Pro)s(of)39 b(of)h(Lemma)f(2.2.)48 b Fs(Under)35 b(our)f(assumptions)g(on)g Fo(!)t Fs(,)g(w)m(e)i(ha)m(v)m (e)f Fo(V)2900 1670 y Fn(!)2981 1655 y Fl(2)c Fs(\012)3148 1670 y Fn(k)3191 1655 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)3478 1619 y Fj(0)3500 1655 y Fs(\).)48 b(Hence,)118 1789 y(b)m(y)34 b(Lemma)d(5.1,)h(w)m(e)i(can)e(decomp)s(ose)h Fo(V)1656 1804 y Fn(!)1734 1789 y Fs(=)28 b Fo(V)1917 1739 y Fi(\()p Fn(i)p Fi(\))1895 1800 y Fn(!)2022 1789 y Fs(+)22 b Fo(V)2198 1739 y Fi(\()p Fn(r)r Fi(\))2177 1800 y Fn(!)2324 1789 y Fs(where)244 1999 y Fl(\017)41 b Fo(V)414 1958 y Fi(\()p Fn(i)p Fi(\))392 2023 y Fn(!)524 1999 y Fl(2)28 b Fo(L)684 1952 y Fn(p)720 1928 y Fg(0)684 2026 y Fi(lo)r(c,unif)916 1999 y Fs(\(\003)1022 2014 y Fn(L)1074 1999 y Fs(\))33 b(and)f(sup)1481 2022 y Fn(x)p Fj(2)p Fi(\003)1617 2033 y Fh(L)1684 1999 y Fl(k)p Fo(V)1812 1948 y Fi(\()p Fn(i)p Fi(\))1791 2009 y Fn(!)1895 1999 y Fl(k)1945 2025 y Fn(L)1993 2006 y Fh(p)2025 1992 y Fg(0)2052 2025 y Fi(\()p Fn(C)5 b Fi(\()p Fn(x;)p Fi(1\)\))2343 1999 y Fl(\024)28 b Fs(1.)244 2153 y Fl(\017)41 b Fs(for)32 b Fo(x)c Fl(2)g Fs(\003)729 2168 y Fn(L)781 2153 y Fs(,)33 b Fl(j)p Fo(V)947 2102 y Fi(\()p Fn(r)r Fi(\))926 2163 y Fn(!)1040 2153 y Fs(\()p Fo(x)p Fs(\))p Fl(j)28 b(\024)g Fo(C)7 b(k)1524 2072 y Fh(p)p 1473 2087 135 3 v 1473 2132 a(p)p Fg(\000)p Fh(p)1585 2118 y Fg(0)1621 2153 y Fo(L)1687 2117 y Fn(\013)1737 2153 y Fs(.)118 2350 y(F)-8 b(or)32 b Fo(')27 b Fl(2)i(C)537 2314 y Fj(1)531 2374 y Fi(0)612 2350 y Fs(\(\003)718 2365 y Fn(L)769 2350 y Fs(\),)k(this)f(yields)880 2631 y Fl(h)p Fo(H)1008 2590 y Fn(D)1000 2656 y(!)r(;)p Fi(\003)1115 2667 y Fh(L)1165 2631 y Fo(';)17 b(')p Fl(i)27 b Fs(=)h Fl(h\000)p Fs(\001)1704 2590 y Fn(D)1704 2656 y Fi(\003)1753 2667 y Fh(L)1804 2631 y Fo(';)17 b(')p Fl(i)k Fs(+)h Fl(h)p Fo(V)2251 2590 y Fn(i)2230 2656 y(!)2280 2631 y Fo(';)17 b(')p Fl(i)k Fs(+)h Fl(h)p Fo(V)2728 2590 y Fn(r)2706 2656 y(!)2766 2631 y Fo(';)17 b(')p Fl(i)1403 2839 y(\025)1518 2772 y Fs(1)p 1518 2816 49 4 v 1518 2908 a(2)1577 2839 y Fl(h\000)p Fs(\001)1774 2798 y Fn(D)1774 2864 y Fi(\003)1823 2875 y Fh(L)1874 2839 y Fo(';)g(')p Fl(i)k(\000)i Fo(C)2300 2729 y Fm(\020)2359 2839 y Fo(k)2474 2758 y Fh(p)p 2423 2774 135 3 v 2423 2819 a(p)p Fg(\000)p Fh(p)2535 2805 y Fg(0)2572 2839 y Fo(L)2638 2798 y Fn(\013)2709 2839 y Fs(+)g(1)2857 2729 y Fm(\021)2932 2839 y Fl(k)p Fo(')p Fl(k)3096 2798 y Fi(2)118 2751 y Fs(\(2.25\))118 3153 y(Equation)32 b(\(2.25\))g(and)h(the)g(v)-5 b(ariational)29 b(principle)i(for)h(eigen)m(v)-5 b(alues)33 b(immediately)c(imply)i (that)1062 3479 y Fo(\025)1119 3494 y Fn(j)1155 3479 y Fs(\()p Fo(H)1282 3438 y Fn(D)1274 3504 y(!)r(;)p Fi(\003)1389 3515 y Fh(L)1439 3479 y Fs(\))d Fl(\025)1620 3412 y Fs(1)p 1620 3456 49 4 v 1620 3548 a(2)1679 3479 y Fo(\025)1736 3494 y Fn(j)1772 3479 y Fs(\()p Fl(\000)p Fs(\001)1968 3438 y Fn(D)1968 3504 y Fi(\003)2017 3515 y Fh(L)2068 3479 y Fs(\))22 b Fl(\000)h Fo(C)2321 3369 y Fm(\020)2381 3479 y Fo(k)2496 3398 y Fh(p)p 2445 3414 135 3 v 2445 3459 a(p)p Fg(\000)p Fh(p)2557 3445 y Fg(0)2593 3479 y Fo(L)2659 3438 y Fn(\013)2731 3479 y Fs(+)f(1)2878 3369 y Fm(\021)118 3792 y Fs(where)j Fo(\025)448 3807 y Fn(j)485 3792 y Fs(\()p Fo(H)8 b Fs(\))23 b(denotes)j(the)e Fo(j)6 b Fs(-th)24 b(eigen)m(v)-5 b(alue)23 b(of)h Fo(H)32 b Fs(\(ordered)24 b(increasingly)f(coun)m(ting)h(m)m(ultiplicit)m(y\).) 118 3909 y(Hence)171 4332 y Fl(k)p Fs(\()p Fo(i)e Fs(+)g Fo(H)501 4291 y Fn(D)493 4357 y(!)r(;)p Fi(\003)608 4368 y Fh(L)658 4332 y Fs(\))696 4291 y Fj(\000)p Fi(1)791 4332 y Fl(k)841 4285 y Fn(q)841 4359 y Fj(T)880 4367 y Fh(q)946 4332 y Fs(=)1049 4238 y Fm(X)1057 4450 y Fn(j)t Fj(2)p Ff(N)1645 4265 y Fs(1)p 1220 4309 900 4 v 1220 4349 a Fm(\000)1265 4430 y Fs(1)g(+)g(\()p Fo(\025)1529 4445 y Fn(j)1566 4430 y Fs(\()p Fo(H)1693 4395 y Fn(D)1685 4457 y(!)r(;)p Fi(\003)1800 4468 y Fh(L)1850 4430 y Fs(\)\))1926 4401 y Fi(2)1965 4349 y Fm(\001)2011 4371 y Fn(q)r(=)p Fi(2)946 4614 y Fl(\024)1436 4520 y Fm(X)1444 4732 y Fn(j)t Fj(2)p Ff(N)1051 4851 y Fn(\025)1092 4861 y Fh(j)1125 4851 y Fi(\()p Fj(\000)p Fi(\001)1266 4827 y Fh(D)1266 4873 y Fd(\003)1307 4890 y Fh(L)1358 4851 y Fi(\))p Fn(<)p Fi(4)p Fn(C)5 b Fi(\()p Fn(k)1657 4775 y Fh(p)p 1606 4790 135 3 v 1606 4834 a(p)p Fg(\000)p Fh(p)1718 4820 y Fg(0)1755 4851 y Fn(L)1803 4827 y Fh(\013)1847 4851 y Fi(+1\))1981 4614 y Fs(1)22 b(+)1058 5069 y(+)1541 4974 y Fm(X)1549 5186 y Fn(j)t Fj(2)p Ff(N)1156 5305 y Fn(\025)1197 5315 y Fh(j)1230 5305 y Fi(\()p Fj(\000)p Fi(\001)1371 5282 y Fh(D)1371 5328 y Fd(\003)1412 5345 y Fh(L)1463 5305 y Fi(\))p Fj(\025)p Fi(4)p Fn(C)5 b Fi(\()p Fn(k)1763 5229 y Fh(p)p 1711 5244 V 1711 5288 a(p)p Fg(\000)p Fh(p)1823 5274 y Fg(0)1860 5305 y Fn(L)1908 5282 y Fh(\013)1952 5305 y Fi(+1\))2928 5002 y Fs(1)p 2096 5046 1712 4 v 2096 5086 a Fm(\020)2156 5196 y Fs(1)22 b(+)g(\()p Fo(\025)2420 5211 y Fn(j)2456 5196 y Fs(\()p Fl(\000)p Fs(\001)2652 5162 y Fn(D)2652 5223 y Fi(\003)2701 5234 y Fh(L)2752 5196 y Fs(\))g Fl(\000)h Fo(C)7 b Fs(\()p Fo(k)3142 5115 y Fh(p)p 3091 5131 135 3 v 3091 5176 a(p)p Fg(\000)p Fh(p)3203 5162 y Fg(0)3239 5196 y Fo(L)3305 5167 y Fn(\013)3377 5196 y Fs(+)22 b(1\)\))3600 5167 y Fi(2)3639 5086 y Fm(\021)3699 5108 y Fn(q)r(=)p Fi(2)3817 5069 y Fo(:)118 4153 y Fs(\(2.26\))1969 5454 y Fp(17)p eop %%Page: 18 18 18 17 bop 118 241 a Fs(No)m(w)34 b(the)f(eigen)m(v)-5 b(alues)34 b(of)e Fl(\000)p Fs(\001)1284 205 y Fn(D)1284 266 y Fi(\003)1333 277 y Fh(L)1417 241 y Fs(are)i(kno)m(wn)g (explicitly:)43 b(for)33 b Fo(j)h Fs(=)29 b(\()p Fo(j)2759 256 y Fi(1)2798 241 y Fo(;)17 b(:)g(:)g(:)f(;)h(j)3057 256 y Fn(d)3098 241 y Fs(\))28 b Fl(2)h Fs(\()p Fk(N)38 b Fl(n)22 b(f)p Fs(0)p Fl(g)p Fs(\))3651 205 y Fn(d)3691 241 y Fs(,)33 b(one)118 432 y(has)g Fo(\025)349 447 y Fn(j)385 432 y Fs(\()p Fl(\000)p Fs(\001)581 391 y Fn(D)581 456 y Fi(\003)630 467 y Fh(L)682 432 y Fs(\))27 b(=)864 364 y Fo(\031)923 328 y Fi(2)p 861 409 106 4 v 861 500 a Fo(L)927 471 y Fi(2)1030 337 y Fm(X)993 549 y Fi(1)p Fj(\024)p Fn(p)p Fj(\024)p Fn(d)1227 432 y Fo(j)1273 391 y Fi(2)1267 456 y Fn(p)1312 432 y Fs(.)43 b(Hence)34 b(equation)f(\(2.26\))e(tells)h(us)h(that)276 907 y Fl(k)p Fs(\()p Fo(i)22 b Fs(+)g Fo(H)606 866 y Fn(D)598 932 y(!)r(;)p Fi(\003)713 943 y Fh(L)763 907 y Fs(\))801 866 y Fj(\000)p Fi(1)896 907 y Fl(k)946 860 y Fn(q)946 934 y Fj(T)985 942 y Fh(q)1051 907 y Fl(\024)1156 796 y Fm(\020)1215 907 y Fs(4)p Fo(C)7 b(L)1407 866 y Fi(2)1447 907 y Fs(\()p Fo(k)1600 826 y Fh(p)p 1549 842 135 3 v 1549 887 a(p)p Fg(\000)p Fh(p)1661 873 y Fg(0)1697 907 y Fo(L)1763 866 y Fn(\013)1835 907 y Fs(+)22 b(1\))2020 796 y Fm(\021)2079 819 y Fn(d=)p Fi(2)2213 907 y Fs(+)1163 1155 y(+)1617 1061 y Fm(X)1261 1324 y Fj(j)p Fn(j)t Fj(j\025)p Fi(2)1424 1268 y Fj(p)p 1482 1268 55 3 v 1482 1324 a Fn(C)5 b(L)p Fi(\()p Fn(k)1712 1248 y Fh(p)p 1661 1263 135 3 v 1661 1307 a(p)p Fg(\000)p Fh(p)1773 1293 y Fg(0)1809 1324 y Fn(L)1857 1305 y Fh(\013)1901 1324 y Fi(+1\))2018 1304 y Fd(1)p Fh(=)p Fd(2)2866 1088 y Fo(L)2932 1052 y Fi(2)p Fn(q)p 2142 1132 1588 4 v 2142 1172 a Fm(\020)2202 1283 y Fo(L)2268 1254 y Fi(4)2330 1283 y Fs(+)22 b(\()p Fl(j)p Fo(j)6 b Fl(j)2568 1254 y Fi(2)2629 1283 y Fl(\000)22 b Fo(C)7 b(L)2871 1254 y Fi(2)2911 1283 y Fs(\()p Fo(k)3064 1201 y Fh(p)p 3013 1217 135 3 v 3013 1262 a(p)p Fg(\000)p Fh(p)3125 1248 y Fg(0)3161 1283 y Fo(L)3227 1254 y Fn(\013)3299 1283 y Fs(+)22 b(1\)\))3522 1254 y Fi(2)3562 1172 y Fm(\021)3621 1194 y Fn(q)r(=)p Fi(2)1051 1526 y Fl(\024)1156 1415 y Fm(\020)1215 1526 y Fs(4)p Fo(C)7 b(L)1407 1485 y Fi(2)1447 1526 y Fs(\()p Fo(k)1600 1445 y Fh(p)p 1549 1461 V 1549 1506 a(p)p Fg(\000)p Fh(p)1661 1492 y Fg(0)1697 1526 y Fo(L)1763 1485 y Fn(\013)1835 1526 y Fs(+)22 b(1\))2020 1415 y Fm(\021)2079 1438 y Fn(d=)p Fi(2)2213 1526 y Fs(+)1163 1774 y(+)1617 1680 y Fm(X)1261 1943 y Fj(j)p Fn(j)t Fj(j\025)p Fi(2)1424 1887 y Fj(p)p 1482 1887 55 3 v 1482 1943 a Fn(C)5 b(L)p Fi(\()p Fn(k)1712 1867 y Fh(p)p 1661 1882 135 3 v 1661 1926 a(p)p Fg(\000)p Fh(p)1773 1912 y Fg(0)1809 1943 y Fn(L)1857 1924 y Fh(\013)1901 1943 y Fi(+1\))2018 1923 y Fd(1)p Fh(=)p Fd(2)2397 1707 y Fo(L)2463 1671 y Fi(2)p Fn(q)p 2142 1752 649 4 v 2142 1862 a Fs(\()p Fo(L)2246 1833 y Fi(4)2308 1862 y Fs(+)22 b Fl(j)p Fo(j)6 b Fl(j)2508 1833 y Fi(4)2547 1862 y Fo(=)p Fs(2\))2682 1813 y Fn(q)r(=)p Fi(2)1051 2145 y Fl(\024)1156 2035 y Fm(\020)1215 2145 y Fs(4)p Fo(C)h(L)1407 2104 y Fi(2)1447 2145 y Fs(\()p Fo(k)1600 2064 y Fh(p)p 1549 2080 135 3 v 1549 2125 a(p)p Fg(\000)p Fh(p)1661 2111 y Fg(0)1697 2145 y Fo(L)1763 2104 y Fn(\013)1835 2145 y Fs(+)22 b(1\))2020 2035 y Fm(\021)2079 2057 y Fn(d=)p Fi(2)2213 2145 y Fs(+)g Fo(C)2381 2160 y Fn(q)2418 2145 y Fo(L)2484 2104 y Fn(d)1051 2366 y Fl(\024)28 b Fo(C)1226 2381 y Fn(q)1264 2366 y Fo(k)1402 2282 y Fh(pd)p 1328 2297 213 3 v 1328 2342 a Fd(2\()p Fh(p)p Fg(\000)p Fh(p)1494 2328 y Fg(0)1516 2342 y Fd(\))1555 2366 y Fo(L)1621 2324 y Fn(d)p Fi(\(1+)1784 2297 y Fh(\013)p 1785 2309 41 3 v 1790 2351 a Fd(2)1835 2324 y Fi(\))1866 2366 y Fo(:)118 2560 y Fs(This)48 b(completes)g(the)h (pro)s(of)e(of)54 b(\(2.18\).)89 b(The)49 b(pro)s(of)f(of)54 b(\(2.19\))47 b(will)f(follo)m(w)g(from)h(an)h(explicit)118 2676 y(computation.)42 b(As)33 b(a)g(corollary)d(of)i(the)h (Golden-Thompson)f(inequalities)f(\(see)i([25]\),)g(w)m(e)g(ha)m(v)m(e) 655 2872 y Fl(k)p Fs(\(1)21 b Fl(\000)i Fs(\001)994 2831 y Fn(D)994 2897 y Fi(\003)1043 2908 y Fh(L)1094 2872 y Fs(\))1132 2831 y Fj(\000)p Fi(1)1226 2872 y Fx(1)1282 2887 y Fi(\003)1331 2868 y Fg(0)1357 2872 y Fl(k)1407 2825 y Fn(q)1407 2899 y Fj(T)1446 2907 y Fh(q)1512 2872 y Fl(\024)29 b(k)p Fx(1)1724 2887 y Fi(\003)1773 2868 y Fg(0)1799 2872 y Fs(\(1)22 b Fl(\000)g Fs(\001)2088 2831 y Fn(D)2088 2897 y Fi(\003)2137 2908 y Fh(L)2188 2872 y Fs(\))2226 2831 y Fj(\000)p Fn(q)r(=)p Fi(2)2390 2872 y Fl(k)2440 2831 y Fi(2)2440 2897 y Fj(T)2479 2906 y Fd(2)1512 3059 y Fs(=)1724 2965 y Fm(X)1616 3184 y Fn(j)t Fj(2)p Fi(\()p Ff(N)p Fj(nf)p Fi(0)p Fj(g)p Fi(\))1934 3165 y Fh(d)1976 3059 y Fl(h)p Fs(\(1)f Fl(\000)i Fs(\001)2304 3018 y Fn(D)2304 3084 y Fi(\003)2353 3095 y Fh(L)2403 3059 y Fs(\))2441 3018 y Fj(\000)p Fn(q)r(=)p Fi(2)2605 3059 y Fx(1)2661 3074 y Fi(\003)2710 3055 y Fg(0)2736 3059 y Fs(\(1)f Fl(\000)h Fs(\001)3026 3018 y Fn(D)3026 3084 y Fi(\003)3075 3095 y Fh(L)3126 3059 y Fs(\))3164 3018 y Fj(\000)p Fn(q)r(=)p Fi(2)3327 3059 y Fo(')3391 3074 y Fn(j)3427 3059 y Fo(;)17 b(')3535 3074 y Fn(j)3572 3059 y Fl(i)1512 3380 y Fs(=)1724 3285 y Fm(X)1616 3505 y Fn(j)t Fj(2)p Fi(\()p Ff(N)p Fj(nf)p Fi(0)p Fj(g)p Fi(\))1934 3486 y Fh(d)2166 3312 y Fs(1)p 2002 3357 377 4 v 2002 3448 a(\(1)22 b(+)g Fo(\025)2266 3463 y Fn(j)2302 3448 y Fs(\))2340 3419 y Fn(q)2388 3380 y Fl(h)p Fx(1)2483 3395 y Fi(\003)2532 3376 y Fg(0)2559 3380 y Fo(')2623 3395 y Fn(j)2659 3380 y Fo(;)17 b(')2767 3395 y Fn(j)2803 3380 y Fl(i)118 3181 y Fs(\(2.27\))118 3693 y(where,)41 b(for)c(the)h(sak)m(e)h(of)f(simplicit)m(y)-8 b(,)36 b(w)m(e)j(denoted)g(the)f(eigen)m(v)-5 b(alues)38 b(of)f Fl(\000)p Fs(\001)3045 3657 y Fn(D)3045 3718 y Fi(\003)3094 3729 y Fh(L)3183 3693 y Fs(b)m(y)i(\()p Fo(\025)3419 3708 y Fn(j)3455 3693 y Fs(\))3493 3708 y Fn(j)3567 3693 y Fs(and)f(the)118 3813 y(asso)s(ciated)23 b(eigen)m(v)m(ector)h(b)m(y) g(\()p Fo(')1298 3828 y Fn(j)1334 3813 y Fs(\))1372 3828 y Fn(j)1409 3813 y Fs(.)40 b(The)24 b(v)m(ector)g Fo(')2014 3828 y Fn(j)2073 3813 y Fs(b)s(eing)e(explicitly)f(kno)m(wn,)27 b(one)c(easily)f(computes)1497 4085 y Fl(h)p Fx(1)1592 4100 y Fi(\003)1641 4081 y Fg(0)1667 4085 y Fo(')1731 4100 y Fn(j)1767 4085 y Fo(;)17 b(')1875 4100 y Fn(j)1911 4085 y Fl(i)28 b(\024)2083 3945 y Fm(\022)2175 4018 y Fs(2)p 2166 4062 67 4 v 2166 4153 a Fo(L)2242 3945 y Fm(\023)2316 3967 y Fn(d)2373 4085 y Fl(j)p Fs(\003)2469 4044 y Fj(0)2492 4085 y Fl(j)118 4335 y Fs(Plugging)j(this)h(in)m(to)g (equation)g(\(2.27\),)g(w)m(e)i(get)735 4605 y Fl(k)p Fs(\(1)22 b Fl(\000)g Fs(\001)1074 4563 y Fn(D)1074 4629 y Fi(\003)1123 4640 y Fh(L)1174 4605 y Fs(\))1212 4563 y Fj(\000)p Fi(1)1306 4605 y Fx(1)1362 4620 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))1571 4605 y Fl(k)1621 4557 y Fn(q)1621 4632 y Fj(T)1660 4640 y Fh(q)1726 4605 y Fl(\024)1831 4464 y Fm(\022)1924 4537 y Fs(2)p 1915 4582 V 1915 4673 a Fo(L)1991 4464 y Fm(\023)2064 4486 y Fn(d)2229 4510 y Fm(X)2121 4729 y Fn(j)t Fj(2)p Fi(\()p Ff(N)p Fj(nf)p Fi(0)p Fj(g)p Fi(\))2439 4711 y Fh(d)2623 4537 y Fl(j)p Fs(\003)2719 4501 y Fj(0)2741 4537 y Fl(j)p 2508 4582 377 4 v 2508 4673 a Fs(\(1)21 b(+)h Fo(\025)2771 4688 y Fn(j)2808 4673 y Fs(\))2846 4644 y Fn(q)2922 4605 y Fl(\024)28 b Fo(C)3097 4620 y Fn(q)3135 4605 y Fl(j)p Fs(\003)3231 4563 y Fj(0)3253 4605 y Fl(j)118 4906 y Fs(using)k(the)h(explicit)f(form)m(ula)e(giv)m(en)j(ab)s(o)m(v)m(e)g (for)f Fo(\025)1980 4921 y Fn(j)2017 4906 y Fs(.)p 3827 5022 4 66 v 3831 4959 59 4 v 3831 5022 V 3889 5022 4 66 v 118 5259 a(2.2.)56 b Fx(Application)30 b(to)j(the)f(estimation)f (of)i(the)f(in)m(tegrated)g(densit)m(y)g(of)h(states.)49 b Fs(W)-8 b(e)29 b(pro)m(v)m(e)1969 5454 y Fp(18)p eop %%Page: 19 19 19 18 bop 118 241 a Fx(Lemma)37 b(2.3.)49 b Fe(Pick)35 b Fo(\027)g(>)28 b Fs(0)p Fe(.)46 b(Ther)-5 b(e)35 b(exists)g Fo(\014)f(>)28 b Fs(0)35 b Fe(and)g Fo(E)2358 256 y Fn(\027)2430 241 y Fo(>)28 b Fs(0)35 b Fe(such)g(that,)h(for)f Fo(E)f(>)28 b(E)3508 256 y Fn(\027)3587 241 y Fe(and)35 b(for)118 357 y Fo(n)28 b Fl(\025)g Fo(E)387 321 y Fn(\014)434 357 y Fe(,)35 b(one)f(has)648 519 y Fk(E)12 b Fs(\()p Fo(N)824 534 y Fn(!)s(;n)943 519 y Fs(\()p Fl(\000)p Fo(E)29 b Fl(\000)23 b Fs(1\)\))e Fl(\000)i Fo(e)1550 478 y Fj(\000)p Fn(E)1661 454 y Fh(\027)1731 519 y Fl(\024)28 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))29 b Fl(\024)f Fk(E)12 b Fs(\()p Fo(N)2465 534 y Fn(!)r(;n)2584 519 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\)\))f(+)h Fo(e)3187 478 y Fj(\000)p Fn(E)3298 454 y Fh(\027)3341 519 y Fo(:)-3250 b Fs(\(2.28\))118 702 y Fx(Pro)s(of)41 b(of)g(Lemma)g(2.3.)54 b Fs(Pic)m(k)36 b Fo(\027)k(>)33 b Fs(0)i(arbitrary)-8 b(.)52 b(By)37 b(equation)e(\(5.13\))o(,)i(the)f(a-priori)d(estimate) 118 818 y(on)g Fo(N)43 b Fs(giv)m(en)32 b(in)g(Lemma)f(5.2,)i(w)m(e)g (kno)m(w)h(that,)e(for)g(some)h Fo(\034)39 b(>)28 b Fs(0)1613 1023 y Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)1894 982 y Fn(\034)1938 1023 y Fs(\))28 b Fl(\024)2119 955 y Fs(1)p 2119 1000 49 4 v 2119 1091 a(4)2178 1023 y Fo(e)2223 982 y Fj(\000)p Fn(E)2334 958 y Fh(\027)2376 1023 y Fo(:)-2285 b Fs(\(2.29\))118 1220 y(Hence)30 b(w)m(e)f(just)g(ha)m(v)m(e)h(to)e(estimate)f Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))14 b Fl(\000)g Fo(N)c Fs(\()p Fl(\000)p Fo(E)2159 1184 y Fn(\034)2204 1220 y Fs(\).)42 b(Therefore)30 b(in)m(tro)s(duce)e(t)m(w)m(o)h (functions)g Fo(')3839 1235 y Fj(\006)118 1336 y Fs(de\014ned)34 b(b)m(y)1488 1498 y Fo(')1552 1513 y Fj(\006)1638 1498 y Fs(=)28 b Fx(1)1798 1523 y Fi([)p Fj(\000)p Fn(E)1929 1504 y Fh(\034)1967 1523 y Fj(\007)2032 1496 y Fd(1)p 2031 1508 31 3 v 2031 1549 a(2)2072 1523 y Fn(;)p Fj(\000)p Fn(E)t Fj(\006)2268 1496 y Fd(1)p 2267 1508 V 2267 1549 a(2)2307 1523 y Fi(])2353 1498 y Fl(\003)22 b Fo(')2489 1513 y Fi(0)118 1677 y Fs(where)29 b Fo(')459 1692 y Fi(0)526 1677 y Fl(2)f(C)678 1641 y Fj(1)672 1702 y Fi(0)753 1677 y Fs(\()p Fk(R)5 b Fs(\))34 b(is)28 b(a)f(non)h(negativ)m(e)g (Gevrey)h(class)g(function)e(of)g(Gevrey)j(exp)s(onen)m(t)f Fo(\032)f(>)f Fs(1)h(suc)m(h)118 1794 y(that)33 b Fo(')394 1809 y Fi(0)462 1794 y Fl(\021)28 b Fs(1)33 b(on)g([)p Fl(\000)899 1755 y Fi(1)p 899 1771 36 4 v 899 1828 a(4)945 1794 y Fo(;)998 1755 y Fi(1)p 998 1771 V 998 1828 a(4)1044 1794 y Fs(])g(and)g(supp)p Fo(')1558 1809 y Fi(0)1626 1794 y Fl(\032)c Fs([)p Fl(\000)1846 1755 y Fi(1)p 1846 1771 V 1846 1828 a(2)1892 1794 y Fo(;)1946 1755 y Fi(1)p 1946 1771 V 1946 1828 a(2)1991 1794 y Fs(].)45 b(The)34 b(functions)f Fo(')2776 1809 y Fj(\006)2867 1794 y Fs(are)g(then)h (Gevrey)g(class)f(of)118 1911 y(Gevrey)h(exp)s(onen)m(t)g Fo(\032)f Fs(and)f(supp)p Fo(')1411 1926 y Fj(\006)1499 1911 y Fl(\032)c Fs([)p Fl(\000)p Fo(E)1786 1875 y Fn(\034)1852 1911 y Fl(\000)23 b Fs(1)p Fo(;)17 b Fl(\000)p Fo(E)28 b Fs(+)22 b(1])32 b(\(see)i(e.g.)44 b([6)o(]\).)g(Then)34 b(one)e(has)1076 2073 y Fl(h)p Fo(')1179 2088 y Fj(\000)1237 2073 y Fo(;)17 b(dN)10 b Fl(i)28 b(\024)g Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))23 b Fl(\000)g Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)2315 2032 y Fn(\034)2359 2073 y Fs(\))27 b Fl(\024)i(h)p Fo(')2633 2088 y Fi(+)2691 2073 y Fo(;)17 b(dN)10 b Fl(i)p Fo(:)-2822 b Fs(\(2.30\))118 2235 y(Using)36 b(the)h(Gevrey)h(estimates)e(for)g(the)h(deriv)-5 b(ativ)m(es)37 b(and)g(the)g(estimates)f(on)g(the)i(supp)s(ort)e(of)h Fo(')3812 2250 y Fj(\006)3871 2235 y Fs(,)118 2351 y(b)m(y)d(Lemma)d (2.1,)h(w)m(e)i(ha)m(v)m(e)g(that,)e(for)g(some)g Fo(C)j(>)28 b Fs(0,)k(and)h(for)f(all)e Fo(n)e Fl(2)g Fk(N)2823 2315 y Fj(\003)2901 2351 y Fs(and)33 b(all)d Fo(k)h Fl(2)d Fk(N)3468 2315 y Fj(\003)3514 2351 y Fs(,)571 2515 y Fl(jh)p Fo(')702 2530 y Fj(\006)761 2515 y Fo(;)17 b(dN)10 b Fl(i)22 b(\000)g Fk(E)13 b Fs(\()p Fl(h)p Fo(')1306 2530 y Fj(\006)1370 2515 y Fo(;)k(dN)1543 2530 y Fn(!)r(;n)1656 2515 y Fl(i)p Fs(\))p Fl(j)27 b(\024)h Fo(C)7 b(n)2028 2474 y Fj(\000)p Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))p Fn(k)2316 2515 y Fo(e)2361 2474 y Fn(C)e(k)13 b Fi(log)g Fn(k)2612 2515 y Fs(\()p Fo(E)2728 2474 y Fn(\034)2793 2515 y Fs(+)22 b Fo(C)7 b Fs(\))3006 2474 y Fn(\032)p Fi(+)p Fn(k)3140 2515 y Fs(\()p Fo(\032)22 b Fs(+)g Fo(k)s Fs(\))3440 2474 y Fn(\021)r Fi(\()p Fn(\032)p Fi(+)p Fn(k)r Fi(\))3667 2515 y Fo(:)-3576 b Fs(\(2.31\))118 2683 y(W)-8 b(e)33 b(optimize)e(the)i(righ)m(t)e(hand)i(side)g(of)39 b(\(2.31\))32 b(in)g Fo(k)j Fs(to)d(get)865 2897 y Fl(jh)p Fo(')996 2912 y Fj(\006)1054 2897 y Fo(;)17 b(dN)10 b Fl(i)22 b(\000)g Fk(E)13 b Fs(\()p Fl(h)p Fo(')1599 2912 y Fj(\006)1663 2897 y Fo(;)k(dN)1836 2912 y Fn(!)r(;n)1949 2897 y Fl(i)p Fs(\))p Fl(j)27 b(\024)h Fo(e)2231 2856 y Fj(\000)p Fi(\()p Fn(\021)r Fi(+)p Fn(C)5 b Fi(\)\()p Fn(n)2557 2832 y Fd(1)p Fg(\000)p Fh(\013)2681 2856 y Fi(\()p Fn(E)2764 2832 y Fh(\034)2803 2856 y Fi(+)p Fn(C)g Fi(\)\))3027 2794 y Fd(1)p 2978 2802 128 3 v 2978 2839 a Fh(\021)r Fd(+)p Fh(C)3124 2897 y Fo(:)118 3072 y Fs(Hence,)34 b(for)e(some)h Fo(\014)g(>)27 b Fs(0,)33 b(if)e Fo(n)d Fl(\025)g Fo(E)1487 3036 y Fn(\014)1534 3072 y Fs(,)33 b(w)m(e)h(ha)m(v)m(e)1200 3277 y Fl(jh)p Fo(')1331 3292 y Fj(\006)1390 3277 y Fo(;)17 b(dN)10 b Fl(i)21 b(\000)i Fk(E)12 b Fs(\()p Fl(h)p Fo(')1934 3292 y Fj(\006)1999 3277 y Fo(;)17 b(dN)2172 3292 y Fn(!)r(;n)2284 3277 y Fl(i)p Fs(\))p Fl(j)27 b(\024)2532 3209 y Fs(1)p 2532 3254 49 4 v 2532 3345 a(4)2590 3277 y Fo(e)2635 3236 y Fj(\000)p Fn(E)2746 3212 y Fh(\027)2788 3277 y Fo(:)-2697 b Fs(\(2.32\))118 3468 y(Th)m(us)118 3659 y(\(2.33\))82 b Fk(E)13 b Fs(\()p Fo(N)627 3674 y Fn(!)r(;n)746 3659 y Fs(\()p Fl(\000)p Fo(E)28 b Fl(\000)23 b Fs(1\))f Fl(\000)g Fo(N)1347 3674 y Fn(!)r(;n)1460 3659 y Fs(\()p Fl(\000)p Fo(E)1653 3618 y Fn(\034)1719 3659 y Fs(+)g(1\)\))27 b 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Fs(\()p Fl(\000)p Fo(E)c Fs(\))32 b(with)f Fk(E)13 b Fs(\()p Fo(N)3098 4402 y Fn(!)r(;n)3216 4387 y Fs(\()p Fl(\000)p Fo(E)26 b Fl(\006)20 b Fo(\017)p Fs(\)\))32 b(\(for)e Fo(\017)118 4504 y Fs(small\).)41 b(The)34 b(price)e(to)g(pa)m(y)i(to)e(get)h(an)f(error)g(of)h(the)g(same)f(size) h(as)g(in)e(\(2.28\))h(w)m(ould)h(ha)m(v)m(e)h(b)s(een)f(to)118 4620 y(tak)m(e)h Fo(n)f Fs(of)f(size)h Fo(E)795 4584 y Fn(\014)842 4620 y Fo(\017)881 4584 y Fj(\000)p Fn(\020)1009 4620 y Fs(for)f(some)h Fo(\020)i(>)28 b Fs(0.)44 b(This)33 b(w)m(as)h(used)g(in)e([10])h(to)f(get)h(precise)h(asymptotics)e(for) 118 4736 y Fo(N)43 b Fs(at)32 b(high)g(energy)-8 b(.)1476 4948 y(3.)55 b Fr(The)38 b(lo)n(wer)f(bounds)331 5122 y Fs(In)49 b(this)f(section,)53 b(w)m(e)d(will)c(pro)m(v)m(e)k(the)f (asymptotic)f(lo)m(w)m(er)h(b)s(ound)g(on)f(the)h(appro)m(ximated)118 5238 y(densit)m(y)g(of)e(states)h Fo(dN)1017 5253 y Fn(!)r(;n)1177 5238 y Fs(de\014ned)h(in)e(section)h(2.1)f(in)f(the)i(di\013eren)m(t)g (cases)h(considered)f(in)f(the)118 5355 y(in)m(tro)s(duction.)1969 5454 y Fp(19)p eop %%Page: 20 20 20 19 bop 118 241 a Fs(3.1.)56 b Fx(The)38 b(general)f(case.)49 b Fs(W)-8 b(e)33 b(pro)m(v)m(e)h(the)f(follo)m(wing)d(general)i(b)s (ound)118 439 y Fx(Prop)s(osition)k(3.1.)49 b Fe(Under)36 b(the)h(assumptions)e(H1,)i(H2)f(and)g(H3,)h(ther)-5 b(e)37 b(exist)f Fo(\014)3219 454 y Fi(0)3289 439 y Fo(>)30 b Fs(0)36 b Fe(such)g(that,)118 555 y(for)f(any)g Fo(\014)e(>)27 b(\014)707 570 y Fi(0)747 555 y Fe(,)35 b(ther)-5 b(e)34 b(exists)h Fo(E)1393 570 y Fi(0)1460 555 y Fo(>)g Fe(such)g(that,)g (for)g Fo(E)e(>)28 b(E)2459 570 y Fi(0)2533 555 y Fe(and)35 b Fo(n)28 b Fs(=)f([)p Fo(E)3017 519 y Fn(\014)3064 555 y Fs(])p Fe(,)35 b(we)g(have)722 753 y Fs(log)16 b(\()p Fk(E)d Fs(\()p Fo(N)1079 768 y Fn(!)r(;n)1197 753 y Fs(\()p Fl(\000)p Fo(E)29 b Fl(\000)23 b Fs(1\)\)\))k Fl(\025)h(\000)p Fs(\(1)22 b(+)g Fo(\013)2155 712 y Fj(\003)2195 753 y Fo(d)p Fs(\))p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(o)p Fs(\(1\)\))p Fo(:)-3177 b Fs(\(3.1\))118 956 y Fe(Her)-5 b(e)35 b Fs([)p Fl(\001)p Fs(])g Fe(denotes)f(the)h(inte)-5 b(ger)35 b(p)-5 b(art)35 b(of)f Fl(\001)p Fe(.)118 1154 y Fx(Pro)s(of.)74 b Fs(The)43 b(strategy)g(used)h(to)e(pro)m(v)m(e)i(the)f(lo)m(w)m(er)g (b)s(ound)g(is)f(quite)h(ob)m(vious:)63 b(w)m(e)44 b(construct)g(a)118 1270 y(normalized)38 b(v)m(ector)i Fo(')g Fs(suc)m(h)g(that)g Fl(h)p Fs(\()p Fo(H)34 b Fs(+)26 b Fo(V)1823 1285 y Fn(!)1874 1270 y Fs(\))p Fo(';)17 b(')p Fl(i)38 b(\024)i(\000)p Fo(E)34 b Fl(\000)27 b Fs(1,)41 b(this)e(with)g(a)g(su\016cien)m(tly)i (large)118 1387 y(probabilit)m(y)-8 b(.)40 b(The)29 b(righ)m(t)e (candidate)h(will)e(b)s(e)i(an)g(asymptotic)f(ground)h(state)g(for)g Fo(H)8 b Fs(\()p Fo(g)t Fs(\))26 b(for)i Fo(g)j Fs(c)m(hosen)118 1503 y(prop)s(erly)h(\(see)i(section)f(1.2\).)331 1619 y(Pic)m(k)41 b Fo(n)h Fs(=)f([)p Fo(E)879 1583 y Fn(\014)927 1619 y Fs(])f(and)h Fo(l)j Fs(=)e([)p Fl(j)17 b Fs(log)f Fo(E)6 b Fl(j)1703 1583 y Fn(\014)1749 1619 y Fs(].)69 b(Pic)m(k)41 b Fo(\032)h(>)f Fs(1)g(large,)g(1)h Fo(>)f(")g(>)h Fs(0)e(small)f(and)i Fo(k)j(>)e Fs(1)118 1735 y(large.)k(Let)34 b Fo( )637 1750 y Fn(g)711 1735 y Fs(b)s(e)g(an)g(asymptotic)f(ground)h (state)g(for)f Fo(H)8 b Fs(\()p Fo(g)t Fs(\))33 b(suc)m(h)i(that)f Fo(\013)2953 1699 y Fj(\003)2992 1735 y Fs(\()p Fo( )3093 1750 y Fn(g)3133 1735 y Fs(\))c Fo(<)g(\013)3370 1699 y Fj(\003)3409 1735 y Fs(\(1)22 b(+)h Fo(")p Fs(\))34 b(\(see)118 1852 y(section)f(1.2\).)43 b(De\014ne)1634 2050 y(\012)1704 2065 y Fn(k)r(;E)1850 2050 y Fs(=)27 b(\012)2023 2009 y Fi(1)2023 2075 y Fn(k)r(;E)2163 2050 y Fl(\\)c Fs(\012)2322 2009 y Fi(2)2322 2075 y Fn(E)118 2249 y Fs(where)525 2446 y(\012)595 2405 y Fi(1)595 2471 y Fn(k)r(;E)741 2446 y Fs(=)28 b Fl(f)p Fo(!)j Fs(:)60 b Fo(k)s Fs(\(1)22 b(+)g(2)p Fo(")p Fs(\))27 b Fl(\025)h Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\(0)p Fo(;)17 b(k)2094 2405 y Fj(\000)p Fn(\013)2194 2382 y Fg(\003)2229 2405 y Fi(\(1+)p Fn(")p Fi(\))2411 2446 y Fs(\)\))27 