Content-Type: multipart/mixed; boundary="-------------0009201303100" This is a multi-part message in MIME format. ---------------0009201303100 Content-Type: text/plain; name="00-373.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-373.keywords" integrated density of states, random Schr\"odinger operators, Wegner estimate, multiscale analysis, localization, localization, indefinite single site potential ---------------0009201303100 Content-Type: application/postscript; name="body.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="body.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: body.dvi %%CreationDate: Wed Sep 20 18:28:46 2000 %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: E:\TEXMF\MIKTEX\BIN\DVIPS.EXE body %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.09.20:1828 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]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/IEn 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 IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/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 A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A 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/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 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 A 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 A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 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}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 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 A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 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/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/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 %%BeginProcSet: special.pro %! 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(ectrum)g(so)g(w)m(e)g(can)g(en)m(umerate)g(its)f(eigen)m(v)-5 b(alues)35 b Fp(\025)3188 4090 y Fo(i)3216 4075 y Fs(\()p Fp(H)3343 4039 y Fo(l)3335 4100 y(!)3385 4075 y Fs(\))h(in)120 4195 y(non-decreasing)d(order)f(and)h(coun)m(ting)f(m)m(ultiplicities.) 120 4384 y Ft(Theorem)37 b(2)120 4505 y Fg(There)29 b(exists)f(a)f (self-a)m(v)m(eraging)g(in)m(tegrated)g(densit)m(y)i(of)e(states)h (asso)s(ciated)f(with)h(the)g(fam-)120 4625 y(ily)j Fp(H)339 4640 y Fo(!)389 4625 y Fp(;)17 b(!)31 b Fq(2)d Fs(\012)p Fg(.)44 b(This)33 b(means)f(that)h(the)g(normalised)e(eigen)m(v)-5 b(alue)32 b(coun)m(ting)g(functions)1172 4827 y Fp(N)1260 4786 y Fo(l)1250 4852 y(!)1300 4827 y Fs(\()p Fp(E)6 b Fs(\))28 b(=)g Fp(l)1617 4786 y Fn(\000)p Fo(d)1712 4827 y Fs(#)p Fq(f)p Fp(i)p Fq(j)33 b Fp(\025)1994 4842 y Fo(i)2022 4827 y Fs(\()p Fp(H)2149 4786 y Fo(l)2141 4852 y(!)2191 4827 y Fs(\))28 b Fp(<)f(E)6 b Fq(g)927 b Fs(\(4\))120 5029 y Fg(of)32 b Fp(H)320 4993 y Fo(l)312 5054 y(!)395 5029 y Fg(con)m(v)m(erge)i(for)e(almost)f(all)g Fp(!)k Fg(to)e(a)f(limit)1399 5231 y Fp(N)10 b Fs(\()p Fp(E)c Fs(\))28 b(:=)41 b(lim)1799 5294 y Fo(l)q Fn(!1)1979 5231 y Fp(N)2067 5190 y Fo(l)2057 5256 y(!)2107 5231 y Fs(\()p Fp(E)6 b Fs(\))1154 b(\(5\))120 5463 y Fg(whic)m(h)33 b(is)f Fp(!)t Fg(-indep)s(enden)m(t.)1806 5712 y Fs(2)p eop %%Page: 3 3 3 2 bop 266 407 a Fs(W)-8 b(e)28 b(call)d Fp(N)10 b Fs(\()p Fq(\001)p Fs(\))27 b(the)g(in)m(tegrated)g(densit)m(y)h(of)f(states)g (\(IDS\))g(of)g Fp(H)2603 422 y Fo(!)2680 407 y Fs(and)g Fp(N)2952 371 y Fo(l)2942 431 y(!)2992 407 y Fs(\()p Fq(\001)p Fs(\))g(the)g(\(\014nite)120 527 y(v)m(olume\))32 b(IDS)g(of)g Fp(H)890 491 y Fo(l)882 552 y(!)933 527 y Fs(.)120 729 y Ft(Remark)37 b(3)120 850 y Fs(While)d(the)i(t)m(w)m(o) g(ab)s(o)m(v)m(e)g(Theorems)g(1)f(and)g(2)g(follo)m(w)e(from)h(the)i (general)f(theory)-8 b(,)36 b(one)g(can)120 970 y(ask)d(for)f(more)g (sp)s(eci\014c)h(sp)s(ectral)g(prop)s(erties)f(of)g(concrete)i(mo)s (dels)e Fp(H)2754 985 y Fo(!)2804 970 y Fp(;)17 b(!)30 b Fq(2)f Fs(\012,)k(e.g.)239 1198 y(1.)49 b(Whic)m(h)c(sp)s(ectral)g(t) m(yp)s(es)h(can)g(o)s(ccur)f(in)f Fp(\033)t Fs(\()p Fp(H)2105 1213 y Fo(!)2155 1198 y Fs(\)?)81 b(Can)46 b(di\013eren)m(t)f(sp)s (ectral)g(t)m(yp)s(es)364 1319 y(co)s(exist,)29 b(and)f(if)f(y)m(es,)j (can)e(the)g(supp)s(orts)h(of)e(the)i(corresp)s(onding)f(sp)s(ectral)f (measures)364 1439 y(o)m(v)m(erlap?)239 1642 y(2.)49 b(Can)34 b(something)f(b)s(e)h(said)g(ab)s(out)f(the)i(regularit)m(y)e (of)g(the)i(IDS)f Fp(N)44 b Fs(as)34 b(a)g(function)f(of)364 1763 y(the)e(energy)g Fp(E)6 b Fs(?)43 b(Is)31 b(it)e(H\177)-49 b(older)30 b(con)m(tin)m(uous;)i(or)e(its)g Fp(!)t Fs(-a)m(v)m(eraged)g (\014nite)g(b)s(o)m(x)h(analoga)364 1883 y Fl(E)12 b Fs([)p Fp(N)540 1847 y Fo(l)530 1908 y Fn(\017)575 1883 y Fs(]?)266 2112 y(Actually)26 b(b)s(oth)h(questions)g(are)g(related)f (to)h(the)g(transp)s(ort)f(prop)s(erties)h(of)f(the)i(medium)120 2232 y(mo)s(deled)38 b(b)m(y)h Fp(H)736 2247 y Fo(!)786 2232 y Fs(.)62 b(A)38 b(p)s(erfect)i(crystal)e(is)g(describ)s(ed)i(b)m (y)f(a)g(Sc)m(hr\177)-49 b(odinger)38 b(op)s(erator)g(with)120 2352 y(p)s(erio)s(dic)c(p)s(oten)m(tial.)50 b(It)36 b(has)g(purely)g (absolutely)e(con)m(tin)m(uous)j(sp)s(ectrum,)f(whic)m(h)g(re\015ects) 120 2473 y(its)28 b(go)s(o)s(d)f(electric)h(transp)s(ort)g(prop)s (erties.)42 b(In)28 b(con)m(trast)h(to)f(this,)h(it)f(has)g(b)s(een)h (pro)m(v)m(en)h(that)120 2593 y(random)j(p)s(erturbations)g(of)g(this)g (regular)g(structure)i(giv)m(e)e(rise)h(to)f(energy)i(in)m(terv)-5 b(als)32 b(with)120 2713 y(pure)j(p)s(oin)m(t)e(sp)s(ectrum.)49 b(This)35 b(corresp)s(onds)g(to)f(the)h(less)g(e\013ectiv)m(e)g(transp) s(ort)f(prop)s(erties)120 2834 y(of)e(random)g(media.)43 b(The)34 b(existence)g(of)f(pure)g(p)s(oin)m(t)f(sp)s(ectrum)h(in)f (this)h(con)m(text)h(is)e(called)120 2954 y Fh(lo)-5 b(c)g(alisation)p Fs(.)266 3075 y(A)44 b(p)s(o)m(w)m(erful)g(to)s(ol)d (for)i(pro)m(ving)h(lo)s(calisation)39 b(is)k(the)h(so-called)e Fh(multisc)-5 b(ale)45 b(analysis)120 3195 y Fs(\(MSA\),)29 b(an)g(induction)e(argumen)m(t)i(o)m(v)m(er)g(increasing)f(length)g (scales)h Fp(l)2709 3210 y Fo(k)2752 3195 y Fp(;)17 b(k)30 b Fq(2)f Fl(N)8 b Fs(.)49 b(This)28 b(tec)m(h-)120 3315 y(nique)41 b(w)m(as)g(\014rst)g(applied)e(b)m(y)i(F)-8 b(r\177)-49 b(ohlic)m(h)39 b(and)h(Sp)s(encer)i([FS83)o(])f(to)f(the)g (discretisation)f(of)120 3436 y(the)34 b(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erator)e(\(1\))i(and)f(underw)m(en)m(t)j(since)e (then)g(a)f(n)m(um)m(b)s(er)h(of)f(strengthen-)120 3556 y(ings)27 b([MS85,)h(FMSS85],)g(simpli\014cations)c([vDK89])j(and)h (adaptations)e(to)h(the)h(con)m(tin)m(uous)120 3676 y(mo)s(del)41 b(on)i Fp(L)632 3640 y Fm(2)672 3676 y Fs(\()p Fl(R)776 3640 y Fo(d)822 3676 y Fs(\),)j(whic)m(h)d(w)m(e)h(are)f(considering)f ([MH84,)h(Klo95a)n(,)g(CH94].)75 b(More)43 b(re-)120 3797 y(cen)m(tly)34 b(it)e(w)m(as)j(used)f(also)f(for)g(Hamiltonians)d 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y(constan)m(t)37 b(indep)s(enden)m(t)h(of)e(the)h(length)e(scale)i Fp(l)1929 4862 y Fo(k)2008 4847 y Fs(and)f Fp(\036)2259 4862 y Fo(k)2302 4847 y Fs(\()p Fp(x)p Fs(\))e Fq(2)h Fp(C)2645 4811 y Fm(2)2721 4847 y Fs(a)h(function)g(whic)m(h)g(is)120 4967 y(iden)m(tically)29 b(equal)i(to)g(0)g(for)g Fp(x)h Fs(with)e Fq(k)p Fp(x)p Fq(k)1654 4982 y Fn(1)1757 4967 y Fp(>)d(l)1889 4982 y Fo(k)1952 4967 y Fq(\000)20 b Fp(\016)35 b Fs(and)c(iden)m(tically)e(equal)i(to)g(one)h(for)f Fp(x)120 5088 y Fs(with)26 b Fq(k)p Fp(x)p Fq(k)491 5103 y Fn(1)593 5088 y Fp(<)i(l)726 5103 y Fo(k)777 5088 y Fq(\000)8 b Fs(2)p Fp(\016)t Fs(.)42 b(The)26 b(comm)m(utator)f Fp(W)14 b Fs(\()p Fp(\036)1966 5103 y Fo(k)2008 5088 y Fs(\))27 b(:=)h([)p Fq(\000)p Fs(\001)p Fp(;)17 b(\036)2491 5103 y Fo(k)2534 5088 y Fs(])28 b(:=)f Fq(\000)p Fs(\(\001)p Fp(\036)2973 5103 y Fo(k)3017 5088 y Fs(\))8 b Fq(\000)g Fs(2\()p Fq(r)p Fp(\036)3376 5103 y Fo(k)3419 5088 y Fs(\))p Fq(r)120 5208 y Fs(is)37 b(a)g(lo)s(cal)e(op)s(erator)h(acting) g(on)i(functions)f(whic)m(h)h(liv)m(e)e(on)h(a)g(ring)f(of)h(width)g Fp(\016)k Fs(near)d(the)120 5329 y(b)s(oundary)c(of)f(\003)740 5344 y Fo(k)811 5329 y Fs(:=)c(\003)1011 5344 y Fo(l)1032 5356 y Fi(k)1074 5329 y Fs(.)46 b(W)-8 b(e)33 b(sa)m(y)i(that)e(a)g (pair)f(\()p Fp(!)t(;)17 b Fs(\003)2195 5344 y Fo(k)2237 5329 y Fs(\))29 b Fq(2)g Fs(\012)23 b Fq(\002)g(B)s Fs(\()p Fl(R)2765 5292 y Fo(d)2811 5329 y Fs(\))33 b(is)g Fp(m)p Fh(-r)-5 b(e)g(gular)33 b Fs(for)1806 5712 y(3)p eop %%Page: 4 4 4 3 bop 120 407 a Fs(a)32 b(giv)m(en)h(energy)h Fp(E)6 b Fs(,)32 b(if)809 614 y(sup)823 696 y Fo(\017)p Fn(6)p Fm(=0)972 614 y Fq(k)p Fp(W)14 b Fs(\()p Fp(\036)1224 