Content-Type: multipart/mixed; boundary="-------------0010030452795" This is a multi-part message in MIME format. ---------------0010030452795 Content-Type: text/plain; name="00-390.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-390.keywords" random Schr dinger operators, density of states, internal Lifshitz tails, random Fourier series, Newton polygon, Floquet eigenvalues ---------------0010030452795 Content-Type: application/postscript; name="conf-londres.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="conf-londres.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: /home1/klopp/PAPTEX/SOUMIS/CONF-LONDRES-00/conf-londres.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips %+ /home1/klopp/PAPTEX/SOUMIS/CONF-LONDRES-00/conf-londres.dvi -o %+ conf-londres.ps %DVIPSParameters: dpi=600, compressed 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Bracket EndArrow moveto } def /ArrowB { } def [ 227.62195 28.45274 128.03734 28.45274 /Lineto /lineto load def false Line gsave 2.0 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.8 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 56.3363 19.91682 56.3363 65.44139 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.8 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 171.28563 19.91682 171.28563 65.44139 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial 2835 2255 a tx@Dict begin { 0.0 11.84457 6.83331 2.86108 0. Uput UUput } PutCoor PutBegin end 2835 2255 a 2786 2272 a Fs(\006)2846 2284 y Fm(p)2835 2255 y tx@Dict begin PutEnd end 2835 2255 a 1299 2728 a tx@Dict begin { 0.0 7.22223 6.83331 0.0 0. 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Uput UUput } PutCoor PutBegin end 2126 2728 a 2067 2746 a Fr(E)2128 2758 y Fn(\000)2126 2728 y tx@Dict begin PutEnd end 2126 2728 a 1256 2927 a Fg(Figure)32 b(2.)41 b Fs(The)27 b(band)h(edges)f(for)g(\006)g(and)h(\006)2722 2939 y Fm(p)118 3187 y Fs(Our)f(main)h(question)f(then)h(is:)1300 3287 y Fl(\(Q\))85 b(how)30 b(do)l(es)h Fr(N)9 b Fs(\()p Fr(E)c Fs(\))30 b Fl(b)l(ehave)h(ne)l(ar)f Fr(E)2617 3299 y Fn(\006)2673 3287 y Fl(?)118 3387 y Fs(Before)38 b(w)n(e)g(state)h(our)f(result,)j(let)f(us)e(review)g(some)h(of)f(the)h (kno)n(wn)g(results)f(on)g(this)h(question.)71 b(Doing)38 b(so,)j(w)n(e)118 3486 y(will)35 b(assume)f(that)h(w)n(e)f(are)g(w)n (orking)f(in)i(dimension)f Fr(d)p Fs(;)39 b(all)34 b(the)h (de\014nitions)g(and)f(results)h(w)n(e)f(ga)n(v)n(e)f(ab)r(o)n(v)n(e)g (can)h(b)r(e)118 3586 y(generalized)26 b(to)i(this)g(case.)1646 3795 y Ft(2.)47 b(Kno)m(wn)32 b(results)331 3944 y Fs(Whether)c(answ)n (ers)e(to)i(\(Q\))g(are)f(kno)n(wn)h(or)f(not)h(dep)r(ends)g(crucially) 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b(will)g(start)f(with)h(the)g(less) f(precise)g(but)h(more)f(general)f(result.)268 2414 y Ft(3.1.)46 b(The)32 b(densit)m(y)g(of)f(states)h(deca)m(ys)h(exp)s (onen)m(tially)e(at)h(band)h(edges.)40 