b(=)h Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\(0)p Fo(;)17 b(l)r Fs(\)\))26 b Fl(\025)i Fo(k)s Fs(\(1)22 b(+)g Fo(")p Fs(\))p Fl(g)-3573 b Fs(\(3.2\))833 2615 y(\012)903 2574 y Fi(2)903 2640 y Fn(E)991 2615 y Fs(=)27 b Fl(f)p Fo(!)k Fs(:)60 b Fl(8)p Fo(\015)33 b Fl(2)c Fo(r)1601 2630 y Fi(0)1640 2615 y Fk(Z)1709 2574 y Fn(d)1747 2615 y Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b(r)2314 2630 y Fi(0)2352 2615 y Fs(\)\))28 b Fo(<)f(E)2637 2574 y Fn(\032)2678 2615 y Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)2828 2574 y Fn(\013)2873 2551 y Fg(0)2921 2615 y Fs(+)22 b(1\))p Fl(g)p Fo(:)-3065 b Fs(\(3.3\))118 2823 y(Here)28 b Fo(r)387 2838 y Fi(0)454 2823 y Fs(is)f(c)m(hosen)h(suc)m (h)h(that)e(supp)p Fo(V)1531 2838 y Fi(2)1599 2823 y Fl(\032)h Fo(C)7 b Fs(\(0)p Fo(;)17 b(r)1956 2838 y Fi(0)1995 2823 y Fs(\).)42 b(W)-8 b(e)27 b(minorize)f(the)h(probabilit)m(y)f(of)h (\012)3487 2787 y Fi(1)3487 2849 y Fn(k)r(;E)3632 2823 y 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Fl(i)28 b Fs(=)f Fl(h)p Fs(\()p Fl(\000)p Fs(\001)c(+)f Fo(V)2208 3996 y Fi(\()p Fn(i)p Fi(\))2187 4062 y Fn(!)r(;n)2300 4037 y Fs(\))p Fo(\037)2399 4052 y Fn(l)2425 4037 y Fo(')2489 4052 y Fn(k)2531 4037 y Fo(;)17 b(\037)2636 4052 y Fn(l)2662 4037 y Fo(')2726 4052 y Fn(k)2769 4037 y Fl(i)k Fs(+)h Fl(h)p Fo(V)3045 3996 y Fi(\()p Fn(e)p Fi(\))3023 4062 y Fn(!)r(;n)3137 4037 y Fo(\037)3198 4052 y Fn(l)3224 4037 y Fo(')3288 4052 y Fn(k)3330 4037 y Fo(;)17 b(\037)3435 4052 y Fn(l)3461 4037 y Fo(')3525 4052 y Fn(k)3568 4037 y Fl(i)118 4235 y Fs(where)523 4461 y Fo(V)602 4420 y Fi(\()p Fn(i)p Fi(\))580 4486 y Fn(!)713 4461 y Fs(=)816 4325 y Fm(Z)872 4551 y Fn(C)5 b Fi(\(0)p Fn(;l)q Fi(\))1079 4461 y Fo(V)21 b Fs(\()p Fo(x)i Fl(\000)g Fo(y)t Fs(\))p Fo(m)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\))29 b(and)k Fo(V)2133 4420 y Fi(\()p Fn(e)p Fi(\))2112 4486 y Fn(!)2253 4461 y Fs(=)2356 4325 y Fm(Z)2412 4551 y Ff(R)2460 4532 y Fh(d)2496 4551 y Fj(n)p Fn(C)5 b Fi(\(0)p Fn(;l)q Fi(\))2739 4461 y Fo(V)21 b Fs(\()p Fo(x)i Fl(\000)f Fo(y)t Fs(\))p Fo(m)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\))118 4758 y(i.e.)42 b(they)31 b(are)e(the)h(parts)g(of)f Fo(V)1238 4773 y Fn(!)1318 4758 y Fs(with)g(cen)m(ters)j(in)d Fo(C)7 b Fs(\(0)p Fo(;)17 b(l)r Fs(\))29 b(or)g(outside)h(of)f Fo(C)7 b Fs(\(0)p Fo(;)17 b(l)r Fs(\),)30 b(and)f Fo(V)3437 4707 y Fi(\()p Fn(i;e)p Fi(\))3415 4768 y Fn(!)r(;n)3602 4758 y Fs(is)g(built)118 4875 y(from)i(these)j(in)e(the)h(same)f(w)m(a)m(y)i (as)f Fo(V)1499 4890 y Fn(!)r(;n)1644 4875 y Fs(is)f(from)g Fo(V)2030 4890 y Fn(!)2113 4875 y Fs(\(see)h(equation)f(\(2.1\)\).)118 4991 y(As)j Fo(V)321 5006 y Fi(2)395 4991 y Fs(is)f(of)g(compact)g (supp)s(ort,)h(as)g Fo(V)1569 5006 y Fi(1)1643 4991 y Fs(is)f(exp)s(onen)m(tially)f(deca)m(ying)i(and)g(as)f Fo(!)h Fl(2)c Fs(\012)3325 4955 y Fi(2)3325 5017 y Fn(k)r(;E)3443 4991 y Fs(,)k(using)f(the)118 5107 y(supp)s(ort)f(prop)s(erties)g(of)f Fo(\037)1110 5122 y Fn(l)1136 5107 y Fs(,)g(w)m(e)i(get)f(that)1328 5316 y Fl(jh)p Fo(V)1473 5274 y Fi(\()p Fn(e)p Fi(\))1452 5340 y Fn(!)r(;n)1565 5316 y Fo( )1628 5331 y Fn(k)1671 5316 y Fo(;)17 b( )1778 5331 y Fn(k)1821 5316 y Fl(ij)27 b(\024)h Fo(C)7 b(E)2175 5274 y Fn(\032)2215 5316 y Fo(e)2260 5274 y Fj(\000j)12 b Fi(log)h Fn(E)t Fj(j)2526 5251 y Fh(\014)2566 5274 y Fn(=C)2661 5316 y Fo(:)-2570 b Fs(\(3.5\))1969 5454 y Fp(20)p eop %%Page: 21 21 21 20 bop 118 244 a Fs(On)35 b(the)h(other)f(hand,)h(if)e(w)m(e)i(set)f Fo(m)p Fs(\()p Fo(!)t Fs(\))d(=)f Fo(m)p Fs(\()p Fo(!)t(;)17 b(B)5 b Fs(\(0)p Fo(;)17 b(k)2238 208 y Fj(\000)p Fn(\013)2338 184 y Fg(\003)2373 208 y Fi(\(1+)p Fn(")p Fi(\))2555 244 y Fs(\)\))35 b(and)g(de\014ne)h(\()p Fo(x)3235 259 y Fn(i)3264 244 y Fs(\()p Fo(!)t Fs(\)\))3443 259 y Fn(i)3505 244 y Fs(to)f(b)s(e)g(the)118 363 y(p)s(oin)m(ts)d(supp)s(orting)g Fo(m)p Fs(\()p Fo(!)t(;)17 b(dx)p Fs(\))32 b(in)g Fo(B)5 b Fs(\(0)p Fo(;)17 b(k)1688 327 y Fj(\000)p Fn(\013)1788 303 y Fg(\003)1824 327 y Fi(\(1+)p Fn(")p Fi(\))2006 363 y Fs(\),)32 b(w)m(e)i(ha)m(v)m(e)445 532 y Fl(h)p Fs(\()p Fl(\000)p Fs(\001)23 b(+)f Fo(V)880 491 y Fi(\()p Fn(i)p Fi(\))858 557 y Fn(!)963 532 y Fs(\))p Fo( )1064 547 y Fn(k)1107 532 y Fo(;)17 b( )1214 547 y Fn(k)1256 532 y Fl(i)473 801 y Fs(=)28 b Fl(h)p Fs(\()p Fl(\000)p Fs(\001)22 b(+)g Fo(m)p Fs(\()p Fo(!)t Fs(\))p Fo(V)f Fs(\))p Fo( )1337 816 y Fn(k)1380 801 y Fo(;)c( )1487 816 y Fn(k)1530 801 y Fl(i)22 b Fs(+)1689 668 y Fn(m)p Fi(\()p Fn(!)r Fi(\))1699 706 y Fm(X)1714 916 y Fn(i)p Fi(=1)1853 801 y Fl(h)p Fs(\()p Fo(\034)1972 816 y Fn(x)2012 826 y Fh(i)2038 816 y Fi(\()p Fn(!)r Fi(\))2143 801 y Fo(V)44 b Fl(\000)22 b Fo(V)g Fs(\))p Fo( )2523 816 y Fn(k)2566 801 y Fo(;)17 b( )2673 816 y Fn(k)2716 801 y Fl(i)473 1155 y Fs(=)28 b Fl(h)p Fs(\()p Fl(\000)p Fs(\001)22 b(+)g Fo(k)s(V)g Fs(\))p Fo( )1166 1170 y Fn(k)1209 1155 y Fo(;)17 b( )1316 1170 y Fn(k)1359 1155 y Fl(i)22 b Fs(+)1528 1088 y Fo(m)p Fs(\()p Fo(!)t Fs(\))f Fl(\000)i Fo(k)p 1528 1132 402 4 v 1701 1224 a(k)1939 1155 y Fl(h)p Fo(k)s(V)e( )2173 1170 y Fn(k)2216 1155 y Fo(;)c( )2323 1170 y Fn(k)2366 1155 y Fl(i)22 b Fs(+)2525 1023 y Fn(m)p Fi(\()p Fn(!)r Fi(\))2535 1061 y Fm(X)2550 1271 y Fn(i)p Fi(=1)2689 1155 y Fl(h)p Fs(\()p Fo(\034)2808 1171 y Fn(x)2848 1181 y Fh(i)2874 1171 y Fi(\()p Fn(!)r Fi(\))2979 1155 y Fo(V)44 b Fl(\000)22 b Fo(V)g Fs(\))p Fo( )3359 1170 y Fn(k)3402 1155 y Fo(;)17 b( )3509 1170 y Fn(k)3552 1155 y Fl(i)473 1510 y(\024)28 b Fo(E)6 b Fs(\()p Fo(k)s Fs(\))22 b(+)g Fo(o)p Fs(\()p Fo(E)6 b Fs(\()p Fo(k)s Fs(\)\))22 b(+)1367 1443 y Fo(m)p Fs(\()p Fo(!)t Fs(\))g Fl(\000)h Fo(k)p 1367 1487 V 1541 1578 a(k)1779 1510 y Fs(\()p Fo(E)6 b Fs(\()p Fo(k)s Fs(\))22 b(+)g Fo(o)p Fs(\()p Fo(E)6 b Fs(\()p Fo(k)s Fs(\)\)\))22 b(+)2634 1378 y Fn(m)p Fi(\()p Fn(!)r Fi(\))2643 1415 y Fm(X)2658 1625 y Fn(i)p Fi(=1)2814 1510 y Fl(jh)p Fs(\()p Fo(\034)2961 1526 y Fn(x)3001 1536 y Fh(i)3026 1526 y Fi(\()p Fn(!)r Fi(\))3132 1510 y Fo(V)43 b Fl(\000)23 b Fo(V)f Fs(\))p Fo( )3512 1525 y Fn(k)3555 1510 y Fo(;)17 b( )3662 1525 y Fn(k)3704 1510 y Fl(ij)473 1865 y(\024)28 b Fo(E)6 b Fs(\()p Fo(k)s Fs(\))22 b(+)g Fo(o)p Fs(\()p Fo(E)6 b Fs(\()p Fo(k)s Fs(\)\))22 b(+)g Fo(")p Fs(\()p Fo(E)6 b Fs(\()p Fo(k)s Fs(\))22 b(+)g Fo(o)p Fs(\()p Fo(E)6 b Fs(\()p Fo(k)s Fs(\)\)\))22 b(+)2258 1732 y Fn(m)p Fi(\()p Fn(!)r Fi(\))2268 1770 y Fm(X)2283 1980 y Fn(i)p Fi(=1)2438 1865 y Fl(jh)p Fs(\()p Fo(\034)2585 1880 y Fn(x)2625 1890 y Fh(i)2651 1880 y Fi(\()p Fn(!)r Fi(\))2756 1865 y Fo(V)44 b Fl(\000)22 b Fo(V)g Fs(\))p Fo( )3136 1880 y Fn(k)3179 1865 y Fo(;)17 b( )3286 1880 y Fn(k)3329 1865 y Fl(ij)p Fo(:)118 1241 y Fs(\(3.6\))118 2118 y(W)-8 b(e)27 b(used)g(the)g(fact)f(that)h Fo( )1113 2133 y Fn(k)1182 2118 y Fs(is)f(an)g(asymptotic)g(ground)g(state)h(and) g(the)g(fact)f(that)g Fo(k)s Fs(\(1)10 b(+)g Fo(")p Fs(\))26 b Fl(\024)i Fo(m)p Fs(\()p Fo(!)t Fs(\).)118 2234 y(As)33 b Fo(m)p Fs(\()p Fo(!)t Fs(\))27 b Fl(\024)i Fo(k)s Fs(\(1)22 b(+)g(2)p Fo(")p Fs(\))31 b(and)i Fo(\013)1299 2198 y Fj(\003)1338 2234 y Fs(\()p Fo( )1439 2249 y Fn(g)1480 2234 y Fs(\))27 b Fo(<)h(\013)1712 2198 y Fj(\003)1751 2234 y Fs(\(1)22 b(+)g Fo(")p Fs(\),)32 b(the)h(de\014nition)f(of)g Fo(\013)2877 2198 y Fj(\003)2916 2234 y Fs(\()p Fo( )3017 2249 y Fn(g)3057 2234 y Fs(\))h(tells)e(us)i(that)1025 2453 y(1)p 919 2497 262 4 v 919 2588 a Fo(E)991 2603 y Fj(\000)1050 2588 y Fs(\()p Fo(k)s Fs(\))1207 2316 y Fm(\014)1207 2375 y(\014)1207 2435 y(\014)1207 2495 y(\014)1207 2555 y(\014)1207 2615 y(\014)1240 2388 y Fn(m)p Fi(\()p Fn(!)r Fi(\))1250 2425 y Fm(X)1264 2635 y Fn(i)p Fi(=1)1403 2520 y Fl(h)p Fs(\()p Fo(\034)1522 2535 y Fn(x)1562 2545 y Fh(i)1588 2535 y Fi(\()p Fn(!)r Fi(\))1694 2520 y Fo(V)43 b Fl(\000)23 b Fo(V)f Fs(\))p Fo(')2075 2535 y Fn(k)2117 2520 y Fo(;)17 b(')2225 2535 y Fn(k)2267 2520 y Fl(i)2306 2316 y Fm(\014)2306 2375 y(\014)2306 2435 y(\014)2306 2495 y(\014)2306 2555 y(\014)2306 2615 y(\014)2367 2520 y Fl(!)27 b Fs(0)33 b(as)f Fo(k)f Fl(!)c Fs(+)p Fl(1)p Fo(:)118 2803 y Fs(Plugging)k(all)f(this)j(in)m (to)e(\(3.6\),)i(w)m(e)g(get)772 2972 y Fl(h)p Fs(\()p Fl(\000)p Fs(\001)23 b(+)f Fo(V)1206 2931 y Fi(\()p Fn(i)p Fi(\))1185 2997 y Fn(!)1289 2972 y Fs(\))p Fo( )1390 2987 y Fn(k)1433 2972 y Fo(;)17 b( )1540 2987 y Fn(k)1583 2972 y Fl(i)28 b(\024)g Fo(E)6 b Fs(\()p Fo(k)s Fs(\)\(1)22 b(+)g Fo(o)p Fs(\(1\))f(+)h Fo(")p Fs(\))32 b(when)i Fo(k)d Fl(!)c Fs(+)p Fl(1)p Fo(:)-3126 b Fs(\(3.7\))118 3134 y(If)33 b(w)m(e)g(no)m(w)g(c)m(hose)1668 3295 y Fo(k)e Fs(=)c Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(")p Fs(\))-2230 b(\(3.8\))118 3457 y(then,)33 b(for)f(su\016cien)m (tly)i(large)d Fo(k)s Fs(,)i(w)m(e)g(get)1303 3620 y Fl(h)p Fs(\()p Fl(\000)p Fs(\001)22 b(+)g Fo(V)1737 3579 y Fi(\()p Fn(i)p Fi(\))1715 3645 y Fn(!)1820 3620 y Fs(\))p Fo( )1921 3635 y Fn(k)1964 3620 y Fo(;)17 b( )2071 3635 y Fn(k)2114 3620 y Fl(i)27 b(\024)h(\000)p Fs(\()p Fo(E)h Fs(+)22 b(1\))p Fo(:)-2595 b Fs(\(3.9\))118 3782 y(But,)33 b(as)h(an)e(asymptotic)h(ground)f(state,)i Fo( )1758 3797 y Fn(k)1834 3782 y Fs(v)-5 b(anishes)33 b(outside)g(some)g (\014xed)h(cub)s(e)g(indep)s(enden)m(t)g(of)118 3898 y Fo(k)s Fs(;)28 b(hence)g(for)d Fo(E)32 b Fs(su\016cien)m(tly)27 b(large,)f(it)f(v)-5 b(anishes)27 b(in)e(a)h(neigh)m(b)s(orho)s(o)s(d)f (of)g(the)h(b)s(oundary)h(of)e(the)i(cub)s(e)118 4015 y Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))33 b(and)h(can)g(b)s(e)g(con)m (tin)m(ued)h(so)f(as)g(to)f(satisfy)h(an)m(y)g(quasi-p)s(erio)s(dic)e (b)s(oundary)i(conditions)f(on)118 4131 y Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))39 b(\(see)h(section)g(2.1\).)63 b(This)40 b(implies)d(that)i(there)h(exist)g Fo(C)46 b(>)39 b Fs(0)g(indep)s(enden)m(t)i(of)e Fo(n)g Fs(and)h Fo(E)118 4247 y Fs(suc)m(h)34 b(that)e(for)h Fo(!)e Fl(2)d Fs(\012)955 4262 y Fn(k)r(;E)1105 4247 y Fs(and)33 b Fo(k)j Fs(giv)m(en)c(b)m(y)j(\(3.8\),)d(w)m(e)i(ha)m(v)m(e.)1495 4416 y Fo(N)1573 4431 y Fn(!)r(;n)1685 4416 y Fs(\()p Fl(\000)p Fo(E)29 b Fl(\000)23 b Fs(1\))k Fl(\025)h Fo(n)2278 4374 y Fj(\000)p Fn(d)2374 4416 y Fo(=C)r(:)-2404 b Fs(\(3.10\))118 4590 y(T)-8 b(aking)30 b(in)m(to)g(accoun)m(t)i(the)f(probabilit)m(y)e (estimate)h(\(3.4\))o(,)h(if)f Fo(\032)h Fs(is)f(large)g(enough)h(and)g (as)g Fo(k)f Fl(\025)f Fo(E)3713 4554 y Fn(\016)3781 4590 y Fs(for)118 4707 y(some)k Fo(\016)e(>)d Fs(0)k(b)m(y)j(\(5.29\))o (,)e(w)m(e)g(get)882 4938 y Fk(E)13 b Fs(\()p Fo(N)1059 4953 y Fn(!)r(;n)1178 4938 y Fs(\()p Fl(\000)p Fo(E)28 b Fl(\000)23 b Fs(1\)\))k Fl(\025)1774 4871 y Fs(1)p 1760 4915 77 4 v 1760 5007 a Fo(C)1847 4938 y(E)1925 4897 y Fj(\000)p Fn(\014)s(d)2064 4938 y Fo(e)2109 4897 y Fj(\000)p Fn(C)5 b(\026l)2283 4874 y Fh(d)2333 4871 y Fs(\()p Fo(\026k)2484 4835 y Fj(\000)p Fn(d\013)2620 4811 y Fg(\003)2656 4835 y Fi(\(1+)p Fn(")p Fi(\))2838 4871 y Fs(\))2876 4835 y Fn(k)r Fi(\(1+)p Fn(")p Fi(\))p 2333 4915 764 4 v 2474 5007 a Fs(\000\()p Fo(k)s Fs(\(1)22 b(+)g Fo(")p Fs(\)\))3106 4938 y Fo(:)118 5154 y Fs(So)33 b(that,)f(as)h Fo(E)38 b Fs(tends)c(to)e(+)p Fl(1)p Fs(,)g(w)m(e)i(get) 676 5316 y(log[)p Fk(E)12 b Fs(\()p Fo(N)1005 5331 y Fn(!)s(;n)1124 5316 y Fs(\()p Fl(\000)p Fo(E)29 b Fl(\000)22 b Fs(1\)\)])28 b Fl(\025)g(\000)p Fs(\(1)22 b(+)g Fo(")g Fs(+)g Fo(d\013)2288 5274 y Fj(\003)2327 5316 y Fs(\(1)g(+)g Fo(")p Fs(\)\))p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h(8)p Fo(")p Fs(\))p Fo(:)1969 5454 y Fp(21)p eop %%Page: 22 22 22 21 bop 118 241 a Fs(As)33 b(w)m(e)h(can)f(c)m(ho)s(ose)g Fo(")f Fs(as)h(small)d(as)j(w)m(e)h(please,)f(this)f(ends)i(the)f(pro)s (of)e(of)h(Prop)s(osition)f(3.1.)p 3827 357 4 66 v 3831 294 59 4 v 3831 357 V 3889 357 4 66 v 118 473 a(By)37 b(Lemma)e(5.5,)i(w)m(e)g(kno)m(w)h(that,)f(for)f(some)g Fo(C)41 b(>)34 b Fs(0)i(and)h Fo(\013)d(>)g Fs(1,)k Fo(g)t Fs(\()p Fo(E)6 b Fs(\))33 b Fl(\024)h Fo(C)7 b(E)43 b Fs(for)36 b(some)g Fo(C)41 b(>)34 b Fs(0)118 589 y(and)h(su\016cien)m (tly)g(large)e Fo(E)6 b Fs(.)49 b(Th)m(us,)36 b(if)d Fo(\014)j(>)31 b Fs(1)j(is)g(large)f(enough)i(and)f Fo(\027)6 b Fs(,)35 b(giv)m(en)g(b)m(y)g(equation)f(\(2.28\),)118 706 y(satis\014es)f Fo(\027)i(>)27 b Fs(1,)33 b(Lemma)e(2.3)h(and)h (Prop)s(osition)d(3.1)j(immediately)c(imply)118 884 y Fx(Prop)s(osition)36 b(3.2.)49 b Fe(Under)36 b(the)f(assumptions)g(H1,) h(H2)g(and)g(H1',)g(ther)-5 b(e)36 b(exists)f Fo(E)3300 899 y Fi(0)3369 884 y Fo(>)h Fe(such)f(that,)118 1000 y(for)g Fo(E)f(>)27 b(E)555 1015 y Fi(0)595 1000 y Fe(,)34 b(we)h(have)1006 1164 y Fs(log)16 b Fo(N)10 b Fs(\()p Fo(E)c Fs(\))29 b Fl(\025)f(\000)p Fs(\(1)22 b(+)g Fo(\013)1871 1123 y Fj(\003)1910 1164 y Fo(d)p Fs(\))p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(o)p Fs(\(1\)\))p Fo(:)118 1342 y Fs(3.2.)56 b Fx(The)38 b(case)g(when)g Fo(V)59 b Fx(is)37 b(b)s(ounded)i(from)e(b) s(elo)m(w.)48 b Fs(W)-8 b(e)33 b(no)m(w)h(assume)f(that)f Fo(V)55 b Fs(is)32 b(b)s(ounded)118 1459 y(from)f(b)s(elo)m(w)i(i.e.)43 b(that)32 b Fo(V)1061 1474 y Fj(\000)1153 1459 y Fs(is)g(b)s(ounded.)44 b(W)-8 b(e)33 b(then)g(pro)m(v)m(e)118 1637 y Fx(Prop)s(osition)j(3.3.) 49 b Fe(Under)36 b(the)h(assumptions)e(H1,)i(H2)f(and)g(H3,)h(ther)-5 b(e)37 b(exist)f Fo(\014)3219 1652 y Fi(0)3289 1637 y Fo(>)30 b Fs(0)36 b Fe(such)g(that,)118 1753 y(for)f(any)g Fo(\014)e(>)27 b(\014)707 1768 y Fi(0)747 1753 y Fe(,)35 b(ther)-5 b(e)34 b(exists)h Fo(E)1393 1768 y Fi(0)1460 1753 y Fo(>)g Fe(such)g(that,)g(for)g Fo(E)e(>)28 b(E)2459 1768 y Fi(0)2533 1753 y Fe(and)35 b Fo(n)28 b Fs(=)f Fo(E)2990 1717 y Fn(\014)3037 1753 y Fe(,)35 b(we)g(have)920 1917 y Fs(log)17 b(\()p Fk(E)12 b Fs(\()p Fo(N)1277 1932 y Fn(!)s(;n)1396 1917 y Fs(\()p Fl(\000)p Fo(E)29 b Fl(\000)22 b Fs(1\)\)\))28 b Fl(\025)g(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(o)p Fs(\(1\)\))p Fo(:)-2978 b Fs(\(3.11\))118 2096 y Fx(Pro)s(of.)47 b Fs(The)35 b(strategy)f(of)g(the)g(pro)s(of)f (is)g(the)i(one)f(used)h(in)e(the)h(pro)s(of)f(of)g(Prop)s(osition)f (3.1.)47 b(Recall)118 2212 y(that)32 b Fo(v)376 2227 y Fj(\000)467 2212 y Fs(is)f(the)i(essen)m(tial)f(in\014m)m(um)e(of)i Fo(V)21 b Fs(.)43 b(Then,)34 b(for)d(an)m(y)i Fo(")27 b(>)h Fs(0,)j(w)m(e)i(can)g(\014nd)f Fo(\037)c Fl(2)g(C)3415 2176 y Fj(1)3409 2236 y Fi(0)3490 2212 y Fs(\()p Fk(R)3594 2176 y Fn(d)3640 2212 y Fs(\))k(suc)m(h)118 2328 y(that)931 2384 y Fm(Z)986 2609 y Ff(R)1034 2590 y Fh(d)1091 2519 y Fo(V)21 b Fs(\()p Fo(x)p Fs(\))p Fo(\037)1361 2478 y Fi(2)1401 2519 y Fs(\()p Fo(x)p Fs(\))p Fo(dx)28 b Fl(\024)g Fo(v)1818 2534 y Fj(\000)1900 2519 y Fs(+)22 b Fo("=)p Fs(2)32 b(and)2380 2384 y Fm(Z)2435 2609 y Ff(R)2483 2590 y Fh(d)2540 2519 y Fo(\037)2601 2478 y Fi(2)2641 2519 y Fs(\()p Fo(x)p Fs(\))p Fo(dx)c Fs(=)f(1)p Fo(:)118 2747 y Fs(Recall)k(that)h Fo(\034)665 2762 y Fn(a)707 2747 y Fo(\037)p Fs(\()p Fo(x)p Fs(\))d(=)e Fo(\037)p Fs(\()p Fo(x)c Fl(\000)f Fo(a)p Fs(\).)44 b(As)33 b Fo(V)49 b Fl(2)28 b Fo(L)1877 2711 y Fi(1)1917 2747 y Fs(\()p Fk(R)2021 2711 y Fn(d)2068 2747 y Fs(\),)k(there)h(exists)h Fo(\016)d(>)d Fs(0)k(suc)m(h)i(that)1110 2964 y(sup)1101 3050 y Fj(j)p Fn(a)p Fj(j\024)p Fn(\016)1283 2828 y Fm(Z)1338 3054 y Ff(R)1386 3035 y Fh(d)1443 2964 y Fl(j)p Fo(V)22 b Fs(\()p Fo(x)p Fs(\))p Fl(j)g(\001)f(j)p Fo(\034)1850 2979 y Fn(a)1892 2964 y Fo(\037)1953 2923 y Fi(2)1993 2964 y Fs(\()p Fo(x)p Fs(\))h Fl(\000)h Fo(\037)2307 2923 y Fi(2)2346 2964 y Fs(\()p Fo(x)p Fs(\))p Fl(j)p Fo(dx)28 b Fl(\024)g Fo("=)p Fs(2)p Fo(:)118 3201 y Fs(Pic)m(k)33 b(no)m(w)g Fo(n)g Fs(and)g Fo(l)i Fs(as)d(in)g(the)h(pro)s(of)f(of)g (Prop)s(osition)f(3.1;)h(pic)m(k)h Fo(k)j Fs(large)31 b(and)i(de\014ne)1642 3365 y(\012)1712 3380 y Fn(\016)o(;E)1850 3365 y Fs(=)27 b(\012)2023 3324 y Fi(1)2023 3390 y Fn(\016)o(;E)2155 3365 y Fl(\\)22 b Fs(\012)2313 3324 y Fi(2)2313 3390 y Fn(E)118 3539 y Fs(where)34 b(\012)470 3503 y Fi(2)470 3564 y Fn(E)562 3539 y Fs(is)f(de\014ned)h(in)d(\(3.3\))h(and)h(\(see)h (\(3.2\))o(\))723 3704 y(\012)793 3663 y Fi(1)793 3728 y Fn(\016)o(;E)930 3704 y Fs(=)27 b Fl(f)p Fo(!)t Fs(;)49 b Fo(k)s Fs(\(1)22 b(+)g(2)p Fo(")p Fs(\))27 b Fl(\025)h Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\(0)p Fo(;)17 b(\016)t Fs(\)\))26 b(=)h Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\(0)p Fo(;)17 b(l)r Fs(\)\))27 b Fl(\025)h Fo(k)s Fs(\(1)22 b(+)g Fo(")p Fs(\))p Fl(g)p Fo(:)118 3873 y Fs(Using)34 b(\(5.2\))o(,)f(w)m(e)g(get)g(that)f(the)h(probabilit)m(y)e(of)h(\012) 2019 3888 y Fn(\016)o(;E)2161 3873 y Fs(is)g(estimated)g(b)m(y)1086 4110 y Fo(P)14 b Fs(\(\012)1271 4125 y Fn(\016)o(;E)1380 4110 y Fs(\))27 b Fl(\025)1575 4043 y Fs(1)p 1560 4087 77 4 v 1560 4179 a Fo(C)1647 4110 y(e)1692 4069 y Fj(\000)p Fn(C)5 b(\026l)1866 4046 y Fh(d)1939 4043 y Fs(\()p Fo(\026\016)t Fs(\))2121 4007 y Fn(k)r(d)p Fi(\(1+)p Fn(")p Fi(\))p 1917 4087 482 4 v 1917 4179 a Fs(\000\()p Fo(k)s Fs(\(1)22 b(+)g Fo(")p Fs(\)\))2430 4110 y Fl(\000)2678 4043 y Fo(C)p 2540 4087 353 4 v 2540 4179 a Fs(\000\()p Fo(E)2717 4150 y Fn(\032)2757 4179 y Fo(=)p Fs(2\))2903 4110 y Fo(:)-2812 b Fs(\(3.12\))118 4329 y(Pic)m(k)31 b Fo(!)g Fl(2)d Fs(\012)590 4344 y Fn(\016)o(;E)699 4329 y Fs(.)43 b(An)30 b(argumen)m(t)g(similar)d(to)j(that)g(used)h(in)f(the)g(pro)s (of)g(of)f(Prop)s(osition)g(3.1)h(in)f(whic)m(h)118 4451 y Fo(k)172 4415 y Fj(\000)p Fn(\013)272 4391 y Fg(\003)308 4415 y Fi(\(1+)p Fn(")p Fi(\))523 4451 y Fs(is)j(replaced)h(b)m(y)g Fo(\016)k Fs(yields)32 b(\(cf)39 b(\(3.6\)\))507 4724 y Fl(h)p Fs(\()p Fl(\000)p Fs(\001)22 b(+)g Fo(V)919 4739 y Fn(!)r(;n)1032 4724 y Fs(\))p Fo(\037;)17 b(\037)p Fl(i)27 b Fs(=)h Fl(kr)p Fo(\037)p Fl(k)1650 4682 y Fi(2)1711 4724 y Fs(+)1809 4591 y Fn(m)p Fi(\()p Fn(!)r Fi(\))1819 4629 y Fm(X)1834 4839 y Fn(i)p Fi(=1)1990 4588 y Fm(Z)2106 4724 y Fo(\034)2148 4739 y Fn(x)2188 4749 y Fh(i)2214 4739 y Fi(\()p Fn(!)r Fi(\))2320 4724 y Fs(\()p Fo(V)21 b(\037)2497 4682 y Fi(2)2537 4724 y Fs(\))h(+)1888 5078 y(+)1986 4946 y Fn(m)p Fi(\()p Fn(!)r Fi(\))1995 4984 y Fm(X)2010 5194 y Fn(i)p Fi(=1)2166 4943 y Fm(Z)2265 5078 y Fs([)p Fo(\034)2334 5094 y Fn(x)2374 5104 y Fh(i)2401 5094 y Fi(\()p Fn(!)r Fi(\))2506 5078 y Fo(V)g Fs(]\()p Fo(\034)2692 5094 y Fn(x)2732 5104 y Fh(i)2758 5094 y Fi(\()p Fn(!)r Fi(\))2863 5078 y Fo(\037)2924 5037 y Fi(2)2986 5078 y Fl(\000)h Fo(\037)3147 5037 y Fi(2)3186 5078 y Fs(\))f(+)g Fl(h)p Fo(V)3462 5037 y Fi(\()p Fn(e)p Fi(\))3440 5103 y Fn(!)r(;n)3553 5078 y Fo(\037;)17 b(\037)p Fl(i)1302 5316 y(\024)29 b Fo(k)s Fs(\(1)22 b(+)g Fo(")p Fs(\)\()p Fo(v)1838 5331 y Fj(\000)1918 5316 y Fs(+)g Fo(")p Fs(\))g(+)g Fo(C)29 b Fs(+)22 b Fo(O)s Fs(\(1)p Fo(=E)6 b Fs(\))118 4968 y(\(3.13\))1969 5454 y Fp(22)p eop %%Page: 23 23 23 22 bop 118 241 a Fs(Hence,)34 b(if)e(w)m(e)h(tak)m(e)h Fo(k)c Fs(=)e Fo(E)6 b(=)p Fl(j)p Fo(v)1267 256 y Fj(\000)1326 241 y Fl(j)p Fs(\(1)21 b(+)h Fo("=)p Fl(j)p Fo(v)1730 256 y Fj(\000)1788 241 y Fl(j)p Fs(\),)33 b(for)f Fo(E)38 b Fs(su\016cien)m(tly)33 b(large,)f(w)m(e)i(get)e(\(cf)40 b(\(3.9\))o(\))1381 411 y Fl(h)p Fs(\()p Fl(\000)p Fs(\001)23 b(+)f Fo(V)1794 426 y Fn(!)r(;n)1906 411 y Fs(\))p Fo(\037;)17 b(\037)p Fl(i)28 b(\024)g(\000)p Fo(E)g Fl(\000)23 b Fs(1)p Fo(:)118 587 y Fs(This)35 b(inequalit)m(y)f(and)g(the)i (probabilit)m(y)c(estimate)i(\(3.12\))g(giv)m(e,)h(for)g(some)f Fo(C)39 b(>)31 b Fs(0)j(indep)s(enden)m(t)i(of)118 704 y Fo(E)j Fs(and)32 b Fo(")1019 902 y Fs(log)17 b(\()p Fk(E)12 b Fs(\()p Fo(N)1376 917 y Fn(!)s(;n)1495 902 y Fs(\()p Fl(\000)p Fo(E)29 b Fl(\000)22 b Fs(1\)\)\))28 b Fl(\025)g(\000)2235 834 y Fo(E)p 2193 879 162 4 v 2193 970 a Fl(j)p Fo(v)2268 985 y Fj(\000)2327 970 y Fl(j)2365 902 y Fs(\(1)22 b(+)g Fo(C)7 b(")p Fs(\))17 b(log)e Fo(E)6 b(:)118 1127 y Fs(This)33 b(ends)h(the)f(pro)s(of)e(of)h(Prop)s (osition)f(3.3.)p 3827 1243 4 66 v 3831 1180 59 4 v 3831 1243 V 3889 1243 4 66 v 118 1359 a(By)i(the)g(same)g(argumen)m(t)f(as)h (ab)s(o)m(v)m(e,)g(w)m(e)h(get)118 1541 y Fx(Prop)s(osition)i(3.4.)49 b Fe(Under)36 b(the)h(assumptions)e(H1,)i(H2)g(and)f(H1',)h(if)f Fo(V)58 b Fe(is)36 b(b)-5 b(ounde)g(d)36 b(fr)-5 b(om)36 b(b)-5 b(elow,)118 1657 y(ther)g(e)35 b(exists)f Fo(E)699 1672 y Fi(0)767 1657 y Fo(>)h Fe(such)f(that,)h(for)g Fo(E)f(>)27 b(E)1765 1672 y Fi(0)1805 1657 y Fe(,)35 b(we)f(have)1205 1828 y Fs(log)16 b Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b Fl(\025)h(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)22 b(+)g Fo(o)p Fs(\(1\)\))p Fo(:)331 2010 y Fs(This)33 b(ends)i(the)e(pro)s(of)g(of)f(Theorem)i(1.5)f(as)g(the)h(asymptotic)f (upp)s(er)g(b)s(ound)h(giv)m(en)f(b)m(y)h(Theo-)118 2126 y(rem)e(1.4)g(coalesces)i(with)e(the)h(lo)m(w)m(er)g(b)s(ound)g(giv)m (en)f(b)m(y)i(Prop)s(osition)d(3.4.)118 2328 y(3.3.)56 b Fx(The)37 b(case)h(when)f Fo(V)58 b Fx(has)38 b(p)s(o)m(w)m(er)f(la)m (w)f(singularities.)47 b Fs(Let)32 b(us)h(assume)g(that)e Fo(V)3499 2343 y Fi(2)3571 2328 y Fs(satis\014es)118 2444 y(assumption)h(H1".)43 b(W)-8 b(e)33 b(will)d(sho)m(w)k(that)118 2626 y Fx(Lemma)j(3.1.)49 b Fe(Under)35 b(the)g(assumptions)f(H1)h(and) f(H1",)h(we)g(have)1828 2796 y Fo(\013)1891 2755 y Fj(y)1954 2796 y Fl(\025)28 b Fo(\013)2122 2755 y Fj(\003)2161 2796 y Fo(:)118 2984 y Fx(Pro)s(of.)48 b Fs(T)-8 b(o)34 b(estimate)f Fo(\013)1069 2948 y Fj(\003)1143 2984 y Fs(w)m(e)i(will)d(use)j(the)f(asymptotic)g(ground)g(state)g (constructed)i(for)e Fo(H)8 b Fs(\()p Fo(g)t Fs(\))32 b(in)118 3118 y(Lemma)41 b(5.7.)71 b(Pic)m(k)43 b Fo(i)960 3133 y Fi(0)1041 3118 y Fs(and)g Fo( )1308 3067 y Fi(\()p Fn(i)1359 3076 y Fd(0)1394 3067 y Fi(\))1304 3128 y Fn(g)1467 3118 y Fs(as)f(in)g(Lemma)e(5.7.)72 b(Let)42 b Fo(\013)i(>)g(\013)2774 3082 y Fj(y)2809 3118 y Fs(.)72 b(If)42 b(the)h(supp)s(ort)f(of)f Fo(\037)i Fs(\(cf)118 3234 y(Lemma)31 b(5.7\))h(is)g(small)f(enough,)i (then,)g(for)f Fl(j)p Fo(a)p Fl(j)27 b(\024)i Fo(g)2067 3198 y Fj(\000)p Fn(\013)2203 3234 y Fs(and)j Fo(g)k Fs(large)c(enough,)h(w)m(e)g(ha)m(v)m(e)679 3490 y(\010\()p Fo(a)p Fs(\))28 b(=)1017 3423 y Fo(g)t Fl(h)p Fs(\()p Fo(\034)1187 3438 y Fn(a)1228 3423 y Fo(V)44 b Fl(\000)23 b Fo(V)e Fs(\))p Fo( )1612 3372 y Fi(\()p Fn(i)1663 3381 y Fd(0)1698 3372 y Fi(\))1608 3433 y Fn(g)1730 3423 y Fo(;)c( )1841 3372 y Fi(\()p Fn(i)1892 3381 y Fd(0)1927 3372 y Fi(\))1837 3433 y Fn(g)1958 3423 y Fl(i)p 1017 3467 980 4 v 1378 3558 a Fo(E)1450 3573 y Fj(\000)1510 3558 y Fs(\()p Fo(g)t Fs(\))904 3789 y(=)1017 3721 y Fo(g)t Fl(h)p Fs([)p Fo(\034)1176 3736 y Fn(a)1218 3721 y Fs(\()p Fo(W)1348 3736 y Fn(i)1372 3745 y Fd(0)1410 3721 y Fo(V)1467 3736 y Fn(i)1491 3745 y Fd(0)1530 3721 y Fs(\))22 b Fl(\000)h Fo(W)1782 3736 y Fn(i)1806 3745 y Fd(0)1845 3721 y Fo(V)1902 3736 y Fn(i)1926 3745 y Fd(0)1964 3721 y Fs(])p Fo( )2058 3670 y Fi(\()p Fn(i)2109 3679 y Fd(0)2144 3670 y Fi(\))2054 3732 y Fn(g)2176 3721 y Fo(;)17 b( )2287 3670 y Fi(\()p Fn(i)2338 3679 y Fd(0)2373 3670 y Fi(\))2283 3732 y Fn(g)2404 3721 y Fl(i)p 1017 3766 1426 4 v 1601 3857 a Fo(E)1673 3872 y Fj(\000)1733 3857 y Fs(\()p Fo(g)t Fs(\))2475 3789 y(+)2583 3721 y Fo(g)t Fl(h)p Fs([)p Fo(\034)2742 3736 y Fn(a)2783 3721 y Fo(V)2840 3736 y Fi(1)2902 3721 y Fl(\000)22 b Fo(V)3058 3736 y Fi(1)3098 3721 y Fs(])p Fo( )3192 3670 y Fi(\()p Fn(i)3243 3679 y Fd(0)3278 3670 y Fi(\))3188 3732 y Fn(g)3309 3721 y Fo(;)17 b( )3420 3670 y Fi(\()p Fn(i)3471 3679 y Fd(0)3506 3670 y Fi(\))3416 3732 y Fn(g)3538 3721 