629 y Fo(k)1266 614 y Fs(\)\()p Fp(H)1431 573 y Fo(l)1423 638 y(!)1495 614 y Fq(\000)23 b Fp(E)28 b Fs(+)22 b Fp(\017i)p Fs(\))1903 573 y Fn(\000)p Fm(1)1998 614 y Fp(\037)2059 629 y Fo(l)2080 641 y Fi(k)2118 629 y Fo(=)p Fm(3)2193 614 y Fq(k)2243 630 y Fn(L)p Fm(\()p Fo(L)2367 612 y Ff(2)2401 630 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1250 y(in)m(terv)-5 b(al)31 b Fp(I)36 b Fq(\032)28 b Fl(R)43 b Fs(in)32 b(whic)m(h)h(w)m(e) h(w)m(an)m(t)f(to)g(pro)m(v)m(e)g(the)g(existence)i(of)d(pure)h(p)s (oin)m(t)f(sp)s(ectrum.)266 1370 y(\(H1\))h Fq(,)f Fs(There)i(exist)e (constan)m(ts)i Fp(Q)1650 1385 y Fm(1)1690 1370 y Fp(;)17 b(m)28 b Fq(2)p Fs(]0)p Fp(;)17 b Fq(1)p Fs([)p Fp(;)g(q)31 b(>)c Fs(0)33 b(suc)m(h)h(that)1026 1577 y Fl(P)p Fq(f)p Fp(!)t Fq(j)17 b Fs(\()p Fp(!)t(;)g Fs(\003)1460 1592 y Fo(Q)1516 1601 y Ff(1)1553 1577 y Fs(\))32 b(is)g Fp(m)p Fs(-regular)o Fq(g)c(\025)g Fs(1)22 b Fq(\000)h Fp(Q)2568 1530 y Fo(q)2568 1601 y Fm(1)2607 1577 y Fp(:)781 b Fs(\(7\))266 1784 y(\(H2\))33 b Fq(,)f Fs(There)i(exist)e(constan)m(ts)i Fp(Q)1650 1799 y Fm(2)1690 1784 y Fp(;)17 b(\021)1782 1799 y Fm(0)1821 1784 y Fp(;)g Fq(2)p Fs(]0)p Fp(;)g Fq(1)p Fs([)32 b(suc)m(h)i(that)1066 1991 y Fl(P)p Fq(f)p Fp(!)t Fq(j)17 b Fp(d)p Fs(\()p Fp(\033)t Fs(\()p Fp(H)1552 2006 y Fo(!)1602 1991 y Fq(j)1630 2006 y Fm(\003)1682 1991 y Fs(\))p 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b(the)f(MSA)h(has)f(b)s (een)g(accomplished,)i(one)e(pro)s(ceeds)h(to)e(pro)m(v)m(e)i(lo)s (calisation)18 b(using)120 3826 y(the)48 b(sp)s(ectral)g(a)m(v)m (eraging)g(tec)m(hnique)h(and)f(expansion)g(in)g(generalised)f (eigenfunctions,)120 3947 y(cf.)33 b(Sections)g(7)f(and)h(8)f(in)g ([KSS98b].)266 4067 y(Actually)i(for)g(the)h(MSA)h(a)e(v)-5 b(ariet)m(y)34 b(of)h(w)m(eak)m(er)h(b)s(ounds)g(than)e(\(8\))h(is)f (su\016cien)m(t.)51 b(It)34 b(is)120 4188 y(enough)f(to)f(kno)m(w)266 4308 y(\(H2'\))h Fq(,)f Fs(There)i(exist)f(constan)m(ts)g Fp(Q)1677 4323 y Fm(2)1717 4308 y Fp(;)17 b(\021)1809 4323 y Fm(0)1848 4308 y Fp(;)g(a;)g(b)28 b Fq(2)p Fs(]0)p Fp(;)17 b Fq(1)p Fs([)32 b(suc)m(h)i(that)1028 4515 y Fl(P)p Fq(f)p Fp(!)t Fq(j)17 b Fp(d)p Fs(\()p Fp(\033)t Fs(\()p Fp(H)1514 4530 y Fo(!)1564 4515 y Fq(j)1592 4530 y Fm(\003)1644 4515 y Fs(\))p Fp(;)g(E)6 b Fs(\))28 b Fq(\024)g Fp(\021)t Fq(g)f(\024)h Fp(C)2279 4530 y Fo(W)2382 4515 y Fp(\021)2434 4474 y Fo(a)2475 4515 y Fq(j)p Fs(\003)p Fq(j)2599 4474 y Fo(b)3415 4515 y Fs(\(9\))120 4722 y(for)34 b(all)f(b)s(o)m(xes)j(\003)f(with)f(side)h(length)f(larger)g(than)h Fp(Q)2091 4737 y Fm(2)2165 4722 y Fs(and)g(all)e Fp(\021)i Fq(\024)d Fp(\021)2735 4737 y Fm(0)2775 4722 y Fs(.)50 b(Inequalit)m(y)35 b(\(9\))f(is)120 4842 y(implied)h(b)m(y)k(the)f (H\177)-49 b(older)38 b(con)m(tin)m(uit)m(y)g(of)f(the)h(a)m(v)m (eraged)i(\014nite)d(v)m(olume)g(IDS,)h(men)m(tioned)120 4963 y(in)32 b(Remark)g(3:)876 5083 y Fl(E)12 b Fq(f)p Fp(N)1075 5042 y Fo(l)1064 5108 y Fn(\017)1110 5083 y Fs(\()p Fp(E)28 b Fs(+)22 b Fp(\021)t Fs(\))g Fq(\000)g Fp(N)1645 5042 y Fo(l)1635 5108 y Fn(\017)1675 5083 y Fs(\()p Fp(E)28 b Fq(\000)23 b Fp(\021)t Fs(\))p Fq(g)k(\024)h Fp(C)2255 5098 y Fo(W)2357 5083 y Fp(\021)2409 5042 y Fo(a)2451 5083 y Fq(j)p Fs(\003)2547 5098 y Fo(l)2572 5083 y Fq(j)2600 5042 y Fo(b)p Fn(\000)p Fm(1)2757 5083 y Fp(:)583 b Fs(\(10\))p 120 5172 1368 4 v 232 5233 a Fe(1)269 5264 y Fd(Eac)n(h)26 b(h)n(yp)r(othesis)h(in)g(Remark)g(4)f 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b(W)-8 b(e)30 b(discuss)h(brie\015y)f (earlier)f(results)h(on)f(W)-8 b(egner)120 1049 y(estimates)32 b(in)g(non-monotonous)g(situations)f(and)i(state)g(our)f(main)f (theorem.)266 1170 y(F)-8 b(or)36 b(a)h(qualitativ)m(e)e(discussion)i (assume)h(that)e Fp(V)2095 1185 y Fo(!)2182 1170 y Fs(dep)s(ends)j(on)d (some)h(comp)s(onen)m(t)g Fp(\030)3498 1185 y Fo(k)120 1290 y Fs(of)f(the)h(\\randomness")g(in)f(a)g(monotone)g(increasing)f (w)m(a)m(y)j(on)f(some)f(subset)i Fp(A)3123 1305 y Fo(k)3201 1290 y Fq(\032)c Fl(R)3378 1254 y Fo(d)3461 1290 y Fs(of)120 1410 y(con\014guration)f(space.)50 b(This)34 b(is)g(e.g.)h(the)f(case)h (for)f(the)h(p)s(oten)m(tial)d(\(2\))i(if)f Fp(\030)2925 1425 y Fo(k)2998 1410 y Fs(=)d Fp(!)3165 1425 y Fo(k)3208 1410 y Fp(;)17 b(k)33 b Fq(2)e Fl(Z)3502 1374 y Fo(d)120 1531 y Fs(and)47 b(w)m(e)h(consider)f(the)g(particular)e(single)h(site) h(p)s(oten)m(tial)e Fp(u)52 b Fs(=)f Fp(\037)2724 1546 y Fm(\003)2773 1555 y Ff(1)2812 1531 y Fs(.)87 b(\(In)47 b(the)g(sequel)120 1651 y(w)m(e)39 b(will)d(abbreviate)i(the)g(c)m (haracteristic)g(function)f Fp(\037)2170 1666 y Fm(\003)2219 1675 y Ff(1)2254 1666 y Fm(+)p Fo(k)2390 1651 y Fs(of)g(the)i(unit)e (cub)s(e)i(\003)3195 1666 y Fm(1)3260 1651 y Fs(+)25 b Fp(k)41 b Fs(at)120 1772 y(site)47 b Fp(k)56 b Fq(2)d Fl(Z)609 1735 y Fo(d)694 1772 y Fs(b)m(y)48 b Fp(\037)905 1787 y Fo(k)947 1772 y Fs(.\))88 b(Because)48 b(of)f(the)h(linear)d (coupling)h Fp(!)2547 1787 y Fo(k)2606 1772 y Fp(u)p Fs(\()p Fq(\001)31 b(\000)i Fp(k)s Fs(\))47 b(the)g(function)120 1892 y Fp(!)181 1907 y Fo(k)251 1892 y Fq(7!)27 b Fp(V)435 1907 y Fo(!)486 1892 y Fq(j)514 1907 y Fo(A)567 1919 y Fi(k)608 1892 y Fp(;)17 b(A)725 1907 y Fo(k)796 1892 y Fs(=)27 b(\003)967 1907 y Fm(1)1021 1892 y Fs(+)14 b Fp(k)31 b Fs(is)e(then)g(strictly)f(increasing.)41 b(Additionally)26 b(w)m(e)k(assume)f(that)265 2092 y Fq(\017)49 b Fs(the)32 b Fp(A)604 2107 y Fo(k)679 2092 y Fs(form)f(a)h(partition)e(of)h Fl(R)1572 2056 y Fo(d)1651 2092 y 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b(the)g(pro)s(of)f(analyticit)m(y)f(assumes)i(the)g(role)f(usually)g (pla)m(y)m(ed)h(b)m(y)120 1009 y(monotonicit)m(y)-8 b(.)42 b(This)32 b(again)f(implies)g(piecewise)i(monotonicit)m(y)-8 b(.)266 1129 y(W)g(e)44 b(state)g(no)m(w)h(our)e(main)f(result.)76 b(It)44 b(is)f(v)-5 b(alid)42 b(only)h(for)g(a)g(quite)h(restricted)g (class)120 1249 y(of)f(single)f(site)g(p)s(oten)m(tials)g Fp(u)h Fs(but)g(in)f(an)m(y)i(dimension)e Fp(d)g Fs(and)i(for)e(all)f (b)s(ounded)j(energy)120 1370 y(in)m(terv)-5 b(als.)42 b(So)29 b(it)g(ma)m(y)h(b)s(e)g(used)h(as)f(an)f(ingredien)m(t)h(of)f (a)h(pro)s(of)e(of)i(lo)s(calisation)c(at)j(in)m(ternal)120 1490 y(sp)s(ectral)j(band)h(edges)h(of)e(m)m(ultidimensional)c(random)k (Sc)m(hr\177)-49 b(odinger)32 b(op)s(erators.)120 1682 y Ft(Theorem)37 b(5)120 1803 y Fg(Let)31 b Fs(\000)c Fq(\032)h Fl(Z)555 1767 y Fo(d)624 1803 y Fg(b)s(e)i(a)g(\014nite)g (set)i(and)e Fp(\013)e Fs(=)g Fq(f)p Fp(\013)1724 1818 y Fo(k)1767 1803 y Fp(;)17 b(k)30 b Fq(2)e Fs(\000)p 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y(!)395 2600 y Fg(w)m(e)h(ha)m(v)m(e)957 2720 y Fl(E)1034 2639 y Fk(\002)1081 2720 y Fp(N)1169 2679 y Fo(l)1159 2745 y Fn(\017)1199 2720 y Fs(\()p Fp(E)1309 2735 y Fm(2)1349 2720 y Fs(\))22 b Fq(\000)g Fp(N)1596 2679 y Fo(l)1586 2745 y Fn(\017)1626 2720 y Fs(\()p Fp(E)1736 2735 y Fm(2)1775 2720 y Fs(\))1813 2639 y Fk(\003)1883 2720 y Fq(\024)28 b Fg(const)33 b Fq(j)p Fp(E)2343 2735 y Fm(2)2404 2720 y Fq(\000)23 b Fp(E)2576 2735 y Fm(1)2616 2720 y Fq(j)32 b Fp(:)664 b Fs(\(12\))266 2905 y(The)34 b(theorem)e(is)g(pro)m(v)m(en)i(in)e(Sections)h(3)f(to)h(5.)43 b(W)-8 b(e)33 b(call)e Fp(\013)i Fs(the)g Fh(c)-5 b(onvolution)34 b(ve)-5 b(ctor)p Fs(.)120 3097 y Ft(Corollary)36 b(6)120 3217 y Fg(Let)47 b Fp(p)52 b Fs(=)g(2)47 b Fg(if)e Fp(d)52 b Fq(\024)h Fs(3)p Fg(,)d Fp(p)i(>)g Fs(2)46 b Fg(if)g Fp(d)52 b Fs(=)g(4)47 b Fg(and)g Fp(p)52 b Fq(\025)g Fp(d=)p Fs(2)46 b Fg(if)g Fp(d)52 b Fq(\025)g Fs(5)p Fg(.)86 b(By)48 b(the)f(same)120 3337 y(pro)s(of)38 b(Theorem)g(5)g 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b(E)3192 4866 y Fm(2)3232 4851 y Fs([)39 b(b)s(e)g(an)120 4972 y(op)s(en)30 b(energy)h(in)m(terv)-5 b(al,)30 b Fp(P)1117 4935 y Fo(l)1103 4996 y(!)1153 4972 y Fs(\()p Fp(I)8 b Fs(\))30 b(the)g(sp)s(ectral)g(pro)5 b(jection)30 b(of)f Fp(H)2497 4935 y Fo(l)2489 4996 y(!)2570 4972 y Fs(on)m(to)h(the)g(in)m(terv)-5 b(al)29 b Fp(I)38 b Fs(and)120 5092 y(let)33 b(T)-8 b(r\()p Fp(A)p Fs(\))34 b(denote)h(the)f(trace)g(of)g(an)g(op)s(erator)f Fp(A)p Fs(.)47 b(F)-8 b(or)33 b(the)i(\014nite)e(v)m(olume)h(IDS)f(and)h(an)m (y)120 5212 y Fp(\017)28 b(>)g Fs(0)k(w)m(e)i(ha)m(v)m(e)878 5444 y Fl(E)956 5364 y Fk(\002)1003 5444 y Fp(N)1091 5403 y Fo(l)1081 5469 y(!)1132 5444 y Fs(\()p Fp(E)1242 5459 y Fm(2)1281 5444 y Fs(\))22 b Fq(\000)h Fp(N)1529 5403 y Fo(l)1519 5469 y(!)1569 5444 y Fs(\()p Fp(E)1679 5459 y Fm(1)1741 5444 y Fs(+)f Fp(\017)p Fs(\))1916 5364 y Fk(\003)1986 5444 y Fq(\024)2112 5377 y Fs(1)p 2101 5421 72 4 v 2101 5513 a Fp(l)2132 5484 y Fo(d)2182 5444 y