b Fs(W)-7 b(e)28 b(pro)n(v)n(e)331 2564 y Fg(Theorem)k Fs(3.1)p Fg(.)40 b Fl(Assume)29 b(\(H.1\).)39 b(F)-6 b(or)30 b(any)g(e)l(dge)g(of)h(a)f (gap)h(of)f Fs(\006)p Fl(,)h(say)f Fr(E)2720 2576 y Fp(0)2758 2564 y Fl(,)g(one)g(has)1329 2747 y Fs(lim)14 b(sup)1356 2816 y Fm(E)s Fn(!)p Fm(E)1523 2824 y Fi(0)1384 2873 y Fm(E)s Fn(2)p Fp(\006)1607 2691 y Fs(log)g Fo(j)g Fs(log)g Fo(j)p Fr(N)9 b Fs(\()p Fr(E)c Fs(\))19 b Fo(\000)f Fr(N)9 b Fs(\()p Fr(E)2386 2703 y Fp(0)2423 2691 y Fs(\))p Fo(jj)p 1607 2728 895 4 v 1838 2804 a Fs(log)14 b Fo(j)p Fr(E)23 b Fo(\000)18 b Fr(E)2210 2816 y Fp(0)2248 2804 y Fo(j)2535 2747 y Fr(<)k Fs(0)p Fr(:)-2569 b Fs(\(3.1\))118 3002 y(Theorem)23 b(3.1)f(sa)n(ys)g(that)h(the)h(in)n(tegrated)e(densit)n(y) 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3867 y Fs(\))532 3882 y Fm(\015)t Fn(2)p Fj(Z)660 3865 y Fi(2)725 3867 y Fs(is)33 b(a)h(collection)f(of)h(indep)r(enden)n(t)g(iden)n (tically)g(distributed)g(random)f(v)-5 b(ariables;)35 b(the)f(supp)r(ort)g(of)374 3967 y(these)27 b(random)g(v)-5 b(ariables)26 b(is)i(the)g(in)n(terv)-5 b(al)27 b([)p Fr(!)1840 3937 y Fn(\000)1895 3967 y Fr(;)14 b(!)1987 3937 y Fp(+)2042 3967 y Fs(].)268 4067 y(4.)41 b(The)27 b(common)g(distribution)h(of)g(the)g(\()p Fr(!)1651 4079 y Fm(\015)1693 4067 y Fs(\))1725 4082 y Fm(\015)t Fn(2)p Fj(Z)1853 4065 y Fi(2)1912 4067 y Fs(satis\014es)1391 4259 y(lim)1383 4310 y Fm(")p Fn(!)p Fp(0)1538 4203 y Fs(log)14 b Fo(j)g Fs(log)g Fq(P)p Fo(fj)p Fr(!)1986 4215 y Fp(0)2039 4203 y Fo(\000)19 b Fr(!)2178 4173 y Fn(\006)2233 4203 y Fo(j)k(\024)g Fr(")p Fo(gj)p 1538 4240 933 4 v 1885 4316 a(\000)14 b Fs(log)f Fr(")2503 4259 y Fs(=)23 b(0)118 4433 y(W)-7 b(e)28 b(need)g(one)f(more)g (assumption.)36 b(T)-7 b(o)27 b(state)h(it,)g(w)n(e)f(will)h(need)g(to) f(in)n(tro)r(duce)g(some)g(notations.)268 4533 y(3.2.1.)41 b Fl(The)30 b(b)l(ackgr)l(ound)h(op)l(er)l(ator.)43 b Fs(The)27 b(Lifshitz)h(exp)r(onen)n(t)g(is)f(computed)h(using)f(a)g(w)n (ell)g(c)n(hosen)g(bac)n(kground)118 4632 y(p)r(erio)r(dic)i(op)r (erator)e(that)j(can)f(but)g(need)g(not)h(b)r(e)f Fr(H)7 b Fs(.)41 b(This)29 b(bac)n(kground)f(op)r(erator)f(dep)r(ends)i(on)g (the)h(sp)r(ectral)e(edge)118 4732 y Fr(E)179 4744 y Fp(0)252 4732 y Fs(one)34 b(is)h(considering.)57 b(More)34 b(precisely)g(it)h(dep)r(ends)g(on)f(whether)h Fr(E)2486 4744 y Fp(0)2559 4732 y Fs(is)f(the)h(upp)r(er)g(\(resp.)58 b(lo)n(w)n(er\))34 b(edge)g(\(i.e.)118 4832 y(maxim)n(um)23 b(\(resp.)35 b(minim)n(um\)\))25 b(of)e(a)f(gap.)35 b(Recall)23 b(that)g(the)h(essen)n(tial)e(in\014m)n(um)i(\(resp.)35 b(suprem)n(um\))23 b(of)g(the)h(random)118 4931 y(v)-5 b(ariables)35 b(\()p Fr(!)555 4943 y Fm(\015)598 4931 y Fs(\))630 4943 y Fm(\015)709 4931 y Fs(is)i Fr(!)854 4943 y Fp(+)945 4931 y Fs(\(resp.)63 b Fr(!)1264 4943 y Fn(\000)1320 4931 y Fs(\).)g(Let)37 b Fr(H)1672 4901 y Fp(+)1763 4931 y Fs(\(resp.)63 b Fr(H)2106 4901 