y Fl(i)p 2583 3766 994 4 v 2951 3857 a Fo(E)3023 3872 y Fj(\000)3082 3857 y Fs(\()p Fo(g)t Fs(\))904 4087 y(=)1017 4020 y Fo(g)t Fl(h)p Fs([\()p Fo(\034)1214 4035 y Fn(a)1255 4020 y Fo(W)1347 4035 y Fn(i)1371 4044 y Fd(0)1410 4020 y Fo(V)1467 4035 y Fn(i)1491 4044 y Fd(0)1530 4020 y Fs(\))22 b Fl(\000)h Fo(W)1782 4035 y Fn(i)1806 4044 y Fd(0)1845 4020 y Fo(V)1902 4035 y Fn(i)1926 4044 y Fd(0)1964 4020 y Fs(])p Fo( )2058 3969 y Fi(\()p Fn(i)2109 3978 y Fd(0)2144 3969 y Fi(\))2054 4030 y Fn(g)2176 4020 y Fo(;)17 b( )2287 3969 y Fi(\()p Fn(i)2338 3978 y Fd(0)2373 3969 y Fi(\))2283 4030 y Fn(g)2404 4020 y Fl(i)p 1017 4065 1426 4 v 1601 4156 a Fo(E)1673 4171 y Fj(\000)1733 4156 y Fs(\()p Fo(g)t Fs(\))2475 4087 y(+)22 b Fo(o)p Fs(\(1\))118 4318 y(as)33 b Fo(V)295 4333 y Fi(1)367 4318 y Fs(is)f(b)s(ounded)h(and)g Fo(g)e Fs(=)c Fo(o)p Fs(\()p Fo(E)1391 4333 y Fj(\000)1450 4318 y Fs(\()p Fo(g)t Fs(\)\).)118 4434 y(Hence)34 b(to)e(estimate)g(\010\()p Fo(a)p Fs(\),)h(it)f(is)g(enough)h(to)f(estimate)g(the)h(expression)602 4622 y Fo(g)t Fl(h)p Fs([)p Fo(\034)761 4637 y Fn(a)802 4622 y Fs(\()p Fo(W)932 4637 y Fn(i)956 4646 y Fd(0)995 4622 y Fo(V)1052 4637 y Fn(i)1076 4646 y Fd(0)1114 4622 y Fs(\))22 b Fl(\000)h Fo(W)1366 4637 y Fn(i)1390 4646 y Fd(0)1429 4622 y Fo(V)1486 4637 y Fn(i)1510 4646 y Fd(0)1548 4622 y Fs(])p Fo( )1642 4571 y Fi(\()p Fn(i)1693 4580 y Fd(0)1729 4571 y Fi(\))1638 4632 y Fn(g)1760 4622 y Fo(;)17 b( )1871 4571 y Fi(\()p Fn(i)1922 4580 y Fd(0)1957 4571 y Fi(\))1867 4632 y Fn(g)1989 4622 y Fl(i)p 602 4667 V 1186 4758 a Fo(E)1258 4773 y Fj(\000)1317 4758 y Fs(\()p Fo(g)t Fs(\))619 4956 y(=)836 4888 y Fo(g)p 733 4933 258 4 v 733 5024 a(E)805 5039 y Fj(\000)864 5024 y Fs(\()p Fo(g)t Fs(\))1017 4820 y Fm(Z)1072 5046 y Fj(j)p Fn(x)p Fj(j)p Fn()f(\013)2730 794 y Fj(y)2801 830 y Fs(and)j Fl(j)p Fo(a)p Fl(j)d(\024)h Fo(g)3293 794 y Fj(\000)p Fn(\013)3396 830 y Fs(.)52 b(Moreo)m(v)m(er,)118 958 y(b)m(y)32 b(Lemma)e(5.7,)h Fo(E)854 973 y Fj(\000)913 958 y Fs(\()p Fo(g)t Fs(\))17 b Fl(\030)f Fo(C)7 b(g)1278 921 y Fi(2)p Fn(\013)1358 898 y Fg(y)1426 958 y Fs(\(for)31 b(some)g Fo(C)7 b Fs(\).)43 b(Hence)32 b(to)f(sho)m(w)h(that)f Fo(\013)e(>)e(\013)3153 921 y Fj(\003)3192 958 y Fs(,)32 b(w)m(e)g(just)g(need)g(to)118 1074 y(sho)m(w)d(that)e Fo(I)8 b Fs(\()p Fo(a)p Fs(\))28 b Fl(!)f Fs(0)g(when)i Fo(g)i Fl(!)c Fs(+)p Fl(1)p Fs(.)41 b(This)28 b(is)f(easily)g(seen)i(cutting)e Fo(I)8 b Fs(\()p Fo(a)p Fs(\))27 b(in)m(to)g(t)m(w)m(o)h(parts.)42 b(In)28 b(the)118 1193 y(\014rst)33 b(one,)f(w)m(e)h(in)m(tegrate)e(o)m(v)m(er) i(some)f(small)e(neigh)m(b)s(orho)s(o)s(d)g(of)i(0)f(and)h(this)g(is)f (small)f(as)i Fo(\037)g Fs(and)g Fo(')3780 1157 y Fi(\()p Fn(i)3831 1166 y Fd(0)3866 1157 y Fi(\))118 1309 y Fs(are)h(b)s(ounded) g(and)g(the)g(singularities)d(of)i Fo(V)1748 1324 y Fn(i)1772 1333 y Fd(0)1843 1309 y Fs(are)g(in)m(tegrable;)g(outside)g(of)g(this)h (neigh)m(b)s(orho)s(o)s(d,)e(the)118 1428 y(p)s(oten)m(tials)36 b Fo(V)630 1443 y Fn(i)654 1452 y Fd(0)729 1428 y Fs(and)h Fo(V)980 1443 y Fn(i)1004 1452 y Fd(0)1043 1428 y Fs(\()p Fl(\001)24 b(\000)i Fo(b)p Fs(\))37 b(are)g(con)m(tin)m(uous)g(and)g (as)g Fo(')2393 1391 y Fi(\()p Fn(i)2444 1400 y Fd(0)2479 1391 y Fi(\))2548 1428 y Fs(is)f(in)g Fo(L)2834 1391 y Fi(2)2874 1428 y Fs(,)i(w)m(e)g(can)f(conclude)g(using)118 1544 y(Leb)s(esgue's)f(Dominated)c(Con)m(v)m(ergence)37 b(Theorem.)48 b(This)34 b(implies)e(that)i Fo(\013)c Fl(\025)h Fo(\013)3166 1508 y Fj(\003)3205 1544 y Fs(.)48 b(As)34 b(it)f(holds)h(for)118 1660 y(an)m(y)f Fo(\013)c(>)e(\013)559 1624 y Fj(y)594 1660 y Fs(,)33 b(w)m(e)h(get)e(the)h(result)g(of)f (Lemma)f(3.1.)p 3827 1776 4 66 v 3831 1714 59 4 v 3831 1776 V 3889 1776 4 66 v 118 1893 a(Com)m(bining)g(Lemma)g(3.1)h(with)h (Theorem)f(1.4,)h(w)m(e)g(get)118 2078 y Fx(Prop)s(osition)j(3.5.)49 b Fe(Under)35 b(the)f(assumptions)g(H1)i(and)e(H1",)h(for)f Fo(E)41 b Fe(lar)-5 b(ge)35 b(enough,)f(one)g(has)983 2254 y Fl(\000)p Fs(\(1)22 b(+)g Fo(\013)1330 2213 y Fj(y)1365 2254 y Fo(d)p Fs(\))p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(o)p Fs(\(1\)\))27 b Fl(\024)i Fs(log)16 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))331 2440 y(In)35 b(the)g(section)g (4.3)g(w)m(e)h(impro)m(v)m(e)e(on)h(the)h(general)e(upp)s(er)h(b)s (ound)g(giv)m(en)h(in)e(Theorem)h(1.4)f(so)118 2556 y(that)e(the)h(new) h(upp)s(er)f(b)s(ound)g(coalesces)g(with)g(the)g(lo)m(w)m(er)f(b)s (ound)h(obtained)f(here.)1488 2809 y(4.)55 b Fr(The)37 b(upper)g(bounds)331 2984 y Fs(In)47 b(this)f(section,)k(w)m(e)e(will)d (pro)m(v)m(e)j(the)f(asymptotic)f(upp)s(er)h(b)s(ound)g(on)f(the)h (appro)m(ximated)118 3100 y(densit)m(y)i(of)e(states)h Fo(dN)1017 3115 y Fn(!)r(;n)1177 3100 y Fs(de\014ned)h(in)e(section)h (2.1)f(in)f(the)i(di\013eren)m(t)g(cases)h(considered)f(in)f(the)118 3216 y(in)m(tro)s(duction.)118 3429 y(4.1.)56 b Fx(The)38 b(general)f(case.)49 b Fs(W)-8 b(e)33 b(pro)m(v)m(e)h(the)f(follo)m (wing)d(general)i(b)s(ound)118 3615 y Fx(Prop)s(osition)k(4.1.)49 b Fe(Under)36 b(the)h(assumptions)e(H1,)i(H2)f(and)g(H3,)h(ther)-5 b(e)37 b(exist)f Fo(\014)3219 3630 y Fi(0)3289 3615 y Fo(>)30 b Fs(0)36 b Fe(such)g(that,)118 3731 y(for)j(any)g Fo(\014)i(>)36 b(\014)732 3746 y Fi(0)771 3731 y Fe(,)41 b(ther)-5 b(e)39 b(exists)g Fo(E)1432 3746 y Fi(0)1507 3731 y Fo(>)g Fe(such)g(that,)h(for)f Fo(E)j(>)36 b(E)2540 3746 y Fi(0)2618 3731 y Fe(and)j Fo(n)d Fs(=)f(2[)p Fo(E)3171 3695 y Fn(\014)3219 3731 y Fs(])25 b Fl(\001)g Fs([)p Fl(j)17 b Fs(log)f Fo(E)6 b Fl(j)3644 3695 y Fn(\014)3691 3731 y Fs(])p Fe(,)40 b(we)118 3847 y(have)921 4023 y Fs(log)17 b(\()p Fk(E)12 b Fs(\()p Fo(N)1278 4038 y Fn(!)r(;n)1397 4023 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\)\)\))27 b Fl(\024)h(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b(+)h Fo(o)p Fs(\(1\)\))p Fo(:)-2977 b Fs(\(4.1\))331 4226 y(By)35 b(Lemma)f(5.5,)i(w)m(e)g(kno)m (w)h(that,)e(for)g(some)g Fo(C)k(>)33 b Fs(0)h(and)i Fo(\013)d(>)f Fs(1,)k Fo(g)t Fs(\()p Fo(E)6 b Fs(\))31 b Fl(\025)i Fo(C)7 b(E)3379 4190 y Fi(1)p Fn(=\013)3534 4226 y Fs(for)35 b(some)118 4342 y Fo(C)44 b(>)37 b Fs(0)g(and)i Fo(E)44 b Fs(large)36 b(enough.)61 b(So)37 b(that,)j(if)d(w)m(e)i(pic)m (k)f Fo(\014)k(>)37 b Fs(1)h(large)f(enough)h(so)g(that)g Fo(\027)45 b Fs(giv)m(en)38 b(b)m(y)118 4458 y(equation)32 b(\(2.28\))g(satis\014es)i Fo(\027)g(>)27 b Fs(1,)33 b(Lemma)e(2.3)h(and)h(Prop)s(osition)e(4.1)h(immediately)d(imply)118 4644 y Fx(Prop)s(osition)36 b(4.2.)49 b Fe(Under)38 b(the)g (assumptions)f(H1,)i(H2)f(and)g(H3,)h(ther)-5 b(e)38 b(exists)f Fo(E)3291 4659 y Fi(0)3365 4644 y Fo(>)h Fe(such)f(that,)118 4760 y(for)e Fo(E)f(>)27 b(E)555 4775 y Fi(0)595 4760 y Fe(,)34 b(we)h(have)1205 4936 y Fs(log)16 b Fo(N)10 b Fs(\()p Fo(E)c Fs(\))28 b Fl(\024)h(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)e Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)22 b(+)g Fo(o)p Fs(\(1\)\))p Fo(:)331 5122 y Fs(T)-8 b(aking)37 b(in)m(to)g(accoun)m(t)i(Prop)s(osition)d(3.2,)j(this)f(ends)h(the)g (pro)s(of)e(of)g(Theorem)h(1.4.)60 b(W)-8 b(e)38 b(no)m(w)118 5238 y(turn)33 b(to)f(the)h(pro)s(of)f(of)g(Prop)s(osition)f(4.1.)1969 5454 y Fp(24)p eop %%Page: 25 25 25 24 bop 118 241 a Fs(4.2.)56 b Fx(Pro)s(of)49 b(of)h(Prop)s(osition)e (4.1.)h Fs(The)44 b(idea)f(of)g(the)g(pro)s(of)g(is)g(to)g(sho)m(w)h (that,)i(if)c Fo(H)3546 256 y Fn(!)r(;n;\022)3756 241 y Fs(has)118 357 y(some)c(lo)m(w)g(energies,)j(then)e(the)f(corresp)s (onding)h(p)s(oten)m(tial)d Fo(V)2462 372 y Fn(!)2551 357 y Fs(m)m(ust)i(ha)m(v)m(e)i(a)e(v)m(ery)i(deep)f(w)m(ell)f(i.e.)118 473 y(the)h(corresp)s(onding)f(realization)d(of)j(the)h(P)m(oisson)f (measure)h(m)m(ust)f(put)h(su\016cien)m(tly)g(man)m(y)f(p)s(oin)m(ts) 118 589 y(inside)i(the)i(cub)s(e)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))40 b(and)h(those)h(p)s(oin)m(ts)e(m)m(ust)h(b)s(e)h (su\016cien)m(tly)f(close)g(to)g(eac)m(h)h(other.)69 b(The)118 706 y(main)31 b(tec)m(hnical)g(di\016cult)m(y)i(comes)f(from) f(the)i(fact)f(that)g(our)g(single)f(site)h(p)s(oten)m(tial)e(is)i(not) g(of)g(\014nite)118 822 y(range;)43 b(so)c(w)m(e)h(need)h(some)e(a)f (priori)g(estimate)g(on)h Fo(N)2160 837 y Fn(!)r(;n)2312 822 y Fs(that)g(tells)f(us)i(that)f(the)h(b)s(eha)m(vior)f(of)f Fo(V)3847 837 y Fn(!)118 938 y Fs(outside)26 b(of)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))26 b(do)s(es)g(not)h(in)m(teract)f(to)s (o)f(m)m(uc)m(h)i(with)f(the)g(one)h(inside,)g(more)e(precisely)-8 b(,)28 b(that)e(this)118 1054 y(in)m(teraction)31 b(can)h(b)s(e)g (large)f(only)g(with)h(a)f(small)f(probabilit)m(y)-8 b(.)41 b(Actually)31 b(w)m(e)i(need)g(this)f(not)f(only)h(on)118 1171 y(the)e(scale)f(of)g(the)g(large)f(cub)s(e)i(used)h(in)d(the)i(p)s (erio)s(dic)d(appro)m(ximation)g(but)j(also)e(on)h(a)g(m)m(uc)m(h)h (smaller)118 1287 y(scale,)37 b(namely)e(the)h(scale)g(of)g(the)g(size) g(of)f(the)i(cub)s(e)f(where)i(w)m(e)e(w)m(an)m(t)h(the)g(P)m(oisson)f (p)s(oin)m(ts)f(to)h(pile)118 1403 y(up.)331 1519 y(Pic)m(k)e Fo(\014)g(>)29 b Fs(0)k(large.)45 b(Set)34 b Fo(n)29 b Fs(=)g(2[)p Fo(E)1620 1483 y Fn(\014)1667 1519 y Fs(])22 b Fl(\001)h Fs([)p Fl(j)17 b Fs(log)f Fo(E)6 b Fl(j)2087 1483 y Fn(\014)2134 1519 y Fs(])33 b(and)g Fo(l)f Fs(=)c([)p Fl(j)17 b Fs(log)f Fo(E)6 b Fl(j)2869 1483 y Fn(\014)2916 1519 y Fs(].)46 b(Pic)m(k)34 b Fo(\032)29 b(>)g Fs(0)k(large)f(and)118 1636 y(let)g(\012)329 1599 y Fi(2)329 1661 y Fn(E)422 1636 y Fs(b)s(e)g(de\014ned)i(as)f(in)f(\(3.3\))o(.)44 b(Then)34 b(w)m(e)f(pro)m(v)m(e)118 1812 y Fx(Lemma)k(4.1.)49 b Fe(F)-7 b(or)34 b(some)g Fo(C)h(>)27 b Fs(0)35 b Fe(and)f (su\016ciently)h(lar)-5 b(ge)35 b Fo(E)6 b Fe(,)35 b(we)f(have)867 2020 y Fk(E)12 b Fs(\()p Fo(N)1043 2035 y Fn(!)r(;n)1162 2020 y Fs(\()p Fl(\000)p Fo(E)28 b Fs(+)22 b(1\)\))28 b Fl(\024)g Fk(E)12 b Fs(\()p Fo(N)1909 2035 y Fn(!)s(;n)2028 2020 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\))p Fx(1)2485 2041 y Fi(\012)2536 2018 y Fd(2)2536 2064 y Fh(E)2591 2020 y Fs(\))g(+)2897 1953 y Fo(C)p 2759 1997 353 4 v 2759 2089 a Fs(\000\()p Fo(E)2936 2060 y Fn(\032)2977 2089 y Fo(=)p Fs(2\))3122 2020 y Fo(:)-3031 b Fs(\(4.2\))118 2252 y Fx(Pro)s(of.)43 b Fs(W)-8 b(e)33 b(decomp)s(ose)1031 2413 y Fk(E)12 b Fs(\()p Fo(N)1207 2428 y Fn(!)r(;n)1326 2413 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\)\))27 b(=)g Fk(E)13 b Fs(\()p Fo(N)2072 2428 y Fn(!)r(;n)2191 2413 y Fs(\()p Fl(\000)p Fo(E)28 b Fs(+)22 b(1\))p Fx(1)2647 2433 y Fi(\012)2698 2411 y Fd(2)2698 2457 y Fh(E)2754 2413 y Fs(\))g(+)g Fo(A)118 2583 y Fs(where)34 b Fo(\013)f Fs(and)g Fo(p)734 2547 y Fj(0)790 2583 y Fs(are)f(as)h(in)f(Lemma)f (5.1,)263 2882 y Fo(A)d Fs(=)467 2787 y Fm(X)475 3000 y Fn(k)r Fj(\025)p Fi(1)628 2882 y Fk(E)12 b Fs(\()p Fo(N)804 2897 y Fn(!)r(;n)923 2882 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\))p Fx(1)1380 2903 y Fi(\012)1431 2915 y Fh(k)1468 2903 y Fi(\()p Fn(\013;)p Fi(1)p Fn(;p)1651 2884 y Fg(0)1674 2903 y Fi(\))p Fj(n)p Fi(\(\012)1814 2880 y Fd(2)1814 2926 y Fh(E)1866 2903 y Fj([)p Fi(\012)1964 2915 y Fh(k)q Fg(\000)p Fd(1)2081 2903 y Fi(\()p Fn(\013;)p Fi(1)p Fn(;p)2264 2884 y Fg(0)2286 2903 y Fi(\)\))2345 2882 y Fs(\))g(+)g Fk(E)13 b Fs(\()p Fo(N)2680 2897 y Fn(!)r(;n)2798 2882 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\))p Fx(1)3255 2903 y Fi(\012)3306 2912 y Fd(0)3340 2903 y Fi(\()p Fn(\013;)p Fi(1)p Fn(;p)3523 2884 y Fg(0)3546 2903 y Fi(\))p Fj(n)p Fi(\012)3659 2880 y Fd(2)3659 2926 y Fh(E)3715 2882 y Fs(\))364 3163 y Fl(\024)489 3068 y Fm(X)469 3280 y Fn(k)r Fj(\025)p Fn(E)619 3261 y Fh(\032)670 3163 y Fk(E)13 b Fs(\()p Fo(N)847 3178 y Fn(!)r(;n)965 3163 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\))p Fx(1)1422 3178 y Fi(\012)1473 3190 y Fh(k)1511 3178 y Fi(\()p Fn(\013;)p Fi(1)p Fn(;p)1694 3159 y Fg(0)1716 3178 y Fi(\))p Fj(n)p Fi(\012)1829 3190 y Fh(k)q Fg(\000)p Fd(1)1946 3178 y Fi(\()p Fn(\013;)p Fi(1)p Fn(;p)2129 3159 y Fg(0)2151 3178 y Fi(\))2183 3163 y Fs(\))118 2730 y(\(4.3\))118 3434 y(as,)29 b(for)d Fo(k)31 b(<)c(E)667 3397 y Fn(\032)708 3434 y Fs(,)h(b)m(y)g(the)g(pro)s(of)e(of)g(Lemma)g(5.1,)i(w)m(e)g(ha)m (v)m(e)h(\012)2361 3449 y Fn(k)2404 3434 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)2691 3397 y Fj(0)2713 3434 y Fs(\))27 b Fl(\032)i Fs(\012)2954 3397 y Fi(2)2954 3459 y Fn(E)3014 3434 y Fs(.)41 b(F)-8 b(or)26 b Fo(!)31 b Fl(2)e Fs(\012)3508 3449 y Fn(k)3551 3434 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)3838 3397 y Fj(0)3860 3434 y Fs(\))118 3550 y(\(see)34 b(Lemma)d(5.1\),)h(w)m(e)i(ha)m(v)m(e)1525 3730 y Fl(k)p Fo(V)1653 3689 y Fi(\()p Fn(r)r Fi(\))1632 3755 y Fn(!)r(;n)1746 3730 y Fl(k)1796 3745 y Fj(1)1898 3730 y Fl(\024)29 b Fo(C)7 b(E)2159 3689 y Fn(\013\014)2251 3730 y Fo(k)2366 3649 y Fh(p)p 2315 3665 135 3 v 2315 3710 a(p)p Fg(\000)p Fh(p)2427 3696 y Fg(0)2464 3730 y Fo(:)118 3897 y Fs(Hence,)34 b(if)e Fo(N)43 b Fs(denotes)33 b(the)g(densit)m(y)h(of)e(states)h(of)g Fl(\000)p Fs(\001,)g(for)f Fo(!)f Fl(2)d Fs(\012)2626 3912 y Fn(k)2669 3897 y Fs(\()p Fo(\013)q(;)17 b Fs(1)p Fo(;)g(p)2956 3861 y Fj(0)2978 3897 y Fs(\),)33 b(w)m(e)g(get)479 4114 y Fo(N)557 4129 y Fn(!)r(;n)670 4114 y Fs(\()p Fl(\000)p Fo(E)c Fs(+)22 b(1\))27 b Fl(\024)h Fo(N)1309 4003 y Fm(\020)1368 4114 y Fl(\000)p Fo(E)h Fs(+)22 b(2)g(+)g Fo(C)7 b(E)1968 4073 y Fn(\013\014)2060 4114 y Fo(k)2175 4033 y Fh(p)p 2124 4048 V 2124 4093 a(p)p Fg(\000)p Fh(p)2236 4079 y Fg(0)2273 4003 y Fm(\021)2360 4114 y Fl(\024)28 b Fo(C)2559 4003 y Fm(\020)2618 4114 y Fl(\000)p Fo(E)h Fs(+)22 b(2)g(+)g Fo(C)7 b(E)3218 4073 y Fn(\013\014)3310 4114 y Fo(k)3425 4033 y Fh(p)p 3374 4048 V 3374 4093 a(p)p Fg(\000)p Fh(p)3486 4079 y Fg(0)3523 4003 y Fm(\021)3582 4025 y Fn(d=)p Fi(2)3710 4114 y Fo(:)118 4306 y Fs(Plugging)31 b(this)h(in)m(to)g(\(4.3\))g(and) h(computing)e(the)i(sum)f(o)m(v)m(er)i Fo(k)s Fs(,)f(w)m(e)g(get)g (Lemma)e(4.1.)p 3827 4423 4 66 v 3831 4360 59 4 v 3831 4423 V 3889 4423 4 66 v 118 4539 a(De\014ne)491 4728 y Fo(V)569 4687 y Fi(\()p Fn(i)p Fi(\))548 4752 y Fn(!)680 4728 y Fs(=)783 4592 y Fm(Z)839 4818 y Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(+2)p Fn(l)q Fi(\))1179 4728 y Fo(V)22 b Fs(\()p Fo(x)g Fl(\000)h Fo(y)t Fs(\))p Fo(m)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\))29 b(and)k Fo(V)2234 4687 y Fi(\()p Fn(e)p Fi(\))2212 4752 y Fn(!)2353 4728 y Fs(=)2457 4592 y Fm(Z)2512 4818 y Ff(R)2560 4799 y Fh(d)2596 4818 y Fj(n)p Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(+2)p Fn(l)q Fi(\))2972 4728 y Fo(V)21 b Fs(\()p Fo(x)i Fl(\000)g Fo(y)t Fs(\))p Fo(m)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\))-3611 b(\(4.4\))118 4988 y(and)43 b(the)h(corresp)s(onding)f(p)s(erio)s(dized)f(p)s(oten)m(tials)g Fo(V)2155 4937 y Fi(\()p Fn(i;e)p Fi(\))2134 4998 y Fn(!)r(;n)2334 4988 y Fs(\(see)i(equation)f(\(2.1\))o(\).)75 b(As)44 b Fo(V)3501 5003 y Fi(2)3583 4988 y Fs(is)f(com-)118 5122 y(pactly)33 b(supp)s(orted,)h Fo(V)979 5071 y Fi(\()p Fn(e)p Fi(\))957 5132 y Fn(!)r(;n)1103 5122 y Fs(is)f(almost)e(surely)j (b)s(ounded)g(for)e Fo(l)k Fs(\(i.e.)44 b Fo(E)6 b Fs(\))33 b(large.)44 b(W)-8 b(e)33 b(will)e(estimate)h(its)118 5238 y(magnitude)f(later.)331 5355 y(The)i(pro)s(of)f(of)g(Prop)s (osition)f(4.1)h(will)e(b)s(e)j(a)f(consequence)k(of)c(the)h(follo)m (wing)d(lemmas.)1969 5454 y Fp(25)p eop %%Page: 26 26 26 25 bop 118 241 a Fx(Lemma)37 b(4.2.)49 b Fe(F)-7 b(or)34 b(any)h Fo(")27 b(>)h Fs(0)p Fe(,)34 b(ther)-5 b(e)35 b(exists)g Fo(E)1979 256 y Fi(0)2053 241 y Fe(and)g Fo(k)2294 256 y Fi(0)2368 241 y Fe(such)f(that,)h(if)g Fo(k)c(>)c(k)3149 256 y Fi(0)3189 241 y Fe(,)34 b Fo(E)g(>)28 b(E)3535 256 y Fi(0)3609 241 y Fe(and)1185 408 y Fl(8)p Fo(x)g Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)51 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\)\))27 b Fl(\024)h Fo(k)s(;)-2696 b Fs(\(4.5\))118 575 y Fe(then,)35 b(for)f(any)h Fo(\022)c Fl(2)d Fo(T)948 539 y Fj(\003)934 600 y Fn(n)987 575 y Fe(,)35 b(we)f(have)593 803 y Fo(H)674 818 y Fn(!)r(;n;\022)869 803 y Fl(\025)28 b(\000)p Fo(E)1123 818 y Fj(\000)1199 663 y Fm(\022)1364 736 y Fo(k)p 1283 781 217 4 v 1283 872 a Fs(1)21 b Fl(\000)i Fo(")1509 663 y Fm(\023)1604 803 y Fl(\000)1733 736 y Fo(C)p 1714 781 117 4 v 1714 872 a("l)1791 843 y Fi(2)1862 803 y Fl(\000)g(k)p Fo(V)2090 762 y Fi(\()p Fn(e)p Fi(\))2069 828 y Fn(!)r(;n)2182 803 y Fl(k)2232 818 y Fj(1)2329 803 y Fl(\000)f Fo(C)7 b Fs(\(\()p Fo(n=)p Fs(2)p Fo(l)r Fs(\))2806 762 y Fn(d)2869 803 y Fs(+)22 b(1\))f Fl(\001)h Fo(k)s(e)3224 762 y Fj(\000)p Fn(l)q(=C)3396 803 y Fo(:)-3305 b Fs(\(4.6\))118 1040 y(and)118 1220 y Fx(Lemma)37 b(4.3.)49 b Fe(Pick)35 b Fo(\016)c(>)d Fs(0)p Fe(.)45 b(If)34 b Fo(E)41 b Fe(is)34 b(lar)-5 b(ge)35 b(enough)f(and)g(if)h Fo(k)c Fl(\025)d Fo(E)2696 1184 y Fn(\016)2734 1220 y Fe(,)35 b(then,)583 1387 y Fs(log)17 b([)p Fo(P)d Fs(\()p Fl(f)p Fo(!)t Fs(;)50 b Fl(9)p Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)52 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\)\))27 b Fo(>)g(k)s Fl(g)p Fs(\)])104 b Fl(\030)2793 1449 y Fn(E)t Fj(!)p Fi(+)p Fj(1)3061 1387 y Fl(\000)p Fo(k)20 b Fs(log)c Fo(k)s(:)-3314 b Fs(\(4.7\))331 1607 y(Before)24 b(pro)m(ving)g(Lemmas) f(4.2)h(and)g(4.3,)i(w)m(e)f(\014nish)f(the)h(pro)s(of)e(of)h(Prop)s (osition)e(4.1.)40 b(Therefore,)118 1723 y(\014x)33 b Fo(")28 b(>)f Fs(0)32 b(and)h(set)g Fo(k)e Fs(=)d Fo(g)t Fs(\()p Fo(E)f Fl(\000)c Fs(2\)\(1)e Fl(\000)i Fo(")p Fs(\).)43 b(F)-8 b(or)32 b Fo(!)f Fl(2)d Fs(\012)2210 1687 y Fi(2)2210 1749 y Fn(E)2270 1723 y Fs(,)k(w)m(e)i(ha)m(v)m(e,)g (for)e(some)h Fo(C)h(>)28 b Fs(0,)1472 1907 y Fl(k)p Fo(V)1600 1866 y Fi(\()p Fn(e)p Fi(\))1579 1931 y Fn(!)r(;n)1692 1907 y Fl(k)1742 1922 y Fj(1)1844 1907 y Fl(\024)g Fo(E)2027 1866 y Fn(\032)2068 1907 y Fo(e)2113 1866 y Fj(\000)p Fi(\(log)13 b Fn(E)t Fi(\))2381 1842 y Fh(\014)2422 1866 y Fn(=C)2517 1907 y Fo(:)-2426 b Fs(\(4.8\))118 2080 y(Hence,)36 b(using)e(\(4.6\))f(and)h(the)g(b)s(ounds)h(kno)m(wn)g(for) e Fo(g)t Fs(\()p Fo(E)6 b Fs(\))33 b(giv)m(en)h(in)f(Lemma)f(5.5,)i(w)m (e)h(get)e(that,)h(for)118 2196 y Fo(!)d Fl(2)d Fs(\012)374 2160 y Fi(2)374 2221 y Fn(E)467 2196 y Fs(suc)m(h)34 b(that)e Fl(8)p Fo(x)d Fl(2)f Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)k Fs(+)h(2)p Fo(l)r Fs(\),)33 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\)\))27 b Fl(\024)h Fo(k)s Fs(,)33 b(for)f(an)m(y)h Fo(\022)e Fl(2)d Fo(T)3116 2160 y Fj(\003)3102 2221 y Fn(n)3155 2196 y Fs(,)33 b(w)m(e)g(ha)m(v)m(e)1593 2363 y Fo(H)1674 2378 y Fn(!)r(;n;\022)1869 2363 y Fl(\025)28 b(\000)p Fo(E)g Fs(+)22 b(3)p Fo(=)p Fs(2)p Fo(:)118 2530 y Fs(In)33 b(other)g(w)m(ords,)g(w)m(e)h(ha)m(v)m(e)201 2727 y Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)32 b(of)h Fo(H)986 2742 y Fn(!)r(;n;\022)1180 2727 y Fl(\024)c(\000)p Fo(E)f Fs(+)22 b(1)p Fl(g)845 2878 y Fs(=)28 b Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)32 b(of)g Fo(H)1733 2893 y Fn(!)r(;n;\022)1928 2878 y Fl(\024)c(\000)p Fo(E)h Fs(+)22 b(1)p Fl(g)p Fx(1)2464 2898 y Fj(f)p Fn(!)r Fj(2)p Fi(\012)2643 2875 y Fd(2)2643 2921 y Fh(E)2695 2898 y Fi(:)32 b Fj(9)p Fn(x)p Fj(2)p Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(+2)p Fn(l)q Fi(\))p Fn(;)33 b(m)p Fi(\()p Fn(!)r(;C)5 b Fi(\()p Fn(\015)t(;)p Fi(2)p Fn(l)q Fi(\)\))p Fn(>k)r Fj(g)3788 2878 y Fo(:)118 3050 y Fs(By)33 b(the)g(de\014nition)f(of)g Fo(N)1062 3065 y Fn(!)r(;n)1175 3050 y Fs(,)g(using)h(F)-8 b(ubini's)31 b(Theorem,)i(w)m(e)h(compute)457 3212 y Fk(E)12 b Fs(\()p Fo(N)634 3227 y Fn(!)r(;n)752 3212 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\))p Fx(1)1209 3232 y Fi(\012)1260 3209 y Fd(2)1260 3255 y Fh(E)1315 3212 y Fs(\))485 3428 y(=)588 3293 y Fm(Z)644 3518 y Fn(T)695 3499 y Fg(\003)685 3535 y Fh(n)751 3428 y Fk(E)13 b Fs(\()p Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)38 b(of)32 b Fo(H)1640 3443 y Fn(!)r(;n;\022)1835 3428 y Fl(\024)c(\000)p Fo(E)h Fs(+)22 b(1)p Fl(g)p Fx(1)2371 3449 y Fi(\012)2422 3426 y Fd(2)2422 3472 y Fh(E)2477 3428 y Fs(\))485 3708 y(=)588 3572 y Fm(Z)644 3798 y Fn(T)695 3779 y Fg(\003)685 3814 y Fh(n)751 3708 y Fk(E)13 b Fs(\()p Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)38 b(of)32 b Fo(H)1640 3723 y Fn(!)r(;n;\022)1835 3708 y Fl(\024)c(\000)p Fo(E)h Fs(+)22 b(1)p Fl(g)p Fx(1)2371 3728 y Fj(f)p Fn(!)r Fj(2)p Fi(\012)2550 3706 y Fd(2)2550 3752 y Fh(E)2601 3728 y Fi(:)33 b Fj(9)p Fn(x)p Fj(2)p Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(+2)p Fn(l)q Fi(\))p Fn(;)32 b(m)p Fi(\()p Fn(!)r(;C)5 b Fi(\()p Fn(\015)t(;)p Fi(2)p Fn(l)q Fi(\)\))p Fn(>k)r Fj(g)3694 3708 y Fs(\))p Fo(:)118 3501 y Fs(\(4.9\))118 3953 y(On)33 b(the)g(other)f(hand,)h(for)f Fo(!)f Fl(2)d Fs(\012)1379 3916 y Fi(2)1379 3978 y Fn(E)1472 3953 y Fs(\(\012)1580 3916 y Fi(2)1580 3978 y Fn(E)1673 3953 y Fs(is)k(de\014ned)i(in)d(\(3.3\)\),)i(w)m(e)g(ha)m(v)m(e)1378 4129 y Fl(k)p Fo(V)1506 4088 y Fi(\()p Fn(i)p Fi(\))1485 4154 y Fn(!)r(;n)1597 4129 y Fl(k)1647 4145 y Fn(L)1695 4126 y Fh(p)1731 4145 y Fi(\()p Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(\)\))2026 4129 y Fl(\024)28 b Fo(C)7 b(E)2286 4088 y Fn(\014)s Fi(\()p Fn(d)p Fi(+)p Fn(\013)2492 4065 y Fg(0)2515 4088 y Fi(\)+)p Fn(\032)118 4296 y Fs(hence,)34 b(using)e(Lemma)g(5.3,)g(there)h(exists)h Fo(\027)g(>)27 b Fs(0)33 b(suc)m(h)h(that)1317 4507 y Fo(H)1398 4522 y Fn(!)r(;n;\022)1592 4507 y Fl(\025)28 b(\000)1784 4439 y Fs(1)p 1784 4484 49 4 v 1784 4575 a(2)1844 4507 y(\001)1925 4522 y Fn(n;\022)2049 4507 y Fl(\000)22 b Fo(E)2226 4465 y Fi(\()p Fn(\014)s Fi(\()p Fn(d)p Fi(+)p Fn(\013)2459 4442 y Fg(0)2483 4465 y Fi(\)+)p Fn(\032)p Fi(\))p Fn(\027)2672 4507 y Fo(:)118 4709 y Fs(where)31 b Fl(\000)p Fs(\001)555 4724 y Fn(n;\022)687 4709 y Fs(is)e(the)h(Laplace)f(op)s(erator)f(on)i Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))29 b(with)g(quasi-p)s(erio)s(dic) f(b)s(oundary)h(conditions.)118 4825 y(This)k(implies)d(that,)i(for)g (some)h Fo(C)i(>)27 b Fs(0,)725 4978 y Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)33 b(of)f Fo(H)1510 4993 y Fn(!)r(;n;\022)1705 4978 y Fl(\024)c(\000)p Fo(E)g Fs(+)22 b(1)p Fl(g)1226 5147 y(\024)28 b Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)32 b(of)54 b Fl(\000)23 b Fs(\001)2237 5162 y Fn(n;\022)2366 5147 y Fl(\024)28 b Fo(C)7 b(E)2626 5106 y Fi(\()p Fn(\014)s Fi(\()p Fn(d)p Fi(+)p Fn(\013)2859 5082 y Fg(0)2883 5106 y Fi(\)+)p Fn(\032)p Fi(\))p Fn(\027)3094 5147 y Fl(\000)23 b Fs(2)p Fo(E)28 b Fs(+)22 b(2)p Fl(g)1698 5316 y(\024)28 b Fo(C)7 b(n)1938 5274 y Fn(d)1979 5316 y Fs(\()p Fo(E)2095 5274 y Fi(\()p Fn(\014)s Fi(\()p Fn(d)p Fi(+)p Fn(\013)2328 5251 y Fg(0)2352 5274 y Fi(\)+)p Fn(\032)p Fi(\))p Fn(\027)2563 5316 y Fl(\000)22 b Fo(E)29 b Fs(+)22 b(1\))2948 5274 y Fn(d=)p Fi(2)3058 5316 y Fo(:)1969 5454 y Fp(26)p eop %%Page: 27 27 27 26 bop 118 241 a Fs(Plugging)31 b(this)h(in)m(to)g(\(4.9\))o(,)h(w)m (e)h(get)201 538 y Fk(E)13 b Fs(\()p Fo(N)378 553 y Fn(!)r(;n)496 538 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\))p Fx(1)953 558 y Fi(\012)1004 535 y Fd(2)1004 581 y Fh(E)1060 538 y Fs(\))572 711 y Fl(\024)28 b Fo(C)7 b Fs(\()p Fo(E)870 670 y Fi(\()p Fn(\014)s Fi(\()p Fn(d)p Fi(+)p Fn(\013)1103 646 y Fg(0)1127 670 y Fi(\)+)p Fn(\032)p Fi(\))p Fn(\027)1338 711 y Fl(\000)23 b Fo(E)28 b Fs(+)22 b(1\))1723 670 y Fn(d=)p