Fl(E)2260 5364 y Fk(\002)2307 5444 y Fs(T)-8 b(r)17 b Fp(P)2501 5403 y Fo(l)2487 5469 y(!)2537 5444 y Fs(\()p Fp(I)8 b Fs(\))2664 5364 y Fk(\003)2754 5444 y Fp(:)586 b Fs(\(14\))1806 5712 y(6)p eop %%Page: 7 7 7 6 bop 120 407 a Fs(Let)297 382 y(~)288 407 y(\003)27 b(:=)h(\003)7 b Fq(\\)g Fl(Z)731 371 y Fo(d)795 407 y Fs(b)s(e)26 b(the)f(lattice)f(p)s(oin)m(ts)h(in)g(\003.)41 b(As)26 b(in)e([FHLM97,)j(Section)e(3])g(w)m(e)i(estimate)916 640 y Fl(E)993 559 y Fk(\002)1041 640 y Fs(T)-8 b(r)16 b Fp(P)1234 599 y Fo(l)1220 665 y(!)1271 640 y Fs(\()p Fp(I)8 b Fs(\))1398 559 y Fk(\003)1467 640 y Fq(\024)28 b Fp(e)1617 599 y Fo(E)1669 608 y Ff(2)1707 640 y Fp(C)1777 655 y Fo(V)1855 545 y Fk(X)1863 773 y Fo(j)t Fn(2)1950 756 y Fm(~)1943 773 y(\003)2032 555 y Fk(\015)2032 615 y(\015)2087 640 y Fl(E)2165 559 y Fk(\002)2212 640 y Fp(\037)2273 655 y Fo(j)2309 640 y Fp(P)2386 599 y Fo(l)2372 665 y(!)2423 640 y Fs(\()p Fp(I)8 b Fs(\))p Fp(\037)2611 655 y Fo(j)2647 559 y Fk(\003)2689 555 y(\015)2689 615 y(\015)3367 640 y Fs(\(15\))120 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b Fq(\000)h Fs(\000)28 b(=)h Fq(f)p Fp(\025)22 b Fq(\000)h Fp(\015)5 b Fq(j)32 b Fp(\025)d Fq(2)1521 5438 y Fs(~)1512 5463 y(\003)o Fp(;)17 b(\015)33 b Fq(2)c Fs(\000)p Fq(g)k Fs(the)g(set)h(of)f(all)e(sites)i Fp(k)j Fs(whose)e(asso)s(ciated)1806 5712 y(7)p eop %%Page: 8 8 8 7 bop 120 407 a Fs(p)s(oten)m(tials)33 b Fp(u)p Fs(\()p Fq(\001)22 b(\000)i Fp(k)s Fs(\))34 b(in\015uence)h(the)g(p)s(oten)m (tial)e Fp(V)1992 422 y Fo(!)2076 407 y Fs(within)g(the)i(cub)s(e)g (\003.)48 b(The)35 b(truncated)120 527 y(matrix)42 b Fp(A)521 542 y Fm(\003)620 527 y Fs(=)j Fq(f)p Fp(\013)853 542 y Fo(j)t Fn(\000)p Fo(k)983 527 y Fq(g)1033 544 y Fo(j;k)r Fn(2)p Fm(\003)1217 525 y Ff(+)1313 527 y Fs(acts)f(on)f Fp(!)1732 542 y Fm(\003)1831 527 y Fs(=)i Fq(f)p Fp(!)2063 542 y Fo(k)2106 527 y Fp(;)17 b(k)48 b Fq(2)e Fs(\003)2429 491 y Fm(+)2488 527 y Fq(g)d Fs(to)g(giv)m(e)g(a)g(v)m(ector)h Fp(\021)3365 542 y Fm(\003)3464 527 y Fs(=)120 648 y Fq(f)p Fp(\021)218 663 y Fo(k)261 648 y Fp(;)17 b(k)30 b Fq(2)e 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b(v)m(ectors)h Fp(A)736 1435 y Fm(\003)789 1420 y Fs(,)i(resp)s(ectiv)m(ely)e Fp(!)1467 1435 y Fm(\003)1519 1420 y Fp(;)17 b(\021)1611 1435 y Fm(\003)1664 1420 y Fs(,)47 b(and)d(write)2214 1395 y(~)2199 1420 y Fp(V)2256 1435 y Fo(\021)2342 1420 y Fs(for)f Fp(V)2559 1435 y Fo(!)2656 1420 y Fs(=)j Fp(V)2835 1437 y Fo(A)2888 1418 y Fa(\000)p Ff(1)2971 1437 y Fo(\021)3056 1420 y Fs(and)3282 1395 y(~)3257 1420 y Fp(H)3346 1384 y Fo(l)3338 1444 y(\021)3423 1420 y Fs(for)120 1540 y Fp(H)209 1504 y Fo(l)201 1565 y(!)279 1540 y Fs(=)27 b Fp(H)471 1504 y Fo(l)463 1571 y(A)516 1552 y Fa(\000)p Ff(1)599 1571 y Fo(\021)673 1540 y Fs(for)32 b(the)h(op)s(erators)g(living)d(on)i (\003)c(=)f(\003)2092 1555 y Fo(l)2118 1540 y Fs(.)43 b(F)-8 b(or)32 b Fp(x)c Fq(2)g Fs(\003)33 b(w)m(e)g(ha)m(v)m(e)1138 1796 y Fp(V)1195 1811 y Fo(!)1245 1796 y Fs(\()p Fp(x)p Fs(\))84 b(=)1639 1702 y Fk(X)1619 1919 y Fo(k)r Fn(2)p Fm(\003)1754 1900 y Ff(+)1820 1796 y Fp(!)1881 1811 y Fo(k)1940 1702 y Fk(X)1956 1914 y Fo(l)q 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bop 120 407 a Fr(5)161 b(Bounds)52 b(on)h(the)g(densit)l(y)120 626 y Fs(The)32 b(structure)h(of)d(the)i(random)e(op)s(erator)1785 601 y(~)1760 626 y Fp(H)1849 590 y Fo(l)1841 650 y(\021)1914 626 y Fs(in)g(the)i(new)g(v)-5 b(ariables)30 b(mak)m(es)i(it)e(easier)h (to)120 746 y(estimate)h(the)h(exp)s(ectation)f(v)-5 b(alue)32 b(on)h(the)g(rhs)g(of)f(\(16\))1142 951 y Fl(E)1220 870 y Fk(\002)1267 951 y Fq(h)p Fp(\036;)17 b(\037)1469 966 y Fo(j)1504 951 y Fs(\()p Fp(H)1631 910 y Fo(l)1623 976 y(!)1696 951 y Fq(\000)22 b Fp(E)29 b Fs(+)22 b Fp(\016)t(i)p Fs(\))2112 910 y Fn(\000)p Fm(1)2206 951 y Fp(\037)2267 966 y Fo(j)2304 951 y Fp(\036)p Fq(i)2401 870 y Fk(\003)2491 951 y Fp(:)849 b Fs(\(24\))120 1170 y(W)-8 b(e)37 b(single)e(out)h(the) h(lattice)e(p)s(oin)m(t)h Fp(j)k Fq(2)1667 1145 y Fs(~)1658 1170 y(\003)c(and)g(consider)h(the)g(one-parameter)f(family)d(of)120 1290 y(op)s(erators)1539 1411 y Fp(\021)1587 1426 y Fo(j)1651 1411 y Fq(7!)1804 1385 y Fs(~)1778 1411 y Fp(H)1867 1369 y Fo(l)1864 1435 y Fm(^)-40 b Fo(\021)1901 1411 y Fs(\()p Fp(\021)1987 1426 y Fo(j)2024 1411 y Fs(\))32 b Fp(;)1246 b Fs(\(25\))120 1595 y(where)38 b(^)-56 b Fp(\021)32 b Fs(=)27 b Fq(f)p Fp(\021)680 1610 y Fo(k)723 1595 y Fp(;)17 b(k)30 b Fq(2)f Fs(\003)1011 1559 y Fm(+)1087 1595 y Fq(n)17 b(f)p Fp(j)6 b Fq(gg)29 b Fs(and)1592 1570 y(~)1567 1595 y Fp(H)1656 1559 y Fo(l)1653 1621 y Fm(^)-40 b Fo(\021)1689 1595 y Fs(\()p Fp(\021)1775 1610 y Fo(j)1812 1595 y Fs(\))28 b(=)2006 1570 y(~)1981 1595 y Fp(H)2070 1559 y Fo(l)2062 1620 y(\021)2104 1595 y Fs(.)43 b(Similarly)26 b(w)m(e)31 b(write)f Fp(k)3030 1610 y Fm(^)-40 b Fo(\021)3067 1595 y Fs(\()p Fp(\021)3153 1610 y Fo(j)3189 1595 y Fs(\))28 b(=)f Fp(k)s Fs(\()p Fp(\021)t Fs(\))120 1716 y(for)32 b(the)h(common)e(densit)m(y)-8 b(.)44 b(Lo)s(cally)30 b(on)j(the)f(cub)s(e)h(\003)2136 1731 y Fm(1)2197 1716 y Fs(+)22 b Fp(j)38 b Fs(the)33 b(dep)s(endence)i(\(25\))c(on)i(the)120 1836 y(parameter)i Fp(\021)637 1851 y Fo(j)710 1836 y 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Fm(\003)1049 2443 y Ff(+)1105 2466 y Fn(j)1157 2502 y Fq(!)27 b Fl(R)1350 2466 y Fn(j)p Fm(\003)1419 2443 y Ff(+)1475 2466 y Fn(j\000)p Fm(1)1589 2502 y Fs(,)1029 2707 y Fp(\031)1104 2626 y Fk(\002)1146 2707 y Fq(f)p Fp(\021)1244 2722 y Fo(k)1286 2707 y Fp(;)17 b(k)31 b Fq(2)d Fs(\003)1574 2666 y Fm(+)1632 2707 y Fq(g)1682 2626 y Fk(\003)1751 2707 y Fs(=)g Fq(f)p Fp(\021)1953 2722 y Fo(k)1996 2707 y Fp(;)17 b(k)30 b Fq(2)e Fs(\003)2283 2666 y Fm(+)2364 2707 y Fq(n)22 b(f)p Fp(j)6 b Fq(gg)735 b Fs(\(27\))120 2912 y(maps)32 b Fp(M)44 b Fs(on)m(to)596 3091 y(^)559 3116 y Fp(M)94 b Fs(=)83 b Fp(\031)t Fs(\()p Fp(M)10 b Fs(\))2222 b(\(28\))747 3319 y(=)906 3178 y Fk(\032)981 3319 y Fp(x)28 b Fs(=)1168 3238 y Fk(\010)1226 3319 y Fp(x)1281 3334 y Fo(k)1324 3319 y Fp(;)17 b(k)30 b Fq(2)e Fs(\003)1611 3278 y Fm(+)1692 3319 y Fq(n)22 b(f)p Fp(j)6 b Fq(g)1910 3238 y Fk(\011)1996 3319 y Fq(2)28 b Fl(R)2156 3278 y Fn(j)p Fm(\003)2225 3254 y Ff(+)2281 3278 y Fn(j\000)p Fm(1)2395 3174 y Fk(\014)2395 3234 y(\014)2395 3294 y(\014)2395 3354 y(\014)906 3621 y Fq(9)p Fp(y)j Fq(2)d Fl(R)39 b Fs(:)28 b Fp(\021)j Fs(=)1472 3511 y Fk(n)1538 3621 y Fp(\021)1586 3636 y Fo(k)1657 3621 y Fs(=)1760 3451 y Fk(\()1857 3548 y Fp(x)1912 3563 y Fo(k)1955 3548 y Fp(;)17 b(k)31 b Fq(6)p Fs(=)c Fp(j)1904 3693 y(y)s(;)17 b(k)31 b Fs(=)c Fp(j)2230 3451 y Fk(\))2327 3621 y Fp(;)17 b(k)30 b Fq(2)e Fs(\003)2614 3580 y Fm(+)2673 3511 y Fk(o)2767 3621 y Fq(2)g Fp(M)2965 3481 y Fk(\033)3073 3621 y Fp(:)120 3930 y Fs(F)-8 b(or)32 b(a)g(giv)m(en)40 b(^)-56 b Fp(\021)31 b Fs(=)d Fq(f)p Fp(\021)912 3945 y Fo(k)954 3930 y Fp(;)17 b(k)31 b Fq(2)d Fs(\003)1242 3894 y Fm(+)1323 3930 y Fq(n)22 b(f)p Fp(j)6 b Fq(gg)27 b(2)h Fl(R)1778 3894 y Fn(j)p Fm(\003)1847 3870 y Ff(+)1903 3894 y Fn(j\000)p Fm(1)2050 3930 y Fs(w)m(e)33 b(de\014ne)794 4222 y Fp(M)893 4237 y Fm(^)-40 b Fo(\021)958 4222 y Fs(=)1061 4051 y Fk(\()1141 4222 y Fp(y)31 b Fq(2)d Fl(R)5 b Fq(j)39 b Fp(\021)31 b Fs(=)1629 4051 y Fk(\()1726 4148 y Fp(\021)1774 4163 y Fo(k)1817 4148 y Fp(;)17 b(k)31 b Fq(2)d Fs(\003)2105 4107 y Fm(+)2186 4148 y Fq(n)21 b(f)p Fp(j)6 b Fq(g)1766 4294 y Fp(y)s(;)17 b(k)31 b Fs(=)c Fp(j)2403 4051 y Fk(\))2511 4222 y Fq(2)h Fp(M)2709 4051 y Fk(\))2839 4222 y Fp(:)501 b Fs(\(29\))120 4518 y(whic)m(h)32 b(is)g(the)g(set)h(of)e(the)h Fp(j)6 b Fs(th)32 b(comp)s(onen)m(ts)h(of)e(the)i(v)m(ectors)g(in)e Fp(\031)2579 4482 y Fn(\000)p Fm(1)2673 4518 y Fs(\()7 b(^)-56 b Fp(\021)t Fs(\))21 b Fq(\\)g Fp(M)10 b Fs(.)44 b(Since)32 b Fp(M)42 b Fs(is)120 4638 y(con)m(v)m(ex,)499 4613 y(^)463 4638 y Fp(M)d Fs(and)29 b Fp(M)881 4653 y Fm(^)-40 b Fo(\021)946 4638 y Fs(are)28 b(this)g(to)s(o,)h(moreo)m(v) m(er)f Fp(M)1999 4653 y Fm(^)-40 b Fo(\021)2065 4638 y Fs(is)27 b(an)i(in)m(terv)-5 b(al.)40 b(By)29 b(F)-8 b(ubini's)27 b(theorem)120 4759 y(w)m(e)34 b(ha)m(v)m(e)g(for)e(\(24\)) f(the)i(upp)s(er)h(b)s(ound)717 5051 y(sup)728 5133 y Fo(\016)r(>)p Fm(0)880 4906 y Fk(\014)880 4966 y(\014)880 5026 y(\014)880 5085 y(\014)913 4915 y(Z)969 5140 y Fo(M)1064 5051 y Fp(d\021)20 b(k)s Fs(\()p Fp(\021)t Fs(\))d Fp(s)p Fs(\()p Fp(\021)t Fs(\))1556 4906 y Fk(\014)1556 4966 y(\014)1556 5026 y(\014)1556 5085 y(\014)1616 5051 y Fq(\024)1721 4915 y Fk(Z)1802 5124 y Fm(^)1777 5140 y Fo(M)1872 5051 y Fp(d)7 b Fs(^)-56 b Fp(\021)37 b Fs(sup)2019 5133 y