y Fn(\000)2162 4931 y Fs(\))37 b(b)r(e)g(the)g(p)r(erio)r(dic)f(Sc)n(hr\177)-42 b(odinger)35 b(op)r(erator)f(de\014ned)118 5031 y(b)n(y)39 b Fr(H)321 5001 y Fn(\000)420 5031 y Fs(=)j Fr(H)34 b Fs(+)25 b Fr(!)772 5043 y Fn(\000)842 4969 y Fk(P)930 5056 y Fm(\015)t Fn(2)p Fj(Z)1058 5039 y Fi(2)1102 5031 y Fr(V)1150 5043 y Fm(\015)1233 5031 y Fs(\(resp.)72 b Fr(H)1585 5001 y Fp(+)1682 5031 y Fs(=)43 b Fr(H)33 b Fs(+)26 b Fr(!)2035 5043 y Fp(+)2104 4969 y Fk(P)2191 5056 y Fm(\015)t Fn(2)p Fj(Z)2319 5039 y Fi(2)2364 5031 y Fr(V)2412 5043 y Fm(\015)2455 5031 y Fs(\).)72 b(Then,)43 b(one)c(writes)g Fr(H)3327 5043 y Fm(!)3418 5031 y Fs(=)k Fr(H)3602 5001 y Fn(\000)3684 5031 y Fs(+)26 b Fr(V)3842 5001 y Fn(\000)3823 5051 y Fm(!)118 5142 y Fs(\(resp.)69 b Fr(H)460 5154 y Fm(!)548 5142 y Fs(=)41 b Fr(H)730 5111 y Fp(+)810 5142 y Fs(+)25 b Fr(V)967 5111 y Fp(+)948 5162 y Fm(!)1022 5142 y Fs(\))39 b(where)e Fr(V)1410 5111 y Fn(\000)1391 5162 y Fm(!)1507 5142 y Fs(=)1613 5079 y Fk(P)1700 5166 y Fm(\015)t Fn(2)p Fj(Z)1828 5150 y Fi(2)1872 5142 y Fr(!)1927 5111 y Fp(+)1924 5162 y Fm(\015)1982 5142 y Fr(V)2030 5154 y Fm(\015)2112 5142 y Fs(and)h Fr(!)2339 5111 y Fp(+)2336 5162 y Fm(\015)2434 5142 y Fs(=)j Fr(!)2592 5154 y Fm(\015)2659 5142 y Fo(\000)25 b Fr(!)2801 5154 y Fn(\000)2895 5142 y Fs(\(resp.)69 b Fr(V)3235 5111 y Fp(+)3216 5162 y Fm(!)3331 5142 y Fs(=)3436 5079 y Fk(P)3524 5166 y Fm(\015)t Fn(2)p Fj(Z)3652 5150 y Fi(2)3696 5142 y Fr(!)3751 5111 y Fn(\000)3748 5162 y Fm(\015)3807 5142 y Fr(V)3855 5154 y Fm(\015)118 5252 y Fs(and)38 b Fr(!)345 5222 y Fn(\000)342 5273 y Fm(\015)440 5252 y Fs(=)i Fr(!)597 5264 y Fm(\015)664 5252 y Fo(\000)25 b Fr(!)806 5264 y Fp(+)861 5252 y Fs(\).)67 b(The)38 b(random)f(v)-5 b(ariables)36 b(\()p Fr(!)1920 5222 y Fp(+)1917 5273 y Fm(\015)1975 5252 y Fs(\))2007 5264 y Fm(\015)2088 5252 y Fs(\(resp.)66 b(\()p Fr(!)2445 5222 y Fn(\000)2442 5273 y Fm(\015)2501 5252 y Fs(\))2533 5264 y Fm(\015)2576 5252 y Fs(\))38 b(are)f(i.i.d.)68 b(and)37 b(non-negativ)n(e)f(\(resp.)118 5355 y(non-p)r(ositiv)n(e\).)g (The)28 b(results)f(of)h([)p Ft(6)o Fs(])g(imply)1991 5454 y Fp(3)p eop %%Page: 4 4 4 3 bop 291 241 a Fo(\017)41 b Fr(E)435 253 y Fp(0)499 241 y Fs(is)27 b(the)g(upp)r(er)g(edge)f(of)h(a)f(gap)g(of)h(\006)f(if) i(and)e(only)h(if)g Fr(E)2201 253 y Fp(0)2265 241 y Fs(is)g(the)g(upp)r (er)g(edge)f(of)h(a)f(gap)g(of)h(the)g(sp)r(ectrum)g(of)374 340 y Fr(H)450 310 y Fn(\000)505 340 y Fs(.)291 440 y Fo(\017)41 b Fr(E)435 452 y Fp(0)501 440 y Fs(is)28 b(the)h(lo)n(w)n (er)e(edge)g(of)i(a)f(gap)f(of)i(\006)f(if)h(and)f(only)g(if)h Fr(E)2203 452 y Fp(0)2269 440 y Fs(is)g(the)f(lo)n(w)n(er)f(edge)h(of)g (a)g(gap)g(of)g(the)h(sp)r(ectrum)g(of)374 540 y Fr(H)450 510 y Fp(+)505 540 y Fs(.)118 665 y(F)-7 b(or)32 b(the)g(sak)n(e)f(of)h (de\014niteness,)h(let)g(us)f(assume)f(that)i Fr(E)1948 677 y Fp(0)2017 665 y Fs(is)f(the)h(upp)r(er)f(edge)g(of)g(a)f(gap)h (of)g(\006.)50 b(Then,)34 b(the)e(correct)118 765 y(bac)n(kground)26 b(op)r(erator)g(will)h(b)r(e)h Fr(H)1246 734 y Fn(\000)1302 765 y Fs(.)268 864 y(3.2.2.)41 b Fl(Flo)l(quet)32 b(the)l(ory.)43 b Fs(W)-7 b(e)30 b(recall)f(some)h(facts)g(on)f(the)i(Flo)r(quet)f(sp)r (ectrum)h(of)f Fr(H)2978 834 y Fn(\000)3064 864 y Fs(\(see)g(e.g.)f([)p Ft(26,)34 b(15,)g(27)o Fs(]\).)118 964 y(The)f(Flo)r(quet)h(sp)r (ectrum)f(of)g Fr(H)1152 934 y Fn(\000)1241 964 y Fs(is)g(the)h(sp)r (ectrum)f(of)g(the)h(di\013eren)n(tial)e(op)r(erator)g Fr(H)2933 934 y Fn(\000)3022 964 y Fs(acting)h(on)f Fr(L)3454 934 y Fp(2)3454 987 y(lo)r(c)3541 964 y Fs(\()p Fq(R)3627 934 y Fp(2)3671 964 y Fs(\))h(with)118 1063 y(quasi-p)r(erio)r(dic)f(b) r(oundary)g(conditions.)53 b(F)-7 b(or)32 b Fr(\022)i Fo(2)e Fq(T)1852 1033 y Fn(\003)1922 1063 y Fs(=)g Fq(R)2073 1033 y Fp(2)2116 1063 y Fr(=)p Fs(\()p Fq(Z)2251 1033 y Fp(2)2282 1063 y Fs(\))2314 1033 y Fn(\003)2386 1063 y Fs(\(here)g(\()p Fq(Z)2697 1033 y Fp(2)2728 1063 y Fs(\))2760 1033 y Fn(\003)2831 1063 y Fs(=)f(2)p Fr(\031)s Fq(Z)3080 1033 y Fp(2)3145 1063 y Fs(is)i(the)g(dual)g(lattice)g(of)118 1163 y Fq(Z)179 1133 y Fp(2)243 1163 y Fs(i.e.)53 b(for)32 b Fr(\015)k Fo(2)c Fq(Z)761 1133 y Fp(2)825 1163 y Fs(and)h Fr(\015)1040 1133 y Fn(\003)1109 1163 y Fo(2)f Fs(\()p Fq(Z)1290 1133 y Fp(2)1321 1163 y Fs(\))1353 1133 y Fn(\003)1392 1163 y Fs(,)i(one)e(has)g Fr(\015)5 b(\015)1855 1133 y Fn(\003)1924 1163 y Fo(2)32 b Fs(2)p Fr(\031)s Fq(Z)o Fs(\),)d(consider)i(the)i(follo)n(wing)f(eigen)n(v)-5 b(alue)32 b(problem)g(on)118 1263 y Fr(L)175 1233 y Fp(2)175 1286 y(lo)r(c)262 1263 y Fs(\()p Fq(R)348 1233 y Fp(2)391 1263 y Fs(\))1190 1349 y Fk(\()1340 1435 y Fr(H)1416 1405 y Fn(\000)1471 1435 y Fr(')24 b Fs(=)e Fr(E)5 b(')1340 1554 y(')p Fs(\()p Fr(x)19 b Fs(+)f Fr(\015)5 b Fs(\))23 b(=)g Fr(e)1805 1524 y Fm(i\015)t(\022)1904 1554 y Fr(')p Fs(\()p Fr(x)p Fs(\))p Fr(;)43 b Fo(8)p Fr(x)22 b Fo(2)i Fq(R)2384 1524 y Fp(2)2427 1554 y Fr(;)41 b Fo(8)p Fr(\015)27 b Fo(2)d Fq(Z)2748 1524 y Fp(2)2779 1554 y Fr(:)118 1491 y Fs(\(3.2\))118 1721 y(As)39 b Fr(H)328 1691 y Fn(\000)424 1721 y Fs(is)g(elliptic,)j(one)d(kno)n(ws)f(that)i(the)g(eigen)n(v)-5 b(alues)38 b(of)45 b(\(3.2\))39 b(are)f(discrete;)45 b(when)39 b(rep)r(eated)g(according)f(to)118 1821 y(m)n(ultiplicit)n(y) -7 b(,)42 b(w)n(e)c(denote)g(them)h(b)n(y)f Fr(E)1414 1833 y Fp(0)1452 1821 y Fs(\()p Fr(\022)r Fs(\))k Fo(\024)e Fr(E)1765 1833 y Fp(1)1803 1821 y Fs(\()p Fr(\022)r Fs(\))i Fo(\024)e(\001)14 b(\001)g(\001)41 b(\024)g Fr(E)2360 1833 y Fm(n)2405 1821 y Fs(\()p Fr(\022)r Fs(\))h Fo(\024)f Fr(:)14 b(:)g(:)g Fs(.)69 b(They)38 b(are)g(called)g(the)h Fl(Flo)l(quet)118 1920 y(eigenvalues)c Fs(of)f Fr(H)734 1890 y Fn(\000)790 1920 y Fs(.)56 b(These)34 b(functions)g(are)f(Lipsc) n(hitz)h(con)n(tin)n(uous)f(in)i(the)f(v)-5 b(ariable)33 b Fr(\022)r Fs(;)k(when)e(simple,)h(they)e(are)118 