Fi(2)1833 711 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)48 b Fl(9)p Fo(x)29 b Fl(2)f Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)k Fs(+)h(2)p Fo(l)r Fs(\))p Fo(;)50 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(2)p Fo(l)r Fs(\)\))26 b Fo(>)i(k)s Fl(g)p Fo(:)118 979 y Fs(Notice)h(that,)h(b)m (y)h(Lemma)d(5.5,)h(our)h(c)m(hoice)g(of)f Fo(k)j Fs(ful\014lls)27 b(the)j(assumptions)g(of)f(Lemma)f(4.3.)42 b(There-)118 1095 y(from)31 b(w)m(e)j(deduce)g(that,)f(if)e Fo(E)39 b Fs(is)32 b(large)f(enough,)i(then)959 1363 y(log)17 b(\()o Fk(E)c Fs(\()p Fo(N)1316 1378 y Fn(!)r(;n)1435 1363 y Fs(\()p Fl(\000)p Fo(E)28 b Fs(+)22 b(1\)\)\))28 b Fl(\024)g(\000)p Fo(g)t Fs(\()p Fo(E)6 b Fs(\))17 b(log)f Fo(g)t Fs(\()p Fo(E)6 b Fs(\)\(1)21 b Fl(\000)h Fs(2)p Fo(")p Fs(\))p Fo(:)118 1631 y Fs(This)33 b(completes)f(the)h(pro)s(of) f(of)g(Prop)s(osition)f(4.1)p 3827 1747 4 66 v 3831 1685 59 4 v 3831 1747 V 3889 1747 4 66 v 118 2143 a(4.2.1.)50 b Fe(The)35 b(pr)-5 b(o)g(of)36 b(of)g(L)-5 b(emma)35 b(4.2.)48 b Fs(Consider)34 b(the)h(partition)d(of)h(the)i(cub)s(e)f Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))34 b(in)m(to)f(cub)s(es)i(of)118 2259 y(side)e(length)f Fo(l)j Fs(i.e.)1510 2543 y Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))27 b(=)2006 2449 y Fm([)1967 2673 y Fn(\015)t Fj(2)p Fn(l)q Ff(Z)2126 2649 y Fh(d)1944 2744 y Fj(j)p Fn(\015)t Fj(j\024)p Fn(n=l)2195 2543 y Fo(C)7 b Fs(\()p Fo(\015)e(;)17 b(l)r Fs(\))p Fo(:)118 3000 y Fs(and)33 b(the)g(co)m(v)m(ering)1484 3284 y Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))28 b Fl(\032)1983 3189 y Fm([)1943 3413 y Fn(\015)t Fj(2)p Fn(l)q Ff(Z)2103 3390 y Fh(d)1921 3485 y Fj(j)p Fn(\015)t Fj(j\024)p Fn(n=l)2172 3284 y Fo(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(2)p Fo(l)r Fs(\))p Fo(:)118 3753 y Fs(Set)33 b Fo(L)28 b Fs(=)g(2[)p Fo(E)638 3717 y Fn(\014)685 3753 y Fs(])k(and)h(consider)g(a)f Fk(Z)1464 3717 y Fn(d)1502 3753 y 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Fs(\()p Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\)\)\))36 b(suc)m(h)i(that)1969 5454 y Fp(27)p eop %%Page: 28 28 28 27 bop 118 241 a Fl(k)p Fo(')p Fl(k)27 b Fs(=)h(1.)43 b(F)-8 b(or)32 b(small)e(p)s(ositiv)m(e)i Fo(")p Fs(,)g(w)m(e)i (compute)232 673 y Fl(h)p Fs(\()p Fl(\000)p Fs(\001)23 b(+)f Fo(V)645 688 y Fn(!)r(;n)758 673 y Fs(\))p Fo(';)17 b(')p Fl(i)27 b Fs(=)g Fl(h)p Fs(\()p Fl(\000)p Fs(\001)c(+)f Fo(V)1571 632 y Fi(\()p Fn(i)p Fi(\))1550 698 y Fn(!)r(;n)1662 673 y Fs(\))p Fo(';)17 b(')p Fl(i)22 b Fs(+)g Fl(h)p Fo(V)2148 632 y Fi(\()p Fn(e)p Fi(\))2127 698 y Fn(!)r(;n)2240 673 y Fo(';)17 b(')p Fl(i)1034 840 y(\025)28 b(kr)p Fo(')p Fl(k)1386 799 y Fi(2)1447 840 y Fs(+)22 b Fl(h)p Fo(V)1662 799 y Fi(\()p Fn(i)p Fi(\))1641 865 y Fn(!)r(;n)1753 840 y Fo(';)17 b(')p Fl(i)22 b(\000)g(k)p Fo(V)2214 799 y Fi(\()p Fn(e)p Fi(\))2192 865 y Fn(!)r(;n)2306 840 y Fl(k)2356 855 y Fj(1)1034 1021 y Fs(=)1326 927 y Fm(X)1137 1146 y Fn(\015)t Fj(2)p Fn(l)q Ff(Z)1297 1127 y Fh(d)1327 1146 y Fj(\\)p 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Fo(n;)j(k)s(;)g(l)r Fs(\))27 b(=)h Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)48 b Fl(9)p Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\)\))27 b Fo(>)g(k)s Fl(g)p Fs(\))p Fo(:)118 3321 y Fs(W)-8 b(e)33 b(will)d(assume)j(that)g Fo(k)s Fs(,)f Fo(n)h Fs(and)g Fo(l)i Fs(are)d(large)g(and)g(that)h (they)g(satisfy)1428 3496 y(\()p Fo(n)22 b Fs(+)g Fo(l)r Fs(\))1713 3455 y Fn(d)p Fi(+1)1844 3496 y Fo(e)1889 3455 y Fn(l)1911 3432 y Fh(d)1979 3496 y Fs(=)27 b Fo(o)p Fs(\()p Fo(l)2198 3455 y Fj(\000)p Fn(k)r(d)2332 3496 y Fs(\000\()p Fo(k)s Fs(\)\))p Fo(:)-2470 b Fs(\(4.14\))118 3667 y(W)-8 b(e)33 b(will)d(pro)m(v)m(e)k(a)e(lo)m(w)m(er)h(and)g(an)f (upp)s(er)h(b)s(ound)g(on)g Fo(P)14 b Fs(\()p Fo(n;)j(k)s(;)g(l)r Fs(\).)331 3783 y(W)-8 b(e)32 b(start)g(with)g(the)g(lo)m(w)m(er)g(b)s (ound.)43 b(Consider)33 b(the)f(partition)e(of)h(the)i(cub)s(e)f Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)k Fs(+)g(2)p Fo(l)r Fs(\))31 b(in)m(to)118 3899 y(cub)s(es)j(of)e(side)h(length)f(2)p Fo(l)i Fs(i.e.)1322 4080 y Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))28 b(=)2082 3986 y Fm([)2025 4210 y Fn(\015)t Fj(2)p Fi(2)p Fn(l)q Ff(Z)2219 4186 y Fh(d)1957 4281 y Fj(j)p Fn(\015)t Fj(j\024)p Fn(n=)p Fi(2)p Fn(l)q Fi(+1)2334 4080 y Fo(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(2)p Fo(l)r Fs(\))p Fo(:)118 4434 y Fs(Using)37 b(the)g(indep)s (endence)i(for)e(disjoin)m(t)f(cub)s(es)i(and)f(the)h(homogeneit)m(y)f (of)f(the)i(P)m(oisson)f(\014eld,)i(w)m(e)118 4550 y(obtain)32 b(that)698 4694 y Fo(P)14 b Fs(\()p Fo(n;)j(k)s(;)g(l)r Fs(\))27 b Fl(\025)h Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)48 b Fl(9)p Fo(\015)33 b Fl(2)28 b Fs(2)p Fo(l)r Fk(Z)1901 4653 y Fn(d)1939 4694 y Fo(;)49 b Fl(j)p Fo(\015)5 b Fl(j)27 b(\024)i Fo(n=)p Fs(2)p Fo(l)24 b Fs(+)e(1;)49 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(2)p Fo(l)r Fs(\)\))26 b Fo(>)i(k)s Fl(g)p Fs(\))1109 4870 y(=)g(1)22 b Fl(\000)g Fs(\(1)g Fl(\000)h Fo(P)14 b Fs(\()p Fo(m)p Fs(\()p Fo(!)t(;)j(C)7 b Fs(\(0)p Fo(;)17 b Fs(2)p Fo(l)r Fs(\)\))25 b Fo(>)j(k)s Fs(\)\))2562 4829 y Fi(\()p Fn(n=)p Fi(2)p Fn(l)q Fi(+1\))2841 4805 y Fh(d)118 4783 y Fs(\(4.15\))118 5116 y(By)g(de\014nition,)f Fo(P)14 b Fs(\()p Fo(m)p Fs(\()p Fo(!)t(;)j(C)7 b Fs(\(0)p Fo(;)17 b Fs(2)p Fo(l)r Fs(\)\))26 b Fo(>)i(k)s Fs(\))f Fl(\025)h Fo(e)1832 5074 y Fj(\000)p Fn(\026)p Fi(\(2)p Fn(l)q Fi(\))2040 5051 y Fh(d)2092 5048 y Fs(\()p Fo(\026)p Fs(\(2)p Fo(l)r Fs(\))2345 5012 y Fn(d)2385 5048 y Fs(\))2423 5012 y Fn(k)p 2092 5093 374 4 v 2238 5184 a Fo(k)s Fs(!)2476 5116 y(.)41 b(Plugging)26 b(this)h(in)m(to)g(\(4.15\))o(,)h(w)m(e)h (get)1298 5316 y(log)17 b Fo(P)d Fs(\()p Fo(n;)j(k)s(;)g(l)r Fs(\))26 b Fl(\025)j(\000)p Fo(k)20 b Fs(log)c Fo(k)s Fs(\(1)22 b(+)g Fo(o)p Fs(\(1\)\))1969 5454 y Fp(29)p eop %%Page: 30 30 30 29 bop 118 241 a Fs(when)34 b Fo(k)s Fs(,)f Fo(n)f Fs(and)h Fo(l)i Fs(tend)e(to)f(+)p Fl(1)g Fs(under)i(the)f(assumption)e (\(4.14\).)331 357 y(T)-8 b(o)29 b(get)g(an)g(upp)s(er)h(b)s(ound,)g(w) m(e)h(consider)e(the)h(partition)d(of)i(the)h(cub)s(e)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)e Fs(+)g(2)p Fo(l)r Fs(\))29 b(in)m(to)f(cub)s(es)118 473 y(of)k(side)h(length)f(4)p Fo(l)i Fs(i.e.)1260 665 y Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)k Fs(+)h(2)p Fo(l)r Fs(\))28 b(=)2082 570 y Fm([)2025 794 y Fn(\015)t Fj(2)p Fi(4)p Fn(l)q Ff(Z)2219 771 y Fh(d)1895 866 y Fj(j)p Fn(\015)t Fj(j\024)p Fn(n=)p Fi(\(4)p Fn(l)q Fi(\)+1)p Fn(=)p Fi(2)2396 665 y Fo(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(4)p Fo(l)r Fs(\))p Fo(:)118 1043 y Fs(F)-8 b(or)35 b(an)m(y)i Fo(x)d Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)24 b Fs(+)g(2)p Fo(l)r Fs(\),)37 b(there)g(exists)f Fo(\015)j Fl(2)34 b Fs(4)p Fo(l)r Fk(Z)2109 1006 y Fn(d)2147 1043 y Fs(,)i Fl(j)p Fo(\015)5 b Fl(j)33 b(\024)h Fo(n=)p Fs(\(4)p Fo(l)r Fs(\))24 b(+)h(1)p Fo(=)p Fs(2)35 b(suc)m(h)i(that)f Fo(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(4)p Fo(l)r Fs(\))24 b Fl(\\)118 1159 y Fo(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\))28 b(=)f Fo(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\).)44 b(Hence,)201 1363 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)48 b Fl(8)p Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\)\))27 b Fl(\024)h Fo(k)s Fl(g)p Fs(\))g Fl(\025)1258 1527 y(\025)g Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)48 b Fl(8)p Fo(\015)33 b Fl(2)28 b Fs(4)p Fo(l)r Fk(Z)2050 1486 y Fn(d)2088 1527 y Fo(;)49 b Fl(j)p Fo(\015)5 b Fl(j)27 b(\024)h Fo(n=)p Fs(2)p Fo(l)d Fs(+)d(1)p Fo(=)p Fs(2;)48 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(4)p Fo(l)r Fs(\)\))27 b Fl(\024)h Fo(k)s Fl(g)p Fs(\))118 1703 y(that)k(is,)h(using)f(the)h (stationarit)m(y)e(of)h(the)h(P)m(oisson)g(pro)s(cess,)737 1865 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)48 b Fl(9)p Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(2)p Fo(l)r Fs(\)\))27 b Fo(>)h(k)s Fl(g)p Fs(\))1237 2046 y Fl(\024)1624 1951 y Fm(X)1343 2171 y Fn(\015)t Fj(2)p Fi(4)p Fn(l)q Ff(Z)1537 2152 y Fh(d)1567 2171 y Fn(;)33 b Fj(j)p Fn(\015)t Fj(j\024)p Fn(n=)p Fi(2)p Fn(l)q Fi(+1)p Fn(=)p Fi(2)2066 2046 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)81 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(4)p Fo(l)r Fs(\)\))26 b Fo(>)i(k)s Fl(g)p Fs(\))1237 2322 y Fl(\024)h Fs(\()p Fo(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))1677 2281 y Fn(d)1717 2322 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)80 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\(0)p Fo(;)17 b Fs(4)p Fo(l)r Fs(\)\))27 b Fo(>)g(k)s Fl(g)p Fs(\))p Fo(:)118 2498 y Fs(On)33 b(the)g(other)f(hand)753 2715 y Fo(P)14 b Fs(\()p Fl(f)p Fo(m)p Fs(\()p Fo(!)t(;)j(C)7 b Fs(\(0)p Fo(;)17 b Fs(4)p Fo(l)r Fs(\)\))25 b Fo(>)j(k)s Fl(g)p Fs(\))f(=)1916 2620 y Fm(X)1925 2833 y Fn(j)t(>k)2076 2715 y Fo(e)2121 2674 y Fj(\000)p Fi(\(4)p Fn(l)q Fi(\))2287 2650 y Fh(d)2324 2674 y Fn(\026)2381 2648 y Fs([)p Fo(\026)p Fs(\(4)p Fo(l)r Fs(\))2623 2611 y Fn(d)2663 2648 y Fs(])2690 2611 y Fn(j)p 2381 2692 346 4 v 2517 2783 a Fo(j)6 b Fs(!)2764 2715 y Fl(\024)2880 2648 y Fs(\()p Fo(\026)p Fs(\(4)p Fo(l)r Fs(\))3133 2611 y Fn(d)3173 2648 y Fs(\))3211 2611 y Fn(k)p 2880 2692 374 4 v 3026 2783 a Fo(k)s Fs(!)118 2715 y(\(4.16\))118 2992 y(This)33 b(then)g(implies)d(that)1298 3168 y(log)17 b Fo(P)d Fs(\()p Fo(n;)j(k)s(;)g(l)r Fs(\))26 b Fl(\024)j(\000)p Fo(k)20 b Fs(log)c Fo(k)s Fs(\(1)22 b(+)g Fo(o)p Fs(\(1\)\))118 3349 y(when)34 b Fo(k)s Fs(,)f Fo(n)f Fs(and)h Fo(l)i Fs(tend)e(to)f(+)p Fl(1)g Fs(under)i(assumption) d(\(4.14\).)p 3827 3466 4 66 v 3831 3403 59 4 v 3831 3466 V 3889 3466 4 66 v 118 3677 a(4.3.)56 b Fx(The)31 b(case)g(when)h Fo(V)52 b Fx(has)32 b(p)s(o)m(w)m(er)f(la)m(w)f (singularities.)47 b Fs(W)-8 b(e)27 b(no)m(w)g(assume)h(that)e(H1")h (holds.)118 3794 y(Ob)m(viously)-8 b(,)40 b(mo)s(difying)c Fo(V)1124 3809 y Fi(1)1163 3794 y Fs(,)k(w)m(e)g(can)f(assume)g(that)f (the)h(functions)f Fo(W)2817 3809 y Fn(i)2884 3794 y Fs(do)h(not)f(c)m(hange)h(sign)f(and)118 3910 y(that)32 b(the)h(supp)s(ort)g(of)g(distinct)e Fo(W)1414 3925 y Fn(i)1475 3910 y Fs(are)i(pairwise)f(disjoin)m(t.)42 b(W)-8 b(e)33 b(then)g(pro)m(v)m(e)118 4095 y Fx(Prop)s(osition)j(4.3.) 49 b Fe(Under)31 b(the)h(assumptions)e(H1)i(and)f(H1",)h(ther)-5 b(e)32 b(exists)f Fo(\014)3081 4110 y Fi(0)3148 4095 y Fo(>)d Fs(0)j Fe(such)g(that,)i(for)118 4211 y(any)g Fo(\014)g(>)28 b(\014)550 4226 y Fi(0)589 4211 y Fe(,)34 b(ther)-5 b(e)33 b(exists)g Fo(E)1231 4226 y Fi(0)1298 4211 y Fo(>)28 b Fs(0)k Fe(such)h(that,)h(for)f Fo(E)h(>)27 b(E)2366 4226 y Fi(0)2439 4211 y Fe(and)33 b Fo(n)27 b Fs(=)h(2[)p Fo(E)2970 4175 y Fn(\014)3017 4211 y Fs(])18 b Fl(\001)g Fs([)p Fl(j)f Fs(log)f Fo(E)6 b Fl(j)3428 4175 y Fn(\014)3475 4211 y Fs(])p Fe(,)34 b(we)e(have)724 4391 y Fs(log)17 b(\()p Fk(E)12 b Fs(\()p Fo(N)1081 4406 y Fn(!)s(;n)1200 4391 y Fs(\()p Fl(\000)p Fo(E)29 b Fs(+)22 b(1\)\)\))27 b Fl(\024)h(\000)p Fs(\(1)23 b(+)f Fo(\013)2157 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(one)i(tak)m(es)h(in)m(to)e(accoun)m(t)h(Prop)s(osition)e(3.5.)1969 5454 y Fp(30)p eop %%Page: 31 31 31 30 bop 118 241 a Fs(4.3.1.)50 b Fe(Pr)-5 b(o)g(of)45 b(of)g(Pr)-5 b(op)g(osition)45 b(4.3.)j Fs(The)d(idea)f(guiding)e(this) i(pro)s(of)f(is)h(essen)m(tially)g(the)g(same)g(as)118 357 y(the)c(one)g(guiding)e(the)i(pro)s(of)e(of)h(Prop)s(osition)f (4.1.)64 b(The)40 b(di\013erence)g(comes)g(from)e(the)i(fact)g(that,) 118 473 y(as)c Fo(E)313 488 y Fj(\000)372 473 y Fs(\()p Fo(g)t Fs(\))e(increases)i(faster)g(than)f(linearly)e(in)i Fo(g)t Fs(,)g(if)f(w)m(e)i(w)m(an)m(t)g(to)f(gather)g Fo(k)k Fs(single)34 b(site)h(p)s(oten)m(tials)118 589 y(su\016cien)m(tly)f(close)f(together)h(so)f(as)g(to)g(get)g(the)h (e\013ect)g(of)f(ha)m(ving)g Fo(k)j Fs(single)c(site)h(p)s(oten)m (tials)f(exactly)118 706 y(at)44 b(the)h(same)f(p)s(oin)m(t,)i(w)m(e)g (need)f(the)g(single)e(site)h(p)s(oten)m(tials)f(to)h(b)s(e)h(roughly)f (at)g(a)g(distance)g(less)118 828 y(than)f Fo(k)410 791 y Fj(\000)p Fn(\013)510 768 y Fg(y)589 828 y Fs(from)e(eac)m(h)j (other.)73 b(Hence,)46 b(the)d(scale)g(on)f(whic)m(h)h(w)m(e)h(w)m(an)m (t)f(the)g(P)m(oisson)g(p)s(oin)m(ts)f(to)118 944 y(concen)m(trate)37 b(is)f(m)m(uc)m(h)g(smaller)e(than)i(the)g(one)g(used)h(to)f(pro)m(v)m (e)h(the)f(upp)s(er)h(b)s(ound)f(in)f(the)h(general)118 1060 y(case.)48 b(This)34 b(leads)f(to)h(some)f(supplemen)m(tary)i(tec) m(hnical)e(di\016culties)g(as)h(the)g(single)e(site)i(p)s(oten)m(tials) 118 1176 y(ha)m(v)m(e)g(a)e(\014nite)h(non)f(zero)h(range.)331 1292 y(Recall)45 b(that)h(\()p Fo(x)957 1307 y Fn(i)986 1292 y Fs(\))1024 1307 y Fi(1)p Fj(\024)p Fn(i)p Fj(\024)p Fn(q)1277 1292 y Fs(are)h(the)g(singularities)d(of)i(the)h(single)f (site)g(p)s(oten)m(tial)f Fo(V)68 b Fs(\(see)48 b(sec-)118 1409 y(tion)32 b(1.2\).)43 b(W)-8 b(e)33 b(de\014ne)1425 1633 y Fo(K)7 b Fs(\()p Fo(x;)17 b(r)s Fs(\))27 b(=)1914 1538 y Fm([)1868 1749 y Fi(1)p Fj(\024)p Fn(i)p Fj(\024)p Fn(q)2087 1633 y Fo(C)7 b Fs(\()p Fo(x)23 b Fl(\000)f Fo(x)2434 1648 y Fn(i)2463 1633 y Fo(;)17 b(r)s Fs(\))118 1935 y(Pic)m(k)33 b Fo(")28 b(>)f Fs(0)32 b(small)f(and)h(de\014ne)i (the)f(ev)m(en)m(ts)660 2113 y(~)650 2138 y(\012)720 2153 y Fi(1)787 2138 y Fs(=)28 b Fl(f)p Fo(!)j Fl(2)d Fs(\012)1197 2097 y Fi(2)1197 2162 y Fn(E)1284 2138 y Fs(:)61 b Fl(9)p Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(K)7 b Fs(\()p Fo(x;)17 b(k)2697 2097 y Fj(\000)p Fi(\()p Fn(\013)2824 2073 y Fg(y)2857 2097 y Fj(\000)p Fi(2)p Fn(")p Fi(\))3011 2138 y Fs(\)\))28 b Fo(>)g(k)s Fl(g)p Fo(;)728 2288 y Fs(~)717 2313 y(\012)787 2328 y Fi(2)854 2313 y Fs(=)g Fl(f)p Fo(!)j Fl(2)d Fs(\012)1264 2272 y Fi(2)1264 2338 y Fn(E)1352 2313 y Fs(:)60 b Fl(9)p Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(1\)\))27 b Fo(>)h(k)3007 2272 y Fi(1+)p Fn("\027)3169 2249 y Fg(y)3205 2313 y Fl(g)p Fo(;)1663 2481 y Fs(and)1864 2456 y(~)1853 2481 y(\012)g(=)2065 2456 y(~)2055 2481 y(\012)2125 2496 y Fi(1)2187 2481 y Fl([)2286 2456 y Fs(~)2275 2481 y(\012)2345 2496 y Fi(2)118 2697 y Fs(Pic)m(k)35 b Fo(\014)i(>)31 b Fs(0)j(large.)48 b(Set)35 b Fo(n)d Fs(=)f(2[)p Fo(E)1423 2661 y Fn(\014)1470 2697 y Fs(])23 b Fl(\001)g Fs([)p Fl(j)17 b Fs(log)f Fo(E)6 b Fl(j)1891 2661 y Fn(\014)1938 2697 y Fs(])35 b(and)f Fo(l)g Fs(=)d([)p Fl(j)17 b Fs(log)f Fo(E)6 b Fl(j)2681 2661 y Fn(\014)2728 2697 y Fs(].)49 b(Pic)m(k)35 b Fo(\032)c(>)g Fs(0)k(large.)48 b(T)-8 b(aking)118 2813 y(in)m(to)44 b(accoun)m(t)h(Lemma)e(4.1,)k(Prop)s(osition)c(4.3)h(is)g (a)g(direct)h(consequence)i(of)d(the)h(follo)m(wing)d(t)m(w)m(o)118 2929 y(lemmas)31 b(\(cf)i(Lemmas)e(4.2)h(and)h(4.3\).)118 3130 y Fx(Lemma)k(4.4.)49 b Fe(F)-7 b(or)36 b(any)g Fo(")30 b(>)h Fs(0)p Fe(,)36 b(ther)-5 b(e)36 b(exists)g Fo(E)1992 3145 y Fi(0)2069 3130 y Fe(and)f Fo(k)2310 3145 y Fi(0)2386 3130 y Fe(such)h(that,)h(if)f Fo(k)e(>)c(k)3178 3145 y Fi(0)3254 3130 y Fe(and)36 b 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(y)g(as)g(Prop)s(osition)e(4.1)h(follo)m(w)m(ed)g(from)f(Lemmas)h(4.1,) g(4.2)g(and)h(4.3.)1969 5454 y Fp(31)p eop %%Page: 32 32 32 31 bop 118 241 a Fs(4.3.2.)50 b Fe(The)37 b(pr)-5 b(o)g(of)38 b(of)g(L)-5 b(emma)38 b(4.4.)48 b Fs(W)-8 b(e)36 b(are)h(going)e(to)h(pro)s(ceed)h(along)e(the)h(same)h(lines)e (as)i(in)e(the)118 369 y(pro)s(of)21 b(of)h(Lemma)f(4.2.)39 b Fo(V)1071 318 y Fi(\()p Fn(i;e)p Fi(\))1049 379 y Fn(!)r(;n)1228 369 y Fs(are)22 b(de\014ned)i(as)e(in)f(\(4.4\).)40 b(Fix)21 b Fo(')27 b Fl(2)h(C)2593 333 y Fj(1)2668 369 y Fs(\()p Fk(R)2772 333 y Fn(d)2819 369 y Fs(\))q Fl(\\)q Fo(L)2991 333 y Fi(2)2991 395 y Fn(\022)3030 369 y Fs(\()p Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\)\))22 b(normalized)118 485 y(b)m(y)34 b Fl(k)p Fo(')p Fl(k)27 b Fs(=)g(1.)44 b(Then)875 664 y Fl(h)p Fo(H)995 679 y Fn(!)r(;n;\022)1162 664 y Fo(';)17 b(')p Fl(i)27 b Fs(=)g Fl(h\000)p Fs(\001)p Fo(';)17 b(')p Fl(i)22 b Fs(+)g Fl(h)p Fo(V)2148 623 y Fi(\()p Fn(i)p Fi(\))2127 688 y Fn(!)r(;n)2240 664 y 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Fl(\000)f Fo(y)t Fs(\))p Fo(m)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\))30 b(and)j Fo(V)2322 1522 y Fi(\()p Fn(i;)p Fi(2\))2301 1588 y Fn(!)2488 1563 y Fs(=)2591 1427 y Fm(Z)2647 1653 y Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(+2)p Fn(l)q Fi(\))2987 1563 y Fo(V)3044 1578 y Fi(2)3083 1563 y Fs(\()p Fo(x)23 b Fl(\000)f Fo(y)t Fs(\))p Fo(m)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\))118 1833 y Fo(V)197 1782 y Fi(\()p Fn(i;)p Fi(2\))175 1843 y Fn(!)r(;n)363 1833 y Fs(con)m(tains)29 b(all)e(the)i(lo)s(cal)d(singularities)h(of)h Fo(V)1989 1782 y Fi(\()p Fn(i)p Fi(\))1968 1843 y Fn(!)r(;n)2109 1833 y Fs(and,)i(as)f Fo(V)2496 1848 y Fi(1)2564 1833 y Fs(is)f(exp)s(onen)m(tially)g(deca)m(ying,)i(there)118 1949 y(exists)j Fo(C)i(>)28 b Fs(0)k(suc)m(h)i(that)e(for)g Fo(!)f Fl(2)d Fs(\012)1513 1913 y Fi(2)1513 1974 y Fn(E)1606 1949 y Fs(satisfying)33 b(\(4.18\))o(,)g(w)m(e)g(ha)m(v)m(e)1568 2136 y Fl(k)p Fo(V)1696 2095 y Fi(\()p Fn(i;)p Fi(1\))1675 2161 y Fn(!)r(;n)1834 2136 y Fl(k)1884 2151 y Fj(1)1986 2136 y Fl(\024)28 b Fo(C)7 b(k)2222 2095 y Fi(1+)p Fn("\027)2384 2071 y Fg(y)2421 2136 y Fo(:)118 2312 y Fs(Hence)34 b(\(4.21\))e(giv)m (es)737 2499 y Fl(h)p Fo(H)857 2514 y Fn(!)r(;n;\022)1024 2499 y Fo(';)17 b(')p Fl(i)27 b(\025)h(h\000)p Fs(\001)p Fo(';)17 b(')p Fl(i)22 b Fs(+)g Fl(h)p Fo(V)2012 2458 y Fi(\()p Fn(i;)p Fi(1\))1991 2524 y Fn(!)r(;n)2150 2499 y Fo(';)17 b(')p Fl(i)k(\000)i Fo(E)2560 2458 y Fn(\032)2600 2499 y Fo(e)2645 2458 y Fj(\000)p Fi(\(log)14 b Fn(E)t Fi(\))2914 2434 y Fh(\014)2955 2458 y Fn(=C)3072 2499 y Fl(\000)22 b Fo(C)7 b(k)3302 2458 y Fi(1+)p Fn("\027)3464 2434 y Fg(y)3501 2499 y Fo(:)-3410 b Fs(\(4.22\))118 2675 y(Consider)33 b(no)m(w)g(a)g(p)s(erio)s(dic)e(partition)f(of)i (the)h(unit)m(y)g(of)f(the)h(cub)s(e)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))32 b(of)g(the)h(form)1662 2867 y Fx(1)1718 2883 y Fn(C)5 b Fi(\(0)p Fn(;n)p Fi(\))1957 2867 y Fs(=)2061 2773 y Fm(X)2067 2984 y Fn(\015)t Fj(2)p Fi(\000)2221 2867 y Fo(\037)2282 2826 y Fi(2)2282 2892 y Fn(\015)2327 2867 y Fo(:)118 3164 y Fs(where)30 b(the)g Fo(\037)622 3179 y Fn(\015)695 3164 y Fs(are)g(supp)s(orted)g(on)f (cells)f(of)h(size)g(roughly)f Fo(k)2344 3128 y Fj(\000)p Fn(\013)2444 3104 y Fg(y)2477 3128 y Fi(+)p Fn(\027)2571 3104 y Fg(y)2604 3128 y Fn("=)p Fi(2)2711 3164 y Fs(.)43 b(These)30 b(cells)f(are)g(cen)m(tered)i(at)118 3294 y(the)h(p)s(oin)m(ts)g(of)f(\000)d(=)f Fo(\016)t Fs(\()p Fo(k)s Fs(\))p Fk(Z)1126 3258 y Fn(d)1184 3294 y Fl(\\)21 b Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)p Fs(\))32 b(where)h Fo(\016)t Fs(\()p Fo(k)s Fs(\))27 b(=)h(1)p Fo(=)p Fs([)p Fo(k)2375 3258 y Fn(\013)2420 3234 y Fg(y)2453 3258 y Fj(\000)p Fn(\027)2547 3234 y Fg(y)2579 3258 y Fn("=)p Fi(2)2687 3294 y Fs(].)43 b(Here)32 b([)p Fl(\001)p Fs(])g(denote)g (the)h(in)m(teger)118 3410 y(part.)43 b(These)35 b(functions)d(can)h (then)g(b)s(e)g(c)m(hosen)h(so)f(that,)g(for)f(some)g Fo(C)j(>)27 b Fs(0,)33 b(w)m(e)g(ha)m(v)m(e)849 3591 y(sup)856 3673 y Fn(\015)t Fj(2)p Fi(\000)1012 3591 y Fl(kr)p Fo(\037)1206 3606 y Fn(\015)1251 3591 y Fl(k)1301 3550 y Fi(2)1301 3616 y Fj(1)1397 3591 y Fs(+)22 b(sup)1503 3673 y Fn(\015)t Fj(2)p Fi(\000)1659 3591 y Fl(k)p Fs(\001)p Fo(\037)1851 3606 y Fn(\015)1895 3591 y Fl(k)1945 3606 y Fj(1)2048 3591 y Fl(\024)28 b Fo(C)7 b(\016)t Fs(\()p Fo(k)s Fs(\))2407 3550 y Fj(\000)p Fi(2)2529 3591 y Fl(\024)28 b Fo(C)7 b(k)2765 3550 y Fi(2\()p Fn(\013)2872 3527 y Fg(y)2906 3550 y Fj(\000)p Fn(\027)3000 3527 y Fg(y)3032 3550 y Fn("=)p Fi(2\))118 3591 y Fs(\(4.23\))118 3827 y(for)32 b(some)h Fo(C)h(>)28 b Fs(1.)118 3944 y(W)-8 b(e)33 b(compute)691 4130 y Fl(h\000)p Fs(\001)p Fo(';)17 b(')p Fl(i)28 b Fs(=)1230 4035 y Fm(X)1236 4247 y Fn(\015)t Fj(2)p Fi(\000)1391 4130 y Fl(kr)p Fs(\()p Fo(\037)1623 4145 y Fn(\015)1667 4130 y Fo(')p Fs(\))p Fl(k)1819 4089 y Fi(2)1880 4130 y Fs(+)22 b Fl(hjr)p Fo(\037)2189 4145 y Fn(\015)2233 4130 y Fl(j)2261 4089 y Fi(2)2300 4130 y Fo(';)17 b(')p Fl(i)22 b(\000)g Fs(2Re\()p Fl(hr)p Fs(\()p Fo(\037)3055 4145 y Fn(\015)3099 4130 y Fo(')p Fs(\))p Fo(;)17 b(')p Fl(r)p Fo(\037)3453 4145 y Fn(\015)3497 4130 y Fl(i)p Fs(\))1127 4452 y Fl(\025)28 b Fs(\(1)22 b(+)g Fo(")p Fs(\))1540 4357 y Fm(X)1545 4568 y Fn(\015)t Fj(2)p Fi(\000)1699 4452 y Fl(kr)p Fs(\()p Fo(\037)1931 4467 y Fn(\015)1976 4452 y Fo(')p Fs(\))p Fl(k)2128 4410 y Fi(2)2189 4452 y Fs(+)2287 4311 y Fm(\022)2360 4452 y Fs(1)g Fl(\000)2541 4384 y Fs(1)p 2541 4429 49 4 v 2543 4520 a Fo(")2599 4311 y Fm(\023)2689 4357 y(X)2696 4568 y Fn(\015)t Fj(2)p Fi(\000)2833 4452 y Fl(hjr)p Fo(\037)3044 4467 y Fn(\015)3088 4452 y Fl(j)3116 4410 y Fi(2)3155 4452 y Fo(';)17 b(')p Fl(i)1127 4773 y(\025)28 b Fs(\(1)22 b Fl(\000)g Fo(")p Fs(\))1541 4678 y Fm(X)1547 4890 y Fn(\015)t Fj(2)p Fi(\000)1701 4773 y Fl(kr)p Fs(\()p Fo(\037)1933 4788 y Fn(\015)1977 4773 y Fo(')p Fs(\))p Fl(k)2129 4732 y Fi(2)2190 4773 y Fs(+)g Fo(C)2382 4633 y Fm(\022)2455 4773 y Fs(1)g Fl(\000)2636 4706 y Fs(1)p 2636 4750 V 2638 4841 a Fo(")2695 4633 y Fm(\023)2784 4773 y Fo(k)2838 4732 y Fi(2\()p Fn(\013)2945 4708 y Fg(y)2979 4732 y Fj(\000)p Fn(\027)3073 4708 y Fg(y)3106 4732 y Fn("=)p Fi(2\))3241 4773 y Fo(:)118 4494 y Fs(\(4.24\))118 5044 y(On)33 b(the)g(other)f(hand)1299 5215 y Fl(h)p Fo(V)1417 5174 y Fi(\()p Fn(i;)p Fi(1\))1395 5240 y Fn(!)r(;n)1555 5215 y Fo(';)17 b(')p Fl(i)26 b Fs(=)1896 5120 y Fm(X)1902 5332 y Fn(\015)t Fj(2)p Fi(\000)2040 5215 y Fl(h)p Fo(V)2157 5174 y Fi(\()p Fn(i;)p Fi(1\))2136 5240 y Fn(!)r(;n)2295 5215 y Fo(\037)2356 5230 y Fn(\015)2401 5215 y Fo(';)17 b(\037)2570 5230 y Fn(\015)2614 5215 y Fo(')p Fl(i)1969 5454 y Fp(32)p eop %%Page: 33 33 33 32 bop 118 249 a Fs(Set)305 224 y(~)291 249 y Fo(V)348 264 y Fn(i)411 249 y Fs(=)35 b Fo(W)614 264 y Fn(i)642 249 y Fo(V)699 264 y Fn(i)727 249 y Fs(.)57 b(As)38 b(the)f Fo(W)1224 264 y Fn(i)1253 249 y Fs('s)g(are)g(of)g(compact)g(supp)s (ort,)h(so)f(are)h(the)2905 224 y(~)2890 249 y Fo(V)2947 264 y Fn(i)2975 249 y Fs(.)57 b(Hence,)40 b(for)c(some)h Fo(R)3858 264 y Fi(0)118 365 y Fs(p)s(ositiv)m(e,)32 b(w)m(e)i(ha)m(v)m(e)603 666 y Fl(h)p Fo(V)720 625 y Fi(\()p Fn(i;)p Fi(1\))699 691 y Fn(!)r(;n)858 666 y Fo(\037)919 681 y Fn(\015)964 666 y Fo(';)17 b(\037)1133 681 y Fn(\015)1177 666 y Fo(')p Fl(i)27 b Fs(=)1645 571 y Fm(X)1520 782 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)r(;)32 b(u)p Fj(\025)p Fi(1)1411 853 y Fn(x)1451 861 y Fh(u)1491 853 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)1686 863 y Fh(j)1720 853 y Fj(2)p Fn(C)5 b Fi(\()p Fn(\015)t(;R)1962 862 y Fd(0)1997 853 y Fi(\))2024 666 y Fl(h)p Fo(\034)2105 681 y Fn(x)2145 689 y Fh(u)2186 681 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)2381 691 y Fh(j)2418 666 y Fs(\()2471 641 y(~)2456 666 y Fo(V)2513 681 y Fn(j)2550 666 y Fs(\))p Fo(\037)2649 681 y Fn(\015)2693 666 y Fo(';)17 b(\037)2862 681 y Fn(\015)2906 666 y Fo(')p Fl(i)1307 1025 y Fs(=)1713 931 y Fm(X)1588 1141 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)r(;)31 b(u)p Fj(\025)p Fi(1)1411 1242 y Fn(x)1451 1250 y