Fo(\016)r(>)p Fm(0)2171 4876 y Fk(\014)2171 4936 y(\014)2171 4996 y(\014)2171 5056 y(\014)2171 5115 y(\014)2204 4915 y(Z)2260 5140 y Fo(M)2333 5152 y Ff(^)-35 b Fi(\021)2386 5051 y Fp(d\021)2485 5066 y Fo(j)2538 5051 y Fp(k)s Fs(\()p Fp(\021)t Fs(\))17 b Fp(s)p Fs(\()p Fp(\021)t Fs(\))2911 4876 y Fk(\014)2911 4936 y(\014)2911 4996 y(\014)2911 5056 y(\014)2911 5115 y(\014)3367 5051 y Fs(\(30\))266 5342 y(W)-8 b(e)47 b(b)s(ound)g(the)g(rhs)g(of)f(\(30\))g(b)m(y)h(the)g (follo)m(wing)d(prop)s(osition)g(whic)m(h)j(is)f([CHM96)q(,)120 5463 y(Theorem)33 b(1.1])f(adopted)h(to)f(our)h(mo)s(del.)1806 5712 y(9)p eop %%Page: 10 10 10 9 bop 120 376 a Ft(Prop)s(osition)36 b(7)120 497 y Fg(Let)320 472 y Fs(~)295 497 y Fp(H)384 461 y Fo(l)376 521 y(\021)418 497 y Fp(;)17 b(s)p Fs(\()p Fp(\021)t Fs(\))p Fp(;)g(M)779 512 y Fm(^)-40 b Fo(\021)847 497 y Fg(and)33 b Fp(k)1093 512 y Fm(^)-40 b Fo(\021)1130 497 y Fs(\()p Fp(\021)1216 512 y Fo(j)1252 497 y Fs(\))33 b Fg(b)s(e)f(de\014ned)i(as)f(ab)s(o)m(v)m(e.)45 b(Then)33 b(w)m(e)h(ha)m(v)m(e)580 800 y Fs(sup)591 882 y Fo(\016)r(>)p Fm(0)743 625 y Fk(\014)743 685 y(\014)743 745 y(\014)743 805 y(\014)743 865 y(\014)776 664 y(Z)832 890 y Fo(M)905 902 y Ff(^)-35 b Fi(\021)958 800 y Fp(d\021)1057 815 y Fo(j)1110 800 y Fp(k)s Fs(\()p Fp(\021)t Fs(\))17 b Fp(s)p Fs(\()p Fp(\021)t Fs(\))1483 625 y Fk(\014)1483 685 y(\014)1483 745 y(\014)1483 805 y(\014)1483 865 y(\014)1543 800 y Fq(\024)28 b(k)17 b Fp(k)1771 815 y Fm(^)-40 b Fo(\021)1807 800 y Fs(\()p Fq(\001)p Fs(\))22 b(+)g Fq(j)p Fp(k)2113 759 y Fn(0)2115 824 y Fm(^)-40 b Fo(\021)2151 800 y Fs(\()p Fq(\001)p Fs(\))p Fq(j)22 b Fs(+)g(2)p Fq(j)p Fp(k)2534 759 y Fn(00)2536 824 y Fm(^)-40 b Fo(\021)2576 800 y Fs(\()p Fq(\001)p Fs(\))p Fq(j)17 b(k)2775 816 y Fo(L)2823 798 y Ff(1)2856 816 y Fm(\()p Fo(M)2956 828 y Ff(^)-35 b Fi(\021)2989 816 y Fm(\))3053 800 y Fp(:)287 b Fs(\(31\))120 1085 y(Using)35 b(F)-8 b(ubini's)35 b(theorem)g(in)g (the)h(rev)m(erse)h(direction)e(and)g(transforming)f(bac)m(k)j(to)e (the)h Fp(!)120 1205 y Fs(v)-5 b(ariables)31 b(w)m(e)j(estimate)e (\(24\))g(b)m(y)916 1321 y Fk(Z)997 1530 y Fm(^)971 1547 y Fo(M)1067 1457 y Fp(d)7 b Fs(^)-56 b Fp(\021)1202 1321 y Fk(Z)1258 1547 y Fo(M)1331 1559 y Ff(^)-35 b Fi(\021)1384 1457 y Fp(d\021)1483 1472 y Fo(j)1536 1376 y Fk(\002)1578 1457 y Fp(k)1634 1472 y Fm(^)-40 b Fo(\021)1670 1457 y Fs(\()p Fp(\021)1756 1472 y Fo(j)1793 1457 y Fs(\))22 b(+)g Fq(j)p Fp(k)2033 1415 y Fn(0)2035 1481 y Fm(^)-40 b Fo(\021)2071 1457 y Fs(\()p Fp(\021)2157 1472 y Fo(j)2194 1457 y Fs(\))p Fq(j)22 b Fs(+)g(2)p Fq(j)p Fp(k)2511 1415 y Fn(00)2513 1481 y Fm(^)-40 b Fo(\021)2553 1457 y Fs(\()p Fp(\021)2639 1472 y Fo(j)2675 1457 y Fs(\))p Fq(j)2741 1376 y Fk(\003)757 1737 y Fs(=)916 1601 y Fk(Z)906 1885 y Fo(M)1032 1737 y Fp(d\021)1151 1656 y Fk(\002)1193 1737 y Fp(k)1249 1752 y Fm(^)g Fo(\021)1285 1737 y Fs(\()p Fp(\021)1371 1752 y Fo(j)1408 1737 y Fs(\))22 b(+)g Fq(j)p Fp(k)1648 1696 y Fn(0)1650 1761 y Fm(^)-40 b Fo(\021)1686 1737 y Fs(\()p Fp(\021)1772 1752 y Fo(j)1809 1737 y Fs(\))p Fq(j)22 b Fs(+)g(2)p Fq(j)p Fp(k)2126 1696 y Fn(00)2128 1761 y Fm(^)-40 b Fo(\021)2168 1737 y Fs(\()p Fp(\021)2254 1752 y Fo(j)2290 1737 y Fs(\))p Fq(j)2356 1656 y Fk(\003)757 2063 y Fs(=)994 1928 y Fk(Z)861 2231 y Fm([0)p Fo(;!)980 2240 y Ff(+)1030 2231 y Fm(])1050 2212 y Fa(j)p Ff(\003)1110 2198 y(+)1160 2212 y Fa(j)1189 2063 y Fp(d!)1320 1983 y Fk(\010)1379 2063 y Fp(F)14 b Fs(\()p Fp(!)t Fs(\))21 b(+)h Fq(j)p Fp(k)1798 2022 y Fn(0)1826 2084 y Fm(^)1795 2101 y Fo(A!)1897 2063 y Fs([\()p Fp(A!)t Fs(\))2138 2078 y Fo(j)2174 2063 y Fs(])p Fq(j)g Fs(+)g(2)p Fq(j)p Fp(k)2480 2022 y Fn(00)2508 2084 y Fm(^)2477 2101 y Fo(A!)2580 2063 y Fs([\()p Fp(A!)t Fs(\))2821 2078 y Fo(j)2857 2063 y Fs(])p Fq(j)2929 1983 y Fk(\011)3367 2063 y Fs(\(32\))120 2443 y(where)446 2418 y(^)402 2443 y Fp(A!)31 b Fs(:=)d Fp(\031)t Fs(\()p Fp(A!)t Fs(\))j(and)i Fp(F)14 b Fs(\()p Fp(!)t Fs(\))27 b(=)1540 2369 y Fk(Q)1634 2473 y Fo(k)r Fn(2)p Fm(\003)1769 2454 y Ff(+)1840 2443 y Fp(f)11 b Fs(\()p Fp(!)1998 2458 y Fo(k)2040 2443 y Fs(\).)266 2564 y(The)40 b(\014rst)e(term)g(giv)m(es)1163 2484 y Fk(R)1246 2564 y Fp(d!)19 b(F)14 b Fs(\()p Fp(!)t Fs(\))37 b(=)g(1,)i(for)f(the)h(others)g(w)m(e)g(ha)m(v)m(e)g(to)f(calculate)g (the)120 2684 y(deriv)-5 b(ativ)m(es)42 b(of)f Fp(k)k Fs(w.r.t.)d Fp(\021)1148 2699 y Fo(j)1185 2684 y Fs(.)71 b(Let)42 b Fp(B)48 b Fs(=)43 b Fq(f)p Fp(b)1799 2699 y Fo(i;j)1880 2684 y Fq(g)1930 2701 y Fo(i;j)t Fn(2)p Fm(\003)2103 2682 y Ff(+)2198 2684 y Fs(denote)g(the)f(in)m(v)m(erse)h (of)e Fp(A)j Fs(=)f Fp(A)3460 2699 y Fm(\003)3513 2684 y Fs(.)120 2805 y(With)32 b(this)g(notation)f(w)m(e)j(ha)m(v)m(e)g(for) e(the)h(deriv)-5 b(ativ)m(es)206 3008 y Fp(k)260 2967 y Fn(0)289 3029 y Fm(^)257 3046 y Fo(A!)360 3008 y Fs([\()p Fp(A!)t Fs(\))601 3023 y Fo(j)637 3008 y Fs(])28 b(=)816 2913 y Fk(X)796 3130 y Fo(k)r Fn(2)p Fm(\003)931 3111 y Ff(+)998 3008 y Fp(f)1057 2967 y Fn(0)1080 3008 y Fs(\()p Fp(!)1179 3023 y Fo(k)1221 3008 y Fs(\))k Fp(b)1332 3023 y Fo(k)r(;j)1468 2913 y Fk(Y)1454 3125 y Fi(i)p Fa(2)p Ff(\003)1559 3111 y(+)1481 3182 y Fi(i)p Fa(6)p Ff(=)p Fi(k)1636 3008 y Fp(f)11 b Fs(\()p Fp(!)1794 3023 y Fo(i)1822 3008 y Fs(\))1507 b(\(33\))206 3415 y Fp(k)260 3373 y Fn(00)289 3435 y Fm(^)257 3452 y Fo(A!)360 3415 y Fs([\()p Fp(A!)t Fs(\))601 3430 y Fo(j)637 3415 y Fs(])28 b(=)837 3320 y Fk(X)806 3532 y Fi(k)q(;l)p Fa(2)p Ff(\003)962 3518 y(+)859 3588 y Fi(k)q Fa(6)p Ff(=)p Fi(l)1039 3415 y Fp(f)1098 3373 y Fn(0)1121 3415 y Fs(\()p Fp(!)1220 3430 y Fo(k)1262 3415 y Fs(\))p Fp(f)1359 3373 y Fn(0)1382 3415 y Fs(\()p Fp(!)1481 3430 y Fo(l)1507 3415 y Fs(\))17 b Fp(b)1603 3430 y Fo(k)r(;j)1697 3415 y Fp(b)1738 3430 y Fo(l)q(;j)1914 3290 y(L)1874 3320 y Fk(Y)1860 3532 y Fi(i)p Fa(2)p Ff(\003)1965 3518 y(+)1867 3588 y Fi(i)p Fa(6)p Ff(=)p Fi(l;k)2042 3415 y Fp(f)11 b Fs(\()p Fp(!)2200 3430 y Fo(i)2228 3415 y Fs(\))22 b(+)2407 3320 y Fk(X)2386 3537 y Fo(k)r Fn(2)p Fm(\003)2521 3518 y Ff(+)2588 3415 y Fp(f)2647 3373 y Fn(00)2689 3415 y Fs(\()p Fp(!)2788 3430 y Fo(k)2830 3415 y Fs(\))17 b Fp(b)2926 3373 y Fm(2)2926 3439 y Fo(k)r(;j)3102 3290 y(L)3062 3320 y Fk(Y)3048 3532 y Fi(i)p Fa(2)p Ff(\003)3153 3518 y(+)3074 3588 y Fi(i)p Fa(6)p Ff(=)p Fi(k)3230 3415 y Fp(f)11 b Fs(\()p Fp(!)3388 3430 y Fo(i)3416 3415 y Fs(\))3367 3701 y(\(34\))120 3897 y(and)33 b(for)f(the)h(corresp)s(onding)f(in)m(tegrals)497 4008 y Fk(Z)363 4311 y Fm([0)p Fo(;!)482 4320 y Ff(+)533 4311 y Fm(])553 4292 y Fa(j)p Ff(\003)613 4278 y(+)663 4292 y Fa(j)692 4143 y Fp(d!)19 b Fq(j)p Fp(k)905 4102 y Fn(0)934 4164 y Fm(^)902 4181 y Fo(A!)1005 4143 y Fs([\()p Fp(A!)t Fs(\))1246 4158 y Fo(j)1282 4143 y Fs(])p Fq(j)83 b(\024)g(k)p Fp(f)1689 4102 y Fn(0)1712 4143 y Fq(k)1762 4160 y Fo(L)1810 4141 y Ff(1)1886 4049 y Fk(X)1865 4265 y Fo(k)r Fn(2)p Fm(\003)2000 4246 y Ff(+)2067 4143 y Fq(j)p Fp(b)2136 4158 y Fo(k)r(;j)2231 4143 y Fq(j)1108 b Fs(\(35\))497 4371 y Fk(Z)363 4675 y Fm([0)p Fo(;!)482 4684 y Ff(+)533 4675 y Fm(])553 4655 y Fa(j)p Ff(\003)613 4641 y(+)663 4655 y Fa(j)692 4507 y Fp(d!)19 b Fq(j)p Fp(k)905 4466 y Fn(00)934 4528 y Fm(^)902 4544 y Fo(A!)1005 4507 y Fs([\()p Fp(A!)t Fs(\))1246 4522 y Fo(j)1282 4507 y Fs(])p Fq(j)83 b(\024)g(k)p Fp(f)1689 4466 y Fn(0)1712 4507 y Fq(k)1762 4466 y Fm(2)1762 4533 y Fo(L)1810 4514 y Ff(1)1907 4412 y Fk(X)1875 4624 y Fi(k)q(;l)p Fa(2)p Ff(\003)2031 4610 y(+)1928 4681 y Fi(k)q Fa(6)p Ff(=)p Fi(l)2108 4507 y Fq(j)p Fp(b)2177 4522 y Fo(k)r(;j)2272 4507 y Fp(b)2313 4522 y Fo(l)q(;j)2392 4507 y Fq(j)21 b Fs(+)h Fq(k)p Fp(f)2648 4466 y Fn(00)2690 4507 y Fq(k)2740 4523 y Fo(L)2788 4504 y Ff(1)2864 4412 y Fk(X)2844 4629 y Fo(k)r Fn(2)p Fm(\003)2979 4610 y Ff(+)3045 4507 y Fp(b)3086 4466 y Fm(2)3086 4531 y Fo(k)r(;j)3214 4507 y Fp(:)126 b Fs(\(36\))120 4867 y(This)33 b(giv)m(es)g(for)f(the)h(exp) s(ectation)f(\(24\))g(the)h(estimate)203 5095 y Fl(E)280 5015 y Fk(\002)328 5095 y Fq(h)p Fp(\036;)17 b(\037)530 5110 y Fo(j)565 5095 y Fs(\()p Fp(H)692 5054 y Fo(l)684 5120 y(!)757 5095 y Fq(\000)22 b Fp(E)28 b Fs(+)23 b Fp(\016)t(i)p Fs(\))1173 5054 y Fn(\000)p Fm(1)1267 5095 y Fp(\037)1328 5110 y Fo(j)1365 5095 y Fp(\036)p Fq(i)1462 5015 y Fk(\003)453 5269 y Fq(\024)28 b Fs(1)22 b(+)g Fq(k)p Fp(f)836 5228 y Fn(0)859 5269 y Fq(k)909 5286 y Fo(L)957 5267 y Ff(1)1032 5175 y Fk(X)1012 5391 y Fo(k)r Fn(2)p Fm(\003)1147 5372 y Ff(+)1214 5269 y Fq(j)p Fp(b)1283 5284 y Fo(k)r(;j)1377 5269 y Fq(j)g Fs(+)g(2)p Fq(k)p Fp(f)1683 5228 y Fn(0)1706 5269 y Fq(k)1756 5228 y Fm(2)1756 5295 y Fo(L)1804 5276 y Ff(1)1900 5175 y