2020 y(ev)n(en)27 b(analytic)g(in)h Fr(\022)r Fs(.)37 b(Moreo)n(v)n(er,)25 b(W)-7 b(eyl's)28 b(la)n(w)f(tells)g(us)h(that)1219 2171 y Fr(E)1280 2183 y Fm(n)1326 2171 y Fs(\()p Fr(\022)r Fs(\))c Fo(!)f Fs(+)p Fo(1)k Fs(as)g Fr(n)c Fo(!)g Fs(+)p Fo(1)k Fs(\(uniformly)h(in)g Fr(\022)r Fs(\))p Fr(:)-2678 b Fs(\(3.3\))331 2331 y(The)29 b(sp)r(ectrum)g(of)f Fr(H)1038 2301 y Fn(\000)1123 2331 y Fs(is)h(giv)n(en)f(b)n(y)g Fr(\033)s Fs(\()p Fr(H)1700 2301 y Fn(\000)1757 2331 y Fs(\))d(=)g Fo([)1959 2343 y Fm(n)p Fn(\025)p Fp(0)2089 2331 y Fr(E)2150 2343 y Fm(n)2196 2331 y Fs(\()p Fq(T)2284 2301 y Fn(\003)2321 2331 y Fs(\).)41 b(Let)29 b Fr(E)34 b Fs(b)r(e)29 b(in)g(the)h(sp)r(ectrum)e(of)h Fr(H)3554 2301 y Fn(\000)3610 2331 y Fs(.)41 b Fr(E)33 b Fs(will)118 2430 y(b)r(e)27 b Fl(simple)g Fs(if)g(the)f(set)h Fo(f)p Fr(p)22 b Fo(2)h Fq(N)t Fs(;)46 b Fo(9)p Fr(\022)26 b Fo(2)d Fq(T)1382 2400 y Fn(\003)1420 2430 y Fr(;)40 b(E)1544 2442 y Fm(p)1583 2430 y Fs(\()p Fr(\022)r Fs(\))24 b(=)e Fr(E)5 b Fo(g)26 b Fs(is)g(reduced)g(to)g(a)g(single)g(in)n(teger.)35 b(Note)26 b(that,)h(b)n(y)g(\(3.3\),)f(this)118 2530 y(set)31 b(of)f(in)n(tegers)f(is)i(alw)n(a)n(ys)d(\014nite.)46 b(An)31 b(edge)f(of)g(a)g(gap)g(of)g(\006)h(will)g(b)r(e)f(called)g (simple)h(if)g(it)g(is)f(simple)h(for)f Fr(H)3608 2500 y Fn(\000)3694 2530 y Fs(\(resp.)118 2630 y Fr(H)194 2599 y Fp(+)249 2630 y Fs(\))f(and)g(it)h(is)f(the)h(upp)r(er)f(edge)g (\(resp.)41 b(the)29 b(lo)n(w)n(er)f(edge\))h(of)g(a)g(gap)f(of)h (\006.)42 b(By)29 b([)p Ft(12)o Fs(],)h(w)n(e)f(kno)n(w)f(that)h(for)g (a)g(generic)118 2729 y(p)r(erio)r(dic)f(p)r(oten)n(tial,)f(edges)g(of) h(gaps)e(are)h(simple.)268 2829 y(3.2.3.)41 b Fl(The)30 b(r)l(esult.)42 b Fs(W)-7 b(e)28 b(pro)n(v)n(e)331 2988 y Fg(Theorem)k Fs(3.2)p Fg(.)40 b Fl(L)l(et)33 b Fr(E)1118 3000 y Fp(0)1190 2988 y Fl(b)l(e)h(an)g(e)l(dge)h(of)g(a)g(gap)g(of)g Fs(\006)f Fl(that)g(is)h(simple.)53 b(Then,)36 b(ther)l(e)e(exists)g Fs(0)c Fr(<)h(\013)g Fo(\024)g Fs(+)p Fo(1)118 3087 y Fl(such)f(that)1360 3266 y Fs(lim)1318 3319 y Fm(E)s Fn(!)p Fm(E)1485 3327 y Fi(0)1346 3376 y Fm(E)s Fn(2)p Fp(\006)1541 3209 y Fs(log)14 b Fo(j)g Fs(log)g Fo(j)p Fr(N)9 b Fs(\()p Fr(E)c Fs(\))19 b Fo(\000)f Fr(N)9 b Fs(\()p Fr(E)2320 3221 y Fp(0)2358 3209 y Fs(\))p Fo(jj)p 1541 3246 895 4 v 1772 3323 a Fs(log)14 b Fo(j)p Fr(E)24 b Fo(\000)18 b Fr(E)2145 3335 y Fp(0)2182 3323 y Fo(j)2469 3266 y Fs(=)23 b Fo(\000)p Fr(\013:)-2580 b Fs(\(3.4\))118 3514 y(The)33 b(exp)r(onen)n(tial)f(deca)n(y)f(on)h(the)h(in)n (tegrated)f(densit)n(y)g(of)h(states)f(at)g(the)h(edges)f(of)g(the)h (sp)r(ectrum)g(\006)f(is)h(kno)n(wn)e(as)118 3613 y(\\Lifshitz)c (tails")e(for)h(the)h(IDS.)g(The)f(exp)r(onen)n(t)h Fr(\013)g Fs(is)f(called)g(the)h Fl(Lifshitz)j(exp)l(onent)p Fs(.)36 b(The)26 b(computation)g(of)h Fr(\013)f Fs(is)h(not)118 3713 y(ob)n(vious;)f(section)i(4)f(is)g(dev)n(oted)g(to)h(its)g (description.)1139 3929 y Ft(4.)47 b(Ho)m(w)32 b(to)g(compute)e(the)i (Lifshitz)g(exp)s(onen)m(t)331 4079 y Fs(W)-7 b(e)30 b(no)n(w)g(assume)g(that)g(\(H.1'\))h(is)f(satis\014ed.)45 b(F)-7 b(or)30 b(the)g(sak)n(e)g(of)g(de\014niteness,)h(let)g(us)f (assume)g(that)g Fr(E)3628 4091 y Fp(0)3696 4079 y Fs(is)g(the)118 4179 y(upp)r(er)e(edge)f(of)h(a)g(gap)f(of)g(\006.)38 b(Assume)28 b Fr(E)1451 4191 y Fp(0)1516 4179 y Fs(is)g(simple.)37 b(Let)28 b Fr(E)2101 4191 y Fm(n)2142 4199 y Fi(0)2179 4179 y Fs(\()p Fo(\001)p Fs(\))h(b)r(e)f(the)g(unique)g(Flo)r(quet)g (eigen)n(v)-5 b(alue)27 b(taking)g(the)118 4278 y(v)-5 b(alue)28 b Fr(E)394 4290 y Fp(0)431 4278 y Fs(.)37 b(De\014ne)28 b(the)g(set)g Fo(S)h Fs(=)23 b Fo(f)p Fr(\022)i Fo(2)e Fq(T)1428 4248 y Fn(\003)1466 4278 y Fs(;)41 b Fr(E)1591 4290 y Fm(n)1632 4298 y Fi(0)1669 4278 y Fs(\()p Fr(\022)r Fs(\))24 b(=)f Fr(E)1947 4290 y Fp(0)1984 4278 y Fo(g)p Fs(.)37 b(Then,)28 b(there)f(exists)g Fr(\021)g(>)22 b Fs(0)27 b(suc)n(h)h(that)291 4404 y Fo(\017)41 b Fs(F)-7 b(or)27 b Fr(n)c(<)f(n)733 4416 y Fp(0)770 4404 y Fs(,)28 b(for)f(all)g Fr(\022)f Fo(2)d Fq(T)1262 4373 y Fn(\003)1300 4404 y Fs(,)k Fr(E)1411 4416 y Fm(n)1457 4404 y Fs(\()p Fr(\022)r Fs(\))d Fr(<)e(E)1734 4416 y Fp(0)1790 4404 y Fo(\000)c Fr(\021)s Fs(.)291 4503 y Fo(\017)41 b Fs(F)-7 b(or)27 b Fr(n)c(>)f(n)733 4515 y Fp(0)770 4503 y Fs(,)28 b(for)f(all)g Fr(\022)f Fo(2)d Fq(T)1262 4473 y Fn(\003)1300 4503 y Fs(,)k Fr(E)1411 4515 y Fm(n)1457 4503 y Fs(\()p Fr(\022)r Fs(\))d Fr(>)e(E)1734 4515 y Fp(0)1790 4503 y Fs(+)c Fr(\021)s Fs(.)118 4628 y(This)29 b(implies)g(that,)h(for)f Fr(\022)i Fs(in)f(some)e(neigh)n(b)r(orho)r(o)r(d)g(of)h Fo(S)6 b Fs(,)30 b(the)g(Flo)r(quet)f(eigen)n(v)-5 b(alue)28 b Fr(E)2945 4640 y Fm(n)2986 4648 y Fi(0)3023 4628 y Fs(\()p Fr(\022)r Fs(\))i(is)f(simple,)h(hence,)f(the)118 4728 y(function)i Fr(\022)f Fo(7!)d Fr(E)686 4740 y Fm(n)727 4748 y Fi(0)764 4728 y Fs(\()p Fr(\022)r Fs(\))k(is)f(real)f(analytic)h (in)g(some)g(neigh)n(b)r(orho)r(o)r(d)f(of)h Fo(S)6 b Fs(.)46 b(The)30 b(Lifshitz)h(exp)r(onen)n(t)f(will)g(dep)r(end)h(on) 118 4828 y(the)d(w)n(a)n(y)e Fr(E)490 4840 y Fm(n)531 4848 y Fi(0)596 4828 y Fs(v)-5 b(anishes)27 b(at)h Fo(S)34 b Fs(and)27 b(on)g(the)h(curv)-5 b(ature)27 b(of)h Fo(S)6 b Fs(.)331 4927 y(T)-7 b(o)32 b(describ)r(e)f(it)i(precisely)-7 b(,)33 b(w)n(e)f(need)g(to)g(in)n(tro)r(duce)g(some)g(ob)5 b(jects)31 b(from)h(analytic)g(geometry)-7 b(.)50 b(If)32 b Fo(E)40 b Fs(is)32 b(a)g(set)118 5027 y(con)n(tained)c(in)h(the)h (closed)e(\014rst)g(quadran)n(t)g(in)h Fq(R)1665 4997 y Fp(2)1737 5027 y Fs(then)h(its)f Fl(exterior)i(c)l(onvex)g(hul)t(l)e Fs(is)g(the)g(con)n(v)n(ex)f(h)n(ull)h(of)f(the)i(union)118 5127 y(of)e(the)g(rectangles)e Fr(R)807 5139 y Fm(xy)907 5127 y Fs(=)d([)p Fr(x;)14 b Fo(1)p Fs(\))19 b Fo(\002)f Fs([)p Fr(y)s(;)c Fo(1)p Fs(\),)28 b(where)f(the)h(union)g(is)f(tak)n (en)g(o)n(v)n(er)f(all)h(\()p Fr(x;)14 b(y)s Fs(\))24 b Fo(2)f(E)7 b Fs(.)331 5226 y(Pic)n(k)26 b Fr(\022)555 5238 y Fp(0)616 5226 y Fo(2)d(S)34 b Fs(and)28 b(consider)e(the)i (Newton)g(diagram)e(of)i Fr(E)2192 5238 y Fm(n)2233 5246 y Fi(0)2297 5226 y Fs(at)g Fr(\022)2438 5238 y Fp(0)2475 5226 y Fs(,)g(i.e.,)268 5352 y(1.)41 b(Express)26 b Fr(E)744 5364 y Fm(n)785 5372 y Fi(0)849 5352 y Fs(as)h(a)g(T)-7 b(a)n(ylor)26 b(series)h(at)g Fr(\022)1644 5364 y Fp(0)1681 5352 y Fs(,)h Fr(E)1793 5364 y Fm(n)1834 5372 y Fi(0)1871 5352 y Fs(\()p Fr(\022)1944 5321 y Fp(1)1982 5352 y Fr(;)14 b(\022)2060 5321 y Fp(2)2097 5352 y Fs(\))24 b(=)2240 5289 y Fk(P)2328 5377 y Fm(ij)2400 5352 y Fr(a)2444 5364 y Fm(ij)2503 5352 y Fs(\()p Fr(\022)2576 5321 y Fp(1)2632 5352 y Fo(\000)18 b Fr(\022)2756 5321 y Fp(1)2754 5372 y(0)2793 5352 y Fs(\))2825 5321 y Fm(i)2853 5352 y Fs(\()p Fr(\022)2926 5321 y Fp(2)2983 5352 y Fo(\000)g Fr(\022)3107 5321 y Fp(2)3105 5372 y(0)3144 5352 y Fs(\))3176 5321 y Fm(j)3211 5352 y Fs(,)28 b Fr(\022)d Fs(=)e(\()p Fr(\022)3487 5321 y Fp(1)3525 5352 y Fr(;)14 b(\022)3603 5321 y Fp(2)3640 5352 y Fs(\).)1991 5454 y Fp(4)p eop %%Page: 5 5 5 4 bop 268 241 a Fs(2.)41 b(F)-7 b(orm)30 b(the)i(exterior)d(con)n(v)n (ex)h(h)n(ull)h(of)g(the)g(p)r(oin)n(ts)g(\()p Fr(i;)14 b(j)5 b Fs(\))31 b(with)h Fr(a)2429 253 y Fm(ij)2516 241 y Fo(6)p Fs(=)c(0.)47 b(This)31 b(is)g(a)g(con)n(v)n(ex)e(p)r (olygon,)i(called)374 340 y(the)d Fl(Newton)h(p)l(olygon)p Fs(.)268 440 y(3.)41 b(The)27 b(b)r(oundary)g(of)h(the)g(p)r(olygon)e (is)i(the)g Fl(Newton)h(diagr)l(am)p Fs(.)118 570 y(The)36 b(Newton)g(deca)n(y)f(exp)r(onen)n(t)h(is)g(then)g(de\014ned)g(as)g (follo)n(ws.)61 b(The)36 b(Newton)g(diagram)e(consists)h(of)h(certain)f (line)118 669 y(segmen)n(ts.)46 b(Extend)30 b(eac)n(h)g(to)h(a)f (complete)h(line)g(and)f(in)n(tersect)h(it)g(with)g(the)g(diagonal)e (line)i Fr(\022)3150 639 y Fp(1)3216 669 y Fs(=)d Fr(\022)3350 639 y Fp(2)3387 669 y Fs(.)47 b(This)31 b(giv)n(es)e(a)118 769 y(collection)i(of)h(p)r(oin)n(ts)g(\()p Fr(a)920 781 y Fm(k)961 769 y Fr(;)14 b(a)1042 781 y Fm(k)1082 769 y Fs(\),)33 b(one)f(for)f(eac)n(h)g(b)r(oundary)g(segmen)n(t.)48 b(T)-7 b(ak)n(e)31 b(the)h(recipro)r(cal)e(of)i(the)g(largest)e Fr(a)3691 781 y Fm(k)3764 769 y Fs(and)118 869 y(call)f(this)h(n)n(um)n (b)r(er)37 b(~)-50 b Fr(\013)p Fs(\()p Fr(E)886 881 y Fm(n)927 889 y Fi(0)964 869 y Fr(;)14 b(\022)1040 881 y Fp(0)1077 869 y Fs(\);)31 b(it)f(is)f(the)h Fl(Newton)h(de)l(c)l(ay)h (exp)l(onent)p Fs(.)42 b(De\014ne)30 b Fr(\013)p Fs(\()p Fr(E)2792 881 y Fm(n)2833 889 y Fi(0)2870 869 y Fr(;)14 b(\022)2946 881 y Fp(0)2983 869 y Fs(\))27 b(=)e(min)q Fo(f)8 b Fs(~)-50 b Fr(\013)p Fs(\()p Fr(E)3459 881 y Fm(n)3500 889 y Fi(0)3556 869 y Fo(\016)19 b Fr(T)3666 881 y Fp(0)3703 869 y Fr(;)14 b(\022)3779 881 y Fp(0)3816 869 y Fs(\))27 b(:)118 968 y Fr(T)167 980 y Fp(0)204 968 y Fs(\()p Fo(\001)p Fs(\))d(=)e