Fh(u)1491 1242 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)1686 1252 y Fh(j)1720 1242 y Fj(2)p Fn(C)5 b Fi(\()p Fn(\015)t(;k)1948 1218 y Fd(2)p Fh(")p Fg(\000)p Fh(\013)2095 1197 y Fg(y)2132 1242 y Fi(\))2159 1025 y Fl(h)p Fo(\034)2240 1041 y Fn(x)2280 1049 y Fh(u)2321 1041 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)2516 1051 y Fh(j)2554 1025 y Fs(\()2607 1000 y(~)2592 1025 y Fo(V)2649 1040 y Fn(j)2685 1025 y Fs(\))p Fo(\037)2784 1040 y Fn(\015)2828 1025 y Fo(';)17 b(\037)2997 1040 y Fn(\015)3042 1025 y Fo(')p Fl(i)1538 1414 y Fs(+)2085 1319 y Fm(X)1959 1529 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)r(;)32 b(u)p Fj(\025)p Fi(1)1636 1630 y Fn(x)1676 1638 y Fh(u)1716 1630 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)1911 1640 y Fh(j)1945 1630 y Fj(2)p Fn(C)5 b Fi(\()p Fn(\015)t(;R)2187 1639 y Fd(0)2222 1630 y Fi(\))p Fj(n)p Fn(C)g Fi(\()p Fn(\015)t(;k)2465 1607 y Fd(2)p Fh(")p Fg(\000)p Fh(\013)2612 1586 y Fg(y)2650 1630 y Fi(\))2677 1414 y Fl(h)p Fo(\034)2758 1429 y Fn(x)2798 1437 y Fh(u)2839 1429 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)3034 1439 y Fh(j)3072 1414 y Fs(\()3125 1388 y(~)3110 1414 y Fo(V)3167 1429 y Fn(j)3203 1414 y Fs(\))p Fo(\037)3302 1429 y Fn(\015)3346 1414 y Fo(';)17 b(\037)3515 1429 y Fn(\015)3559 1414 y Fo(')p Fl(i)118 1136 y Fs(\(4.25\))118 1911 y(where)34 b(the)f(\()p Fo(x)661 1926 y Fn(u)706 1911 y Fs(\()p Fo(!)t Fs(\)\))885 1926 y Fn(u)p Fj(\025)p Fi(1)1052 1911 y Fs(are)g(the)g(supp)s(ort)g(of)f (the)h(P)m(oisson)g(measure)g Fo(m)p Fs(\()p Fo(!)t(;)17 b(dy)t Fs(\).)118 2040 y(F)-8 b(or)32 b Fl(j)p Fo(\015)27 b Fl(\000)22 b Fo(x)553 2055 y Fn(u)599 2040 y Fs(\()p Fo(!)t Fs(\))f Fl(\000)i Fo(x)916 2055 y Fn(j)953 2040 y Fl(j)k(\025)h Fo(k)1167 2004 y Fn(")p Fj(\000)p Fn(\013)1300 1980 y Fg(y)1369 2040 y Fs(and)33 b Fo(x)g Fs(in)f(the)h(supp)s(ort)g (of)f Fo(\037)2462 2055 y Fn(\015)2506 2040 y Fs(,)h(one)g(has)1350 2344 y Fl(j)p Fo(\034)1420 2359 y Fn(x)1460 2367 y Fh(u)1501 2359 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)1696 2369 y Fh(j)1733 2344 y Fs(\()1786 2319 y(~)1771 2344 y Fo(V)1828 2359 y Fn(j)1865 2344 y Fs(\)\()p Fo(x)p Fs(\))p Fl(j)27 b(\024)h Fo(C)7 b(k)2325 2303 y Fn("\027)2397 2279 y Fg(y)2430 2303 y Fj(\000)p Fn(\013)2530 2279 y Fg(y)2563 2303 y Fn(\027)2602 2279 y Fg(y)2639 2344 y Fo(:)118 2654 y Fs(As)33 b(w)m(e)h(assumed)f Fl(8)p Fo(x)c Fl(2)f Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)k Fs(+)h(2)p Fo(l)r Fs(\),)33 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b Fs(1\)\))27 b Fl(\024)h Fo(k)2350 2618 y Fi(1+)p Fn("\027)2512 2594 y Fg(y)2549 2654 y Fs(,)k(w)m(e)i(obtain)1071 2865 y Fm(X)946 3076 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)r(;)d(u)p Fj(\025)p Fi(1)622 3177 y Fn(x)662 3185 y Fh(u)703 3177 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)898 3187 y Fh(j)931 3177 y Fj(2)p Fn(C)5 b Fi(\()p Fn(\015)t(;R)1173 3186 y Fd(0)1208 3177 y Fi(\))p Fj(n)p Fn(C)g Fi(\()p Fn(\015)t(;k)1451 3153 y Fd(2)p Fh(")p Fg(\000)p Fh(\013)1598 3132 y Fg(y)1636 3177 y Fi(\))1664 2960 y Fl(h)p Fo(\034)1745 2975 y Fn(x)1785 2983 y Fh(u)1826 2975 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)2021 2985 y Fh(j)2058 2960 y Fs(\()2111 2935 y(~)2096 2960 y Fo(V)2153 2975 y Fn(j)2189 2960 y Fs(\))p Fo(\037)2288 2975 y Fn(\015)2333 2960 y Fo(';)17 b(\037)2502 2975 y Fn(\015)2546 2960 y Fo(')p Fl(i)27 b(\025)h(\000)p Fo(C)7 b(k)2989 2919 y Fj(\000)p Fi(2)p Fn("\027)3151 2895 y Fg(y)3184 2919 y Fi(+)p Fn(\013)3284 2895 y Fg(y)3317 2919 y Fn(\027)3356 2895 y Fg(y)3389 2919 y Fi(+1+)p Fn("\027)3606 2895 y Fg(y)2676 3343 y Fs(=)28 b Fl(\000)p Fo(C)7 b(k)2988 3302 y Fi(2)p Fn(\013)3068 3278 y Fg(y)3102 3302 y Fj(\000)p Fn("\027)3229 3278 y Fg(y)118 3143 y Fs(\(4.26\))118 3633 y(Hence,)32 b(to)e(estimate)f Fl(h)p Fo(H)1058 3648 y Fn(!)r(;n;\022)1225 3633 y Fo(';)17 b(')p Fl(i)30 b Fs(using)f(equations)i(\(4.24\))o(,)f(\(4.25\))g(and)g (\(4.26\))o(,)h(w)m(e)g(only)f(need)h(to)118 3749 y(estimate)651 4058 y Fo(A)c Fs(:=)h Fl(kr)p Fs(\()p Fo(\037)1114 4073 y Fn(\015)1158 4058 y Fo(')p Fs(\))p Fl(k)1310 4017 y Fi(2)1372 4058 y Fs(+)1563 3990 y(1)p 1480 4035 217 4 v 1480 4126 a(1)22 b Fl(\000)g Fo(")1966 3963 y Fm(X)1841 4174 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)r(;)31 b(u)p Fj(\025)p Fi(1)1722 4274 y Fn(x)1762 4282 y Fh(u)1803 4274 y Fi(\()p Fn(!)r Fi(\))p Fj(2)p Fn(K)5 b Fi(\()p Fn(\015)t(;k)2141 4251 y Fd(2)p Fh(")p Fg(\000)p Fh(\013)2288 4230 y Fg(y)2326 4274 y Fi(\))2353 4058 y Fl(h)p Fo(\034)2434 4073 y Fn(x)2474 4081 y Fh(u)2515 4073 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)2710 4083 y Fh(j)2747 4058 y Fs(\()2800 4033 y(~)2785 4058 y Fo(V)2842 4073 y Fn(j)2879 4058 y Fs(\))p Fo(\037)2978 4073 y Fn(\015)3022 4058 y Fo(';)17 b(\037)3191 4073 y Fn(\015)3235 4058 y Fo(')p Fl(i)p Fo(:)118 4552 y Fs(Therefore)34 b(w)m(e)g(notice)f(that,)g(as)h(the)f (\()p Fo(x)1613 4567 y Fn(j)1650 4552 y Fs(\))1688 4567 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)1936 4552 y Fs(are)h(distinct,)e(for)h Fo(k)j Fs(su\016cien)m(tly)e(large,)e(for)g (an)m(y)i Fo(!)t Fs(,)118 4683 y(an)m(y)e Fo(\015)k Fs(and)31 b(an)m(y)h Fo(u)p Fs(,)f(there)h(is)e(at)h(most)f(a)h(single)f(1)e Fl(\024)g Fo(j)34 b Fl(\024)28 b Fo(q)35 b Fs(suc)m(h)d(that)f Fo(x)2844 4698 y Fn(u)2889 4683 y Fs(\()p Fo(!)t Fs(\))c Fl(2)h Fo(C)7 b Fs(\()p Fo(\015)24 b Fl(\000)c Fo(x)3493 4698 y Fn(j)3530 4683 y Fo(;)d(k)3628 4647 y Fi(2)p Fn(")p Fj(\000)p Fn(\013)3796 4623 y Fg(y)3833 4683 y Fs(\).)118 4799 y(Hence,)34 b(w)m(e)g(get)e(a)h(partition)224 5216 y Fo(Q)28 b Fs(:=)g Fl(f)p Fo(x)565 5231 y Fn(u)610 5216 y Fs(;)17 b Fo(x)709 5231 y Fn(u)754 5216 y Fs(\()p Fo(!)t Fs(\))27 b Fl(2)h Fo(K)7 b Fs(\()p Fo(\015)e(;)17 b(k)1298 5175 y Fi(2)p Fn(")p Fj(\000)p Fn(\013)1466 5151 y Fg(y)1503 5216 y Fs(\))p Fl(g)27 b Fs(=)1772 5121 y Fm([)1722 5332 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)1933 5216 y Fl(f)p Fo(x)2038 5231 y Fn(u)2083 5216 y Fs(;)17 b Fo(x)2182 5231 y Fn(u)2228 5216 y Fs(\()p Fo(!)t Fs(\))26 b Fl(2)i Fo(C)7 b Fs(\()p Fo(\015)28 b Fl(\000)22 b Fo(x)2837 5231 y Fn(j)2874 5216 y Fo(;)17 b(k)2972 5175 y Fi(2)p Fn(")p Fj(\000)p Fn(\013)3140 5151 y Fg(y)3177 5216 y Fs(\))p Fl(g)27 b Fs(=:)3473 5121 y Fm([)3423 5332 y Fi(1)p Fj(\024)p Fn(j)t Fj(\024)p Fn(q)3651 5216 y Fo(Q)3728 5231 y Fn(j)3765 5216 y Fo(:)1969 5454 y Fp(33)p eop %%Page: 34 34 34 33 bop 118 252 a Fs(Set)33 b Fo(m)p Fs(\()p Fo(!)t Fs(\))27 b(=)h Fo(m)p Fs(\()p Fo(!)t(;)17 b(K)7 b Fs(\()p Fo(\015)e(;)17 b(k)1157 216 y Fi(2)p Fn(")p Fj(\000)p Fn(\013)1325 192 y Fg(y)1361 252 y Fs(\)\))32 b(and)h Fo(q)1702 267 y Fn(j)1767 252 y Fs(=)27 b Fo(]Q)1985 267 y Fn(j)2022 252 y Fs(.)43 b(W)-8 b(e)33 b(compute)695 549 y Fo(A)27 b Fs(=)h Fl(kr)p Fs(\()p Fo(\037)1131 564 y Fn(\015)1175 549 y Fo(')p Fs(\))p Fl(k)1327 508 y Fi(2)1388 549 y Fs(+)1580 482 y(1)p 1496 526 217 4 v 1496 617 a(1)22 b Fl(\000)h Fo(")1794 420 y Fn(q)1739 454 y Fm(X)1750 664 y Fn(j)t Fi(=1)1900 454 y Fm(X)1928 666 y Fn(Q)1984 676 y Fh(j)2044 549 y Fl(h)p Fo(\034)2125 564 y Fn(x)2165 572 y Fh(u)2205 564 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)2400 574 y Fh(j)2438 549 y Fs(\()2491 524 y(~)2476 549 y Fo(V)2533 564 y Fn(j)2569 549 y Fs(\))p Fo(\037)2668 564 y Fn(\015)2713 549 y Fo(';)17 b(\037)2882 564 y Fn(\015)2926 549 y Fo(')p Fl(i)795 937 y Fs(=)954 809 y Fn(q)899 842 y Fm(X)910 1052 y Fn(j)t Fi(=1)1158 870 y Fs(1)p 1069 914 226 4 v 1069 1005 a Fo(m)p Fs(\()p Fo(!)t Fs(\))1322 737 y Fm(0)1322 916 y(@)1409 937 y Fo(q)1452 952 y Fn(j)1488 937 y Fl(kr)p Fs(\()p Fo(\037)1720 952 y Fn(\015)1765 937 y Fo(')p Fs(\))p Fl(k)1917 896 y Fi(2)1978 937 y Fs(+)2086 870 y Fo(m)p Fs(\()p Fo(!)t Fs(\))p 2086 914 V 2091 1005 a(1)22 b Fl(\000)g Fo(")2338 842 y Fm(X)2366 1054 y Fn(Q)2422 1064 y Fh(j)2482 937 y Fl(h)p Fo(\034)2563 953 y Fn(x)2603 961 y Fh(u)2644 953 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)2839 963 y Fh(j)2876 937 y Fs(\()2929 912 y(~)2914 937 y Fo(V)2971 952 y Fn(j)3007 937 y Fs(\))p Fo(\037)3106 952 y Fn(\015)3151 937 y Fo(';)17 b(\037)3320 952 y Fn(\015)3364 937 y Fo(')p Fl(i)3467 737 y Fm(1)3467 916 y(A)795 1296 y Fs(=)954 1167 y Fn(q)899 1201 y Fm(X)910 1411 y Fn(j)t Fi(=1)1158 1229 y Fs(1)p 1069 1273 V 1069 1364 a Fo(m)p Fs(\()p Fo(!)t Fs(\))1322 1201 y Fm(X)1349 1413 y Fn(Q)1405 1423 y Fh(j)1482 1156 y Fm(\022)1555 1296 y Fl(kr)p Fs(\()p Fo(\037)1787 1311 y Fn(\015)1832 1296 y Fo(')p Fs(\))p Fl(k)1984 1255 y Fi(2)2045 1296 y Fs(+)2153 1229 y Fo(m)p Fs(\()p Fo(!)t Fs(\))p 2153 1273 V 2158 1364 a(1)k Fl(\000)i Fo(")2388 1296 y Fl(h)p Fo(\034)2469 1312 y Fn(x)2509 1320 y Fh(u)2550 1312 y Fi(\()p Fn(!)r Fi(\)+)p Fn(x)2745 1322 y Fh(j)2782 1296 y Fs(\()2835 1271 y(~)2820 1296 y Fo(V)2877 1311 y Fn(j)2914 1296 y Fs(\))p Fo(\037)3013 1311 y Fn(\015)3057 1296 y Fo(';)17 b(\037)3226 1311 y Fn(\015)3270 1296 y Fo(')p Fl(i)3373 1156 y Fm(\023)795 1684 y Fl(\025)901 1484 y Fm(2)901 1663 y(4)1022 1556 y Fn(q)967 1590 y Fm(X)978 1800 y Fn(j)t Fi(=1)1210 1617 y Fo(q)1253 1632 y Fn(j)p 1137 1661 V 1137 1753 a Fo(m)p Fs(\()p Fo(!)t Fs(\))1390 1590 y Fm(X)1417 1801 y Fn(Q)1473 1811 y Fh(j)1573 1659 y Fs(~)1550 1684 y Fo(E)1622 1699 y Fn(j)1675 1544 y Fm(\022)1792 1617 y Fo(m)p Fs(\()p Fo(!)t Fs(\))p 1759 1661 293 4 v 1759 1753 a(\(1)22 b Fl(\000)g Fo(")p Fs(\))2061 1544 y Fm(\023)2134 1484 y(3)2134 1663 y(5)2217 1684 y Fl(k)p Fo(\037)2328 1699 y Fn(\015)2372 1684 y Fo(')p Fl(k)2486 1643 y Fi(2)118 2032 y Fs(where)423 2007 y(~)400 2032 y Fo(E)472 2047 y Fn(j)509 2032 y Fs(\()p Fo(g)t Fs(\))31 b(is)i(the)g(ground)f(state)h(of)f Fl(\000)p Fs(\001)23 b(+)f Fo(g)1959 2007 y Fs(~)1944 2032 y Fo(V)2001 2047 y Fn(j)2037 2032 y Fs(.)44 b(In)33 b(section)f(5.4,)h(w)m(e)g(pro) m(v)m(e)h(that)f(the)g(lo)m(w)m(est)g(of)118 2148 y(these)f(ground)f (states)h(is)e(asymptotic)g(to)h Fo(E)6 b Fs(\()p Fo(g)t Fs(\))30 b(as)h Fo(g)g Fl(!)c Fs(+)p Fl(1)p Fs(.)42 b(Hence,)33 b(for)d Fo(k)k Fs(su\016cien)m(tly)d(large,)g(w)m(e)118 2264 y(ha)m(v)m(e)735 2511 y Fo(A)d Fl(\025)g(\000)p Fs(\(1)22 b(+)g Fo(")p Fs(\))p Fo(E)1381 2526 y Fj(\000)1457 2370 y Fm(\022)1540 2443 y Fo(m)p Fs(\()p Fo(!)t Fs(\))p 1540 2488 226 4 v 1545 2579 a(1)f Fl(\000)i Fo(")1775 2370 y Fm(\023)1865 2511 y Fl(k)p Fo(\037)1976 2526 y Fn(\015)2021 2511 y Fo(')p Fl(k)2135 2470 y Fi(2)2201 2511 y Fl(\025)29 b(\000)p Fs(\(1)22 b(+)g Fo(")p Fs(\))p Fo(E)2747 2526 y Fj(\000)2822 2370 y Fm(\022)2987 2443 y Fo(k)p 2906 2488 217 4 v 2906 2579 a Fs(1)g Fl(\000)g Fo(")3132 2370 y Fm(\023)3222 2511 y Fl(k)p Fo(\037)3333 2526 y Fn(\015)3377 2511 y Fo(')p Fl(k)3491 2470 y Fi(2)118 2511 y Fs(\(4.27\))118 2782 y(b)m(y)39 b(\(4.18\))o(.)54 b(Com)m(bining)35 b(this)h(with)g(equations)g(\(4.24\))o(,)h(\(4.25\))e (and)h(\(4.26\),)h(w)m(e)g(end)g(the)g(pro)s(of)e(of)118 2898 y(Lemma)c(4.4.)p 3827 3015 4 66 v 3831 2952 59 4 v 3831 3015 V 3889 3015 4 66 v 118 3295 a(4.3.3.)50 b Fe(The)30 b(pr)-5 b(o)g(of)30 b(of)h(L)-5 b(emma)30 b(4.5.)48 b Fs(W)-8 b(e)29 b(\014rst)f(pro)m(v)m(e)i(an)e(asymptotic)f(lo)m(w)m (er)i(b)s(ound)f(for)g Fo(P)14 b Fs(\()3557 3270 y(~)3546 3295 y(\012)3616 3310 y Fi(1)3655 3295 y Fs(\).)42 b(W)-8 b(e)118 3411 y(notice)32 b(that,)h(for)f Fo(E)38 b Fs(large)32 b(enough,)201 3661 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)30 b Fl(2)e Fs(\012)621 3620 y Fi(2)621 3686 y Fn(E)709 3661 y Fs(:)60 b Fl(9)p Fo(x)29 b Fl(2)f Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(K)7 b Fs(\()p Fo(x;)17 b(k)2122 3620 y Fj(\000)p Fi(\()p Fn(\013)2249 3596 y Fg(y)2282 3620 y Fj(\000)p Fi(2)p Fn(")p Fi(\))2436 3661 y Fs(\)\))28 b Fo(>)f(k)s Fl(g)p Fs(\))h Fl(\025)1112 3836 y(\025)g Fo(P)14 b Fs(\()p Fl(f)p Fo(!)30 b Fl(2)e Fs(\012)1637 3795 y Fi(2)1637 3861 y Fn(E)1725 3836 y Fs(:)60 b Fl(9)p Fo(x)28 b Fl(2)h Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)k Fs(+)h(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(x;)17 b(k)3124 3795 y Fj(\000)p Fi(\()p Fn(\013)3251 3772 y Fg(y)3284 3795 y Fj(\000)p Fi(2)p Fn(")p Fi(\))3439 3836 y Fs(\)\))27 b Fo(>)h(k)s Fl(g)p Fs(\))p Fo(:)118 4047 y Fs(Hence,)34 b(using)e(the)h(pro)s(of)f(of)g (the)h(lo)m(w)m(er)g(estimate)f(in)f(Lemma)h(4.3,)g(w)m(e)i(get)1235 4327 y(lim)17 b(inf)1244 4389 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1806 4259 y Fs(log)f Fo(P)e Fs(\()2074 4234 y(~)2063 4259 y(\012)2133 4274 y Fi(1)2173 4259 y Fs(\))p 1532 4304 954 4 v 1532 4395 a(\(1)21 b(+)h(\()p Fo(\013)1839 4366 y Fj(y)1897 4395 y Fl(\000)g Fs(2)p Fo(")p Fs(\))p Fo(d)p Fs(\))p Fo(k)d Fs(log)e Fo(k)2522 4327 y Fl(\025)29 b(\000)p Fs(1)p Fo(:)-2663 b Fs(\(4.28\))118 4613 y(Let)29 b(us)h(pro)m(v)m(e)g(the)f(asymptotic)g(upp)s(er)g(b)s(ound)g(for)g Fo(P)14 b Fs(\()2177 4588 y(~)2166 4613 y(\012)2236 4628 y Fi(1)2275 4613 y Fs(\).)42 b(W)-8 b(e)29 b(partition)e Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)e Fs(+)g(3)p Fo(l)r Fs(\))28 b(in)m(to)g(cub)s(es)118 4741 y(of)45 b(side)g(length)g Fo(k)815 4705 y Fj(\000)p Fi(\()p Fn(\013)942 4681 y Fg(y)975 4705 y Fj(\000)p Fi(2)p Fn(")p Fi(\))1130 4741 y Fs(,)j(the)e(cub)s(es)g(b)s(eing)f(indexed)h(b)m(y)g(\004)2529 4756 y Fn(k)2621 4741 y Fs(=)j Fo(k)2800 4705 y Fj(\000)p Fi(\()p Fn(\013)2927 4681 y Fg(y)2960 4705 y Fj(\000)p Fi(2)p Fn(")p Fi(\))3115 4741 y Fk(Z)3184 4705 y Fn(d)3252 4741 y Fl(\\)32 b Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)30 b Fs(+)h(3)p Fo(l)r Fs(\).)118 4868 y(F)-8 b(or)39 b(1)h Fl(\024)g Fo(i)h Fl(\024)f Fo(q)t Fs(,)i(let)d Fo(\015)1012 4883 y Fn(i)1080 4868 y Fs(b)s(e)h(the)g(pro)5 b(jection)40 b(of)f Fo(x)2039 4883 y Fn(i)2107 4868 y Fs(on)h(the)g(lattice)f Fo(k)2785 4832 y Fj(\000)p Fi(\()p Fn(\013)2912 4809 y Fg(y)2945 4832 y Fj(\000)p Fi(2)p Fn(")p Fi(\))3100 4868 y Fk(Z)3169 4832 y Fn(d)3206 4868 y Fs(.)66 b(Then,)43 b(for)c(an)m(y)118 4984 y Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\),)33 b(there)g(exists)g Fo(\015)g Fl(2)28 b Fs(\004)1620 4999 y Fn(k)1695 4984 y Fs(suc)m(h)34 b(that)1066 5216 y Fo(K)7 b Fs(\()p Fo(x;)17 b(k)1347 5175 y Fj(\000)p Fi(\()p Fn(\013)1474 5151 y Fg(y)1507 5175 y Fj(\000)p Fi(2)p Fn(")p Fi(\))1662 5216 y Fs(\))27 b Fl(\032)1879 5121 y Fm([)1833 5332 y Fi(1)p Fj(\024)p Fn(i)p Fj(\024)p Fn(q)2052 5216 y Fo(C)7 b Fs(\()p Fo(\015)27 b Fl(\000)c Fo(\015)2396 5231 y Fn(i)2424 5216 y Fo(;)17 b Fs(4)p Fo(k)2571 5175 y Fj(\000)p Fi(\()p Fn(\013)2698 5151 y Fg(y)2731 5175 y Fj(\000)p Fi(2)p Fn(")p Fi(\))2885 5216 y Fs(\))p Fo(:)1969 5454 y Fp(34)p eop %%Page: 35 35 35 34 bop 118 241 a Fs(Hence)606 393 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)30 b Fl(2)e Fs(\012)1026 352 y Fi(2)1026 418 y Fn(E)1114 393 y Fs(:)60 b Fl(9)p Fo(x)28 b Fl(2)g Fo(C)7 b Fs(\(0)p Fo(;)17 b(n)22 b Fs(+)g(2)p Fo(l)r Fs(\))p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(K)7 b Fs(\()p Fo(x;)17 b(k)2526 352 y Fj(\000)p Fi(\()p Fn(\013)2653 329 y Fg(y)2686 352 y Fj(\000)p Fi(2)p Fn(")p Fi(\))2841 393 y Fs(\)\))27 b Fo(>)h(k)s Fl(g)p Fs(\))536 644 y Fl(\024)655 550 y Fm(X)641 761 y Fn(\015)t Fj(2)p Fi(\004)775 773 y Fh(k)830 644 y Fo(P)923 474 y Fm( \()1082 644 y Fo(!)j Fl(2)d Fs(\012)1338 603 y Fi(2)1338 669 y Fn(E)1398 644 y Fs(;)49 b Fo(m)1576 474 y Fm( )1655 644 y Fo(!)t(;)1809 550 y Fm([)1764 760 y Fi(1)p Fj(\024)p Fn(i)p Fj(\024)p Fn(q)1983 644 y Fo(C)7 b Fs(\()p Fo(\015)2149 659 y Fn(i)2199 644 y Fs(+)22 b Fo(\015)5 b(;)17 b Fs(4)p Fo(k)2500 603 y Fj(\000)p Fi(\()p Fn(\013)2627 580 y Fg(y)2660 603 y Fj(\000)p Fi(2)p Fn(")p Fi(\))2814 644 y Fs(\))2852 474 y Fm(!)2958 644 y Fo(>)28 b(k)3116 474 y Fm(\)!)536 996 y Fl(\024)g Fs(\(\()p Fo(n)22 b Fs(+)g(3)p Fo(l)r Fs(\))p Fo(k)1067 954 y Fi(\()p Fn(\013)1139 931 y Fg(y)1173 954 y Fj(\000)p Fi(2)p Fn(")p Fi(\))1327 996 y Fs(\))1365 954 y Fn(d)1405 996 y Fo(P)1498 825 y Fm( \()1658 996 y Fo(!)31 b Fl(2)d Fs(\012)1914 954 y Fi(2)1914 1020 y Fn(E)2001 996 y Fs(:)61 b Fo(m)2191 825 y Fm( )2269 996 y Fo(!)t(;)2424 901 y Fm([)2378 1111 y Fi(1)p Fj(\024)p Fn(i)p Fj(\024)p Fn(q)2597 996 y Fo(C)7 b Fs(\()p Fo(\015)2763 1011 y Fn(i)2791 996 y Fo(;)17 b Fs(4)p Fo(k)2938 954 y Fj(\000)p Fi(\()p Fn(\013)3065 931 y Fg(y)3098 954 y Fj(\000)p Fi(2)p Fn(")p Fi(\))3252 996 y Fs(\))3290 825 y Fm(!)3397 996 y Fo(>)27 b(k)3554 825 y Fm(\)!)3730 996 y Fo(:)118 1270 y Fs(As)33 b Fo(k)e(>)d(E)526 1234 y Fn(\016)596 1270 y Fs(and)33 b(as)f Fo(n)h Fs(and)g Fo(l)i Fs(are)d(at)g(most)g(of)h(p)s(olynomial)c(size)j(in)g Fo(E)6 b Fs(,)33 b(this)f(giv)m(es)1220 1502 y(lim)17 b(sup)1244 1583 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1820 1434 y Fs(log)g Fo(P)d Fs(\()2089 1409 y(~)2078 1434 y(\012)2148 1449 y Fi(1)2187 1434 y Fs(\))p 1546 1479 954 4 v 1546 1570 a(\(1)22 b(+)g(\()p Fo(\013)1854 1541 y Fj(y)1911 1570 y Fl(\000)h Fs(2)p Fo(")p Fs(\))p Fo(d)p Fs(\))p Fo(k)18 b Fs(log)f Fo(k)2537 1502 y Fl(\024)28 b(\000)p Fs(1)p Fo(:)118 1728 y Fs(Com)m(bined)k(with)h(\(4.28\))o(,)g(w)m(e)g (get)1134 1898 y(log)17 b Fo(P)d Fs(\()1403 1873 y(~)1392 1898 y(\012)1462 1913 y Fi(1)1501 1898 y Fs(\))104 b Fl(\030)1556 1960 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1824 1898 y Fl(\000)p Fs(\(1)22 b(+)g(\()p Fo(\013)2209 1857 y Fj(y)2266 1898 y Fl(\000)h Fs(2)p Fo(")p Fs(\))p Fo(d)p Fs(\))p Fo(k)c Fs(log)d Fo(k)s(:)-2763 b Fs(\(4.29\))118 2099 y(On)33 b(the)g(other)f(hand,)h(Lemma)f(4.3)g(tells)f(us)i(that) 1188 2271 y(log)16 b Fo(P)e Fs(\()1456 2246 y(~)1445 2271 y(\012)1515 2286 y Fi(2)1554 2271 y Fs(\))104 b Fl(\030)1609 2333 y Fn(E)t Fj(!)p Fi(+)p Fj(1)1877 2271 y Fl(\000)p Fs(\(1)22 b(+)g Fo("\027)2261 2230 y Fj(y)2297 2271 y Fs(\))p Fo(k)2389 2230 y Fi(1+)p Fn("\027)2551 2207 y Fg(y)2604 2271 y Fs(log)17 b Fo(k)s(:)118 2495 y Fs(So)35 b(that)h Fo(P)14 b Fs(\()597 2470 y(~)586 2495 y(\012)656 2510 y Fi(2)695 2495 y Fs(\))35 b(is)g(negligible)d (with)j(resp)s(ect)i(to)e Fo(P)14 b Fs(\()2115 2470 y(~)2104 2495 y(\012)2174 2510 y Fi(1)2213 2495 y Fs(\).)52 b(In)36 b(view)g(of)42 b(\(4.29\))o(,)36 b(this)f(completes)g(the)118 2611 y(pro)s(of)d(of)g(Lemma)f(4.5.)p 3827 2728 4 66 v 3831 2665 59 4 v 3831 2728 V 3889 2728 4 66 v 1711 2938 a(5.)55 b Fr(Appendix)118 3112 y Fs(5.1.)h Fx(The)51 b(structure)f(of)h(the)g(P)m(oisson)g(p)s(oten)m(tial.)c Fs(Let)e Fo(V)69 b Fl(2)48 b Fo(L)2843 3076 y Fn(p)2883 3112 y Fs(\()p Fk(R)2987 3076 y Fn(d)3033 3112 y Fs(\))c(and)h(b)s(e)f (compactly)118 3229 y(supp)s(orted)k(b)s(e)g(a)f(p)s(oten)m(tial)e (satisfying)i(assumptions)g(H1,)k(H2)c(and)g(H3.)87 b(Let)48 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(dx)p Fs(\))46 b(b)s(e)i(a)118 3345 y(P)m(oisson)31 b(measure)h(of)e(concen)m(tration)h Fo(\026)p Fs(.)42 b(Consider)32 b(the)f(p)s(oten)m(tial)e Fo(V)2731 3360 y Fn(!)2812 3345 y Fs(de\014ned)j(b)m(y)i(\(1.1\))o(.)43 b(One)31 b(has)118 3522 y Fx(Lemma)37 b(5.1.)49 b Fe(F)-7 b(or)40 b(any)g Fo(\013)f Fl(2)g Fs(\(0)p Fo(;)17 b Fs(1\))p Fe(,)40 b Fo(p)1661 3486 y Fj(0)1723 3522 y Fo(<)e(p)i Fe(and)g Fs(0)e Fo(<)g(b)g(<)g Fs(1)p Fe(,)k(ther)-5 b(e)40 b(exists)g Fo(C)45 b(>)38 b Fs(0)p Fe(,)k(such)e(that)118 3638 y Fo(!)t Fe(-almost)34 b(sur)-5 b(ely,)1594 3816 y Fo(V)1651 3831 y Fn(!)1729 3816 y Fl(2)1832 3721 y Fm([)1823 3933 y Fn(k)r Fj(\025)p Fi(1)1969 3816 y Fs(\012)2039 3831 y Fn(k)2082 3816 y Fs(\()p Fo(\013)q(;)17 b(b;)g(p)2361 3774 y Fj(0)2384 3816 y Fs(\))118 4090 y Fe(wher)-5 b(e,)38 b(for)g Fo(k)f Fl(\025)d Fs(1)p Fe(,)k Fs(\012)972 4105 y Fn(k)1016 4090 y Fs(\()p Fo(\013)q(;)17 b(b;)g(p)1295 4054 y Fj(0)1317 4090 y Fs(\))38 b Fe(is)g(the)g(set)g(of)g(me)-5 b(asur)g(able)37 b(functions)h Fo(V)55 b Fs(:)34 b Fk(R)3105 4054 y Fn(d)3185 4090 y Fl(!)f Fk(C)64 b Fe(that)38 b(c)-5 b(an)38 b(b)-5 b(e)118 4206 y(written)35 b(in)g(the)g(form)1676 4370 y Fo(V)50 b Fs(=)27 b Fo(V)1965 4329 y Fi(\()p Fn(r)r Fi(\))2080 4370 y Fs(+)22 b Fo(V)2256 4329 y Fi(\()p Fn(i)p Fi(\))118 4549 y Fe(wher)-5 b(e)34 b Fo(V)472 4512 y Fi(\()p Fn(r)r Fi(\))600 4549 y Fe(and)g Fo(V)867 4512 y Fi(\()p Fn(i)p Fi(\))985 4549 y Fe(satisfy)214 4728 y(1.)41 b Fl(j)p Fo(V)441 4687 y Fi(\()p Fn(r)r Fi(\))534 4728 y Fs(\()p Fo(x)p Fs(\))p Fl(j)28 b(\024)g Fo(C)7 b(k)1018 4647 y Fh(p)p 967 4663 135 3 v 967 4708 a(p)p Fg(\000)p Fh(p)1079 4694 y Fg(0)1115 4728 y Fo(b)1183 4647 y Fg(\000)p Fh(p)1263 4626 y Fg(0)p 1167 4663 V 1167 4708 a Fh(p)p Fg(\000)p Fh(p)1279 4694 y Fg(0)1315 4728 y Fs(\()p Fl(j)p Fo(x)p Fl(j)1464 4687 y Fn(\013)1536 4728 y Fs(+)22 b(1\))34 b Fe(for)h Fo(x)28 b Fl(2)g Fk(R)2154 4692 y Fn(d)2201 4728 y Fe(,)214 4860 y(2.)41 b Fo(V)414 4819 y Fi(\()p Fn(i)p Fi(\))524 4860 y Fl(2)28 b Fo(L)684 4813 y Fn(p)720 4789 y Fg(0)684 4888 y Fc(lo)l(c,unif)952 4860 y Fe(and)34 b Fl(k)p Fo(V)1269 4819 y Fi(\()p Fn(i)p Fi(\))1352 4860 y Fl(k)1402 4900 y Fn(L)1450 4868 y Fh(p)1482 4854 y Fg(0)1450 4924 y Fb(lo)m(c,unif)1675 4860 y Fl(\024)28 b Fo(b)36 b Fe(\(her)-5 b(e)34 b Fl(k)22 b(\001)g(k)2279 4880 y Fn(L)2327 4847 y Fh(p)2327 4903 y Fb(lo)m(c,unif)2552 4860 y Fs(=)39 b(sup)2655 4949 y Fn(x)p Fj(2)p Ff(R)2790 4931 y Fh(d)2843 4860 y Fl(k)22 b(\001)g(k)3015 4875 y Fn(L)3063 4857 y Fh(p)3099 4875 y Fi(\()p Fn(C)5 b Fi(\()p Fn(x;)p Fi(1\)\))3362 4860 y Fe(\).)118 5064 y(Mor)-5 b(e)g(over,)35 b(ther)-5 b(e)34 b(exists)h Fo(C)g(>)27 b Fs(0)35 b Fe(such)f(that)1332 5286 y Fk(P)p Fs(\()p Fl(f)p Fo(V)1536 5301 y Fn(!)1615 5286 y Fl(62)28 b Fs(\012)1779 5301 y Fn(k)1823 5286 y Fs(\()p Fo(\013)q(;)17 b(b;)g(p)2102 5245 y Fj(0)2124 5286 y Fs(\))p Fl(g)p Fs(\))28 b Fl(\024)2393 5219 y Fs(\()p Fo(C)7 b(\026)p Fs(\))2605 5183 y Fn(k)p 2393 5263 255 4 v 2479 5355 a Fo(k)s Fs(!)2657 5286 y Fo(:)-2566 b Fs(\(5.1\))1969 5454 y Fp(35)p eop %%Page: 36 36 36 35 bop 118 241 a Fx(Pro)s(of.)43 b Fs(Fix)29 b Fo(\013)q Fs(,)i Fo(p)806 205 y Fj(0)860 241 y Fs(and)g Fo(b)g Fs(as)f(in)g(the)h(lemma.)41 b(Pic)m(k)31 b(0)c Fo(<)h(\013)2326 205 y Fj(0)2377 241 y Fo(<)f(\013)q Fs(.)43 b(As)31 b Fo(V)2812 256 y Fi(2)2882 241 y Fs(is)f(compactly)g(supp)s(orted,)118 357 y(there)43 b(exists)g Fo(r)700 372 y Fi(0)783 357 y Fo(>)h Fs(1)e(suc)m(h)i(that)e(supp)p Fo(V)1702 372 y Fi(2)1786 357 y Fl(\032)j Fo(C)7 b Fs(\(0)p Fo(;)17 b(r)2160 372 y Fi(0)2198 357 y Fs(\).)73 b(Let)42 b(\000)i(=)g Fo(r)2789 372 y Fi(0)2828 357 y Fk(Z)2897 321 y Fn(d)2935 357 y Fs(.)72 b(Pic)m(k)43 b Fo(k)k(>)d Fs(0.)72 b(De\014ne)118 477 y Fo(P)181 492 y Fn(k)r(;\013)285 474 y Fg(0)339 477 y Fs(:=)27 b Fo(P)14 b Fs(\()p Fl(f9)p Fo(\015)32 b Fl(2)d Fs(\000;)49 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b(r)1495 492 y Fi(0)1533 477 y Fs(\)\))28 b Fl(\025)g Fo(k)s Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)1946 441 y Fn(\013)1991 418 y Fg(0)2039 477 y Fs(+)22 b(1\))p Fl(g)p Fs(\))32 b(Using)i(\(4.16\))o(,)f(w)m(e)g (compute)977 674 y Fo(P)1040 689 y Fn(k)r(;\013)1144 670 y Fg(0)1198 674 y Fl(\024)1303 579 y Fm(X)1309 791 y Fn(\015)t Fj(2)p Fi(\000)1464 674 y Fo(P)14 b Fs(\()p Fl(f)p Fo(!)t Fs(;)47 b Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b(r)2259 689 y Fi(0)2298 674 y Fs(\)\))27 b Fl(\025)h Fo(k)s Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)2710 633 y Fn(\013)2755 609 y Fg(0)2804 674 y Fs(+)22 b(1\))p Fl(g)1198 1027 y(\024)1303 932 y Fm(X)1309 1143 y Fn(\015)t Fj(2)p Fi(\000)1488 959 y Fs(\()p Fo(\026r)1632 923 y Fn(d)1629 984 y Fi(0)1672 959 y Fs(\))1710 923 y Fn(k)r Fi(\()p Fj(j)p