Fk(X)1869 5387 y Fi(k)q(;l)p Fa(2)p Ff(\003)2025 5373 y(+)1922 5443 y Fi(k)q Fa(6)p Ff(=)p Fi(l)2102 5269 y Fq(j)p Fp(b)2171 5284 y Fo(k)r(;j)2266 5269 y Fp(b)2307 5284 y Fo(l)q(;j)2385 5269 y Fq(j)g Fs(+)g(2)p Fq(k)p Fp(f)2691 5228 y Fn(00)2733 5269 y Fq(k)2783 5286 y Fo(L)2831 5267 y Ff(1)2907 5175 y Fk(X)2886 5391 y Fo(k)r Fn(2)p Fm(\003)3021 5372 y Ff(+)3088 5269 y Fp(b)3129 5228 y Fm(2)3129 5294 y Fo(k)r(;j)3256 5269 y Fp(:)84 b Fs(\(37\))1781 5712 y(10)p eop %%Page: 11 11 11 10 bop 120 407 a Fs(By)33 b(the)g(b)s(ound)g(\(18\))f(on)g(the)h(in) m(v)m(erse)h(of)e Fp(A)h Fs(w)m(e)h(kno)m(w)705 577 y Fk(X)684 793 y Fo(k)r Fn(2)p Fm(\003)819 774 y Ff(+)886 671 y Fq(j)p Fp(b)955 686 y Fo(k)r(;j)1050 671 y Fq(j)60 b(\024)g(k)p Fp(B)5 b Fq(k)1022 b(\024)2703 604 y Fs(1)p 2592 648 273 4 v 2592 740 a(1)21 b Fq(\000)i Fp(\013)2825 711 y Fn(\003)3367 671 y Fs(\(38\))565 907 y Fk(X)533 1119 y Fi(k)q(;l)p Fa(2)p Ff(\003)689 1105 y(+)586 1176 y Fi(k)q Fa(6)p Ff(=)p Fi(l)767 1002 y Fq(j)p Fp(b)836 1017 y Fo(k)r(;j)930 1002 y Fp(b)971 1017 y Fo(l)q(;j)1050 1002 y Fq(j)60 b(\024)1296 907 y Fk(X)1275 1124 y Fo(k)r Fn(2)p Fm(\003)1410 1105 y Ff(+)1477 1002 y Fq(j)p Fp(b)1546 1017 y Fo(k)r(;j)1641 1002 y Fq(j)1698 907 y Fk(X)1686 1124 y Fo(l)q Fn(2)p Fm(\003)1804 1105 y Ff(+)1870 1002 y Fq(j)p Fp(b)1939 1017 y Fo(l)q(;j)2018 1002 y Fq(j)430 b(\024)2582 861 y Fk(\022)2777 934 y Fs(1)p 2665 979 V 2665 1070 a(1)22 b Fq(\000)g Fp(\013)2898 1041 y Fn(\003)2947 861 y Fk(\023)3021 884 y Fm(2)3367 1002 y Fs(\(39\))760 1325 y Fk(X)740 1542 y Fo(k)r Fn(2)p Fm(\003)875 1523 y Ff(+)941 1419 y Fp(b)982 1378 y Fm(2)982 1444 y Fo(k)r(;j)1138 1419 y Fq(\024)1275 1249 y Fk( )1375 1325 y(X)1354 1542 y Fo(k)r Fn(2)p Fm(\003)1489 1523 y Ff(+)1556 1419 y Fq(j)p Fp(b)1625 1434 y Fo(k)r(;j)1720 1419 y Fq(j)1748 1249 y Fk(!)1826 1271 y Fm(2)2476 1419 y Fq(\024)2582 1279 y Fk(\022)2777 1352 y Fs(1)p 2665 1396 V 2665 1488 a(1)g Fq(\000)g Fp(\013)2898 1459 y Fn(\003)2947 1279 y Fk(\023)3021 1301 y Fm(2)3109 1419 y Fp(:)231 b Fs(\(40\))120 1738 y(So)32 b(\(24\))g(is)h(b)s(ounded)g(b)m(y)602 2032 y Fp(C)672 2047 y Fo(\013)749 2032 y Fs(:=)28 b(1)22 b(+)g Fq(k)p Fp(f)1158 1991 y Fn(0)1181 2032 y Fq(k)1231 2049 y Fo(L)1279 2030 y Ff(1)1439 1965 y Fs(1)p 1327 2009 V 1327 2100 a(1)g Fq(\000)h Fp(\013)1561 2072 y Fn(\003)1632 2032 y Fs(+)f(2)1796 1951 y Fk(\000)1841 2032 y Fq(k)p Fp(f)1950 1991 y Fn(0)1973 2032 y Fq(k)2023 1991 y Fm(2)2023 2058 y Fo(L)2071 2039 y Ff(1)2131 2032 y Fs(+)g Fq(k)p Fp(f)2338 1991 y Fn(00)2380 2032 y Fq(k)2430 2049 y Fo(L)2478 2030 y Ff(1)2517 1951 y Fk(\001)2579 1892 y(\022)2774 1965 y Fs(1)p 2662 2009 V 2662 2100 a(1)g Fq(\000)h Fp(\013)2896 2072 y Fn(\003)2945 1892 y Fk(\023)3018 1914 y Fm(2)3367 2032 y Fs(\(41\))120 2309 y(whic)m(h)31 b(is)f(indep)s(enden)m(t)i(of)e Fp(I)8 b(;)17 b(\016)n(;)g Fs(\003)1401 2324 y Fo(l)1458 2309 y Fs(and)30 b Fp(j)6 b Fs(.)43 b(The)32 b(a)m(v)m(erage)f(trace)g(in)f (\(15\))g(is)g(th)m(us)i(b)s(ounded)120 2430 y(b)m(y)1476 2539 y Fp(e)1521 2503 y Fo(E)1573 2512 y Ff(2)1612 2539 y Fp(C)1682 2554 y Fo(V)1743 2539 y Fp(C)1813 2554 y Fo(\013)p 1476 2584 386 4 v 1640 2675 a Fp(\031)1888 2606 y Fq(j)p Fp(I)8 b Fq(j)17 b(j)p Fs(\003)p Fq(j)31 b Fp(:)1173 b Fs(\(42\))120 2819 y(Note)36 b(that)f Fq(j)p Fs(\003)p Fq(j)c Fs(=)i Fq(j)p Fs(\003)933 2834 y Fo(l)958 2819 y Fq(j)f Fs(=)h Fp(l)1158 2783 y Fo(d)1198 2819 y Fs(.)52 b(Th)m(us)37 b(w)m(e)g(pro)m(v)m(ed)g(that)e(the)h(a)m(v)m (eraged)g(\014nite)f(v)m(olume)g(IDS)120 2940 y(is)d(Lipsc)m(hitz)h (con)m(tin)m(uous)796 3160 y Fl(E)874 3079 y Fk(\002)921 3160 y Fp(N)1009 3119 y Fo(l)999 3184 y Fn(\017)1038 3160 y Fs(\()p Fp(E)1148 3175 y Fm(2)1188 3160 y Fs(\))22 b Fq(\000)g Fp(N)1435 3119 y Fo(l)1425 3184 y Fn(\017)1465 3160 y Fs(\()p Fp(E)1575 3175 y Fm(1)1637 3160 y Fs(+)g Fp(\017)p Fs(\))1812 3079 y Fk(\003)1881 3160 y Fq(\024)28 b Fp(C)c Fq(j)p Fp(E)2180 3175 y Fm(2)2241 3160 y Fq(\000)f Fp(E)2413 3175 y Fm(1)2453 3160 y Fq(j)p Fp(;)49 b Fq(8)32 b Fp(\017)d(>)e Fs(0)503 b(\(43\))120 3402 y(with)38 b Fp(C)45 b Fs(:=)613 3360 y Fo(e)646 3337 y Fi(E)691 3352 y Ff(2)729 3360 y Fo(C)779 3371 y Fi(V)833 3360 y Fo(C)883 3368 y Fi(\013)p 613 3379 315 4 v 749 3436 a Fo(\031)937 3402 y Fs(.)61 b(By)39 b(the)1369 3377 y(\024)1358 3402 y(Ceb)m(y)-5 b(\024)-44 b(sev)41 b(inequalit)m(y)d (the)h(W)-8 b(egner)39 b(estimate)e(\(H2\))h(no)m(w)120 3522 y(follo)m(ws.)120 3855 y Fr(6)161 b(Discussion)120 4074 y Fs(In)32 b(this)g(section)g(w)m(e)h(consider)f(some)g (generalisations)e(of)i(Theorem)g(5)g(and)g(discuss)h(appli-)120 4195 y(cations.)120 4377 y Ft(Remark)k(8)g(\(Existence)f(of)i(the)f (densit)m(y)g(of)h(states\))120 4498 y Fs(Note)23 b(that)g(the)g (estimate)f(\(43\))g(on)h(the)g(\014nite)f(v)m(olume)g(IDS)h(for)f(a)h (b)s(ounded)g(energy)h(in)m(terv)-5 b(al)120 4618 y(is)30 b(indep)s(enden)m(t)h(of)f Fq(j)p Fs(\003)971 4633 y Fo(l)996 4618 y Fq(j)p Fs(.)43 b(So)30 b(one)h(can)f(tak)m(e)h(the)g (limit)c Fp(l)j Fq(!)d(1)j Fs(and)g(obtain)f(the)i(Lipsc)m(hitz)120 4738 y(con)m(tin)m(uit)m(y)i(of)f(the)h(\(in\014nite)e(v)m(olume\))h (IDS)1069 4958 y Fp(N)10 b Fs(\()p Fp(E)1267 4973 y Fm(2)1307 4958 y Fs(\))22 b Fq(\000)h Fp(N)10 b Fs(\()p Fp(E)1665 4973 y Fm(1)1705 4958 y Fs(\))28 b Fq(\024)g Fs(const)33 b Fq(j)p Fp(E)2231 4973 y Fm(2)2292 4958 y Fq(\000)23 b Fp(E)2464 4973 y Fm(1)2504 4958 y Fq(j)32 b Fp(:)776 b Fs(\(44\))120 5178 y(By)33 b(Rademac)m(her's)f(theorem)g(it)f(follo)m (ws)g(that)h(the)h(deriv)-5 b(ativ)m(e)31 b Fp(dN)10 b Fs(\()p Fp(E)c Fs(\))32 b(exists)h(for)f(almost)120 5299 y(all)e Fp(E)39 b Fs(and)33 b(is)f(b)s(ounded)h(b)m(y)1198 5257 y Fo(C)1248 5268 y Fi(V)1301 5257 y Fo(C)1351 5265 y Fi(\013)p 1198 5276 199 4 v 1275 5333 a Fo(\031)1406 5299 y Fp(e)1451 5263 y Fo(E)1511 5299 y Fs(.)43 b Fp(dN)10 b Fs(\()p Fp(E)c Fs(\))33 b(is)f(called)f(the)i(densit)m(y)h(of)e (states.)1781 5712 y(11)p eop %%Page: 12 12 12 11 bop 120 382 a Ft(Remark)37 b(9)g(\(Discrete)f(mo)s(del\))120 502 y Fs(One)d(could)f(also)g(consider)h(the)g(discrete)g(Sc)m(hr\177) -49 b(odinger)33 b(op)s(erator)1246 734 y Fp(h)1302 749 y Fo(!)1380 734 y Fs(=)28 b Fq(\000)p Fs(\001)1642 749 y Fm(disc)1787 734 y Fs(+)22 b Fp(V)1942 749 y Fo(!)2025 734 y Fs(on)32 b Fp(l)2191 692 y Fm(2)2231 734 y Fs(\()p Fl(Z)2338 692 y Fo(d)2376 734 y Fs(\))953 b(\(45\))120 954 y(where)34 b([)p Fq(\000)p Fs(\001)587 969 y Fm(disc)710 954 y Fp(\036)p Fs(]\()p Fp(i)p Fs(\))29 b(=)1037 879 y Fk(P)1142 983 y Fn(j)p Fo(i)p Fn(\000)p Fo(n)p Fn(j)p Fm(=1)1397 954 y Fs(\()p Fp(\036)p Fs(\()p Fp(i)p Fs(\))23 b Fq(\000)g Fp(\036)p Fs(\()p Fp(n)p Fs(\)\))32 b(and)h(in)g(the)g (de\014nition)f(of)h(the)g(m)m(ultipli-)120 1082 y(cation)g(op)s (erator)g Fp(V)865 1097 y Fo(!)949 1082 y Fs(the)h(c)m(haracteristic)g (function)f(of)h(the)g(unit)f(cub)s(e)i Fp(\037)2884 1097 y Fm(0)2923 1082 y Fs(\()p Fp(x)p Fs(\))f(is)g(replaced)120 1203 y(b)m(y)27 b(the)g(Kronec)m(k)m(er)i(sym)m(b)s(ol)d Fp(\016)1240 1218 y Fm(0)1279 1203 y Fs(\()p Fp(i)p Fs(\).)42 b(An)27 b(additional)c Fl(Z)2134 1167 y Fo(d)2172 1203 y Fs(-p)s(erio)s(dic)h(p)s(oten)m(tial)h Fp(V)3035 1218 y Fm(0)3101 1203 y Fs(w)m(ould)i(just)120 1323 y(shift)32 b(the)h(Hamiltonian)c Fp(h)1125 1338 y Fo(!)1208 1323 y Fs(b)m(y)k(a)g(constan)m(t.)266 1443 y(Our)41 b(pro)s(of)f(w)m(orks)i (for)e(this)g(mo)s(del,)h(to)s(o,)h(since)f(the)h(results)f(in)f ([CH94,)j(Section)d(4])120 1564 y(and)g([CHM96,)i(Theorem)d(1.1])h(are) f(form)m(ulated)f(for)h(abstract)h(one-parameter)f(families)120 1684 y(of)c(op)s(erators.)53 b(In)36 b(Sections)g(3)g(to)f(5)h(of)f (this)h(pap)s(er)g(w)m(e)g(w)m(ould)g(just)g(ha)m(v)m(e)h(to)f(c)m (hange)h(the)120 1805 y(notations.)120 1987 y Ft(Remark)g(10)g (\(Correlated)g(p)s(oten)m(tials\))120 2108 y Fs(W)-8 b(e)23 b(can)g(regard)f(Theorems)h(5)g(as)f(a)g(result)h(ab)s(out)f (allo)m(y-t)m(yp)s(e)f(Sc)m(hr\177)-49 b(odinger)23 b(op)s(erators)f (with)120 2228 y(non-negativ)m(e)30 b(single)g(site)g(p)s(oten)m(tial)f Fp(u)g 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Fp(;)g(:)g(:)g(:)e(;)i Fs(0\))27 b Fq(2)i Fl(Z)1148 3433 y Fo(d)1218 3469 y Fs(and)k Fp(u)28 b Fs(=)f Fp(\037)1656 3484 y Fm(0)1718 3469 y Fq(\000)c Fp(\037)1879 3484 y Fo(e)1949 3469 y Fs(b)s(e)33 b(the)g(single)e(site)i(p)s(oten)m(tial)e (of)h Fp(H)3305 3484 y Fo(!)3388 3469 y Fs(\(1\).)120 3590 y(This)h(corresp)s(onds)h(to)f Fp(\013)1060 3605 y Fm(0)1128 3590 y Fs(=)28 b(1)p Fp(;)17 b(\013)1387 3605 y Fo(e)1452 3590 y Fs(=)28 b Fq(\000)p Fs(1)33 b(and)g Fp(\013)1967 3605 y Fo(k)2038 3590 y Fs(=)28 b(0)33 b(otherwise.)45 b(The)34 b(truncations)f Fp(A)3487 3605 y Fm(\003)120 3710 y Fs(ha)m(v)m(e)k(in)m(v)m(erses)h Fp(B)787 3725 y Fm(\003)874 3710 y Fs(:=)c Fq(f)p Fp(b)1102 3725 y Fo(j;k)1193 3710 y Fq(g)1243 3726 y Fo(j;k)r Fn(2)p Fm(\003)1427 3707 y Ff(+)1516 3710 y Fs(with)i(en)m(tries)g Fp(b)2098 3725 y Fo(j;k)2223 3710 y Fs(=)d(1)j(for)g Fp(j;)17 b(k)37 b Fq(2)d Fs(\003)2910 3674 y Fm(+)2968 3710 y Fs(,)j Fp(k)3083 3725 y Fm(1)3156 3710 y Fq(\024)d Fp(j)3307 3725 y Fm(1)3383 3710 y Fs(and)120 3830 y Fp(k)171 3845 y Fo(i)227 3830 y Fs(=)27 b Fp(j)370 3845 y Fo(i)425 3830 y Fs(for)e Fp(i)j Fs(=)g(2)p Fp(;)17 b(:)g(:)g(:)e(;)i(d)26 b Fs(and)g Fp(b)1300 3845 y Fo(j;k)1419 3830 y Fs(=)i(0)e(otherwise.)41 b(Here)27 b(for)f(a)g Fp(i)i Fs(=)f(1)p Fp(;)17 b(:)g(:)g(:)f(;)h(d)25 b Fs(the)i(n)m(um)m(b)s(ers)120 3951 y Fp(k)171 3966 y Fo(i)237 3951 y Fs(and)38 b Fp(j)472 3966 y Fo(i)538 3951 y Fs(denote)g(the)g Fp(i)p Fs(-th)g(comp)s(onen)m(ts)g(of)g(the)g (v)m(ectors)h Fp(k)s(;)17 b(j)42 b Fq(2)37 b Fl(Z)2746 3915 y Fo(d)2784 3951 y Fs(.)59 b(In)38 b(this)g(case)h(the)120 4071 y Fp(B)194 4086 y Fm(\003)277 4071 y Fs(are)30 b(not)g(uniformly)e (b)s(ounded)j(in)e(\003.)42 b(Ho)m(w)m(ev)m(er,)33 b(the)e(terms)f(in)f (\(37\))h(dep)s(ending)g(on)g Fp(B)120 4191 y Fs(can)j(b)s(e)g (estimated)f(b)m(y)h(putting)f(all)e Fp(b)1535 4206 y Fo(k)r(;j)1658 4191 y Fs(=)e(1:)1256 4337 y Fk(X)1236 4554 y Fo(k)r Fn(2)p Fm(\003)1371 4535 y Ff(+)1437 4432 y Fp(b)1478 4391 y Fm(2)1478 4456 y Fo(k)r(;j)1573 4432 y Fp(;)1638 4337 y Fk(X)1617 4554 y Fo(k)r Fn(2)p Fm(\003)1752 4535 y Ff(+)1819 4432 y Fq(j)p Fp(b)1888 4447 y Fo(k)r(;j)1983 4432 y Fq(j)f(\024)h(j)p Fs(\003)2239 4391 y Fm(+)2239 4460 y Fo(l)2298 4432 y Fq(j)1041 b Fs(\(46\))1498 4611 y Fk(X)1466 4823 y Fi(k)q(;l)p Fa(2)p Ff(\003)1622 4809 y(+)1519 4880 y Fi(k)q Fa(6)p Ff(=)p Fi(l)1700 4706 y Fq(j)p Fp(b)1769 4721 y Fo(k)r(;j)1863 4706 y Fp(b)1904 4721 y Fo(l)q(;j)1983 4706 y Fq(j)27 b(\024)h(j)p Fs(\003)2239 4665 y Fm(+)2239 4734 y Fo(l)2298 4706 y Fq(j)2326 4665 y Fm(2)2397 4706 y Fp(:)943 b Fs(\(47\))120 5090 y(Since)33 b Fq(j)p Fs(\003)471 5049 y Fm(+)471 5118 y Fo(l)529 5090 y Fq(j)27 b(\024)i Fs(\()p Fp(l)24 b Fs(+)e Fp(g)t Fs(\))968 5054 y Fo(d)1035 5090 y Fq(\024)28 b Fp(C)1210 5105 y Fm(\000)1275 5090 y Fp(l)1306 5054 y Fo(d)1379 5090 y Fs(where)34 b Fp(g)d Fs(=)d(diam)15 b(\000)32 b(w)m(e)i(obtain)d(the)i(estimate)973 5310 y Fl(E)1051 5229 y Fk(\002)1098 5310 y Fp(N)1186 5269 y Fo(l)1176 5334 y Fn(\017)1215 5310 y Fs(\()p Fp(E)1325 5325 y Fm(2)1365 5310 y Fs(\))22 b Fq(\000)h Fp(N)1613 5269 y Fo(l)1603 5334 y Fn(\017)1642 5310 y Fs(\()p Fp(E)1752 5325 y Fm(1)1792 5310 y Fs(\))1830 5229 y Fk(\003)1899 5310 y Fq(\024)2026 5285 y Fs(~)2004 5310 y Fp(C)h Fq(j)p Fp(E)2198 5325 y Fm(2)2259 5310 y Fq(\000)f Fp(E)2431 5325 y Fm(1)2470 5310 y Fq(jj)p Fs(\003)2594 5325 y Fo(l)2620 5310 y Fq(j)2648 5269 y Fm(2)3367 5310 y Fs(\(48\))1781 5712 y(12)p eop %%Page: 13 13 13 12 bop 120 418 a Fs(where)423 393 y(~)401 418 y Fp(C)35 b Fs(=)619 365 y Fo(e)652 341 y Fi(E)697 356 y Ff(2)735 365 y Fo(C)785 376 y Fi(V)839 365 y Fo(C)894 341 y Ff(2)889 387 y(\000)p 619 395 312 4 v 754 452 a Fo(\031)958 418 y Fs(\(1)20 b(+)g Fq(k)p Fp(f)1270 382 y Fn(0)1293 418 y Fq(k)1343 435 y Fo(L)1391 416 y Ff(1)1450 418 y Fs(+)h(2)p Fq(k)p Fp(f)1705 382 y Fn(0)1727 418 y Fq(k)1777 382 y Fm(2)1777 449 y Fo(L)1825 430 y Ff(1)1885 418 y Fs(+)f(2)p Fq(k)p Fp(f)2139 382 y Fn(00)2181 418 y Fq(k)2231 435 y Fo(L)2279 416 y Ff(1)2317 418 y Fs(\))32 b(The)h(same)e(W)-8 b(egner)33 b(estimate)120 538 y(extends)i(to)d(similar)d(single)j(site) g(p)s(oten)m(tials,)f(e.g.)818 758 y Fp(u)c Fs(=)h Fp(\037)1066 774 y Fm(\(0)p Fo(;:::)o(;)p Fm(0\))1316 758 y Fs(+)22 b Fp(\037)1475 774 y Fm(\(1)p Fo(;)p Fm(1)p Fo(;)p Fm(0)p Fo(;:::)o(;)p Fm(0\))1835 758 y Fq(\000)g Fp(\037)1995 774 y Fm(\(1)p Fo(;)p Fm(0)p Fo(;:::)o(;)p Fm(0\))2300 758 y Fq(\000)h Fp(\037)2461 774 y Fm(\(0)p Fo(;)p Fm(1)p Fo(;)p Fm(0)p Fo(;:::)o(;)p Fm(0\))2815 758 y Fp(:)525 b Fs(\(49\))120 978 y(Note)33 b(that)g(while)f(w)m(e)i(ha)m(v)m(e)g (pro)m(v)m(en)h(\(10\))d(with)h(the)g(exp)s(onen)m(t)h Fp(b)29 b Fs(=)g(3)j(w)m(e)i(cannot)f(deduce)120 1099 y(the)44 b(Lipsc)m(hitz)f(con)m(tin)m(uit)m(y)h(of)f(the)g(IDS)h(b)s (ecause)g(of)f(the)h(div)m(ergen)m(t)g(term)f Fq(j)p Fs(\003)3154 1114 y Fo(l)3180 1099 y Fq(j)3208 1063 y Fm(2)3247 1099 y Fs(.)76 b(This)120 1219 y(example)30 b(illustrates)g(that)h(\(H2'\))f(can)i(b)s(e)f(pro)m(v)m(en)h(for)f Fp(A)g Fs(\(resp)s(ectiv)m(ely)h Fp(\013)q Fs(\))e(with)h(in)m(v)m (erses)120 1340 y(whose)26 b(norms)f(gro)m(w)h(at)e(most)h(p)s (olynomially)c(in)k Fq(j)p Fs(\003)p Fq(j)p Fs(.)40 b(It)25 b(w)m(ould)g(b)s(e)g(desirable)g(to)g(get)g(a)g(nice)120 1460 y(description)37 b(of)h(this)f(class)h(of)f(T)-8 b(o)s(eplitz)37 b(matrices)g(in)g(terms)g(of)h(the)g(con)m(v)m(olution) f(v)m(ector)120 1580 y Fp(\013)q Fs(.)120 1763 y Ft(Remark)g(13)g (\(Other)g(single)g(site)f(densities\))120 1883 y Fs(The)h(set)f(of)g (admissible)d(single)i(site)h(densities)g Fp(f)43 b Fq(2)34 b Fp(C)2195 1847 y Fm(2)2188 1908 y(0)2234 1883 y Fs(\(]0)p Fp(;)17 b(!)2453 1898 y Fm(+)2512 1883 y Fs([\))35 b(is)h(quite)g (restrictiv)m(e.)53 b(It)120 2018 y(can)22 b(b)s(e)g(replaced)g(b)m(y)h (the)g(Sob)s(olev)e(space)i Fp(W)1775 1971 y Fm(2)p Fo(;)p Fm(1)1761 2042 y(0)1869 2018 y Fs(\()p Fl(R)5 b Fs(\))28 b(since)22 b(the)h(existence)g(of)f(the)g(deriv)-5 b(ativ)m(es)120 2138 y(is)36 b(only)f(needed)j(for)d(partial)f(in)m(tegration.)52 b(Here)37 b(the)g(index)f(2)g(stands)h(for)e(second)j(order)120 2259 y(deriv)-5 b(ativ)m(es)33 b(and)f(the)h(index)g(1)f(for)g(the)h (exp)s(onen)m(t)h(of)f(the)g(in)m(tegration.)266 2379 y(There)h(is)d(a)h(completely)f(di\013eren)m(t)h Fp(f)43 b Fs(w)m(e)33 b(can)f(cop)s(e)g(with,)g(namely)f(the)i(uniform)d(den-) 120 2499 y(sit)m(y)-8 b(,)33 b(as)g(the)g(follo)m(wing)c(example)j(sho) m(ws)j([V)-8 b(es99].)120 2707 y Ft(Example)36 b(14)i(\(Uniform)e (densit)m(y\))120 2827 y Fs(Consider)i(again)e(the)i(single)e(site)h(p) s(oten)m(tial)f Fp(u)g Fs(=)g Fp(\037)2112 2842 y Fm(0)2177 2827 y Fq(\000)26 b Fp(\037)2341 2842 y Fo(e)2415 2827 y Fs(but)38 b(no)m(w)g(with)f(the)h(uniform)120 2948 y(densit)m(y)c Fp(f)11 b Fs(\()p Fp(x)p Fs(\))29 b(=)820 2909 y Fm(1)p 790 2925 95 4 v 790 2982 a Fo(!)834 2991 y Ff(+)894 2948 y Fp(\037)955 2963 y Fm([0)p Fo(;!)1074 2972 y Ff(+)1125 2963 y Fm(])1181 2948 y Fs(for)k(the)g(coupling)f (constan)m(ts)i Fp(!)2383 2963 y Fo(k)2426 2948 y Fp(;)17 b(k)31 b Fq(2)e Fl(Z)2716 2912 y Fo(d)2754 2948 y Fs(.)45 b(The)34 b(reasoning)e(of)120 3068 y(Sections)h(3)g(and)g(4)g(remain)f (v)-5 b(alid)31 b(for)i(this)f(case,)i(to)s(o.)45 b(Section)33 b(5)f(has)i(to)e(b)s(e)i(replaced)f(b)m(y)120 3188 y(explicit)h (estimates)h(on)g(the)h(v)m(olume)e(of)h(the)h(in)m(tegration)e(domain) f Fp(M)46 b Fs(and)36 b(the)f(common)120 3309 y(densit)m(y)f Fp(k)h Fs(of)d(the)h(transformed)f(v)-5 b(ariables)32 b(using)g Fq(j)17 b Fs(supp)g Fp(f)11 b Fq(j)17 b(k)p Fp(f)11 b Fq(k)2552 3324 y Fn(1)2653 3309 y Fs(=)27 b(1)33 b(to)f(get)866 3529 y Fl(E)944 3448 y Fk(\002)991 3529 y Fp(N)1079 3488 y Fo(l)1069 3554 y Fn(\017)1108 3529 y Fs(\()p Fp(E)1218 3544 y Fm(2)1258 3529 y Fs(\))22 b Fq(\000)h Fp(N)1506 3488 y Fo(l)1496 3554 y Fn(\017)1535 3529 y Fs(\()p Fp(E)1645 3544 y Fm(1)1685 3529 y Fs(\))1723 3448 y Fk(\003)1792 3529 y Fq(\024)60 b Fs(const)34 b Fq(j)p Fp(E)2285 3544 y Fm(2)2346 3529 y Fq(\000)23 b Fp(E)2518 3544 y Fm(1)2557 3529 y Fq(jj)p Fs(\003)2681 3544 y Fo(l)2707 3529 y Fq(j)32 b Fp(:)573 b Fs(\(50\))120 3749 y(Unfortunately)45 b(w)m(e)i(cannot)f(deal)f(with)g(the)h(sup)s (erp)s(osition)e(of)i(the)g(uniform)d(and)j Fp(C)3468 3713 y Fm(2)3461 3774 y(0)3507 3749 y Fs(-)120 3869 y(densities)39 b(due)h(to)f(the)g(transformation)e Fp(A)1758 3833 y Fn(\000)p Fm(1)1891 3869 y Fs(whic)m(h)j(app)s(ears)g(in)e(the)h (common)f(densit)m(y)120 3990 y Fp(k)s Fs(.)120 4322 y Fr(7)161 b(Lo)t(calisation)120 4541 y Fs(An)33 b(imp)s(ortan)m(t)e (application)f(of)j(W)-8 b(egner's)34 b(estimate)e(is)g(the)h(pro)s(of) f(of)g(lo)s(calisation.)41 b(This)120 4662 y(raises)d(the)g(question)g (whether)h(there)f(is)f(a)g(class)h(of)f(single)g(site)g(p)s(oten)m (tials)f Fp(u)h Fs(for)g(whic)m(h)120 4782 y(Theorem)30 b(5)g(is)g(v)-5 b(alid)28 b(and)j(additionally)c(an)j(initial)c(scale)k (estimate)g(\(H1\))g(can)g(b)s(e)g(pro)m(v)m(en.)120 4903 y(As)44 b(men)m(tioned)f(b)s(efore,)j(for)d(non-negativ)m(e)g Fp(u)p Fs(,)i(Lifshitz)d(tails)g(can)i(b)s(e)f(used)h(to)f(deduce)120 5023 y(\(H1\))d(near)g(the)g(in\014m)m(um)e(of)i(the)g(sp)s(ectrum)g (of)g