Fn(\015)t Fj(j)1856 900 y Fh(\013)1896 879 y Fg(0)1922 923 y Fi(+1\))p 1474 1004 586 4 v 1474 1095 a Fs(\()p Fo(k)s Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)1716 1066 y Fn(\013)1761 1047 y Fg(0)1809 1095 y Fs(+)22 b(1\)\)!)1198 1353 y Fl(\024)28 b Fo(C)1390 1286 y Fs(\()p Fo(\026r)1534 1250 y Fn(d)1531 1311 y Fi(0)1574 1286 y Fs(\))1612 1250 y Fn(k)p 1390 1330 265 4 v 1482 1422 a Fo(k)s Fs(!)1697 1353 y(for)k(some)g Fo(C)j(>)28 b Fs(0)p Fo(:)118 1020 y Fs(\(5.2\))118 1554 y(W)-8 b(e)33 b(write)1666 1726 y Fo(V)1723 1741 y Fn(!)1801 1726 y Fs(=)28 b Fo(V)1962 1741 y Fi(1)p Fn(;!)2089 1726 y Fs(+)22 b Fo(V)2244 1741 y Fi(2)p Fn(;!)118 1726 y Fs(\(5.3\))118 1965 y(where)34 b Fo(V)457 1980 y Fi(1)p Fn(;!)562 1965 y Fs(\()p Fo(x)p Fs(\))28 b(=)825 1829 y Fm(Z)880 2055 y Ff(R)928 2036 y Fh(d)985 1965 y Fo(V)1042 1980 y Fi(1)1081 1965 y Fs(\()p Fo(x)23 b Fl(\000)f Fo(y)t Fs(\))p Fo(dm)p Fs(\()p Fo(!)t(;)17 b(y)t Fs(\))30 b(and)j Fo(V)2036 1980 y Fi(2)p Fn(;!)2141 1965 y Fs(\()p Fo(x)p Fs(\))28 b(=)2403 1829 y Fm(Z)2459 2055 y Ff(R)2507 2036 y Fh(d)2564 1965 y Fo(V)2621 1980 y Fi(2)2660 1965 y Fs(\()p Fo(x)22 b Fl(\000)h Fo(y)t Fs(\))p Fo(dm)p Fs(\()p Fo(!)t(;)17 b(y)t Fs(\).)118 2132 y(No)m(w)33 b(assume)g(that)g Fo(!)j Fs(is)c(suc)m(h)i(that)1200 2304 y Fl(8)p Fo(\015)f Fl(2)28 b Fs(\000)p Fo(;)49 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b(r)2061 2319 y Fi(0)2100 2304 y Fs(\)\))27 b Fo(<)h(k)s Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)2511 2263 y Fn(\013)2556 2240 y Fg(0)2604 2304 y Fs(+)22 b(1\))p Fo(:)-2698 b Fs(\(5.4\))118 2482 y(and)42 b(\()p Fo(x)410 2497 y Fn(n)457 2482 y Fs(\()p Fo(!)t Fs(\)\))636 2497 y Fn(n)p Fj(2)p Ff(N)819 2482 y Fs(the)g(p)s(oin)m(ts)g(in)f(the)h(supp)s(ort)g(of)f Fo(m)p Fs(\()p Fo(!)t(;)17 b(dx)p Fs(\).)70 b(W)-8 b(e)42 b(will)e(estimate)h Fo(V)3390 2497 y Fi(1)p Fn(;!)3537 2482 y Fs(and)g Fo(V)3792 2497 y Fi(2)p Fn(;!)118 2598 y Fs(separately)-8 b(.)44 b Fo(V)669 2613 y Fi(1)p Fn(;!)807 2598 y Fs(can)32 b(b)s(e)h(estimated)f(as)h(in)f([9].)43 b(One)33 b(has)1074 2795 y Fl(j)p Fo(V)1159 2810 y Fi(1)p Fn(;!)1263 2795 y Fs(\()p Fo(x)p Fs(\))p Fl(j)28 b(\024)1602 2700 y Fm(X)1555 2920 y Fn(\014)s Fj(2)p Fn(r)1677 2929 y Fd(0)1712 2920 y Ff(Z)1762 2901 y Fh(d)1972 2700 y Fm(X)1809 2916 y Fn(x)1849 2924 y Fh(n)1891 2916 y Fi(\()p Fn(!)r Fi(\))p Fj(2)p Fn(C)5 b Fi(\()p Fn(\014)s(;r)2215 2925 y Fd(0)2251 2916 y Fi(\))2295 2795 y Fl(j)p Fo(V)2380 2810 y Fi(1)2419 2795 y Fs(\()p Fo(x)22 b Fl(\000)h Fo(x)2689 2810 y Fn(n)2736 2795 y Fs(\()p Fo(!)t Fs(\)\))p Fl(j)1450 3085 y(\024)28 b Fo(C)1695 2990 y Fm(X)1649 3210 y Fn(\014)s Fj(2)p Fn(r)1771 3219 y Fd(0)1805 3210 y Ff(Z)1856 3191 y Fh(d)1902 3085 y Fo(e)1947 3044 y Fj(\000j)p Fn(x)p Fj(\000)p Fn(\014)s Fj(j)p Fn(=C)2274 3085 y Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b(r)2765 3100 y Fi(0)2803 3085 y Fs(\)\))1450 3374 y Fl(\024)28 b Fo(C)1695 3280 y Fm(X)1649 3499 y Fn(\014)s Fj(2)p Fn(r)1771 3508 y Fd(0)1805 3499 y Ff(Z)1856 3480 y Fh(d)1902 3374 y Fo(e)1947 3333 y Fj(\000j)p Fn(x)p Fj(\000)p Fn(\014)s Fj(j)p Fn(=C)2274 3374 y Fo(k)s Fs(\()p Fl(j)p Fo(\014)6 b Fl(j)2483 3333 y Fn(\013)2528 3310 y Fg(0)2576 3374 y Fs(+)22 b(1\))1450 3652 y Fl(\024)28 b Fo(C)7 b(k)s Fs(\()p Fl(j)p Fo(x)p Fl(j)1835 3611 y Fn(\013)1880 3587 y Fg(0)1929 3652 y Fs(+)22 b(1\))p Fo(:)118 3216 y Fs(\(5.5\))118 3837 y(As)33 b Fo(V)319 3852 y Fi(2)386 3837 y Fl(2)28 b Fo(L)546 3801 y Fn(p)586 3837 y Fs(\()p Fk(R)690 3801 y Fn(d)737 3837 y Fs(\),)k(for)g Fo(K)j 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Fh(p)p 2061 4292 135 3 v 2101 4337 a(p)2133 4323 y Fg(0)2209 4357 y Fl(k)p Fo(V)2316 4372 y Fi(2)2355 4357 y Fl(k)2405 4306 y Fn(p=p)2512 4283 y Fg(0)2405 4384 y Fn(L)2453 4365 y Fh(p)2538 4357 y Fo(:)118 4186 y Fs(\(5.6\))118 4561 y(Let)k(us)g(in)m(tro)s(duce)g(a)g(notation:)42 b(tak)m(e)34 b(a)e(function)g Fo(W)47 b Fs(and)33 b(decomp)s(ose)g(it)f (in)m(to)g Fo(W)41 b Fs(=)28 b Fo(W)3496 4510 y Fi(\()p Fn(r)r Fi(\))3482 4588 y Fn(K)3611 4561 y Fs(+)22 b Fo(W)3815 4510 y Fi(\()p Fn(i)p Fi(\))3801 4588 y Fn(K)118 4677 y Fs(where)1050 4849 y Fo(W)1156 4798 y Fi(\()p Fn(r)r Fi(\))1142 4876 y Fn(K)1276 4849 y Fs(:=)28 b Fx(1)1463 4865 y Fj(j)p Fn(W)10 b Fi(\()p Fn(x)p Fi(\))p Fj(j\024)p Fn(K)1796 4849 y Fo(W)46 b Fs(and)33 b Fo(W)2230 4798 y Fi(\()p Fn(i)p Fi(\))2216 4876 y Fn(K)2340 4849 y Fs(:=)28 b Fx(1)2527 4865 y Fj(j)p Fn(W)10 b Fi(\()p Fn(x)p Fi(\))p Fj(j)p Fn(>K)2860 4849 y Fo(W)-2834 b Fs(\(5.7\))118 5027 y(Fix)31 b(a)g(sequence)i(of)e(p)s(ositiv)m(e)g(n)m(um)m(b)s(ers)h (\()p Fo(K)1757 5042 y Fn(\015)1801 5027 y Fs(\))1839 5042 y Fn(\015)t Fj(2)p Fi(\000)1975 5027 y Fs(.)43 b(As)32 b(supp)p Fo(V)2445 5042 y Fi(2)2513 5027 y Fl(\032)c Fo(C)7 b Fs(\(0)p Fo(;)17 b(r)2870 5042 y Fi(0)2909 5027 y Fs(\),)31 b Fo(V)3062 5042 y Fi(2)p Fn(;!)3198 5027 y Fs(can)h(b)s(e)f(rewritten)118 5144 y(as)1634 5316 y Fo(V)1691 5331 y Fi(2)p Fn(;!)1824 5316 y Fs(=)d Fo(V)2006 5265 y Fi(\()p Fn(i)p Fi(\))1985 5340 y(2)p Fn(;!)2112 5316 y Fs(+)22 b Fo(V)2289 5265 y Fi(\()p Fn(r)r Fi(\))2267 5340 y(2)p Fn(;!)118 5316 y Fs(\(5.8\))1969 5454 y Fp(36)p eop %%Page: 37 37 37 36 bop 118 241 a Fs(where)34 b(w)m(e)g(de\014ne)691 494 y Fo(V)770 443 y Fi(\()p Fn(i;r)r Fi(\))748 518 y(2)p Fn(;!)906 494 y Fs(\()p Fo(x)p Fs(\))28 b(:=)1196 399 y Fm(X)1202 611 y Fn(\015)t Fj(2)p Fi(\000)1356 494 y Fx(1)1412 509 y Fn(C)5 b Fi(\()p Fn(\015)t(;r)1586 518 y Fd(0)1621 509 y Fi(\))1653 494 y Fs(\()p Fo(x)p Fs(\))1801 294 y Fm(0)1801 473 y(@)2067 399 y(X)1888 615 y Fn(x)1928 623 y Fh(n)1970 615 y Fi(\()p Fn(!)r Fi(\))p Fj(2)p Fn(C)g Fi(\()p Fn(\015)t(;)p Fi(2)p Fn(r)2326 624 y Fd(0)2362 615 y Fi(\))2390 494 y Fs(\()p Fo(V)2485 509 y Fi(2)2524 494 y Fs(\))2562 443 y Fi(\()p Fn(i;r)r Fi(\))2562 521 y Fn(K)2622 529 y Fh(\015)2698 494 y Fs(\()p Fo(x)23 b Fl(\000)f Fo(x)2968 509 y Fn(n)3016 494 y Fs(\()p Fo(!)t Fs(\)\))3195 294 y Fm(1)3195 473 y(A)3298 494 y Fo(:)118 792 y Fs(Pic)m(k)27 b Fo(a)h(>)g Fs(0,)g Fo(K)700 807 y Fn(\015)772 792 y Fs(=)f Fo(a)p Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)1076 756 y Fn(\014)1133 792 y Fs(+)10 b(1\))27 b(and)g(set)h Fo(\014)33 b Fs(=)1866 749 y Fn(p)1902 725 y Fg(0)p 1866 769 58 4 v 1877 827 a Fn(p)1933 792 y Fo(\013)q Fs(,)28 b Fo(\013)2114 756 y Fj(0)2165 792 y Fs(=)2278 749 y Fn(p)p Fj(\000)p Fn(p)2405 725 y Fg(0)p 2278 769 149 4 v 2335 827 a Fn(p)2437 792 y Fo(\013)q Fs(.)41 b(Then,)29 b(for)d(some)h Fo(C)35 b(>)27 b Fs(0,)h(b)m(y)h (\(5.6\),)118 917 y(w)m(e)34 b(ha)m(v)m(e)822 1096 y Fl(k)p Fx(1)928 1111 y Fn(C)5 b Fi(\()p Fn(\015)t(;r)1102 1120 y Fd(0)1137 1111 y Fi(\))1168 1096 y Fo(V)1247 1045 y Fi(\()p Fn(i)p Fi(\))1225 1120 y(2)p Fn(;!)1330 1096 y Fl(k)1380 1122 y Fn(L)1428 1103 y Fh(p)1460 1089 y Fg(0)1518 1096 y Fl(\024)29 b Fo(C)7 b(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b Fs(2)p Fo(r)2241 1111 y Fi(0)2279 1096 y Fs(\)\)\()p Fo(a)p Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)2594 1055 y Fn(\014)2662 1096 y Fs(+)22 b(1\)\))2895 1015 y Fh(p)2927 994 y Fg(0)2949 1015 y(\000)p Fh(p)p 2895 1030 135 3 v 2935 1075 a(p)2967 1061 y Fg(0)1518 1302 y Fl(\024)29 b Fo(C)7 b(k)s(a)1816 1221 y Fh(p)1848 1200 y Fg(0)1870 1221 y(\000)p Fh(p)p 1816 1236 V 1856 1281 a(p)1888 1267 y Fg(0)1964 1302 y Fs(\()p Fl(j)p Fo(\015)e Fl(j)2114 1251 y Fn(\013)2159 1228 y Fg(0)2181 1251 y Fi(+)p Fn(\014)2289 1221 y Fh(p)2321 1200 y Fg(0)2344 1221 y(\000)p Fh(p)p 2289 1236 V 2329 1281 a(p)2361 1267 y Fg(0)2460 1302 y Fs(+)22 b(1\))27 b Fl(\024)h Fs(2)p Fo(C)7 b(k)s(a)3018 1221 y Fh(p)3050 1200 y Fg(0)3073 1221 y(\000)p Fh(p)p 3018 1236 V 3058 1281 a(p)3090 1267 y Fg(0)3167 1302 y Fo(:)118 1178 y Fs(\(5.9\))118 1459 y(and)906 1616 y Fl(k)p Fx(1)1012 1631 y Fn(C)e Fi(\()p Fn(\015)t(;r)1186 1640 y Fd(0)1221 1631 y Fi(\))1252 1616 y Fo(V)1331 1565 y Fi(\()p Fn(r)r Fi(\))1309 1640 y(2)p Fn(;!)1424 1616 y Fl(k)1474 1631 y Fj(1)1576 1616 y Fl(\024)28 b Fo(C)7 b(k)s(a)p Fs(\()p Fl(j)p Fo(\015)e Fl(j)2013 1575 y Fn(\013)2058 1551 y Fg(0)2080 1575 y Fi(+)p Fn(\014)2204 1616 y Fs(+)22 b(1\))28 b Fl(\024)g Fo(C)7 b(k)s(a)p Fs(\()p Fl(j)p Fo(\015)e Fl(j)2854 1575 y Fn(\013)2925 1616 y Fs(+)22 b(1\))-2992 b(\(5.10\))118 1833 y(Pic)m(k)34 b Fo(a)c Fs(=)f(\(2)610 1792 y Fj(\000)p Fn(d)705 1833 y Fo(b)p Fs(\))834 1752 y Fh(p)866 1731 y Fg(0)p 794 1768 V 794 1813 a Fh(p)826 1799 y Fg(0)848 1813 y(\000)p Fh(p)943 1833 y Fs(\()p Fo(C)7 b(k)s Fs(\))1199 1752 y Fh(p)1231 1731 y Fg(0)p 1159 1768 V 1159 1813 a Fh(p)p Fg(\000)p Fh(p)1271 1799 y Fg(0)1341 1833 y Fs(\(where)35 b Fo(C)40 b Fs(is)33 b(the)h(constan)m(t)h(app)s(earing)d (in)h(\(5.9\))o(\).)46 b(As)34 b Fo(r)3503 1848 y Fi(0)3572 1833 y Fo(>)29 b Fs(1,)34 b(w)m(e)118 1967 y(ha)m(v)m(e)g Fl(k)p Fo(V)471 1916 y Fi(\()p Fn(i)p Fi(\))450 1992 y(2)p Fn(;!)555 1967 y Fl(k)605 2008 y Fn(L)653 1975 y Fh(p)685 1961 y Fg(0)653 2031 y Fd(lo)r(c,unif)887 1967 y Fl(\024)28 b Fs(2)1041 1926 y Fn(d)1098 1967 y Fs(sup)1106 2049 y Fn(\015)t Fj(2)p Fi(\000)1262 1967 y Fl(k)p Fx(1)1368 1983 y Fn(C)5 b Fi(\()p Fn(\015)t(;r)1542 1992 y Fd(0)1576 1983 y Fi(\))1608 1967 y Fo(V)1686 1916 y Fi(\()p Fn(i)p Fi(\))1665 1992 y(2)p Fn(;!)1770 1967 y Fl(k)1820 1994 y Fn(L)1868 1975 y Fh(p)1900 1961 y Fg(0)1930 1967 y Fs(.)44 b(Hence,)34 b(b)m(y)h(\(5.9\))o(,)e(w)m(e)g (get)1649 2221 y Fl(k)p Fo(V)1777 2170 y Fi(\()p Fn(i)p Fi(\))1756 2246 y(2)p Fn(;!)1861 2221 y Fl(k)1911 2262 y Fn(L)1959 2229 y Fh(p)1991 2215 y Fg(0)1959 2285 y Fd(lo)r(c,unif)2193 2221 y Fl(\024)28 b Fo(b:)-2248 b Fs(\(5.11\))118 2426 y(Moreo)m(v)m(er,)34 b(b)m(y)h(\(5.10\))o(,)e(for) f(some)h Fo(C)h(>)28 b Fs(0,)k(for)g Fo(x)c Fl(2)g Fk(R)2121 2390 y Fn(d)2168 2426 y Fs(,)k(w)m(e)i(ha)m(v)m(e)1315 2630 y Fl(j)p Fo(V)1421 2579 y Fi(\()p Fn(r)r Fi(\))1400 2654 y(2)p Fn(;!)1514 2630 y Fs(\()p Fo(x)p Fs(\))p Fl(j)28 b(\024)g Fo(C)7 b(k)1998 2549 y Fh(p)p 1947 2564 V 1947 2609 a(p)p Fg(\000)p Fh(p)2059 2595 y Fg(0)2096 2630 y Fo(b)2163 2549 y Fg(\000)p Fh(p)2243 2528 y Fg(0)p 2147 2564 V 2147 2609 a Fh(p)p Fg(\000)p Fh(p)2259 2595 y Fg(0)2295 2630 y Fs(\()p Fl(j)p Fo(x)p Fl(j)2444 2589 y Fn(\013)2516 2630 y Fs(+)22 b(1\))-2583 b(\(5.12\))118 2792 y(No)m(w)45 b(putting)e(together)h(equations)g(\(5.3\),)g(\(5.5\)) o(,)g(\(5.8\),)g(\(5.11\))f(and)h(\(5.12\))o(,)j(w)m(e)e(see)g(that)f (if)e Fo(!)118 2913 y Fs(is)d(suc)m(h)i(that)e Fl(8)p Fo(\015)45 b Fl(2)40 b Fs(\000,)h Fo(m)p Fs(\()p Fo(!)t(;)17 b(C)7 b Fs(\()p Fo(\015)e(;)17 b(r)1545 2928 y Fi(0)1584 2913 y Fs(\)\))39 b Fo(<)g(k)s Fs(\()p Fl(j)p Fo(\015)5 b Fl(j)2018 2877 y Fn(\013)2063 2853 y Fg(0)2117 2913 y Fs(+)26 b(1\),)41 b(then)f Fo(V)2660 2928 y Fn(!)2750 2913 y Fl(2)g Fs(\012)2926 2928 y Fn(k)2969 2913 y Fs(\()p Fo(\013)q(;)17 b(b;)g(p)3248 2877 y Fj(0)3271 2913 y Fs(\).)64 b(T)-8 b(aking)39 b(in)m(to)118 3029 y(accoun)m(t)34 b(equation)e(\(5.2\))o(,)h(w)m(e)h(get)e(Lemma)f(5.1.)p 3827 3145 4 66 v 3831 3083 59 4 v 3831 3145 V 3889 3145 4 66 v 118 3320 a(5.2.)56 b Fx(An)38 b(a-priori)f(estimate)g(on)h(the)g (densit)m(y)g(of)g(states.)49 b Fs(Recall)32 b(that)h Fo(N)44 b Fs(denote)34 b(the)f(in)m(te-)118 3436 y(grated)g(densit)m(y) g(of)f(states)i(of)e Fo(H)1339 3451 y Fn(!)1389 3436 y Fs(.)44 b(W)-8 b(e)32 b(pro)m(v)m(e)118 3610 y Fx(Lemma)37 b(5.2.)49 b Fe(Under)29 b(assumption)f(H1,)j(H2)e(and)f(H3,)i(ther)-5 b(e)29 b(exists)g Fo(C)35 b(>)27 b Fs(0)i Fe(such)g(that,)h(for)f Fo(E)k(>)28 b Fs(1)p Fe(,)118 3726 y(we)35 b(have)1360 3907 y Fs(log)17 b Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))28 b Fl(\024)h(\000)2057 3840 y Fs(1)p 2043 3884 77 4 v 2043 3975 a Fo(C)2130 3907 y(E)2218 3830 y Fd(2)p Fh(p)p Fg(\000)p Fh(d)p 2218 3846 143 3 v 2258 3887 a Fd(2)p Fh(p)2391 3907 y Fs(log)17 b Fo(E)6 b(;)-2521 b Fs(\(5.13\))118 4094 y Fe(and,)34 b(for)h(any)g Fo(n)28 b Fl(\025)g Fs(1)34 b Fe(and)h Fo(E)e(>)28 b Fs(1)p Fe(,)35 b(we)f(have)1246 4294 y Fs(log)17 b Fk(E)12 b Fs(\()p Fo(N)1565 4309 y Fn(!)s(;n)1684 4294 y Fs(\()p Fl(\000)p Fo(E)6 b Fs(\)\))28 b Fl(\024)g(\000)2188 4226 y Fs(1)p 2173 4271 77 4 v 2173 4362 a Fo(C)2261 4294 y(E)2349 4216 y Fd(2)p Fh(p)p Fg(\000)p Fh(d)p 2349 4232 143 3 v 2389 4273 a Fd(2)p Fh(p)2522 4294 y Fs(log)16 b Fo(E)6 b(:)-2651 b Fs(\(5.14\))118 4503 y Fx(Pro)s(of.)47 b Fs(Equations)35 b(\(5.14\))e(and)h(\(5.13\))f(are)h(pro)m(v)m(ed)h (along)e(the)h(same)g(lines.)47 b(W)-8 b(e)34 b(will)d(only)j(write)118 4619 y(the)f(details)f(for)g(\(5.13\))o(.)43 b(By)33 b([7],)g(w)m(e)h(kno)m(w)f(that,)g(for)f(an)m(y)h(cub)s(e)g(\003)g(and) f Fo(E)i(>)28 b Fs(0,)k(w)m(e)i(ha)m(v)m(e)212 4960 y Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))29 b Fl(\024)f Fk(E)742 4819 y Fm(\022)949 4892 y Fs(1)p 831 4937 285 4 v 831 5028 a(V)-8 b(ol)o(\(\003\))1126 4960 y Fo(])p Fl(f)p Fs(eigen)m(v)j(alues)32 b(of)g Fo(H)1918 4919 y Fn(N)1910 4984 y(!)r(;)p Fi(\003)2061 4960 y Fs(smaller)f(than)i(or)f (equal)g(to)55 b Fl(\000)22 b Fo(E)6 b Fl(g)3371 4819 y Fm(\023)240 5223 y Fs(=)471 5156 y(1)p 353 5200 V 353 5292 a(V)-8 b(ol)o(\(\003\))664 5129 y Fm(X)667 5340 y Fn(n)p Fj(2)p Ff(N)825 5223 y Fk(E)902 5143 y Fm(\000)954 5223 y Fx(1)1010 5239 y Fj(f)p Fn(!)r Fi(;)32 b Fn(m)p Fi(\()p Fn(!)r(;)p Fi(\003\)=)p Fn(n)p Fj(g)1513 5223 y Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)32 b(of)g Fo(H)2305 5182 y Fn(N)2297 5248 y(!)r(;)p Fi(\003)2448 5223 y Fs(smaller)f(than)h (or)h(equal)f(to)54 b Fl(\000)23 b Fo(E)6 b Fl(g)3758 5143 y Fm(\001)118 4768 y Fs(\(5.15\))1969 5454 y Fp(37)p eop %%Page: 38 38 38 37 bop 118 241 a Fs(where)34 b Fo(H)489 205 y Fn(N)481 266 y(!)r(;)p Fi(\003)632 241 y Fs(denotes)g(the)f(restriction)f(of)g Fo(H)1812 256 y Fn(!)1894 241 y Fs(to)h(\003)f(with)g(Neumann)h(b)s (oundary)g(conditions.)118 357 y(One)g(has)g(the)118 571 y Fx(Lemma)k(5.3.)49 b Fe(Pick)i Fo(p)57 b(>)f(d=)p Fs(2)p Fe(.)92 b(If)50 b Fo(V)78 b Fl(2)58 b Fo(L)1927 535 y Fn(p)1967 571 y Fs(\()p Fk(R)2071 535 y Fn(d)2117 571 y Fs(\))51 b Fe(then,)j(for)c Fo(\013)58 b(>)2929 503 y(d)p 2906 548 98 4 v 2906 639 a Fs(2)p Fo(p)3064 571 y Fe(and)50 b Fo(E)63 b(>)57 b Fs(0)p Fe(,)d Fs(\()p Fo(E)40 b Fl(\000)118 753 y Fs(\001\))237 717 y Fj(\000)p Fn(\013=)p Fi(2)412 753 y Fo(V)22 b Fs(\()p Fo(E)28 b Fl(\000)23 b Fs(\001\))848 717 y Fj(\000)p Fn(\013=)p Fi(2)1058 753 y Fe(is)34 b(b)-5 b(ounde)g(d)34 b(on)h Fo(L)1739 717 y Fi(2)1779 753 y Fs(\()p Fk(R)1882 717 y Fn(d)1929 753 y Fs(\))g Fe(and)f(for)h(some)f Fo(C)2666 768 y Fn(\013;p)2798 753 y Fo(>)27 b Fs(0)p Fe(,)35 b(one)f(has)732 940 y Fl(k)p Fs(\()p Fo(E)28 b Fl(\000)23 b Fs(\001\))1139 898 y Fj(\000)p Fn(\013=)p Fi(2)1314 940 y Fo(V)e Fs(\()p Fo(E)28 b Fl(\000)23 b Fs(\001\))1749 898 y Fj(\000)p Fn(\013=)p Fi(2)1924 940 y Fl(k)1974 956 y Fj(L)p Fi(\()p Fn(L)2098 937 y Fd(2)2133 956 y Fi(\))2192 940 y Fl(\024)28 b Fo(C)2367 955 y Fn(\013;p)2471 940 y Fl(k)p Fo(V)22 b Fl(k)2650 959 y Fn(L)2698 940 y Fh(p)2734 959 y Fi(\()p Ff(R)2809 940 y Fh(d)2845 959 y Fi(\))2877 940 y Fo(E)2955 898 y Fj(\000)p Fn(\013)p Fi(+)p Fn(d=)p Fi(2)p Fn(p)3257 940 y Fo(:)-3166 b Fs(\(5.16\))118 1135 y Fx(Pro)s(of.)46 b Fs(Using)33 b(classical)e(prop)s(erties)j(of)e(the)i(F)-8 b(ourier)32 b(transform)h(from)f Fo(L)2941 1099 y Fn(p)3014 1135 y Fs(to)h Fo(L)3200 1099 y Fn(q)3272 1135 y Fs(\()p Fo(q)3357 1099 y Fj(\000)p Fi(1)3473 1135 y Fs(+)23 b Fo(p)3621 1099 y Fj(\000)p Fi(1)3744 1135 y Fs(=)29 b(1)118 1254 y(and)34 b(1)c Fl(\024)h Fo(p)f Fl(\024)g Fs(2\))k(\(see)h(e.g.)48 b([6]\),)34 b(one)g(sho)m(ws)i(\()p Fo(E)29 b Fl(\000)23 b Fs(\001\))2216 1218 y Fj(\000)p Fn(\013=)p Fi(2)2425 1254 y Fs(is)34 b(b)s(ounded)h(from)d Fo(L)3222 1218 y Fi(2)3262 1254 y Fs(\()p Fk(R)3366 1218 y Fn(d)3413 1254 y Fs(\))h(to)h Fo(L)3671 1218 y Fn(r)3709 1254 y Fs(\()p Fk(R)3813 1218 y Fn(d)3860 1254 y Fs(\))118 1419 y(for)277 1351 y(1)p 277 1396 49 4 v 278 1487 a Fo(r)364 1419 y(<)477 1351 y(\013)p 477 1396 63 4 v 484 1487 a Fs(2)572 1419 y(+)681 1351 y(1)p 680 1396 51 4 v 680 1487 a Fo(d)773 1419 y Fs(with)e(the)h(b)s(ound)1175 1667 y Fl(k)p Fs(\()p Fo(E)28 b Fl(\000)23 b Fs(\001\))1582 1626 y Fj(\000)p Fn(\013=)p Fi(2)1757 1667 y Fl(k)1807 1684 y Fn(L)1855 1665 y Fd(2)1890 1684 y Fj(!)p Fn(L)2009 1665 y Fh(r)2075 1667 y Fl(\024)28 b Fo(C)2250 1682 y Fn(\013;r)2352 1667 y Fo(E)2430 1626 y Fj(\000)2495 1599 y Fh(\013)p 2495 1611 41 3 v 2500 1652 a Fd(2)2545 1626 y Fi(+)2610 1592 y Fh(d)p Fd(\(2)p Fg(\000)p Fh(r)r Fd(\))p 2610 1611 189 3 v 2674 1652 a(4)p Fh(r)2813 1667 y Fo(:)118 1843 y Fs(One)33 b(concludes)h(using)e(dualit)m(y)f(and)i(H\177)-49 b(older's)33 b(inequalit)m(y)-8 b(.)p 3827 1960 4 66 v 3831 1897 59 4 v 3831 1960 V 3889 1960 4 66 v 118 2076 a(Then,)34 b(b)m(y)f(Lemma)f(5.3,)g(there)h(exists)g Fo(C)i(>)28 b Fs(0)k(suc)m(h)i(that,)e(for)g Fo(\036)c Fl(2)g Fo(H)2752 2040 y Fi(1)2791 2076 y Fs(\()p Fk(R)2895 2040 y Fn(d)2941 2076 y Fs(\))33 b(and)f Fo(\017)c(>)g Fs(0,)k(w)m(e)i(ha)m(v)m(e)1097 2277 y Fl(hj)p Fo(V)21 b Fl(j)p Fo(\036;)c(\036)p Fl(i)26 b(\024)i Fo(\017)p Fl(kr)p Fo(\036)p Fl(k)1880 2236 y Fi(2)1880 2305 y Fn(L)1928 2286 y Fd(2)1963 2305 y Fi(\()p Ff(R)2038 2286 y Fh(d)2074 2305 y Fi(\))2128 2277 y Fs(+)22 b Fo(C)7 b(\017)2407 2203 y Fh(d)p 2352 2215 143 3 v 2352 2256 a(d)p Fg(\000)p Fd(2)p Fh(p)2509 2277 y Fl(k)p Fo(\036)p Fl(k)2667 2236 y Fi(2)2667 2305 y Fn(L)2715 2286 y Fd(2)2749 2305 y Fi(\()p Ff(R)2824 2286 y Fh(d)2860 2305 y Fi(\))2892 2277 y Fo(:)-2801 b Fs(\(5.17\))118 2469 y(This)34 b(iden)m(tit)m(y)g (can)h(b)s(e)f(carried)f(o)m(v)m(er)j(to)d Fo(H)1765 2433 y Fi(1)1804 2469 y Fs(\(\003\))h(in)f(the)i(follo)m(wing)c(w)m(a)m (y)-8 b(.)49 b(Let)34 b Fo(\036)c Fl(2)h Fo(H)3362 2433 y Fi(1)3401 2469 y Fs(\(\003\);)j(w)m(e)h(can)118 2594 y(extend)h Fo(\036)e Fs(to)g Fk(R)716 2558 y Fn(d)797 2594 y Fs(and)g(w)m(e)i(denote)f(this)f(extension)h(b)m(y)2225 2568 y(~)2213 2594 y Fo(\036)f Fs(\(see)h([29,)f(32]\).)49 b(By)35 b(de\014nition,)3557 2568 y(~)3545 2594 y Fo(\036)3603 2610 y Fj(j)p Fi(\003)3706 2594 y Fs(=)c Fo(\036)p Fs(.)118 2711 y(Moreo)m(v)m(er)j(w)m(e)g(kno)m(w)g(that,)e(for)g(some)g Fo(C)j(>)28 b Fs(0,)841 2890 y Fl(k)904 2864 y Fs(~)891 2890 y Fo(\036)o Fl(k)998 2909 y Fn(L)1046 2891 y Fd(2)1081 2909 y Fi(\()p Ff(R)1156 2891 y Fh(d)1192 2909 y Fi(\))1251 2890 y Fl(\024)h Fo(C)1427 2905 y Fi(0)1466 2890 y Fl(k)p Fo(\036)p Fl(k)1624 2907 y Fn(L)1672 2888 y Fd(2)1706 2907 y Fi(\(\003\))1846 2890 y Fs(and)k Fl(kr)2182 2864 y Fs(~)2169 2890 y Fo(\036)p Fl(k)2277 2909 y Fn(L)2325 2891 y Fd(2)2359 2909 y Fi(\()p Ff(R)2434 2891 y Fh(d)2470 2909 y Fi(\))2530 2890 y Fl(\024)28 b Fo(C)2705 2905 y Fi(0)2744 2890 y Fl(kr)p Fo(\036)p Fl(k)2985 2907 y Fn(L)3033 2888 y Fd(2)3067 2907 y Fi(\(\003\))118 2890 y Fs(\(5.18\))118 3080 y(Pic)m(k)33 b Fo(x)391 3095 y Fi(0)459 3080 y Fl(2)28 b Fk(R)619 3043 y Fn(d)698 3080 y Fs(and)k(de\014ne)i Fo(V)1226 3095 y Fn(x)1266 3104 y Fd(0)1305 3080 y Fs(\()p Fo(x)p Fs(\))28 b(=)f Fo(V)22 b Fs(\()p Fo(x)g Fl(\000)h Fo(x)1916 3095 y Fi(0)1956 3080 y Fs(\).)43 b(Then,)34 b(for)e Fo(\036)27 b Fl(2)h Fo(H)2763 3043 y Fi(1)2802 3080 y Fs(\(\003\))k(and)h Fo(\017)28 b(>)g Fs(0,)k(b)m(y)j(\(5.17\))913 3265 y Fl(hj)p Fo(V)1037 3280 y Fn(x)1077 3289 y Fd(0)1114 3265 y Fl(j)p Fo(\036;)17 b(\036)p Fl(i)1341 3282 y Fn(L)1389 3263 y Fd(2)1423 3282 y Fi(\(\003\))1558 3265 y Fs(=)28 b Fl(hj)p Fo(V)1786 3280 y Fn(x)1826 3289 y Fd(0)1864 3265 y Fl(j)p Fx(1)1948 3280 y Fi(\003)2013 3239 y Fs(~)2000 3265 y Fo(\036;)2114 3239 y Fs(~)2102 3265 y Fo(\036)p Fl(i)2199 3284 y Fn(L)2247 3265 y Fd(2)2281 3284 y Fi(\()p Ff(R)2356 3265 y Fh(d)2392 3284 y Fi(\))1558 3449 y Fl(\024)g Fo(\017)p Fl(kr)1848 3423 y Fs(~)1835 3449 y Fo(\036)p Fl(k)1943 3408 y Fi(2)1943 3478 y Fn(L)1991 3459 y Fd(2)2026 3478 y Fi(\()p Ff(R)2101 3459 y Fh(d)2137 3478 y Fi(\))2191 3449 y Fs(+)22 b Fo(C)7 b(\017)2471 3376 y Fh(d)p 2415 3388 V 2415 3429 a(d)p Fg(\000)p Fd(2)p Fh(p)2572 3449 y Fl(k)2635 3423 y Fs(~)2622 3449 y Fo(\036)o Fl(k)2729 3408 y Fi(2)2729 3478 y Fn(L)2777 3459 y Fd(2)2812 3478 y Fi(\()p Ff(R)2887 3459 y Fh(d)2923 3478 y Fi(\))1558 3640 y Fl(\024)28 b Fo(C)1733 3655 y Fi(0)1773 3640 y Fo(\017)p Fl(kr)p Fo(\036)p Fl(k)2053 3599 y Fi(2)2053 3666 y Fn(L)2101 3647 y Fd(2)2135 3666 y Fi(\(\003\))2265 3640 y Fs(+)22 b Fo(C)2433 3655 y Fi(0)2472 3640 y Fo(C)7 b(\017)2654 3566 y Fh(d)p 2599 3578 V 2599 3619 a(d)p Fg(\000)p Fd(2)p Fh(p)2755 3640 y Fl(k)p Fo(\036)p Fl(k)2913 3599 y Fi(2)2913 3666 y Fn(L)2961 3647 y Fd(2)2995 3666 y Fi(\(\003\))118 3833 y Fs(Hence,)34 b(there)f(exists)h Fo(C)g(>)28 b Fs(0)k(suc)m(h)i(that,)f(for)f(an)m(y)h Fo(x)2090 3848 y Fi(0)2158 3833 y Fl(2)28 b Fk(R)2318 3796 y Fn(d)2364 3833 y Fs(,)33 b Fo(\017)28 b(>)f Fs(0)33 b(and)f Fo(\036)c Fl(2)g Fo(H)3134 3796 y Fi(1)3173 3833 y Fs(\(\003\),)k(w)m(e)h(ha)m(v)m(e)1022 4034 y Fl(hj)p Fo(V)1146 4049 y Fn(x)1186 4058 y Fd(0)1224 4034 y Fl(j)p Fo(\036;)17 b(\036)p Fl(i)1451 4050 y Fn(L)1499 4031 y Fd(2)1532 4050 y Fi(\(\003\))1667 4034 y Fl(\024)29 b Fo(\017)p Fl(kr)p Fo(\036)p Fl(k)2053 3992 y Fi(2)2053 4059 y Fn(L)2101 4041 y Fd(2)2135 4059 y Fi(\(\003\))2265 4034 y Fs(+)22 b Fo(C)7 b(\017)2545 3960 y Fh(d)p 2489 3972 V 2489 4013 a(d)p Fg(\000)p Fd(2)p Fh(p)2646 4034 y Fl(k)p Fo(\036)p Fl(k)2804 3992 y Fi(2)2804 4059 y Fn(L)2852 4041 y Fd(2)2886 4059 y Fi(\(\003\))118 4034 y Fs(\(5.19\))118 4223 y(No)m(w)33 b(pic)m(k)g Fo(!)j Fs(suc)m(h)e(that)e Fo(m)p Fs(\()p Fo(!)t(;)17 b Fs(\003\))27 b(=)h Fo(n)p Fs(.)43 b(By)35 b(\(5.19\))o(,)e(for)f Fo(\036)27 b Fl(2)h Fo(H)2552 4187 y Fi(1)2591 4223 y Fs(\(\003\),)k(w)m(e)i(ha)m (v)m(e)952 4424 y Fl(h)p Fo(V)1048 4439 y Fn(!)1098 4424 y Fo(\036;)17 b(\036)p Fl(i)1297 4441 y Fn(L)1345 4422 y Fd(2)1379 4441 y Fi(\(\003\))1514 4424 y Fl(\025)29 b(\000)p Fo(n\017)p Fl(kr)p Fo(\036)p Fl(k)2035 4383 y Fi(2)2035 4450 y Fn(L)2083 4431 y Fd(2)2118 4450 y Fi(\(\003\))2248 4424 y Fl(\000)22 b Fo(C)7 b(n\017)2587 4350 y Fh(d)p 2532 4362 V 2532 4404 a(d)p Fg(\000)p Fd(2)p Fh(p)2688 4424 y Fl(k)p Fo(\036)p Fl(k)2846 4383 y Fi(2)2846 4450 y Fn(L)2894 4431 y Fd(2)2928 4450 y Fi(\(\003\))3036 4424 y Fo(:)118 4666 y Fs(Pic)m(k)33 b Fo(\017)28 b Fs(=)546 4599 y(1)p 517 4643 108 4 v 517 4734 a(2)p Fo(n)666 4666 y Fs(to)33 b(get)848 4934 y Fl(h)p Fs(\()p Fl(\000)p Fs(\001)22 b(+)g Fo(V)1260 4949 y Fn(!)1311 4934 y Fs(\))p Fo(\036;)17 b(\036)p Fl(i)1548 4951 y Fn(L)1596 4932 y Fd(2)1629 4951 y Fi(\(\003\))1765 4934 y Fl(\025)1880 4867 y Fs(1)p 1880 4911 49 4 v 1880 5002 a(2)1939 4934 y Fl(kr)p Fo(\036)p Fl(k)2180 4893 y Fi(2)2180 4960 y Fn(L)2228 4941 y Fd(2)2262 4960 y Fi(\(\003\))2392 4934 y Fl(\000)22 b Fo(C)7 b(n)2676 4857 y Fd(2)p Fh(p)p 2636 4872 143 3 v 2636 4913 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)2793 4934 y Fl(k)p Fo(\036)p Fl(k)2951 4893 y Fi(2)2951 4960 y Fn(L)2999 4941 y Fd(2)3033 4960 y Fi(\(\003\))3141 4934 y Fo(:)-3050 b Fs(\(5.20\))118 5138 y(Hence)1638 5316 y Fo(H)1727 5274 y Fn(N)1719 5340 y(!)r(;)p Fi(\003)1866 5316 y Fl(\025)28 b(\000)p Fo(C)7 b(n)2234 5238 y Fd(2)p Fh(p)p 2194 5254 V 2194 5295 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)2350 5316 y Fo(:)-2259 b Fs(\(5.21\))1969 5454 y Fp(38)p eop %%Page: 39 39 39 38 bop 118 241 a Fs(As)33 b(a)g(direct)f(consequence)k(of)j (\(5.20\))o(,)32 b(w)m(e)i(get)f(that)713 428 y Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)33 