Fp(H)2073 5038 y Fo(!)2123 5023 y Fs(.)65 b(No)m(w,)43 b(for)c Fp(u)g Fs(with)h(c)m(hanging)f(sign)120 5143 y(there)e(are)g(no)f(results)h(on)g(Lifshitz)e(asymptotics.)56 b(The)37 b(pro)s(ofs)f(are)h(not)f(stable)h(under)g(a)120 5264 y(\(ev)m(en)e(small\))30 b(negativ)m(e)k(p)s(erturbation)e(of)g Fp(u)p Fs(.)45 b(Ho)m(w)m(ev)m(er,)35 b(while)e(the)g(standard)h (deduction)120 5384 y(of)39 b(the)g(asymptotic)g(b)s(eha)m(viour)g(is)g (based)h(on)f(a)g(sequence)j(of)d(inequalities)e(for)i(the)g(\014rst) 1781 5712 y(13)p eop %%Page: 14 14 14 13 bop 120 407 a Fs(Neumann)33 b(eigen)m(v)-5 b(alue)32 b(of)h Fp(H)1229 371 y Fo(l)1221 431 y(!)1304 407 y Fs(on)g (arbitrarily)d(large)h(cub)s(es)k(\003)2485 422 y Fo(l)2510 407 y Fs(,)e(\(H1\))g(is)f(implied)e(b)m(y)k(this)120 527 y(inequalit)m(y)e(on)g(a)g(su\016cien)m(tly)i(large,)d(but)i (\014xed)h(\003)2026 542 y Fo(Q)2082 551 y Ff(1)2120 527 y Fs(.)266 648 y(W)-8 b(e)39 b(will)c(sho)m(w)k(that)f(the)g(basic) g(estimate)f(on)h(the)g(\014rst)g(Neumann)g(eigen)m(v)-5 b(alue)37 b(on)h(a)120 768 y(\014xed)27 b(scale)f(\003)644 783 y Fo(Q)700 792 y Ff(1)763 768 y Fs(is)g(stable)f(under)i(a)e (negativ)m(e)h(p)s(erturbation)f(of)g Fp(u)g Fs(as)h(long)f(as)h(it)f (is)g(coupled)120 888 y(with)34 b(a)f(small)f(parameter)i Fp(\017)p Fs(.)48 b(The)35 b(dep)s(endence)h(\(see)f(also)f(Remark)f (20\))h Fp(\017)c Fs(=)g Fp(\017)p Fs(\()p Fp(Q)3253 903 y Fm(1)3293 888 y Fs(\))g Fq(!)g Fs(0)120 1009 y(for)i Fp(Q)346 1024 y Fm(1)413 1009 y Fq(!)c(1)k Fs(explains)g(wh)m(y)i(our)e (estimate)g(is)g(no)h(go)s(o)s(d)e(for)h(pro)m(ving)g(Lifshitz)g (tails.)120 1211 y Ft(Notation)k(15)i(\(Small)d(negativ)m(e)i(p)s (erturbation)g(uf)h Fp(u)p Ft(\))120 1331 y Fs(W)-8 b(e)24 b(decomp)s(ose)f Fp(u)28 b Fs(=)f Fp(u)1003 1346 y Fm(+)1065 1331 y Fq(\000)s Fp(\017u)1240 1346 y Fn(\000)1322 1331 y Fs(in)m(to)c(a)g(non-negativ)m(e)g Fp(u)2204 1346 y Fm(+)2286 1331 y Fs(and)g(a)g(non-p)s(ositiv)m(e)f(part)h Fq(\000)p Fp(\017u)3453 1346 y Fn(\000)3513 1331 y Fs(,)120 1452 y(with)35 b Fq(k)p Fp(u)451 1467 y Fn(\000)509 1452 y Fq(k)559 1467 y Fn(1)666 1452 y Fq(\024)e Fs(1)p Fp(;)17 b(\017)32 b Fq(2)g Fs([0)p Fp(;)17 b Fs(1])35 b(and)g(supp)18 b Fp(u)32 b Fq(\032)g Fs(\003)1944 1467 y Fo(g)1984 1452 y Fp(;)17 b(g)35 b(>)d Fs(0.)52 b(W)-8 b(e)35 b(set)h Fp(N)43 b Fs(=)32 b Fq(k)2967 1377 y Fk(P)3072 1481 y Fo(k)r Fn(2)p Fj(Z)3207 1462 y Fi(d)3258 1452 y Fp(u)3314 1467 y Fn(\000)3373 1452 y Fs(\()p Fq(\001)24 b(\000)120 1572 y Fp(k)s Fs(\))p Fq(k)262 1587 y Fn(1)337 1572 y Fs(.)266 1692 y(The)35 b(follo)m(wing)30 b(argumen)m(ts)k(are)f (adaptations)g(of)g(inequalit)m(y)f(\(2\))h(and)h(Prop)s(osition)d(3) 120 1813 y(in)40 b([KS86])h(to)f Fp(u)h Fs(with)f(c)m(hanging)h(sign.) 67 b(The)42 b(restriction)e(of)h Fp(H)2585 1828 y Fo(!)2676 1813 y Fs(and)g Fp(H)2955 1828 y Fm(0)3036 1813 y Fs(=)g Fq(\000)p Fs(\001)29 b(+)f Fp(V)3501 1828 y Fm(0)120 1933 y Fs(to)39 b(\003)314 1948 y Fo(l)378 1933 y Fs(with)g(Neumann)g (b.c.)h(will)d(b)s(e)i(denoted)h(b)m(y)g Fp(H)2182 1897 y Fo(l)q(;N)2174 1958 y(!)2329 1933 y Fs(and)g Fp(H)2615 1886 y Fo(l)q(;N)2607 1957 y Fm(0)2762 1933 y Fs(resp)s(ectiv)m(ely)-8 b(.)64 b Fp(N)3431 1897 y Fo(l)q(;N)3421 1958 y(!)120 2053 y Fs(is)35 b(the)g(IDS)g(of)g Fp(H)793 2017 y Fo(l)q(;N)785 2078 y(!)901 2053 y Fs(.)51 b(Assume)36 b(that)f Fp(V)1614 2068 y Fm(0)1689 2053 y Fs(is)g(symmetric)f(under)i(the)f(re\015ection) h(along)e(the)120 2174 y(co)s(ordinate)d(axes.)45 b(Let)32 b Fp(\036)g Fs(b)s(e)h(the)g(ground)f(state)h(of)f Fp(H)2186 2127 y Fm(1)p Fo(;N)2178 2198 y Fm(0)2339 2174 y Fs(and)h(\010)g(its)e (p)s(erio)s(dic)g(extension)120 2294 y(to)k Fl(R)308 2258 y Fo(d)354 2294 y Fs(.)51 b(Then)37 b(for)d Fp(l)g Fq(2)f Fl(N)9 b Fs(,)42 b Fp( )1201 2309 y Fo(l)1259 2294 y Fs(=)32 b Fq(j)p Fs(\003)1463 2309 y Fo(l)1488 2294 y Fq(j)1516 2258 y Fn(\000)p Fm(1)p Fo(=)p Fm(2)1698 2294 y Fs(\010)17 b Fp(\037)1846 2309 y Fm(\003)1895 2321 y Fi(l)1958 2294 y Fs(is)35 b(the)g(ground)g(state)h(of)f(b)s(oth) g Fp(H)3240 2258 y Fo(l)q(;N)3232 2319 y(!)3383 2294 y Fs(and)120 2415 y Fp(H)209 2378 y Fo(l)q(;)p Fm(p)r(er)201 2439 y Fo(!)354 2415 y Fs(,)e(where)h(\\p)s(er")e(stands)h(for)f(p)s (erio)s(dic)f(b.c.)44 b(So)33 b(w)m(e)h(ha)m(v)m(e)727 2662 y(inf)22 b Fp(\033)t Fs(\()p Fp(H)1039 2677 y Fm(0)1078 2662 y Fs(\))28 b(=)g Fp(\025)1305 2677 y Fm(1)1361 2581 y Fk(\000)1406 2662 y Fp(H)1495 2614 y Fm(1)p Fo(;)p Fm(p)r(er)1487 2686 y(0)1654 2581 y Fk(\001)1727 2662 y Fs(=)g Fp(\025)1888 2677 y Fm(1)1944 2551 y Fk(\020)2003 2662 y Fp(H)2092 2614 y Fo(l)q(;)p Fm(p)r(er)2084 2686 y(0)2237 2551 y Fk(\021)2324 2662 y Fs(=)g Fp(\025)2485 2677 y Fm(1)2541 2551 y Fk(\020)2600 2662 y Fp(H)2689 2614 y Fo(l)q(;N)2681 2686 y Fm(0)2798 2551 y Fk(\021)2906 2662 y Fp(:)434 b Fs(\(51\))120 2904 y(By)33 b(adding)f(a)g(constan)m (t)i(w)m(e)f(get)1515 3124 y(inf)23 b Fp(\033)t Fs(\()p Fp(H)1828 3139 y Fm(0)1867 3124 y Fs(\))28 b(=)f(0)33 b Fp(:)1222 b Fs(\(52\))120 3344 y(Set)44 b Fp(m)384 3359 y Fm(1)471 3344 y Fs(=)594 3263 y Fk(R)677 3344 y Fp(dx)17 b(u)p Fs(\()p Fp(x)p Fs(\)\010)1057 3307 y Fm(2)1096 3344 y Fs(\()p Fp(x)p Fs(\))45 b(and)f(assume)g(that)g Fp(\017)g Fs(is)g(so)g(small)d(that)j Fp(m)2942 3359 y Fm(1)3029 3344 y Fp(>)j Fs(0.)77 b(F)-8 b(or)43 b(a)120 3464 y(giv)m(en)48 b(energy)h Fp(E)60 b Fq(2)p Fs(]0)p Fp(;)17 b Fs(1[)47 b(and)h(a)g(parameter)f Fp(\014)60 b(>)53 b Fs(0)48 b(c)m(ho)s(ose)h(the)f(length)f(scale)h Fp(l)56 b Fs(:=)120 3584 y([\()p Fp(\014)6 b(E)g Fs(\))362 3548 y Fn(\000)p Fm(1)p Fo(=)p Fm(2)527 3584 y Fs(])554 3599 y Fm(Gau\031)723 3584 y Fs(.)266 3771 y(By)33 b(Diric)m (hlet-Neumann)d(brac)m(k)m(eting)k(and)1938 3746 y(\024)1927 3771 y(Ceb)m(y)-5 b(\024)-44 b(sev's)36 b(inequalit)m(y)c(w)m(e)i(kno)m (w)358 3991 y Fl(P)p Fq(f)p Fp(\033)t Fs(\()p Fp(H)653 3950 y Fo(l)645 4016 y(!)697 3991 y Fs(\))17 b 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1771 y Fs(=)f(0)p Fg(.)60 b(Then)40 b(there)f(exist)f Fp(\017)g(>)f Fs(0)h Fg(and)120 1902 y Fp(E)198 1865 y Fn(\003)265 1902 y Fp(>)28 b Fs(0)f(=)h(inf)22 b Fp(\033)t Fs(\()p Fp(H)861 1917 y Fm(0)901 1902 y Fs(\))32 b Fg(suc)m(h)i(that)e (if)g Fs(min)1654 1862 y Fo(d)1654 1926 y(i)p Fm(=1)1789 1902 y Fp(\013)1851 1917 y Fo(i)1907 1902 y Fq(\025)c(\000)p Fp(\017)34 b Fg(then)1095 2122 y Fq(;)27 b(6)p Fs(=)h Fp(\033)t Fs(\()p Fp(H)1454 2137 y Fo(!)1504 2122 y Fs(\))17 b Fq(\\)g Fs(])22 b Fq(\000)h(1)p Fp(;)17 b(E)2013 2080 y Fn(\003)2051 2122 y Fs([)p Fq(\032)29 b Fp(\033)2239 2137 y Fo(pp)2314 2122 y Fs(\()p Fp(H)2433 2137 y Fo(!)2483 2122 y Fs(\))17 b Fp(:)802 b Fs(\(61\))266 2342 y(W)-8 b(e)36 b(turn)f(our)g(atten)m(tion)g(no)m(w)h(to)f(lo)s(calisation)c(a) m(w)m(a)m(y)37 b(from)d(the)h(in\014m)m(um)f(of)h Fp(\033)t Fs(\()p Fp(H)3425 2357 y Fo(!)3475 2342 y Fs(\).)120 2462 y(The)30 b(sp)s(ectrum)e(of)h(the)g(p)s(erio)s(dic)e(Sc)m(hr\177) -49 b(odinger)28 b(op)s(erator)g Fp(H)2379 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y(As)d(a)f(substitute)i(for)d(the)i(Lifshitz)f(asymptotic)f(a)h(sp)s (ecial)g(disorder)g(regime)f(has)i(b)s(een)120 3425 y(assumed)39 b(in)f(sev)m(eral)i(pap)s(ers)f([BCH97,)g(KSS98b].)62 b(In)39 b(this)f(case)h(the)g(densit)m(y)h Fp(f)49 b Fs(of)39 b(the)120 3545 y(coupling)31 b(constan)m(ts)j(is)e(required)h (to)g(satisfy)239 3692 y Fk(Z)339 3719 y Fo(!)383 3728 y Fa(\000)435 3719 y Fm(+)p Fo(\016)294 3918 y(!)338 3927 y Fa(\000)544 3828 y Fp(f)11 b Fs(\()p Fp(x)p Fs(\))17 b Fp(dx)28 b Fq(\024)g Fp(\016)1037 3787 y Fo(\034)1112 3828 y Fs(or)1248 3692 y Fk(Z)1348 3719 y Fo(!)1392 3728 y Ff(+)1303 3918 y Fo(!)1347 3927 y Ff(+)1398 3918 y Fn(\000)p Fo(\016)1507 3828 y Fp(f)11 b Fs(\()p Fp(x)p Fs(\))45 b Fq(\024)28 b Fp(\016)1894 3787 y Fo(\034)1970 3828 y Fs(for)k(some)g Fp(\034)39 b(>)28 b(d=)p Fs(2)j(and)i(small)d Fp(\016)t Fs(.)120 b(\(62\))120 4111 y(The)35 b(\014rst)f(condition)f (is)g(needed)j(when)f(considering)e(lo)m(w)m(er)h(band)g(edges,)i(the)e (second)h(for)120 4231 y(upp)s(er)k(band)h(edges.)63 b(In)40 b(this)e(case)i(w)m(e)g(can)f(pro)m(v)m(e)h(using)f(the)g (argumen)m(ts)g(of)g([KSS98b,)120 4352 y(pp.10-11]:)120 4554 y Ft(Prop)s(osition)d(18)120 4674 y Fg(Let)41 b Fp(H)384 4689 y Fo(!)434 4674 y Fp(;)17 b(u;)g(\013)q(;)g(!)746 4689 y Fm(+)845 4674 y Fg(b)s(e)41 b(as)g(in)f(Theorem)i(17,)h(the)e (densit)m(y)h Fp(f)52 b Fg(as)41 b(in)f(\(62\))h(and)g Fp(E)47 b Fg(a)41 b(lo)m(w)m(er)120 4804 y(sp)s(ectral)32 b(band)h(edge.)44 b(Let)32 b Fp(p)c Fq(2)p Fs(]0)p Fp(;)17 b Fs(2)p Fp(\034)32 b Fq(\000)22 b Fp(d)p Fs([)32 b Fg(and)g Fp(\030)g Fq(2)p Fs(]0)p Fp(;)17 b Fs(1)22 