b(of)f Fo(H)1506 387 y Fn(N)1498 452 y(!)r(;)p Fi(\003)1649 428 y Fs(smaller)f(than)h(or)h (equal)f(to)54 b Fl(\000)23 b Fo(E)6 b Fl(g)829 613 y(\024)28 b Fo(])p Fl(f)p Fs(eigen)m(v)-5 b(alues)32 b(of)55 b Fl(\000)22 b Fs(\001)1840 572 y Fn(N)1840 638 y Fi(\003)1940 613 y Fs(smaller)31 b(than)h(or)h(equal)f(to)g Fo(C)7 b(n)3185 536 y Fd(2)p Fh(p)p 3145 552 143 3 v 3145 593 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)3324 613 y Fl(\000)23 b Fo(E)6 b Fl(g)829 799 y(\024)28 b Fo(C)7 b Fs(V)-8 b(ol)o(\(\003\))p Fo(n)1404 722 y Fd(2)p Fh(p)p 1364 738 V 1364 779 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)118 615 y Fs(\(5.22\))118 988 y(Plugging)31 b(this)h(in)m(to)g(\(5.15\))o(,)h (for)f(some)g Fo(C)j(>)27 b Fs(0)33 b(\(dep)s(ending)f(on)h(\003\),)f (w)m(e)i(obtain)764 1224 y Fo(N)10 b Fs(\()p Fl(\000)p Fo(E)c Fs(\))29 b Fl(\024)1331 1157 y Fo(C)p 1227 1201 285 4 v 1227 1292 a Fs(V)-8 b(ol)o(\(\003\))1671 1129 y Fm(X)1538 1396 y Fn(n)p Fj(\025)p Fn(E)1701 1330 y Fd(2)p Fh(p)p Fg(\000)p Fh(d)p 1701 1345 143 3 v 1741 1379 a Fd(2)p Fh(p)1858 1396 y Fn(=C)1965 1224 y Fk(E)2042 1114 y Fm(\020)2107 1224 y Fx(1)2163 1240 y Fj(f)p Fn(!)r Fi(;)33 b Fn(m)p Fi(\()p Fn(!)r(;)p Fi(\003\)=)p Fn(n)p Fj(g)2666 1224 y Fs(V)-8 b(ol)o(\(\003\))p Fo(n)3059 1147 y Fd(2)p Fh(p)p 3019 1163 V 3019 1204 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)3175 1114 y Fm(\021)1112 1637 y Fl(\024)28 b Fo(C)1444 1542 y Fm(X)1311 1809 y Fn(n)p Fj(\025)p Fn(E)1474 1743 y Fd(2)p Fh(p)p Fg(\000)p Fh(d)p 1474 1758 V 1514 1792 a Fd(2)p Fh(p)1630 1809 y Fn(=C)1747 1569 y Fo(n)1855 1492 y Fd(2)p Fh(p)p 1815 1508 V 1815 1549 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)1972 1569 y Fs(\()p Fo(\026)p Fs(V)-8 b(ol)n(\(\003\)\))2391 1533 y Fn(n)p 1747 1614 691 4 v 2050 1705 a Fo(n)p Fs(!)1112 2076 y Fl(\024)28 b Fo(C)1304 2009 y(E)6 b Fs(\()p Fo(\026)p Fs(V)-8 b(ol)o(\(\003\)\))1802 1973 y Fn(E)1867 1907 y Fd(2)p Fh(p)p Fg(\000)p Fh(d)p 1867 1922 143 3 v 1906 1956 a Fd(2)p Fh(p)2023 1973 y Fn(=C)p 1304 2054 814 4 v 1474 2177 a Fs(\()p Fo(E)1600 2100 y Fd(2)p Fh(p)p Fg(\000)p Fh(d)p 1600 2116 143 3 v 1640 2157 a Fd(2)p Fh(p)1757 2177 y Fo(=C)7 b Fs(\)!)2127 2076 y Fo(:)118 2347 y Fs(This)33 b(ends)h(the)f(pro)s(of)e(of)h(Lemma)g(5.2.)p 3827 2463 4 66 v 3831 2401 59 4 v 3831 2463 V 3889 2463 4 66 v 118 2807 a(5.3.)56 b Fx(Exp)s(onen)m(tial)36 b(deca)m(y)i (estimates.)48 b Fs(One)33 b(has)118 2996 y Fx(Lemma)k(5.4.)49 b Fe(L)-5 b(et)44 b Fo(\013)g Fl(2)g Fs(\(0)p Fo(;)17 b Fs(1\))p Fe(,)p Fo(p)43 b(>)h(d=)p Fs(2)e Fe(and)h Fo(q)48 b(>)43 b(d=)p Fs(2)p Fe(.)70 b(Then,)44 b(ther)-5 b(e)44 b(exists)f Fo(C)3330 3011 y Fn(\013;p;q)3531 2996 y Fo(>)h Fs(0)f Fe(and)118 3112 y Fo(\017)157 3127 y Fn(\013;p;q)343 3112 y Fo(>)28 b Fs(0)34 b Fe(such)h(that,)g(for)g(any) g Fo(V)21 b Fe(,)35 b(a)g(function)f(that)i(satis\014es)1731 3301 y Fo(V)50 b Fs(=)27 b Fo(V)2020 3260 y Fn(i)2070 3301 y Fs(+)22 b Fo(V)2246 3260 y Fn(r)118 3484 y Fe(wher)-5 b(e)244 3634 y Fl(\017)41 b Fo(V)414 3593 y Fn(i)470 3634 y Fl(2)28 b Fo(L)630 3587 y Fn(p)630 3662 y Fc(lo)l(c,unif)897 3634 y Fe(and)34 b Fl(k)p Fo(V)1214 3593 y Fn(i)1243 3634 y Fl(k)1293 3654 y Fn(L)1341 3621 y Fh(p)1341 3678 y Fb(lo)m(c,unif)1566 3634 y Fl(\024)28 b Fo(b)35 b Fe(\(for)g(some)f Fo(b)28 b(>)f Fs(0)p Fe(\).)244 3765 y Fl(\017)41 b Fe(for)35 b(some)f Fo(K)h(>)27 b Fs(0)p Fe(,)35 b Fo(V)1153 3729 y Fn(r)1226 3765 y Fe(satis\014es)1219 3948 y Fl(j)27 b Fo(V)1353 3907 y Fn(r)1390 3948 y Fs(\()p Fo(x)p Fs(\))h Fl(j\024)g Fo(K)7 b Fs(\()p Fl(j)p Fo(x)p Fl(j)1921 3907 y Fn(\013)1993 3948 y Fs(+)22 b(1\))34 b Fe(for)h(al)5 b(l)35 b Fo(x)28 b Fl(2)g Fk(R)2751 3907 y Fn(d)118 4145 y Fe(Then,)34 b(ther)-5 b(e)35 b(exists)f Fo(C)981 4160 y Fn(b)1043 4145 y Fo(>)28 b Fs(0)34 b Fe(such)h(that,)g(for)g(any)g Fs(\()p Fo(\015)5 b(;)17 b(\015)2218 4109 y Fj(0)2241 4145 y Fs(\))27 b Fl(2)h Fk(Z)2469 4109 y Fn(d)2529 4145 y Fl(\002)23 b Fk(Z)2698 4109 y Fn(d)2771 4145 y Fe(and)34 b Fo(z)e Fl(2)d Fk(C)48 b Fl(n)22 b Fk(R)5 b Fe(,)40 b(one)35 b(has)118 4416 y Fs(\(5.23\))82 b Fl(k)p Fx(1)556 4432 y Fn(C)5 b Fi(\()p Fn(\015)t(;)p Fi(1\))765 4416 y Fs(\()p Fo(z)27 b Fl(\000)c Fs(\()p Fl(\000)p Fs(\001)g(+)f Fo(V)f Fs(\)\))1446 4375 y Fj(\000)p Fi(1)1541 4416 y Fo(\037)1602 4432 y Fn(C)5 b Fi(\()p Fn(\015)1724 4413 y Fg(0)1747 4432 y Fn(;)p Fi(1\))1833 4416 y Fl(k)1883 4431 y Fj(T)1922 4439 y Fh(q)1988 4416 y Fl(\024)28 b Fo(C)2163 4431 y Fn(\013;p;q)2331 4349 y Fs(\(1)22 b(+)g Fl(j)p Fo(\015)2622 4313 y Fj(0)2645 4349 y Fl(j)p Fs(\))2711 4313 y Fn(\013)p 2331 4394 429 4 v 2351 4485 a Fo(\021)t Fs(\()p Fo(z)t(;)17 b(N)5 b(;)17 b(b)p Fs(\))2770 4416 y Fo(e)2815 4375 y Fj(\000)p Fn(\017)2899 4383 y Fh(\013;p;q)3043 4375 y Fj(\001)p Fn(\021)r Fi(\()p Fn(z)s(;K)q(;b)p Fi(\))p Fj(j)p Fn(\016)3372 4383 y Fh(\013)3416 4375 y Fi(\()p Fn(\015)t Fi(\))p Fj(\000)p Fn(\016)3596 4383 y Fh(\013)3641 4375 y Fi(\()p Fn(\015)3708 4352 y Fg(0)3731 4375 y Fi(\))p Fj(j)1246 4689 y Fe(wher)-5 b(e)35 b Fo(\016)1565 4704 y Fn(\013)1614 4689 y Fs(\()p Fo(x)p Fs(\))28 b(=)g(\(1)22 b(+)g Fo(x)2139 4648 y Fi(2)2179 4689 y Fs(\))2217 4648 y Fi(\(1)p Fj(\000)p Fn(\013)p Fi(\))p Fn(=)p Fi(2)2516 4689 y Fe(and)35 b Fo(\021)t Fs(\()p Fo(z)t(;)17 b(K)r(;)g(b)p Fs(\))27 b(=)3397 4622 y Fl(j)p Fe(Im)n Fo(z)t Fl(j)p 3238 4666 540 4 v 3238 4758 a(j)p Fo(z)t Fl(j)22 b Fs(+)g Fo(K)29 b Fs(+)22 b Fo(C)3743 4773 y Fn(b)3788 4689 y Fo(:)118 4933 y Fe(Her)-5 b(e)30 b Fl(k)11 b(\001)g(k)496 4948 y Fj(T)535 4956 y Fh(q)601 4933 y Fe(denotes)29 b(the)h(norm)f(in)g(the)h Fo(q)t Fe(-th)g(Schatten)f(class)g(and)g (dist)p Fs(\()p Fo(z)t(;)17 b(z)2962 4896 y Fj(0)2986 4933 y Fs(\))30 b Fe(denotes)f(the)g(distanc)-5 b(e)118 5049 y(in)35 b Fk(C)20 b Fe(.)331 5238 y Fs(Up)51 b(to)f(small)f(mo)s (di\014cations,)k(the)f(pro)s(of)e(of)g(this)g(result)h(is)g(the)g (same)g(as)g(the)g(pro)s(of)f(of)118 5355 y(Lemma)31 b(4.1)h(in)g([9].)1969 5454 y Fp(39)p eop %%Page: 40 40 40 39 bop 118 241 a Fs(5.4.)56 b Fx(Some)45 b(useful)g(facts)h(ab)s (out)g(the)f(single)f(site)h(p)s(oten)m(tial)f(Hamiltonian)e Fo(H)3516 256 y Fn(g)3556 241 y Fx(.)49 b Fs(Recall)118 357 y(that,)e(for)d Fo(g)50 b Fl(2)e Fk(R)5 b Fs(,)53 b(in)44 b(section)g(1.2,)j(w)m(e)e(de\014ned)h Fo(H)8 b Fs(\()p Fo(g)t Fs(\))46 b(=)h Fl(\000)p Fs(\001)31 b(+)f Fo(g)t(V)65 b Fs(and)44 b Fl(\000)p Fo(E)3257 372 y Fj(\000)3317 357 y Fs(\()p Fo(g)t Fs(\))f(to)h(b)s(e)g(the)118 473 y(in\014m)m(um)e(of)i(the)g(sp)s(ectrum)g(of)f Fo(H)8 b Fs(\()p Fo(g)t Fs(\).)76 b(It)44 b(is)f(w)m(ell)g(kno)m(wn)i(that,)i (for)c Fo(g)k Fs(large)c(enough,)k Fl(\000)p Fo(E)3729 488 y Fj(\000)3832 473 y Fs(is)118 589 y(a)41 b(simple)f(eigen)m(v)-5 b(alue)40 b(\([23,)h(26]\);)k(hence)e(it)d(is)h(analytic)e(in)i Fo(g)t Fs(.)68 b(Moreo)m(v)m(er)43 b(it)d(is)h(con)m(v)m(ex)i(\(b)m(y)f (the)118 706 y(v)-5 b(ariational)33 b(principle\).)53 b(Its)37 b(\014rst)f(and)h(second)g(deriv)-5 b(ativ)m(e)36 b(are)g(p)s(ositiv)m(e.)54 b(Let)37 b Fo(')3229 721 y Fn(g)3305 706 y Fs(b)s(e)f(the)h(unique)118 822 y(p)s(ositiv)m(e)k (normalized)e(ground)j(state)g(asso)s(ciated)f(to)g Fl(\000)p Fo(E)2333 837 y Fj(\000)2393 822 y Fs(\()p Fo(g)t Fs(\).)69 b(Then)43 b(the)f(eigen)m(v)-5 b(alue)41 b(equation)118 938 y(giv)m(es)1309 1119 y Fl(kr)p Fo(')1506 1134 y Fn(g)1546 1119 y Fl(k)1596 1078 y Fi(2)1657 1119 y Fs(+)22 b Fo(g)t Fl(h)p Fo(V)f(')1987 1134 y Fn(g)2027 1119 y Fo(;)c(')2135 1134 y Fn(g)2175 1119 y Fl(i)27 b Fs(=)h Fl(\000)p Fo(E)2494 1134 y Fj(\000)2553 1119 y Fs(\()p Fo(g)t Fs(\))p Fo(:)118 1306 y Fs(Hence)34 b Fo(E)486 1269 y Fj(0)480 1330 y(\000)539 1306 y Fs(\()p Fo(g)t Fs(\))27 b(=)h Fl(\000h)p Fo(V)21 b(')1055 1321 y Fn(g)1095 1306 y Fo(;)c(')1203 1321 y Fn(g)1243 1306 y Fl(i)p Fs(;)33 b(so)f(that)h Fo(E)1745 1321 y Fj(\000)1804 1306 y Fs(\()p Fo(g)t Fs(\))f(satis\014es)1423 1494 y Fl(kr)p Fo(')1620 1509 y Fn(g)1660 1494 y Fl(k)1710 1453 y Fi(2)1771 1494 y Fs(+)22 b Fo(E)1941 1509 y Fj(\000)2001 1494 y Fs(\()p Fo(g)t Fs(\))k(=)i Fo(g)t(E)2387 1453 y Fj(0)2381 1519 y(\000)2439 1494 y Fs(\()p Fo(g)t Fs(\))p Fo(:)-2475 b Fs(\(5.24\))118 1675 y(Let)33 b Fo(g)j Fs(b)s(e)c(an)h(in) m(v)m(erse)h(of)e Fo(E)1150 1690 y Fj(\000)1242 1675 y Fs(in)g(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h(+)p Fl(1)p Fs(.)43 b(Then,)34 b(one)f(has)g(the)g(follo)m(wing)118 1863 y Fx(Lemma)k(5.5.)49 b Fe(L)-5 b(et)38 b Fo(V)53 b Fl(2)32 b Fo(L)1183 1827 y Fn(p)1223 1863 y Fs(\()p Fk(R)1327 1827 y Fn(d)1373 1863 y Fs(\))37 b Fe(wher)-5 b(e)37 b Fo(p)g Fe(is)g(chosen)f(as)g(in)h(assumption)f(2.)51 b(Then,)37 b(ther)-5 b(e)37 b(exists)118 1979 y Fo(C)e(>)27 b Fs(0)35 b Fe(such)g(that,)g(for)f Fo(g)39 b Fe(and)34 b Fo(E)41 b Fe(su\016ciently)35 b(lar)-5 b(ge,)34 b(one)g(has)1536 2136 y Fs(1)p 1522 2180 77 4 v 1522 2272 a Fo(C)1609 2203 y(g)d Fl(\024)d Fo(E)1864 2218 y Fj(\000)1924 2203 y Fs(\()p Fo(g)t Fs(\))e Fl(\024)j Fo(C)7 b(g)2360 2126 y Fd(2)p Fh(p)p 2320 2142 143 3 v 2320 2183 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)2476 2203 y Fo(:)-2385 b Fs(\(5.25\))1019 2384 y Fo(E)1091 2399 y Fj(\000)1151 2384 y Fs(\()p Fo(g)t Fs(\))p 1019 2428 258 4 v 1123 2519 a Fo(g)1322 2451 y Fe(is)34 b(b)-5 b(ounde)g(d)34 b(if)h(and)f(only)h(if)g Fo(V)2445 2466 y Fj(\000)2538 2451 y Fe(is)g(b)-5 b(ounde)g(d)o Fo(:)-2888 b Fs(\(5.26\))1082 2651 y Fe(In)34 b(this)h(c)-5 b(ase,)34 b(we)h(have:)44 b Fo(E)2116 2666 y Fj(\000)2175 2651 y Fs(\()p Fo(g)t Fs(\))93 b Fl(\030)2319 2711 y Fn(g)r Fj(!)p Fi(+)p Fj(1)2566 2651 y Fl(k)p Fo(V)2673 2666 y Fj(\000)2732 2651 y Fl(k)2782 2666 y Fj(1)2856 2651 y Fo(g)t(:)686 2906 y Fl(9)p Fo(R)29 b(>)e Fs(0)35 b Fe(such)g(that)1468 2771 y Fm(Z)1524 2996 y Fj(j)p Fn(x)p Fj(j)p Fn(>R)1732 2906 y Fl(j)p Fo(')1824 2921 y Fn(g)1864 2906 y Fl(j)1892 2865 y Fi(2)1931 2906 y Fo(dx)22 b Fs(+)2157 2771 y Fm(Z)2212 2996 y Fj(j)p Fn(x)p Fj(j)p Fn(>R)2420 2906 y Fl(jr)p Fo(')2595 2921 y Fn(g)2635 2906 y Fl(j)2663 2865 y Fi(2)2702 2906 y Fo(dx)28 b Fl(!)f Fs(0)35 b Fe(as)g Fo(g)c Fl(!)c Fs(+)p Fl(1)p Fo(:)-3462 b Fs(\(5.27\))1618 3160 y Fl(kr)p Fo(')1815 3175 y Fn(g)1855 3160 y Fl(k)1905 3119 y Fi(2)1972 3160 y Fl(\024)28 b Fo(C)7 b(g)2254 3083 y Fd(2)p Fh(p)p 2214 3098 143 3 v 2214 3139 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)2371 3160 y Fo(:)-2280 b Fs(\(5.28\))1535 3300 y(1)p 1521 3345 77 4 v 1521 3436 a Fo(C)1608 3368 y(E)1696 3290 y Fd(2)p Fh(p)p Fg(\000)p Fh(d)p 1696 3306 143 3 v 1736 3347 a Fd(2)p Fh(p)1880 3368 y Fl(\024)28 b Fo(g)t Fs(\()p Fo(E)6 b Fs(\))27 b Fl(\024)h Fo(C)7 b(E)f(:)-2386 b Fs(\(5.29\))867 3543 y Fl(9)p Fo(C)35 b(>)27 b Fs(0)p Fo(;)52 b(E)1330 3558 y Fi(0)1397 3543 y Fo(>)28 b Fs(0)34 b Fe(such)h(that)g Fl(8)p 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Fo(E)3088 205 y Fj(0)3082 265 y(\000)3182 241 y Fs(is)f(increasing,)i(for)118 357 y Fo(")27 b(>)h Fs(0,)k(w)m(e)i(ha) m(v)m(e)1049 520 y Fo(E)1121 535 y Fj(\000)1180 520 y Fs(\()p Fo(g)t Fs(\(1)21 b(+)h Fo(")p Fs(\)\))g Fl(\000)h Fo(E)1791 535 y Fj(\000)1850 520 y Fs(\()p Fo(g)t Fs(\))k Fl(\025)h Fo("g)t(E)2284 478 y Fj(0)2278 544 y(\000)2336 520 y Fs(\()p Fo(g)t Fs(\))f Fl(\025)h Fo(")p Fl(kr)p Fo(')2838 535 y Fn(g)2878 520 y Fl(k)2928 478 y Fi(2)118 688 y Fs(b)m(y)34 b(equation)e(\(5.24\))o(.)44 b(\(5.28\))31 b(is)i(then)g(an)f(immediate)e(consequence)36 b(of)j(\(5.25\))o(.)118 804 y(Let)33 b(us)h(no)m(w)g(pro)m(v)m(e)g(\(5.27\))o(.)45 b(W)-8 b(e)34 b(will)d(distinguish)g(t)m(w)m(o)j(cases)h(when)f Fo(V)2798 819 y Fj(\000)2890 804 y Fs(is)e(b)s(ounded)i(and)f(when)i (it)118 920 y(is)29 b(not.)42 b(Let)29 b(us)h(start)f(with)f(assuming)h Fo(V)1642 935 y Fj(\000)1729 920 y Fs(b)s(ounded)h(and)f(let)g Fo(v)2495 935 y Fj(\000)2583 920 y Fs(b)s(e)g(its)g(essen)m(tial)g (in\014m)m(um.)40 b(Then,)118 1037 y(b)m(y)35 b(\(5.26\))o(,)e(as)g Fo(g)e Fl(!)c(1)p Fs(,)32 b(w)m(e)i(ha)m(v)m(e)1197 1252 y Fl(kr)p Fo(')1394 1267 y Fn(g)1434 1252 y Fl(k)1484 1211 y Fi(2)1546 1252 y Fs(+)22 b Fo(g)1711 1116 y Fm(Z)1827 1252 y Fo(V)f Fl(j)p Fo(')1997 1267 y Fn(g)2037 1252 y Fl(j)2065 1211 y Fi(2)2104 1252 y Fo(dx)28 b Fs(=)g Fo(g)t(v)2440 1267 y Fj(\000)2520 1252 y Fs(+)22 b Fo(o)p Fs(\()p Fo(g)t Fs(\))p Fo(:)118 1460 y Fs(Hence,)1358 1670 y(0)27 b Fl(\024)1539 1534 y Fm(Z)1639 1670 y Fs(\()p Fo(V)44 b Fl(\000)22 b Fo(v)1924 1685 y Fj(\000)1984 1670 y Fs(\))p Fl(j)p Fo(')2114 1685 y Fn(g)2153 1670 y Fl(j)2181 1629 y Fi(2)2220 1670 y Fo(dx)28 b Fl(\024)g Fo(o)p Fs(\(1\))p Fo(:)-2540 b Fs(\(5.31\))118 1885 y(By)37 b(assumptions)g(H1)f(and)h(H2,)g(there)g(exists)h Fo(\016)g(>)c Fs(0)j(suc)m(h)h(that)e(for)g Fl(j)p Fo(x)p Fl(j)e(\025)h Fs(1)p Fo(=\016)t Fs(,)i Fo(V)22 b Fs(\()p Fo(x)p Fs(\))j Fl(\000)g Fo(v)3618 1900 y Fj(\000)3712 1885 y Fl(\025)35 b Fo(\016)t Fs(.)118 2001 y(Then,)f(as)f Fo(V)43 b Fl(\000)23 b Fo(v)767 2016 y Fj(\000)859 2001 y Fs(is)32 b(non)g(negativ)m(e,)h (equation)g(\(5.31\))f(tells)f(us)i(that)1533 2081 y Fm(Z)1588 2306 y Fj(j)p Fn(x)p Fj(j\025)p Fi(1)p Fn(=\016)1847 2216 y Fl(j)p Fo(')1939 2231 y Fn(g)1978 2216 y Fl(j)2006 2175 y Fi(2)2046 2216 y Fo(dx)27 b Fl(\024)i Fo(o)p Fs(\(1\))p Fo(:)118 2452 y Fs(If)c Fo(V)47 b Fs(is)25 b(not)g(b)s(ounded)h(from)e (b)s(elo)m(w,)j(let)d Fo(\037)i Fs(b)s(e)f(a)g Fl(C)1958 2416 y Fj(1)1952 2477 y Fi(0)2058 2452 y Fs(cut-o\013)g(for)g(the)g (cub)s(e)h Fo(C)7 b Fs(\(0)p Fo(;)17 b(R)q Fs(\).)41 b(The)26 b(eigen)m(v)-5 b(alue)118 2568 y(equation)32 b(for)g Fo(')729 2583 y Fn(g)802 2568 y Fs(giv)m(es)h(us)690 2731 y(\()p Fl(\000)p Fs(\001)22 b Fl(\000)h Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\)[\(1)21 b Fl(\000)i Fo(\037)p Fs(\))p Fo(')1649 2746 y Fn(g)1689 2731 y Fs(])28 b(=)f Fo(g)t(V)21 b Fs(\(1)h Fl(\000)g Fo(\037)p Fs(\))p Fo(')2347 2746 y Fn(g)2410 2731 y Fs(+)g(2)p Fl(r)p Fo(\037)g Fl(\001)g(r)p Fo(')2920 2746 y Fn(g)2982 2731 y Fs(+)g(\001)p Fo(\037')3286 2746 y Fn(g)118 2894 y Fs(So)33 b(that)1480 3056 y(\(1)22 b Fl(\000)h Fo(\037)p Fs(\))p Fo(')1852 3071 y Fn(g)1920 3056 y Fs(=)k Fo(a)2074 3071 y Fi(1)2136 3056 y Fs(+)22 b Fo(a)2285 3071 y Fi(2)2347 3056 y Fs(+)g Fo(a)2496 3071 y Fi(3)118 3219 y Fs(where)1250 3381 y Fo(a)1301 3396 y Fi(1)1369 3381 y Fs(=)27 b Fo(g)t Fs(\()p Fl(\000)p Fs(\001)22 b Fl(\000)h Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))2084 3340 y Fj(\000)p Fi(1)2177 3381 y Fs([)p Fo(V)22 b Fs(\(1)g Fl(\000)h Fo(\037)p Fs(\))p Fo(')2655 3396 y Fn(g)2695 3381 y Fs(])p Fo(;)1334 3542 y(a)1385 3557 y Fi(2)1452 3542 y Fs(=)28 b(2\(\001)22 b(+)g Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))2087 3501 y Fj(\000)p Fi(1)2181 3542 y Fs([)p Fl(r)p Fo(\037)22 b Fl(\001)g(r)p Fo(')2571 3557 y Fn(g)2611 3542 y Fs(])p Fo(;)1445 3702 y(a)1496 3717 y Fi(3)1563 3702 y Fs(=)28 b(\(\001)22 b(+)g Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))2149 3661 y Fj(\000)p Fi(1)2243 3702 y Fs([\001)p Fo(\037')2476 3717 y Fn(g)2516 3702 y Fs(])p Fo(:)118 3870 y Fs(F)-8 b(or)35 b Fo(R)h Fs(large)e(enough,)j Fo(V)56 b Fs(is)35 b(b)s(ounded)h(on)g(the)f(supp)s(ort)h(of)f(1)24 b Fl(\000)g Fo(\037)p Fs(;)37 b(this)e(and)g(\(5.26\))g(implies)e(that) 118 3987 y Fl(k)p Fo(a)219 4002 y Fi(1)259 3987 y Fl(k)22 b Fs(+)g Fl(k)p Fo(a)530 4002 y Fi(3)569 3987 y Fl(k)27 b(!)h Fs(0)k(as)h Fo(g)e Fl(!)c Fs(+)p Fl(1)p Fs(.)43 b(W)-8 b(e)33 b(write)847 4151 y Fo(a)898 4166 y Fi(2)965 4151 y Fs(=)28 b(2)1135 4070 y Fm(\000)1180 4151 y Fs(\(\001)22 b(+)g Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))1662 4110 y Fj(\000)p Fi(1)1756 4151 y Fs([\001)p Fo(;)17 b Fl(r)p Fo(\037)p Fs(])22 b(+)g Fl(r)p Fo(\037)2343 4070 y Fm(\001)2406 4151 y Fs(\(\001)g(+)g Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))2888 4110 y Fj(\000)p Fi(1)2982 4151 y Fl(r)p Fo(')3129 4166 y Fn(g)118 4323 y Fs(As)30 b(\(\001)15 b(+)g Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))727 4287 y Fj(\000)p Fi(1)821 4323 y Fl(r)28 b(!)f Fs(0)i(in)f Fo(L)1313 4287 y Fi(2)1353 4323 y Fs(-op)s(erator)g(norm)g(when)j Fo(g)g Fl(!)c Fs(0,)j(w)m(e)g(ha)m(v)m(e)g Fl(k)p Fo(a)3051 4338 y Fi(2)3091 4323 y Fl(k)d(!)h Fs(0)g(as)i Fo(g)h Fl(!)c Fs(+)p Fl(1)p Fs(.)118 4440 y(Hence)34 b Fl(k)p Fs(\(1)22 b Fl(\000)g Fo(\037)p Fs(\))p Fo(')829 4455 y Fn(g)869 4440 y Fl(k)28 b(!)f Fs(0)32 b(as)h Fo(g)e Fl(!)c Fs(+)p Fl(1)p Fs(.)331 4556 y(W)-8 b(e)33 b(compute)448 4714 y Fl(kr)p Fs([\(1)22 b Fl(\000)g Fo(\037)p Fs(\))p Fo(')979 4729 y Fn(g)1019 4714 y Fs(])p Fl(k)1096 4673 y Fi(2)1163 4714 y Fs(=)28 b(2)p Fl(h)p Fs(\(1)21 b Fl(\000)i Fo(\037)p Fs(\))p Fl(r)p Fo(')1809 4729 y Fn(g)1849 4714 y Fo(;)17 b(')1957 4729 y Fn(g)1997 4714 y Fl(r)p Fo(\037)p Fl(i)22 b Fs(+)g Fl(h)p Fs(\001)p Fo(\037')2545 4729 y Fn(g)2585 4714 y Fo(;)17 b(\037')2754 4729 y Fn(g)2794 4714 y Fl(i)1163 4874 y Fs(=)28 b(2)p Fl(hr)p Fs([\(1)21 b Fl(\000)i Fo(\037)p Fs(\))p Fo(')1836 4889 y Fn(g)1876 4874 y Fs(])p Fo(;)17 b(')2011 4889 y Fn(g)2051 4874 y Fl(r)p Fo(\037)p Fl(i)22 b(\000)h Fs(2)p Fl(h)p Fs(\()p Fl(r)p Fo(\037)p Fs(\))2664 4833 y Fi(2)2703 4874 y Fo(')2767 4889 y Fn(g)2807 4874 y Fo(;)17 b(')2915 4889 y Fn(g)2954 4874 y Fl(r)p Fo(\037)p Fl(i)22 b Fs(+)g Fl(h)p Fs(\001)p Fo(\037')3502 4889 y Fn(g)3542 4874 y Fo(;)17 b(\037')3711 4889 y Fn(g)3751 4874 y Fl(i)p Fo(:)118 5037 y Fs(F)-8 b(or)32 b Fo(R)i Fs(su\016cien)m(tly)f(large,)f(b)m(y)h(what)g(w)m(e)h(ha)m(v)m(e)g (just)f(pro)m(v)m(ed,)i(the)e(last)f(t)m(w)m(o)h(terms)g(in)f(the)h (equation)118 5153 y(ab)s(o)m(v)m(e)h(tend)f(to)f(0)g(as)h Fo(g)e Fl(!)c Fs(+)p Fl(1)p Fs(.)43 b(Using)32 b(the)h(Cauc)m(h)m(y-Sc) m(h)m(w)m(artz)k(inequalit)m(y)-8 b(,)31 b(this)h(implies)929 5316 y Fl(kr)p Fs([\(1)22 b Fl(\000)h Fo(\037)p Fs(\))p Fo(')1461 5331 y Fn(g)1501 5316 y Fs(])p Fl(k)1578 5274 y Fi(2)1645 5316 y Fl(\024)28 b Fs(2)p Fl(kr)p Fs([\(1)21 b Fl(\000)i Fo(\037)p Fs(\))p Fo(')2330 5331 y Fn(g)2370 5316 y Fs(])p Fl(kk)p Fo(')2561 5331 y Fn(g)2601 5316 y Fl(r)p Fo(\037)p Fl(k)f Fs(+)g Fo(o)p Fs(\(1\))1969 5454 y Fp(41)p eop %%Page: 42 42 42 41 bop 118 241 a Fs(As)35 b Fl(k)p Fo(')378 256 y Fn(g)418 241 y Fl(r)p Fo(\037)p Fl(k)c(!)g Fs(0)k(when)h Fo(g)e Fl(!)d Fs(+)p Fl(1)p Fs(,)k(w)m(e)h(get)e(that)h Fl(kr)p Fs([\(1)23 b Fl(\000)h Fo(\037)p Fs(\))p Fo(')2623 256 y Fn(g)2663 241 y Fs(])p Fl(k)31 b(!)g Fs(0)k(when)h Fo(g)e Fl(!)d Fs(+)p Fl(1)p Fs(.)49 b(This)118 357 y(completes)32 b(the)h(pro)s(of)f(of)g(equation)h(\(5.27\))e(hence)j(of)e(Lemma)g (5.5.)p 3827 473 4 66 v 3831 411 59 4 v 3831 473 V 3889 473 4 66 v 118 589 a(As)h(a)g(corollary)d(of)i(this)h(lemma,)d(w)m(e)k (get)118 774 y Fx(Lemma)j(5.6.)49 b Fe(L)-5 b(et)38 b Fo(V)53 b Fl(2)32 b Fo(L)1183 738 y Fn(p)1223 774 y Fs(\()p Fk(R)1327 738 y Fn(d)1373 774 y Fs(\))37 b Fe(wher)-5 b(e)37 b Fo(p)g Fe(is)g(chosen)f(as)g(in)h(assumption)f(2.)51 b(Then,)37 b(ther)-5 b(e)37 b(exists)118 891 y Fo(R)192 906 y Fi(0)274 891 y Fo(>)42 b Fs(0)h Fe(such)f(that,)j(if)e Fo(\037)f Fl(2)h(C)1325 854 y Fj(1)1319 915 y Fi(0)1400 891 y Fs(\()p Fk(R)1504 854 y Fn(d)1550 891 y Fs(\))p Fe(,)i Fs(0)d Fl(\024)h Fo(\037)f Fl(\024)h Fs(1)p Fe(,)h Fo(\037)p Fs(\()p Fo(x)p Fs(\))f(=)f(1)h Fe(if)f Fl(j)p Fo(x)p Fl(j)g(\024)h Fo(R)3114 906 y Fi(0)3196 891 y Fe(and)f Fo(\037)p Fs(\()p Fo(x)p Fs(\))h(=)f(0)h Fe(if)118 1007 y Fl(j)p Fo(x)p Fl(j)28 b(\025)g Fs(2)p Fo(R)485 1022 y Fi(0)524 1007 y Fe(,)35 b(then)g Fo( )869 1022 y Fn(g)937 1007 y Fs(=)27 b Fo(\037)c Fl(\001)e Fo(')1237 1022 y Fn(g)1277 1007 y Fo(=)p Fl(k)p Fo(\037)h Fl(\001)g Fo(')1573 1022 y Fn(g)1613 1007 y Fl(k)35 b Fe(is)f(an)h(asymptotic)g (gr)-5 b(ound)34 b(state)h(and)g(we)f(have)812 1106 y Fm(\()892 1277 y Fo(\013)28 b(>)g Fs(0;)99 b(lim)1213 1336 y Fn(g)r Fj(!)p Fi(+)p Fj(1)1518 1277 y Fs(sup)1462 1363 y Fj(j)p Fn(a)p Fj(j\024)p Fn(g)1630 1344 y Fg(\000)p Fh(\013)1738 1136 y Fm(\024)1801 1209 y Fo(g)t Fl(jh)p Fs(\()p Fo(\034)1999 1224 y Fn(a)2039 1209 y Fo(V)44 b Fl(\000)23 b Fo(V)e Fs(\))p Fo( )2419 1224 y Fn(g)2460 1209 y Fo(;)c( )2567 1224 y Fn(g)2607 1209 y Fl(ij)p 1801 1254 873 4 v 2108 1345 a Fo(E)2180 1360 y Fj(\000)2239 1345 y Fs(\()p Fo(g)t Fs(\))2683 1136 y Fm(\025)2763 1277 y Fs(=)28 b(0)2916 1106 y Fm(\))3024 1277 y Fl(6)p Fs(=)f Fl(;)p Fo(:)118 1547 y Fx(Pro)s(of.)43 b Fs(If)33 b(w)m(e)h(pic)m(k)e Fo(\037)h Fs(as)g(ab)s(o)m(v)m(e)g(for)f Fo(R)1624 1562 y Fi(0)1696 1547 y Fs(large)g(enough,)h(b)m(y)h(Lemma)d (5.5,)h(w)m(e)i(kno)m(w)f(that)1027 1723 y Fl(k)p Fs(\(1)21 b Fl(\000)i Fo(\037)p Fs(\))p Fo(')1448 1738 y Fn(g)1488 1723 y Fl(k)f Fs(+)g Fl(k)p Fs(\(1)g Fl(\000)g Fo(\037)p Fs(\))p Fl(r)p Fo(')2162 1738 y Fn(g)2202 1723 y Fl(k)28 b(!)f Fs(0)32 b(as)h Fo(g)e Fl(!)d Fs(+)p Fl(1)-2872 b Fs(\(5.32\))118 1904 y(Then,)34 b(as)f Fo(')584 1919 y Fn(g)656 1904 y Fs(is)f(the)h(ground)g(state)g(of)f Fo(H)8 b Fs(\()p Fo(g)t Fs(\),)31 b(w)m(e)j(compute)565 2074 y Fl(h)p Fs(\()p Fo(H)8 b Fs(\()p Fo(g)t Fs(\))20 b Fl(\000)j Fo(E)6 b Fs(\()p Fo(g)t Fs(\)\))p Fo( )1284 2089 y Fn(g)1324 2074 y Fo(;)17 b( )1431 2089 y Fn(g)1471 2074 y Fl(i)27 b Fs(=)h Fl(h)p Fs(\()p Fo(H)8 b 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Fo(g)t Fs(\))f Fl(\000)i Fs(\001\))3516 4604 y Fj(\000)p Fi(1)p Fn(=)p Fi(2)3681 4645 y Fl(k)849 4816 y(\024)28 b Fo(C)7 b Fs([)p Fo(E)1130 4831 y Fj(\000)1189 4816 y Fs(\()p Fo(g)t Fs(\)])1343 4775 y Fj(\000)p Fi(1+)p Fn(\016)r Fi(+)p Fn(d=)p Fi(2)p Fn(p)1739 4816 y Fs(\()p Fl(j)p Fo(a)p Fl(j)p Fo(E)1956 4831 y Fj(\000)2015 4816 y Fs(\()p Fo(g)t Fs(\)\))2179 4767 y Fn(\016)118 5004 y Fs(writing)31 b(\()p Fo(\034)534 5019 y Fn(a)598 5004 y Fl(\000)23 b Fs(1\)\()p Fo(E)895 5019 y Fj(\000)954 5004 y Fs(\()p Fo(g)t Fs(\))e Fl(\000)i Fs(\001\))1321 4968 y Fj(\000)p Fn(\016)1446 5004 y Fs(as)33 b(a)f(F)-8 b(ourier)32 b(m)m(ultiplier.)40 b(Hence)717 5241 y(sup)660 5328 y Fj(j)p Fn(a)p Fj(j\024)p Fn(g)828 5309 y Fg(\000)p Fh(\013)937 5101 y Fm(\024)999 5174 y Fo(g)t Fl(jh)p Fs(\()p Fo(\034)1197 5189 y Fn(a)1238 5174 y Fo(V)k Fl(\000)22 b Fo(V)g Fs(\))p Fo(')1619 5189 y Fn(g)1659 5174 y Fo(;)17 b(')1767 5189 y Fn(g)1806 5174 y Fl(ij)p 999 5218 874 4 v 1307 5310 a Fo(E)1379 5325 y Fj(\000)1439 5310 y Fs(\()p Fo(g)t Fs(\))1883 5101 y Fm(\025)1963 5241 y Fl(\024)28 b Fo(C)7 b(g)t Fs([)p Fo(E)2295 5256 y Fj(\000)2354 5241 y Fs(\()p Fo(g)t Fs(\)])2508 5200 y Fj(\000)p Fi(2+2)p Fn(\016)r Fi(+)p Fn(d=)p Fi(2)p Fn(p)2923 5241 y Fo(g)2974 5195 y Fj(\000)p Fn(\013\016)r Fi(+)3212 5164 y Fd(2)p Fh(p)p 3172 5180 143 3 v 3172 5221 a Fd(2)p Fh(p)p Fg(\000)p Fh(d)3328 5241 y Fo(:)1969 5454 y Fp(42)p eop %%Page: 43 43 43 42 bop 118 241 a Fs(Using)34 b(\(5.25\))d(and)i(pic)m(king)f Fo(\013)h Fs(large)f(enough,)h(w)m(e)h(get)1191 556 y(lim)1143 616 y Fn(g)r Fj(!)p Fi(+)p Fj(1)1448 556 y Fs(sup)1391 643 y Fj(j)p Fn(a)p Fj(j\024)p Fn(g)1559 624 y Fg(\000)p Fh(\013)1667 416 y Fm(\024)1730 489 y Fo(g)t Fl(jh)p Fs(\()p Fo(\034)1928 504 y Fn(a)1969 489 y Fo(V)43 b Fl(\000)23 b Fo(V)e Fs(\))p Fo(')2349 504 y Fn(g)2389 489 y Fo(;)c(')2497 504 y Fn(g)2537 489 y Fl(ij)p 1730 533 874 4 v 2038 625 a Fo(E)2110 640 y Fj(\000)2169 625 y Fs(\()p Fo(g)t Fs(\))2614 416 y Fm(\025)2694 556 y Fs(=)27 b(0)p Fo(:)118 890 y Fs(No)m(w)41 b(using)g(equations)g (\(5.25\))o(,)g(\(5.32\))e(and)i(the)g(fact)g(that,)h(for)f Fo(R)2718 905 y Fi(0)2798 890 y Fs(large)e(enough,)44 b Fo(\034)3460 905 y Fn(a)3502 890 y Fo(V)49 b Fl(\000)28 b Fo(V)62 b Fs(is)118 1006 y(b)s(ounded)33 b(outside)g Fl(fj)p Fo(x)p Fl(j)27 b(\024)h Fs(2)p Fo(R)1269 1021 y Fi(0)1309 1006 y Fl(g)p Fs(,)k(w)m(e)i(get)1191 1322 y(lim)1143 1382 y Fn(g)r Fj(!)p Fi(+)p Fj(1)1448 1322 y Fs(sup)1392 1409 y Fj(j)p Fn(a)p Fj(j\024)p Fn(g)1560 1390 y Fg(\000)p Fh(\013)1668 1181 y Fm(\024)1730 1254 y Fo(g)t Fl(jh)p Fs(\()p Fo(\034)1928 1269 y Fn(a)1969 1254 y Fo(V)44 b Fl(\000)22 b Fo(V)g Fs(\))p Fo( )2349 1269 y Fn(g)2389 1254 y Fo(;)17 b( )2496 1269 y Fn(g)2537 1254 y Fl(ij)p 1730 1299 873 4 v 2038 1390 a Fo(E)2110 1405 y Fj(\000)2169 1390 y Fs(\()p Fo(g)t Fs(\))2613 1181 y Fm(\025)2693 1322 y Fs(=)28 b(0)p Fo(:)118 1650 y Fs(This)33 b(ends)h(the)f(pro)s(of)e(of)h(Lemma)g(5.6.)p 3827 1767 4 66 v 3831 1704 59 4 v 3831 1767 V 3889 1767 4 66 v 118 1883 a(T)-8 b(o)25 b(end)g(this)g(section)g(w)m(e)g(describ) s(e)h(the)f(ground)g(state)g(energy)g(of)g Fo(H)8 b Fs(\()p Fo(g)t Fs(\))23 b(and)i(an)g(asymptotic)f(ground)118 1999 y(state)33 b(in)f(the)h(case)g(when)h Fo(V)54 b Fs(satis\014es)33 b(assumption)f(H1".)118 2115 y(F)-8 b(or)30 b(1)e Fl(\024)g Fo(i)g Fl(\024)g Fo(q)t Fs(,)j(de\014ne)i Fo(H)1114 2079 y Fn(i)1106 2140 y(g)1173 2115 y Fs(=)28 b Fl(\000)p Fs(\001)20 b(+)e Fo(g)t(V)1657 2130 y Fn(i)1716 2115 y Fs(where)32 b Fo(V)2053 2130 y Fn(i)2112 2115 y Fs(is)f(de\014ned)h(in)e(equation)h(\(1.10\).)43 b Fo(H)3461 2079 y Fn(i)3453 2140 y(g)3523 2115 y Fs(is)31 b(a)g(form)118 2231 y(b)s(ounded)40 b(p)s(erturbation)e(of)h Fl(\000)p Fs(\001)h(with)e(relativ)m(e)h(b)s(ound)g(0.)62 b(By)40 b(the)g(homogeneit)m(y)e(prop)s(erties)h(of)118 2348 y Fo(V)175 2363 y Fn(i)203 2348 y Fs(,)31 b(when)g Fo(E)585 2363 y Fn(i)641 2348 y Fo(<)c Fs(0)j(\()p Fo(E)933 2363 y Fn(i)991 2348 y Fs(is)f(the)i(ground)f(state)g(energy)h(of)e Fo(H)2322 2312 y Fn(i)2314 2372 y Fi(1)2354 2348 y Fs(\),)h(it)f(is)g (ob)m(vious)h(that)g(the)g(ground)g(state)118 2464 y(energy)k(of)e Fo(H)630 2428 y Fn(i)622 2489 y(g)694 2464 y Fs(is)g(giv)m(en)h(b)m(y) 1660 2759 y Fo(E)1732 2774 y Fn(i)1761 2759 y Fs(\()p Fo(g)t Fs(\))26 b(=)i Fo(E)2090 2774 y Fn(i)2118 2759 y Fo(g)2231 2680 y Fd(2)p 2179 2692 136 3 v 2179 2733 a(2)p Fg(\000)p Fh(\027)2288 2749 y(i)2328 2759 y Fo(:)-2237 b Fs(\(5.33\))118 3045 y(De\014ne)34 b Fo(E)493 3060 y Fj(y)529 3045 y Fs(\()p Fo(g)t Fs(\))28 b(=)h(inf)23 b Fo(E)996 3060 y Fn(i)1024 3045 y Fs(\()p Fo(g)t Fs(\).)45 b(Moreo)m(v)m(er,)36 b(if)c Fo(')1834 3060 y Fn(i)1895 3045 y Fs(is)h(the)h(ground)g(state)f(of)g Fo(H)2935 3009 y Fn(i)2927 3070 y Fi(1)3000 3045 y Fs(then)h Fo(')3287 2994 y Fi(\()p Fn(i)p Fi(\))3287 3055 y Fn(g)3370 3045 y Fs(,)g(the)g(ground)118 3161 y(state)f(of)f Fo(H)557 3125 y Fn(i)549 3186 y(g)622 3161 y Fs(has)h(the)g(follo)m(wing)d(form) 1404 3460 y Fo(')1468 3418 y Fi(\()p Fn(i)p 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Fs(\))p Fe(,)36 b(then,)f(if)h Fo(\037)2780 4538 y Fn(i)2804 4547 y Fd(0)2879 4523 y Fe(is)f(a)h Fl(C)3128 4487 y Fj(1)3122 4548 y Fi(0)3239 4523 y Fe(cut-o\013)g (function)118 4660 y(for)i(a)g(su\016ciently)g(smal)5 b(l)38 b(neighb)-5 b(orho)g(o)g(d)36 b(of)i Fo(x)1878 4675 y Fn(i)1902 4684 y Fd(0)1941 4660 y Fe(,)h Fo( )2077 4609 y Fi(\()p Fn(i)2128 4618 y Fd(0)2163 4609 y Fi(\))2073 4670 y Fn(g)2195 4660 y Fs(\()p Fl(\001)p Fs(\))33 b(=)h Fo(\037)2503 4675 y Fn(i)2527 4684 y Fd(0)2565 4660 y Fo(')2629 4609 y Fi(\()p Fn(i)2680 4618 y Fd(0)2715 4609 y Fi(\))2629 4670 y Fn(g)2747 4660 y Fs(\()p Fl(\001)24 b(\000)h Fo(x)2994 4675 y Fn(i)3018 4684 y Fd(0)3057 4660 y Fs(\))p Fo(=)p Fl(k)p Fo(\037)3255 4675 y Fn(i)3279 4684 y Fd(0)3318 4660 y Fo(')3382 4609 y Fi(\()p Fn(i)3433 4618 y Fd(0)3468 4609 y Fi(\))3382 4670 y Fn(g)3499 4660 y Fs(\()p Fl(\001)g(\000)g Fo(x)3747 4675 y Fn(i)3771 4684 y Fd(0)3810 4660 y Fs(\))p Fl(k)118 4776 y Fe(is)35 b(an)f(asymptotic)h(gr)-5 b(ound)34 b(state)i(for)e Fo(H)8 b Fs(\()p Fo(g)t Fs(\))p Fe(.)118 5006 y Fx(Pro)s(of.)56 b Fs(Fix)36 b Fo(")e(>)h Fs(0)h(small.)54 b(As)37 b(in)f(section)h (4.3,)h(at)e(the)h(cost)h(of)e(adding)g(a)g(term)g(to)h Fo(V)3467 5021 y Fi(1)3506 5006 y Fs(,)h(w)m(e)g(ma)m(y)118 5122 y(assume)45 b(that)f(the)h(functions)g Fo(W)1395 5137 y Fn(i)1468 5122 y Fs(sta)m(y)g(non-negativ)m(e)f(and)h(smaller)d (than)j(1)30 b(+)g Fo(")44 b Fs(and)h(that)f(the)118 5238 y(supp)s(ort)33 b(of)g(the)g Fo(W)851 5253 y Fn(i)912 5238 y Fs(are)g(t)m(w)m(o)h(b)m(y)f(t)m(w)m(o)h(disjoin)m(t.)43 b(F)-8 b(or)32 b(1)c Fl(\024)h Fo(i)f Fl(\024)h Fo(q)t Fs(,)k(let)f Fo(\037)2800 5253 y Fn(i)2861 5238 y Fs(b)s(e)h(a)g Fl(C)3134 5202 y Fj(1)3128 5263 y Fi(0)3242 5238 y Fs(cut-o\013)f (function)118 5355 y(of)41 b(a)f(neigh)m(b)s(orho)s(o)s(d)g(of)g Fo(x)1122 5370 y Fn(i)1192 5355 y Fs(suc)m(h)i(that,)h(on)e(the)g(supp) s(ort)g(of)g Fo(\037)2539 5370 y Fn(i)2567 5355 y Fs(,)i(the)e (function)g Fo(W)3296 5370 y Fn(i)3324 5355 y Fs(\()p Fl(\001)28 b(\000)g Fo(x)3578 5370 y Fn(i)3606 5355 y Fs(\))41 b(sta)m(ys)1969 5454 y Fp(43)p eop %%Page: 44 44 44 43 bop 118 241 a Fs(larger)32 b(than)g(1)22 b Fl(\000)h Fo(")p Fs(.)43 b(De\014ne)33 b Fo(\037)1272 205 y Fi(2)1272 265 y(0)1339 241 y Fs(=)28 b(1)22 b Fl(\000)1613 166 y Fm(P)1718 270 y Fi(1)p Fj(\024)p Fn(i)p Fj(\024)p Fn(q)1942 241 y Fo(\037)2003 205 y Fi(2)2003 265 y Fn(i)2043 241 y Fs(.)43 b(Pic)m(k)33 b Fo(')28 b Fl(2)g Fo(H)2606 205 y Fi(1)2645 241 y Fs(\()p Fk(R)2749 205 y Fn(d)2795 241 y Fs(\).)44 b(Then)487 508 y Fl(h)p Fo(H)8 b Fs(\()p Fo(g)t Fs(\))p Fo(';)17 b(')p Fl(i)25 b Fs(=)1137 379 y Fn(q)1082 413 y Fm(X)1097 623 y Fn(i)p Fi(=0)1226 508 y Fl(h\000)p Fs(\001)p Fo(\037)1484 467 y Fi(2)1484 533 y Fn(i)1524 508 y Fo(';)17 b(')p Fl(i)k Fs(+)1909 379 y Fn(q)1854 413 y Fm(X)1869 623 y Fn(i)p Fi(=0)2015 508 y Fo(g)t Fl(h)p Fo(V)f(\037)2243 467 y Fi(2)2243 533 y Fn(i)2283 508 y Fo(';)d(')p Fl(i)978 838 y Fs(=)1137 709 y Fn(q)1082 743 y Fm(X)1097 953 y Fn(i)p Fi(=0)1242 838 y Fl(kr)p Fs(\()p Fo(\037)1474 853 y Fn(i)1503 838 y Fo(')p Fs(\))p Fl(k)1654 789 y Fi(2)1716 838 y Fl(\000)22 b Fs(2Re)p Fl(h)p Fo(\037)2079 853 y Fn(i)2107 838 y Fl(r)p Fo(';)17 b(')p Fl(r)p Fo(\037)2506 853 y Fn(i)2534 838 y Fl(i)22 b(\000)h(hjr)p Fo(\037)2906 853 y Fn(i)2933 838 y Fl(j)2961 797 y Fi(2)3001 838 y Fo(';)17 b(')p Fl(i)1682 1167 y Fs(+)1835 1039 y Fn(q)1780 1073 y Fm(X)1794 1283 y Fn(i)p Fi(=1)1940 1167 y Fo(g)t Fl(h)p Fo(\034)2072 1182 y Fn(x)2112 1192 y Fh(i)2142 1167 y Fs(\()p Fo(W)2272 1182 y Fn(i)2300 1167 y Fo(V)2357 1182 y Fn(i)2385 1167 y Fs(\))p Fo(\037)2484 1126 y Fi(2)2484 1192 y Fn(i)2523 1167 y Fo(';)g(')p Fl(i)22 b Fs(+)g Fo(g)t Fl(h)p Fo(V)3001 1182 y Fi(0)3039 1167 y Fo(';)17 b(')p Fl(i)978 1497 y(\025)29 b Fs(\(1)21 b Fl(\000)i Fo(")p Fs(\))1447 1369 y Fn(q)1392 1403 y Fm(X)1407 1613 y Fn(i)p Fi(=1)1553 1357 y Fm(\022)1626 1497 y Fl(kr)p Fs(\()p Fo(\037)1858 1512 y Fn(i)1886 1497 y Fo(')p Fs(\))p Fl(k)2038 1449 y Fi(2)2099 1497 y Fs(+)2290 1430 y Fo(g)p 2207 1474 217 4 v 2207 1566 a Fs(1)f Fl(\000)h Fo(")2433 1497 y Fl(h)p Fo(\034)2514 1512 y Fn(x)2554 1522 y Fh(i)2585 1497 y Fs(\()p Fo(W)2715 1512 y Fn(i)2743 1497 y Fo(V)2800 1512 y Fn(i)2828 1497 y Fs(\))p Fo(\037)2927 1512 y Fn(i)2955 1497 y Fo(';)17 b(\037)3124 1512 y Fn(i)3152 1497 y Fo(')p Fl(i)3255 1357 y Fm(\023)2154 1827 y Fs(+)22 b Fo(g)t Fl(h)p Fo(V)2399 1842 y Fi(0)2437 1827 y Fo(';)17 b(')p Fl(i)22 b Fs(+)2768 1687 y Fm(\022)2841 1827 y Fs(1)g Fl(\000)3022 1760 y Fs(1)p 3022 1804 49 4 v 3024 1895 a Fo(")3080 1687 y Fm(\023)3225 1698 y Fn(q)3170 1732 y Fm(X)3185 1942 y Fn(i)p Fi(=0)3331 1827 y Fl(kr)p Fo(\037)3525 1842 y Fn(i)3575 1827 y Fl(\001)g Fo(')p Fl(k)3739 1779 y Fi(2)118 1171 y Fs(\(5.35\))118 2085 y(where)29 b Fo(V)452 2100 y Fi(0)519 2085 y Fs(is)e(some)h(b)s(ounded)g(p)s(oten)m(tial)e (\(dep)s(ending)i(on)g Fo(")p Fs(\).)41 b(On)28 b(the)g(other)g(hand,)h (on)f(the)g(supp)s(ort)118 2201 y(of)k Fo(\037)290 2216 y Fn(i)318 2201 y Fs(,)h(w)m(e)h(ha)m(v)m(e)1131 2421 y Fo(W)1223 2436 y Fn(i)1251 2421 y Fo(V)1308 2436 y Fn(i)1364 2421 y Fl(\025)28 b Fo(V)1526 2436 y Fn(i)1576 2421 y Fl(\000)23 b Fo(")p Fl(j)p Fo(V)e Fl(j)1856 2436 y Fn(i)1916 2421 y Fs(where)34 b Fl(j)p Fo(V)21 b Fl(j)2332 2436 y Fn(i)2360 2421 y Fs(\()p Fl(\001)p Fs(\))27 b(:=)2632 2353 y Fl(j)p Fo(h)2716 2368 y Fn(i)2744 2353 y Fs(\()p Fl(\001)p Fs(\))p Fl(j)p 2632 2398 244 4 v 2657 2489 a(j)22 b(\001)g(j)2785 2460 y Fn(\027)2820 2470 y Fh(i)118 2640 y Fs(Hence)218 2838 y Fl(kr)p Fs(\()p Fo(\037)450 2853 y Fn(i)478 2838 y Fo(')p Fs(\))p Fl(k)629 2789 y Fi(2)691 2838 y Fs(+)882 2770 y Fo(g)p 799 2815 217 4 v 799 2906 a Fs(1)g Fl(\000)h Fo(")1025 2838 y Fl(h)p Fo(\034)1106 2853 y Fn(x)1146 2863 y Fh(i)1176 2838 y Fs(\()p Fo(W)1306 2853 y Fn(i)1335 2838 y Fo(V)1392 2853 y Fn(i)1420 2838 y Fs(\))p Fo(\037)1519 2796 y Fi(2)1519 2862 y Fn(i)1558 2838 y Fo(';)17 b(')p Fl(i)27 b(\025)1000 3062 y(\025)i Fs(\(1)21 b Fl(\000)i Fo(")p Fs(\))17 b Fl(k)o(r)p Fs(\()p Fo(\037)1646 3077 y Fn(i)1674 3062 y Fo(')p Fs(\))p Fl(k)1826 3013 y Fi(2)1888 3062 y Fs(+)2079 2995 y Fo(g)p 1996 3039 V 1996 3130 a Fs(1)k Fl(\000)i Fo(")2222 3062 y Fl(h)p Fo(\034)2303 3077 y Fn(x)2343 3087 y Fh(i)2373 3062 y Fs(\()p Fo(V)2468 3077 y Fn(i)2496 3062 y Fs(\))p Fo(\037)2595 3077 y Fn(i)2623 3062 y Fo(';)17 b(\037)2792 3077 y Fn(i)2820 3062 y Fo(')p Fl(i)22 b Fs(+)1877 3320 y(+)g Fo(")2038 3180 y Fm(\022)2110 3320 y Fl(kr)p Fs(\()p Fo(\037)2342 3335 y Fn(i)2371 3320 y Fo(')p Fs(\))p Fl(k)2522 3272 y Fi(2)2584 3320 y Fl(\000)2776 3253 y Fo(g)p 2693 3297 V 2693 3389 a Fs(1)g Fl(\000)h Fo(")2919 3320 y Fl(h)p Fo(\034)3000 3335 y Fn(x)3040 3345 y Fh(i)3071 3320 y Fs(\()p Fl(j)p Fo(V)e Fl(j)3243 3335 y Fn(i)3271 3320 y Fs(\))p Fo(\037)3370 3335 y Fn(i)3398 3320 y Fo(';)c(\037)3567 3335 y Fn(i)3595 3320 y Fo(')p Fl(i)3698 3180 y Fm(\023)3788 3320 y Fo(:)118 3548 y Fs(By)33 b(equation)g(\(5.35\))o(,)f(w)m(e)i(get)f(that)703 3775 y Fl(kr)p Fs(\()p Fo(\037)935 3790 y Fn(i)963 3775 y Fo(')p Fs(\))p Fl(k)1114 3727 y Fi(2)1176 3775 y Fs(+)1367 3708 y Fo(g)p 1284 3752 V 1284 3844 a Fs(1)22 b Fl(\000)h Fo(")1510 3775 y Fl(h)p Fo(\034)1591 3790 y Fn(x)1631 3800 y Fh(i)1661 3775 y Fs(\()p Fo(W)1791 3790 y Fn(i)1820 3775 y Fo(V)1877 3790 y Fn(i)1905 3775 y Fs(\))p Fo(\037)2004 3734 y Fi(2)2004 3800 y Fn(i)2043 3775 y Fo(';)17 b(')p Fl(i)27 b(\025)h Fs(\(1)22 b(+)g Fo("C)7 b Fs(\))p Fo(E)2826 3790 y Fn(i)2871 3635 y Fm(\022)3037 3708 y Fo(g)p 2954 3752 V 2954 3844 a Fs(1)22 b Fl(\000)h Fo(")3180 3635 y Fm(\023)3270 3775 y Fl(k)p Fo(\037)3381 3790 y Fn(i)3409 3775 y Fo(')p Fl(k)3523 3734 y Fi(2)118 3775 y Fs(\(5.36\))118 4003 y(for)40 b(some)h(constan)m(t)g Fo(C)48 b Fs(\(indep)s(enden)m(t) 42 b(of)e Fo(")p Fs(\))g(giv)m(en)h(b)m(y)h(the)f(lo)m(w)m(est)g(eigen) m(v)-5 b(alue)40 b(of)h Fl(\000)p Fs(\001)28 b Fl(\000)g Fo(g)t Fl(j)p Fo(V)21 b Fl(j)3843 4018 y Fn(i)3871 4003 y Fs(.)118 4119 y(Here)29 b(w)m(e)h(used)g(the)f(fact)f(that)h(this)f (eigen)m(v)-5 b(alue)28 b(has)h(the)g(same)g(gro)m(wth)g(rate)f(in)g Fo(g)k Fs(as)d Fo(E)3383 4134 y Fn(i)3411 4119 y Fs(\()p Fo(g)t Fs(\);)g(indeed,)118 4235 y(in)35 b(the)g(presen)m(t)i(case,)g (this)e(gro)m(wth)g(rate)g(only)g(dep)s(ends)i(on)e(the)h(homogeneit)m (y)e(prop)s(erties)h(of)g(the)118 4351 y(p)s(oten)m(tial)c Fl(j)p Fo(V)21 b Fl(j)664 4366 y Fn(i)725 4351 y Fs(\(as)32 b(can)h(b)s(e)g(seen)h(b)m(y)f(a)g(scaling)e(argumen)m(t\).)118 4468 y(Putting)39 b(equations)h(\(5.36\))e(and)i(\(5.35\))e(together,)k (w)m(e)e(obtain)f(that,)i(there)f(exists)g Fo(C)46 b(>)39 b Fs(0,)i(suc)m(h)118 4584 y(that,)33 b(for)f(an)m(y)h Fo(")27 b(>)h Fs(0)k(small)e(and)j Fo(')28 b Fl(2)g Fo(H)1668 4548 y Fi(1)1707 4584 y Fs(\()p Fk(R)1810 4548 y Fn(d)1857 4584 y Fs(\),)k(w)m(e)i(ha)m(v)m(e)972 4811 y Fl(h)p Fo(H)8 b Fs(\()p Fo(g)t Fs(\))p Fo(';)17 b(')p Fl(i)26 b(\025)1569 4671 y Fm(\024)1621 4811 y Fs(\(1)c Fl(\000)h Fo(")p Fs(\))p Fo(E)1986 4826 y Fj(y)2038 4671 y Fm(\022)2204 4744 y Fo(g)p 2121 4788 V 2121 4879 a Fs(1)f Fl(\000)h Fo(")2347 4671 y Fm(\023)2443 4811 y Fs(+)f Fo(O)s Fs(\()p Fo(g)t Fs(\))2746 4671 y Fm(\025)2813 4811 y Fl(k)p Fo(')p Fl(k)2977 4770 y Fi(2)3016 4811 y Fo(:)118 5033 y Fs(This)33 b(pro)m(v)m(es)h(that)1619 5253 y(lim)17 b(sup)1652 5332 y Fn(g)r Fj(!)p Fi(+)p Fj(1)1959 5185 y Fo(E)6 b Fs(\()p Fo(g)t Fs(\))p 1944 5230 234 4 v 1944 5321 a Fo(E)2016 5336 y Fj(y)2052 5321 y Fs(\()p Fo(g)t Fs(\))2216 5253 y Fl(\024)28 b Fs(1)p Fo(:)1969 5454 y Fp(44)p eop %%Page: 45 45 45 44 bop 118 241 a Fs(Pic)m(k)38 b(1)c Fl(\024)i Fo(i)570 256 y Fi(0)645 241 y Fl(\024)f Fo(q)41 b Fs(suc)m(h)d(that)f Fo(E)1353 256 y Fn(i)1377 265 y Fd(0)1416 241 y Fs(\()p Fo(g)t Fs(\))d(=)h Fo(E)1760 256 y Fj(y)1796 241 y Fs(\()p Fo(g)t Fs(\))h(and)h Fo(\037)g Fs(a)f Fl(C)2394 205 y Fj(1)2388 265 y Fi(0)2506 241 y Fs(cut-o\013)h(function)f(for)g(a)h (su\016cien)m(tly)118 378 y(small)d(neigh)m(b)s(orho)s(o)s(d)i(of)g Fo(x)1164 393 y Fn(i)1188 402 y Fd(0)1227 378 y Fs(.)56 b(By)37 b(the)g(de\014nition)e(of)i Fo(')2256 327 y Fi(\()p Fn(i)2307 336 y Fd(0)2342 327 y Fi(\))2256 388 y Fn(g)2373 378 y Fs(,)h(it)d(is)i(immediate)c(that)k Fl(k)p Fo(\037')3513 327 y Fi(\()p Fn(i)3564 336 y Fd(0)3599 327 y Fi(\))3513 388 y Fn(g)3630 378 y Fl(k)e(!)f Fs(1)118 494 y(as)f Fo(g)e Fl(!)c Fs(+)p Fl(1)p Fs(.)43 b(An)33 b(immediate)d(computations) h(giv)m(es)i(that)1421 752 y(lim)1373 811 y Fn(g)r Fj(!)p Fi(+)p Fj(1)1632 684 y Fl(h)p Fo(H)8 b Fs(\()p Fo(g)t Fs(\))p Fo(\037')2012 633 y Fi(\()p Fn(i)2063 642 y Fd(0)2096 633 y Fi(\))2012 694 y Fn(g)2128 684 y Fo(;)17 b(\037')2297 633 y Fi(\()p Fn(i)2348 642 y Fd(0)2382 633 y Fi(\))2297 694 y Fn(g)2414 684 y Fl(i)p 1632 729 822 4 v 1912 820 a Fo(E)1984 835 y Fn(i)2008 844 y Fd(0)2047 820 y Fs(\()p Fo(g)t Fs(\))2490 752 y(=)28 b(1)118 984 y(This)33 b(implies)d(that) 1633 1216 y(lim)17 b(inf)1652 1276 y Fn(g)r Fj(!)p Fi(+)p Fj(1)1945 1149 y Fo(E)6 b Fs(\()p Fo(g)t Fs(\))p 1930 1193 234 4 v 1930 1284 a Fo(E)2002 1299 y Fj(y)2038 1284 y Fs(\()p Fo(g)t Fs(\))2201 1216 y Fl(\025)29 b Fs(1)p Fo(:)118 1452 y Fs(Hence)34 b(it)e(completes)g(the)h(pro)s(of)f(of)g (Lemma)f(5.7.)1717 1710 y Fr(References)160 1868 y Fw([1])41 b(J.)j(M.)g(Com)n(b)r(es)f(and)h(P)-7 b(.)44 b(D.)h(Hislop.)e(Lo)r (calization)g(for)h(some)f(con)n(tin)n(uous)g(random)g(hamiltonians)h (in)g(d-)289 1967 y(dimensions.)27 b Fq(Journal)j(of)g(F)-6 b(unctional)30 b(A)n(nalysis)p Fw(,)e(124:149{180,)23 b(1994.)160 2067 y([2])41 b(H.L.)25 b(Cycon,)h(R.G.)f(F)-7 b(ro)r(ese,)25 b(W.)h(Kirsc)n(h,)f(and)g(B.)g(Simon.)g Fq(Schr\177)-42 b(odinger)30 b(Op)l(er)l(ators)p Fw(.)25 b(Springer)f(V)-7 b(erlag,)25 b(Berlin,)289 2167 y(1987.)160 2266 y([3])41 b(M.)35 b(Donsk)n(er)e(and)i(S.R.S.)g(V)-7 b(aradhan.)34 b(Asymptotics)g(for)g(the)h(wiener)g(sausage.)d Fq(Communic)l(ations)37 b(on)g(Pur)l(e)289 2366 y(and)30 b(Applie)l(d)h(Mathematics)p Fw(,)f(28:525{565,)23 b(1975.)160 2465 y([4])41 b(M.)31 b(Efros)f(and)h(B.)h(Shlo)n(vski.)e Fq(Ele)l(ctr)l(onic)k(pr)l(op)l(erties)g(of)g(dop)l(e)l(d)g(semi-c)l (onductors)p Fw(.)e(Springer)e(V)-7 b(erlag,)32 b(Heidel-)289 2565 y(b)r(erg,)27 b(1984.)160 2665 y([5])41 b(B.)f(Hel\013er)g(and)g (J.)g(Sj\177)-42 b(ostrand.)40 b(On)f(diamagnetism)h(and)g(the)g(De)h (Haas-V)-7 b(an)39 b(Alphen)h(e\013ect.)h Fq(A)n(nnales)g(de)289 2764 y(l'Institut)29 b(Henri)g(Poinc)l(ar)n(\023)-40 b(e,)32 b(s)n(\023)-40 b(erie)30 b(Physique)i(Th)n(\023)-40 b(eorique)p Fw(,)29 b(52:303{375,)24 b(1990.)160 2864 y([6])41 b(L.)23 b(H\177)-42 b(ormander.)22 b Fq(The)27 b(analysis)g(of)g(line)l(ar)f(p)l(artial)h(di\013er)l(ential)g(e)l (quations.)g(I)p Fw(,)c(v)n(olume)g(256)e(of)j Fq(Grund)t(lehr)l(en)i (der)289 2964 y(Mathematischen)31 b(Wissenschaften)p Fw(.)e(Springer)e(V)-7 b(erlag,)26 b(1990.)160 3063 y([7])41 b(W.)28 b(Kirsc)n(h.)f(Random)h(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erators.)g(In)i(A.)h(Jensen)e(H.)i(Holden,)f(editor,)g Fq(Schr\177)-42 b(odinger)31 b(Op)l(er)l(ators)p Fw(,)289 3163 y(n)n(um)n(b)r(er)j(345)e(in)j(Lecture)f(Notes)f(in)i(Ph)n(ysics,) f(Berlin,)i(1989.)c(Springer)h(V)-7 b(erlag.)33 b(Pro)r(ceedings,)h (Sonderb)r(org,)289 3262 y(Denmark)27 b(1988.)160 3362 y([8])41 b(F.)26 b(Klopp.)g(An)g(asymptotic)g(expansion)f(for)g(the)i (densit)n(y)e(of)h(states)g(of)g(a)g(random)f(Sc)n(hr\177)-42 b(odinger)24 b(op)r(erator)g(with)289 3462 y(Bernoulli)j(disorder.)f Fq(R)l(andom)k(Op)l(er)l(ators)g(and)g(Sto)l(chastic)g(Equations)p Fw(,)f(3\(4\):315{332,)24 b(1995.)160 3561 y([9])41 b(F.)20 b(Klopp.)e(A)i(lo)n(w)f(concen)n(tration)f(asymptotic)h(expansion)f (for)h(the)h(densit)n(y)f(of)h(states)f(of)g(a)g(random)g(Sc)n(hr\177) -42 b(odinger)289 3661 y(op)r(erator)26 b(with)i(P)n(oisson)d (disorder.)i Fq(Journal)i(of)i(F)-6 b(unctional)29 b(A)n(nalysis)p Fw(,)g(145:267{295,)23 b(1995.)118 3761 y([10])41 b(F.)d(Klopp.)g (Precise)e(high)i(energy)f(asymptotics)g(for)h(the)g(in)n(tegrated)f (densit)n(y)h(of)g(states)g(of)g(an)f(un)n(b)r(ounded)289 3860 y(random)26 b(Jacobi)h(matrix.)g(Preprin)n(t,)f(Univ)n(ersit)n (\023)-39 b(e)26 b(Paris-Nord,)g(Villetaneuse,)h(1997.)118 3960 y([11])41 b(F.)34 b(Klopp.)f(Band)h(edge)f(b)r(eha)n(viour)g(for)g (the)i(in)n(tegrated)e(densit)n(y)g(of)h(states)g(of)g(random)f(Jacobi) f(matrices)h(in)289 4059 y(dimension)27 b(1.)h Fq(Journal)h(of)i (Statistic)l(al)f(Physics)p Fw(,)f(90\(3-4\):927{947,)23 b(1998.)118 4159 y([12])41 b(F.)32 b(Klopp.)f(In)n(ternal)g(Lifshits)h (tails)f(for)h(random)e(p)r(erturbations)h(of)h(p)r(erio)r(dic)f(Sc)n (hr\177)-42 b(odinger)30 b(op)r(erators.)g Fq(Duke)289 4259 y(Mathematic)l(al)i(Journal)p Fw(,)27 b(1998.)f(to)h(app)r(ear.) 118 4358 y([13])41 b(F.)28 b(Klopp)f(and)g(L.)h(P)n(astur.)e(in)i (progress.)118 4458 y([14])41 b(P)-7 b(.)31 b(Kuc)n(hmen)n(t.)h Fq(Flo)l(quet)h(the)l(ory)h(for)h(p)l(artial)f(di\013er)l(ential)h(e)l (quations)p Fw(,)e(v)n(olume)e(60)g(of)h Fq(Op)l(er)l(ator)h(The)l (ory:)48 b(A)l(d-)289 4558 y(vanc)l(es)30 b(and)g(Applic)l(ations)p Fw(.)g(Birkh\177)-42 b(auser,)25 b(Basel,)i(1993.)118 4657 y([15])41 b(L.)j(D.)h(Landau)f(and)g(E.)g(M.)h(Lifshitz.)g Fq(M)n(\023)-40 b(e)l(c)l(anique)46 b(quantique,)j(th)n(\023)-40 b(eorie)46 b(non-r)l(elativiste)p Fw(.)f(Editions)f(MIR,)289 4757 y(Moscou,)26 b(1966.)118 4856 y([16])41 b(I.)c(M.)g(Lifshitz.)h (Structure)f(of)g(the)g(energy)f(sp)r(ectrum)h(of)g(impurit)n(y)g (bands)g(in)g(disordered)f(solid)g(solutions.)289 4956 y Fq(Soviet)30 b(Physics)h(JETP)p Fw(,)e(17:1159{1170,)22 b(1963.)118 5056 y([17])41 b(I.)d(M.)h(Lifshitz.)g(Energy)e(sp)r (ectrum)h(structure)g(and)h(quan)n(tum)f(states)g(of)g(disordered)f (condensed)h(systems.)289 5155 y Fq(Soviet)30 b(Physics)h(Usp)l(ekhi)p Fw(,)e(7:549{573,)24 b(1965.)118 5255 y([18])41 b(I.M.)c(Lifshitz,)j (S.A.)d(Gredeskul,)i(and)e(L.A.)g(P)n(astur.)e Fq(Intr)l(o)l(duction)j (to)g(the)h(the)l(ory)g(of)g(disor)l(der)l(e)l(d)h(systems)p Fw(.)289 5355 y(Wiley)-7 b(,)28 b(New-Y)-7 b(ork,)26 b(1988.)1969 5454 y Fp(45)p eop %%Page: 46 46 46 45 bop 118 241 a Fw([19])41 b(J.)24 b(N.)h(Mather.)g(On)f(Niren)n(b) r(erg's)g(pro)r(of)g(of)g(Malgrange's)f(preparation)g(theorem.)h(In)h Fq(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)g(Liverp)l(o)l(ol)289 340 y(Singularities-Symp)l(osium)40 b(I)p Fw(,)d(n)n(um)n(b)r(er)g(192) g(in)g(Lecture)h(Notes)f(in)h(Mathematics,)h(Berlin,)h(1971.)c (Springer)289 440 y(V)-7 b(erlag.)118 540 y([20])41 b(L.)19 b(P)n(astur.)f(Beha)n(viour)g(of)i(some)f(Wiener)g(in)n(tegrals)f(as)h Fu(t)k Ft(!)g Fw(+)p Ft(1)d Fw(and)f(the)h(densit)n(y)f(of)h(states)f (of)h(the)g(Sc)n(hr\177)-42 b(odinger)289 639 y(equation)27 b(with)h(a)f(random)g(p)r(oten)n(tial.)g Fq(T)-6 b(e)l(or.-Mat.-Fiz)p Fw(,)30 b(32:88{95,)24 b(1977.)i(\(in)i(russian\).)118 739 y([21])41 b(L.)20 b(P)n(astur)e(and)i(A.)h(Figotin.)f Fq(Sp)l(e)l(ctr)l(a)i(of)i(R)l(andom)f(and)g(A)n(lmost-Perio)l(dic)h (Op)l(er)l(ators)p Fw(.)d(Springer)e(V)-7 b(erlag,)20 b(Berlin,)289 839 y(1992.)118 938 y([22])41 b(M.)31 b(Reed)h(and)f(B.)g (Simon.)g Fq(Metho)l(ds)j(of)g(Mo)l(dern)g(Mathematic)l(al)h(Physics,)h (V)-6 b(ol)33 b(IV:)h(Analysis)f(of)h(Op)l(er)l(ators)p Fw(.)289 1038 y(Academic)27 b(Press,)f(New-Y)-7 b(ork,)27 b(1978.)118 1137 y([23])41 b(M.)g(Reed)h(and)f(B.)g(Simon.)g Fq(Metho)l(ds)i(of)h(Mo)l(dern)f(Mathematic)l(al)h(Physics,)j(V)-6 b(ol)43 b(I:)f(Functional)h(Analysis)p Fw(.)289 1237 y(Academic)27 b(Press,)f(New-Y)-7 b(ork,)27 b(1980.)118 1337 y([24])41 b(M.A.)19 b(Sh)n(ubin.)g(Sp)r(ectral)g(theory)f(and)g (index)h(of)g(elliptic)g(op)r(erators)e(with)i(almost)f(p)r(erio)r(dic) h(co)r(e\016cien)n(ts.)f Fq(R)n(ussian)289 1436 y(Mathematic)l(al)32 b(Surveys)p Fw(,)27 b(34:109{157,)d(1979.)118 1536 y([25])41 b(B.)27 b(Simon.)h Fq(T)-6 b(r)l(ac)l(e)30 b(ide)l(als)h(and)g(their)f (applic)l(ations)p Fw(.)g(Cam)n(bridge)c(Univ)n(ersit)n(y)h(Press,)f (1979.)118 1636 y([26])41 b(B.)24 b(Simon.)h(Sc)n(hr\177)-42 b(odinger)22 b(semigroups.)h Fq(Bul)t(letin)k(of)h(the)f(A)n(meric)l (an)g(Mathematic)l(al)i(So)l(ciety)p Fw(,)c(7:447{526,)d(1982.)118 1735 y([27])41 b(B.)27 b(Simon.)h(Lifshits)g(tails)f(for)g(the)h (Anderson)f(mo)r(del.)h Fq(Journal)i(of)g(Statistic)l(al)g(Physics)p Fw(,)g(38:65{76,)24 b(1985.)118 1835 y([28])41 b(J.)28 b(Sj\177)-42 b(ostrand.)28 b(Microlo)r(cal)f(analysis)h(for)g(p)r(erio) r(dic)g(magnetic)g(Sc)n(hr\177)-42 b(odinger)27 b(equation)h(and)g (related)g(questions.)289 1934 y(In)e Fq(Micr)l(olo)l(c)l(al)k (analysis)g(and)f(applic)l(ations)p Fw(,)g(v)n(olume)c(1495)f(of)i Fq(L)l(e)l(ctur)l(e)h(Notes)h(in)h(Mathematics)p Fw(,)f(Berlin,)e (1991.)289 2034 y(Springer)g(V)-7 b(erlag.)118 2134 y([29])41 b(E.)26 b(Stein.)i Fq(Singular)i(inte)l(gr)l(als)f(and)h(Di\013er)l (entiability)g(pr)l(op)l(erties)h(of)f(functions)p Fw(.)d(Princeton)f (Univ)n(ersit)n(y)g(Press,)289 2233 y(Princeton,)g(N.J.,)i(1970.)118 2333 y([30])41 b(A.)22 b(Sznitman.)h(Lifshitz)f(tails)g(and)g(Wiener)f (sausages.)f(I.)i Fq(Journal)j(of)g(F)-6 b(unctional)25 b(A)n(nalysis)p Fw(,)f(94:223{246,)19 b(1990.)118 2433 y([31])41 b(A.)31 b(Sznitman.)h(Fluctuations)e(of)h(principal)g(eigen)n (v)-5 b(alues)30 b(and)g(random)g(scales.)g Fq(Communic)l(ations)k(in)f (Mathe-)289 2532 y(matic)l(al)d(Physics)p Fw(,)g(189:337{363,)23 b(1997.)118 2632 y([32])41 b(M.)28 b(T)-7 b(a)n(ylor.)25 b Fq(Partial)32 b(di\013er)l(ential)f(e)l(quations)p Fw(.)d(Springer,)e(New-Y)-7 b(ork,)27 b(Berlin,)g(1996.)331 2819 y(\(F)-7 b(r)n(\023)-39 b(ed)n(\023)g(eric)16 b(Klopp\))j Fv(D)1016 2812 y(\023)1016 2819 y(ep)-6 b(ar)g(tement)24 b(de)f(Ma)-6 b(th)1872 2812 y(\023)1872 2819 y(ema)g(tique,)25 b(Institut)d(Galil)2902 2812 y(\023)2902 2819 y(ee,)h(U.M.R)f(7539)g (C.N.R.S,)118 2918 y(Universit)521 2911 y(\023)521 2918 y(e)32 b(de)f(P)-7 b(aris-Nord,)30 b(A)-10 b(venue)32 b(J.-B.)e(Cl)1927 2911 y(\023)1927 2918 y(ement,)i(F-93430)f(Villet)-6 b(aneuse,)31 b(France)331 3018 y Fq(E-mail)f(addr)l(ess)7 b Fw(:)38 b Fa(klopp@math.univ-p)o(ar)o(is1)o(3.)o(fr)331 3187 y Fw(\(Leonid)22 b(P)n(astur\))g Fv(D)995 3180 y(\023)995 3187 y(ep)-6 b(ar)g(tement)28 b(de)e(Ma)-6 b(th)1858 3180 y(\023)1858 3187 y(ema)g(tique,)28 b(Universit)2693 3180 y(\023)2693 3187 y(e)f(P)-7 b(aris)25 b(VI)r(I,)g(2,)i(Pla)n(ce)f (Jussieu,)118 3287 y(F-75005)31 b(P)-7 b(aris,)31 b(France)331 3386 y Fq(E-mail)f(addr)l(ess)7 b Fw(:)38 b Fa(pastur@math.jussi)o(eu)o (.fr)1969 5454 y Fp(46)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9811040152360--