b Fq(\000)2407 4765 y Fo(d)p 2388 4781 75 4 v 2388 4838 a Fm(2)p Fo(\034)2473 4804 y Fs([)p Fg(.)43 b(Then)33 b(there)g(exists)g(a)f Fp(Q)3500 4819 y Fm(1)120 4941 y Fg(suc)m(h)i(that)e(for)g(all)f Fp(l)f Fq(\025)e Fp(Q)1077 4956 y Fm(1)1149 4941 y Fg(and)33 b Fp(\017)28 b Fq(\024)1532 4902 y Fo(l)1554 4879 y Fi(p)p Fa(\000)p Ff(2)p 1521 4918 158 4 v 1521 4976 a Fo(!)1565 4985 y Ff(+)1616 4976 y Fo(N)1721 4941 y Fg(w)m(e)34 b(ha)m(v)m(e)594 5156 y Fo(d)531 5242 y Fs(min)555 5302 y Fo(i)p Fm(=1)710 5242 y Fp(\013)772 5257 y Fo(i)828 5242 y Fq(\025)28 b(\000)p Fp(\017)h Fs(=)-17 b Fq(\))55 b Fl(P)p Fq(f)p Fp(\033)t Fs(\()p Fp(H)1587 5201 y Fo(l)1579 5266 y(!)1631 5242 y Fs(\))22 b Fq(\\)h Fs([)p Fp(E)28 b Fq(\000)22 b Fp(l)2037 5201 y Fo(p)p Fn(\000)p Fm(2)2168 5242 y Fp(;)17 b(E)27 b Fs(+)22 b Fp(l)2440 5201 y Fo(p)p Fn(\000)p Fm(2)2571 5242 y Fs(])27 b Fq(6)p Fs(=)h Fq(;g)f(\024)h Fp(l)2992 5201 y Fn(\000)p Fo(\030)3102 5242 y Fp(:)238 b Fs(\(63\))1781 5712 y(16)p eop %%Page: 17 17 17 16 bop 266 407 a Fs(No)m(w)33 b(one)f(pro)s(ceeds)i(as)e(in)f ([KSS98b])h(using)g(Com)m(b)s(es-Thomas)g(argumen)m(ts)g(to)g(pro)m(v)m (e)120 527 y(an)41 b(initial)d(scale)j(estimate)g(and)h(thereb)m(y)h (lo)s(calisation.)66 b(Note)42 b(that)f(in)f(the)i(men)m(tioned)120 648 y(pap)s(er)49 b(the)g(b)s(ound)f(\(H1\))h(is)f(sligh)m(tly)f(more)h (\015exible,)k(i.e.)c(w)m(eak)m(er,)55 b(then)49 b(the)g(one)g(w)m(e) 120 768 y(presen)m(ted)35 b(in)d(Section)g(1.)120 954 y Ft(Theorem)37 b(19)120 1075 y Fg(Let)d Fp(H)377 1090 y Fo(!)428 1075 y Fp(;)17 b(u;)g(\013)q(;)g(!)740 1090 y Fm(+)831 1075 y Fg(b)s(e)34 b(as)h(in)e(Theorem)i(17,)f(the)g(densit) m(y)i Fp(f)44 b Fg(as)35 b(in)e(\(62\))h(and)g Fp(E)40 b Fg(a)34 b(sp)s(ectral)120 1205 y(band)h(edge.)50 b(There)35 b(exist)g Fp(r)m(;)17 b(\017)31 b(>)g Fs(0)j Fg(suc)m(h)i(that)e(if)g Fs(min)2182 1165 y Fo(d)2182 1230 y(i)p Fm(=1)2317 1205 y Fp(\013)2379 1220 y Fo(i)2438 1205 y Fq(\025)e(\000)p Fp(\017)j Fg(the)g(sp)s(ectrum)g(of)f Fp(H)3490 1220 y Fo(!)120 1325 y Fg(in)e(the)h(in)m(terv)-5 b(al)31 b Fs([)p Fp(E)d Fq(\000)23 b Fp(r)m(;)17 b(e)22 b Fs(+)g Fp(r)s Fs(])33 b Fg(is)f(pure)h(p)s(oin)m(t.)120 1517 y Ft(Remark)k(20)120 1638 y Fs(F)-8 b(or)42 b(the)i(lo)s(calisation)c (Theorems)k(17)e(and)i(19)e(w)m(e)j(had)e(to)g(c)m(ho)s(ose)h Fp(\017)g Fs(small)d(dep)s(ending)120 1758 y(on)35 b(the)g(initial)30 b(scale)35 b Fp(Q)1024 1773 y Fm(1)1064 1758 y Fs(.)49 b(It)35 b(migh)m(t)e(seem)i(irritating)d(that)i Fp(Q)2480 1773 y Fm(1)2554 1758 y Fs(in)g(turn)h(dep)s(ends)h(on)f Fp(V)3463 1773 y Fo(!)3513 1758 y Fs(,)120 1878 y(i.e.)43 b(implicitly)38 b(on)43 b Fp(\017)p Fs(.)75 b(Ho)m(w)m(ev)m(er)46 b(an)d(admissible)e(initial)e(scale)k Fp(Q)2667 1893 y Fm(1)2750 1878 y Fs(for)f Fp(\017)k Fs(=)f(1,)h(i.e.)d(the)120 1999 y(p)s(oten)m(tial)36 b Fp(V)615 1963 y Fo(\017)p Fm(=1)594 2023 y Fo(!)738 1999 y Fs(\()p Fp(x)p Fs(\))h(=)1018 1924 y Fk(P)1123 2028 y Fo(k)r Fn(2)p Fj(Z)1259 2009 y Fi(d)1310 1999 y Fp(!)1371 2014 y Fo(k)1430 1999 y Fs(\()p Fp(u)1524 2014 y Fm(+)1608 1999 y Fq(\000)26 b Fp(u)1767 2014 y Fn(\000)1825 1999 y Fs(\)\()p Fp(x)g Fq(\000)g Fp(k)s Fs(\))38 b(is)f(admissible)e(for)i(the)h(p)s(oten)m (tials)120 2119 y(for)j(other)g(v)-5 b(alues)41 b(of)g Fp(\017)i Fq(2)f Fs([0)p Fp(;)17 b Fs(1],)43 b(also.)69 b(In)41 b(this)g(sense)i(the)f(dep)s(endence)h(of)e Fp(Q)3169 2134 y Fm(1)3250 2119 y Fs(on)g Fp(\017)g Fs(is)120 2240 y(uniform)27 b(for)i Fp(\017)f Fq(2)g Fs([0)p Fp(;)17 b Fs(1].)42 b(This)29 b(means)g(that)g(c)m(ho)s(osing)g Fp(\017)g Fs(closer)g(to)g(0)g(do)s(es)h(not)f(c)m(hange)h(the)120 2360 y(initial)f(scale)j Fp(Q)712 2375 y Fm(1)752 2360 y Fs(.)266 2639 y Ft(Ac)m(kno)m(wledgemen)m(ts:)266 2760 y Fs(W)-8 b(e)42 b(are)f(grateful)f(for)h(stim)m(ulating)d(discussions) k(with)f(W.)g(Kirsc)m(h,)j(K.)d(V)-8 b(eseli)m(\023)-46 b(c)41 b(and)120 2880 y(S.)f(B\177)-49 b(oc)m(k)m(er,)44 b(as)c(w)m(ell)f(as)h(for)f(v)-5 b(aluable)39 b(commen)m(ts)h (concerning)g(this)g(w)m(ork)g(b)m(y)h(E.)g(Giere,)120 3000 y(T.)26 b(Hupfer,)i(C.)e(Riebling)d(and)j(S.)g(W)-8 b(arzel.)40 b(W)-8 b(e)26 b(thank)g(the)h(Ruth-und-Gert-Massen)m(b)s (erg-)120 3121 y(Stiftung)k(for)g(\014nancial)f(supp)s(ort)i(and)g(the) g(MaPh)m(ySto)h(Cen)m(tre,)g(Denmark,)f(for)f(hospital-)120 3241 y(it)m(y)-8 b(,)32 b(where)i(part)e(of)h(this)f(w)m(ork)h(w)m(as)h (done.)120 3570 y Fr(References)120 3789 y Fs([BCH97])124 b(J.)32 b(M.)g(Barbaroux,)g(J.)g(M.)g(Com)m(b)s(es,)h(and)f(P)-8 b(.)32 b(D.)f(Hislop.)41 b(Lo)s(calization)29 b(near)608 3910 y(band)c(edges)h(for)e(random)g(Sc)m(hr\177)-49 b(odinger)25 b(op)s(erators.)30 b Fh(Helv.)e(Phys.)f(A)-5 b(cta)p Fs(,)27 b(70\(1-)608 4030 y(2\):16{43,)39 b(1997.)60 b(P)m(ap)s(ers)40 b(honouring)e(the)h(60th)f(birthda)m(y)h(of)f(Klaus)g (Hepp)608 4151 y(and)32 b(of)g(W)-8 b(alter)32 b(Hunzik)m(er,)i(P)m (art)f(I)s(I)f(\(Z)s(\177)-51 b(uric)m(h,)31 b(1995\).)120 4347 y([BS98])213 b(D.)55 b(Busc)m(hmann)i(and)e(G.)h(Stolz.)111 b(Tw)m(o-parameter)56 b(sp)s(ectral)f(a)m(v)m(eraging)608 4468 y(and)29 b(lo)s(calization)24 b(for)29 b(non-monotoneous)f(random) g(Sc)m(hr\177)-49 b(odinger)29 b(op)s(erators.)608 4588 y(Preprin)m(t)j(No.:)44 b(98-617)31 b(h)m(ttp://www.ma.utexas.edu/mp)p 2868 4588 30 4 v 37 w(arc/,)i(1998.)120 4785 y([CH94])193 b(J.-M.)45 b(Com)m(b)s(es)h(and)g(P)-8 b(.D.)45 b(Hislop.)81 b(Lo)s(calization)42 b(for)j(some)g(con)m(tin)m(uous,)608 4905 y(random)26 b(Hamiltionians)c(in)k(d-dimensions.)33 b Fh(J.)d(F)-7 b(unct.)29 b(A)n(nal.)p Fs(,)e(124:149{180,)608 5025 y(1994.)120 5222 y([CHM96])104 b(J.)47 b(M.)h(Com)m(b)s(es,)k(P)-8 b(.)48 b(D.)f(Hislop,)j(and)e(E.)g(Mourre.)88 b(Sp)s(ectral)47 b(a)m(v)m(eraging,)608 5342 y(p)s(erturbation)21 b(of)g(singular)g(sp)s (ectra,)j(and)e(lo)s(calization.)g Fh(T)-7 b(r)i(ans.)24 b(A)n(mer.)h(Math.)608 5463 y(So)-5 b(c.)p Fs(,)32 b (348\(12\):4883{4894,)d(1996.)1781 5712 y(17)p eop %%Page: 18 18 18 17 bop 120 407 a Fs([CHT99])123 b(J.)32 b(M.)h(Com)m(b)s(es,)g(P)-8 b(.)32 b(D.)g(Hislop,)f(and)i(A.)f(Tip.)43 b(Band)33 b(edge)g(lo)s(calization)28 b(and)608 527 y(the)c(densit)m(y)g(of)g (states)g(for)f(acoustic)h(and)g(electromagnetic)e(w)m(a)m(v)m(es)k(in) d(random)608 648 y(media.)42 b Fh(A)n(nn.)34 b(Inst.)g(H.)h(Poinc)-5 b(ar)n(\023)-47 b(e)34 b(Phys.)g(Th)n(\023)-47 b(eor.)p Fs(,)31 b(70\(4\):381{428,)f(1999.)120 848 y([CL90])205 b(R.)31 b(Carmona)g(and)g(J.)h(Lacroix.)41 b Fh(Sp)-5 b(e)g(ctr)g(al)33 b(The)-5 b(ory)34 b(of)f(R)-5 b(andom)33 b(Schr\177)-50 b(odinger)608 969 y(Op)-5 b(er)g(ators)p Fs(.)42 b(Birkh\177)-49 b(auser,)34 b(Boston,)e(1990.)120 1169 y([FHLM97])49 b(W)-8 b(erner)46 b(Fisc)m(her,)k(Thomas)c(Hupfer,)k (Ha)5 b(jo)46 b(Lesc)m(hk)m(e,)52 b(and)46 b(P)m(eter)h(M)s(\177)-51 b(uller.)608 1290 y(Existence)46 b(of)e(the)h(densit)m(y)h(of)e(states) h(for)g(m)m(ulti-dimensional)39 b(con)m(tin)m(uum)608 1410 y(Sc)m(hr\177)-49 b(odinger)54 b(op)s(erators)g(with)f(Gaussian)h (random)f(p)s(oten)m(tials.)106 b Fh(Comm.)608 1530 y(Math.)35 b(Phys.)p Fs(,)d(190\(1\):133{141,)e(1997.)120 1731 y([FK96])196 b(Alexander)33 b(Figotin)d(and)j(Ab)s(el)f(Klein.)42 b(Lo)s(calization)30 b(of)i(classical)g(w)m(a)m(v)m(es.)j(I.)608 1851 y(Acoustic)d(w)m(a)m(v)m(es.)46 b Fh(Comm.)34 b(Math.)h(Phys.)p Fs(,)e(180\(2\):439{482,)d(1996.)120 2052 y([FK97])196 b(Alexander)29 b(Figotin)e(and)i(Ab)s(el)g(Klein.)36 b(Lo)s(calization)26 b(of)i(classical)g(w)m(a)m(v)m(es.)k(II.)608 2172 y(Electromagnetic)f(w)m(a)m(v)m(es.)46 b Fh(Comm.)33 b(Math.)i(Phys.)p Fs(,)e(184\(2\):411{441,)d(1997.)120 2373 y([FLM98])122 b(W.)26 b(Fisc)m(her,)j(H.)d(Lesc)m(hk)m(e,)31 b(and)26 b(P)-8 b(.)27 b(M)s(\177)-51 b(uller.)32 b(Asp)s(ects)c(of)e (a)g(lo)s(calization)d(pro)s(of)608 2493 y(for)41 b(con)m(tin)m(uum)g (Sc)m(hr\177)-49 b(odinger)42 b(op)s(erators)f(with)h(Gaussian)f (random)f(p)s(oten-)608 2614 y(tials.)h(T)-8 b(alk)33 b(giv)m(en)f(in)g(Prague,)h(at)f(the)h(conference)i(QMath7,)d(June)h (1998.)120 2814 y([FMSS85])75 b(J.)30 b(F)-8 b(r\177)-49 b(ohlic)m(h,)30 b(F.)g(Martinelli,)e(E.)j(Scopp)s(ola,)f(and)g(T.)h(Sp) s(encer.)42 b(Constructiv)m(e)608 2935 y(pro)s(of)f(of)h(lo)s (calization)c(in)j(the)i(Anderson)g(tigh)m(t)f(binding)f(mo)s(del.)70 b Fh(Comm.)608 3055 y(Math.)35 b(Phys.)p Fs(,)d(101\(1\):21{46,)f (1985.)120 3256 y([FS83])218 b(J.)45 b(F)-8 b(r\177)-49 b(ohlic)m(h)43 b(and)i(T.)h(Sp)s(encer.)81 b(Absence)47 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b(from)e(the)i(German)f(b)m(y)h(Joseph)608 1860 y(Sz)s(\177)-51 b(ucs.)1781 5712 y(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0009201303100--