Fr(\022)441 980 y Fp(0)497 968 y Fs(+)c Fr(T)12 b Fs(\()p Fo(\001)18 b(\000)g Fr(\022)836 980 y Fp(0)873 968 y Fs(\))p Fr(;)42 b(T)34 b Fo(2)23 b Fr(S)5 b(L)p Fs(\(2)p Fr(;)14 b Fq(R)p Fs(\))p Fo(g)p Fs(.)331 1068 y(Similarly)-7 b(,)38 b(de\014ne)e Fr(\013)p Fs(\()p Fr(E)1105 1080 y Fm(n)1146 1088 y Fi(0)1184 1068 y Fr(;)14 b(\022)r Fs(\))37 b(if)g Fr(\022)h Fs(is)f(an)n(y)e (other)h(p)r(oin)n(t)g(in)h Fo(S)6 b Fs(,)39 b(the)e(zero)e(set)h(of)h Fr(E)3065 1080 y Fm(n)3106 1088 y Fi(0)3143 1068 y Fs(.)63 b(Then,)39 b(the)d Fl(Lifshitz)118 1167 y(exp)l(onent)27 b Fr(\013)h Fs(is)g(de\014ned)g(b)n(y)1671 1327 y Fr(\013)23 b Fs(=)g(min)1842 1381 y Fm(\022)r Fn(2S)1987 1327 y Fr(\013)p Fs(\()p Fr(E)2133 1339 y Fm(n)2174 1347 y Fi(0)2212 1327 y Fr(;)14 b(\022)r Fs(\))p Fr(:)-2227 b Fs(\(4.1\))118 1525 y(The)28 b(Lifshitz)g(exp)r(onen)n(t)f Fr(\013)h Fs(is)f(p)r(ositiv)n(e)g(as)g Fr(\022)e Fo(7!)e Fr(\013)p Fs(\()p Fr(E)1834 1537 y Fm(n)1875 1545 y Fi(0)1913 1525 y Fr(;)14 b(\022)r Fs(\))28 b(is)f(a)g(p)r(ositiv)n(e,)g(lo)n(w)n(er)f (semi-con)n(tin)n(uous)g(function)i(and)f Fo(S)118 1624 y Fs(is)h(compact.)331 1724 y(T)-7 b(o)21 b(conclude)g(this)g (subsection,)h(let)g(us)f(compare)f Fr(N)30 b Fs(to)22 b Fr(N)2157 1736 y Fm(p)2195 1724 y Fs(,)g(the)g(densit)n(y)f(of)g (states)g(of)g(the)h(underlying)f(p)r(erio)r(dic)118 1823 y(op)r(erator)26 b Fr(H)7 b Fs(.)36 b(Up)28 b(to)f(a)g(constan)n (t)g(factor,)f Fr(N)1556 1835 y Fm(p)1594 1823 y Fs(\()p Fr(E)1687 1835 y Fp(0)1743 1823 y Fs(+)17 b Fr(")p Fs(\))h Fo(\000)g Fr(N)2064 1835 y Fm(p)2102 1823 y Fs(\()p Fr(E)2195 1835 y Fp(0)2233 1823 y Fs(\))27 b(is)g(the)h(v)n(olume)f(of)g Fo(f)p Fr(\022)e Fo(2)e Fq(T)3138 1793 y Fn(\003)3176 1823 y Fs(;)41 b Fr(E)3301 1835 y Fm(n)3342 1843 y Fi(0)3379 1823 y Fs(\()p Fr(\022)r Fs(\))24 b Fo(\024)f Fr(E)3657 1835 y Fp(0)3712 1823 y Fs(+)17 b Fr(")p Fo(g)p Fs(.)118 1923 y(F)-7 b(rom)28 b(the)h(equiv)-5 b(alence)28 b(b)r(et)n(w)n(een)h (\(2.2\))f(and)g(\(2.3\))h(\(see)f(section)g(2.2\),)h(one)f(ma)n(y)g (ha)n(v)n(e)f(conjectured)h(that,)h(at)g(band)118 2026 y(edges)c Fr(N)34 b Fs(b)r(eha)n(v)n(es)24 b(roughly)g(lik)n(e)h Fr(e)1235 1996 y Fn(\000)p Fp(1)p Fm(=)l(N)1403 2004 y Ff(p)1466 2026 y Fs(\(if)h(one)f(tak)n(es)g(the)g(double)g(logarithm) f(of)i(b)r(oth)f(terms\).)36 b(W)-7 b(e)26 b(see)f(here)f(that)118 2125 y(this)d(will)g(not)g(b)r(e)g(true)g(in)g(general)e(if)i(the)g (set)g Fo(S)28 b Fs(has)20 b(some)g(curv)-5 b(ature.)34 b(F)-7 b(or)20 b(example,)i(if)f(w)n(e)f(assume)g(that)h Fr(E)3541 2137 y Fm(n)3582 2145 y Fi(0)3619 2125 y Fs(\()p Fr(\022)r Fs(\))5 b Fo(\000)g Fr(E)3860 2137 y Fp(0)118 2225 y Fs(v)-5 b(anishes)29 b(to)h(constan)n(t)e(order)g Fr(q)33 b Fs(on)c(a)g(curv)n(e,)g(and)h(if)g(the)g(maxim)n(um)f(order)f (of)i(v)-5 b(anishing)29 b(of)g(the)h(curv)-5 b(ature)29 b(of)g(the)118 2325 y(curv)n(e)f(is)i Fr(m)f Fs(\(th)n(us)h(non-v)-5 b(anishing)28 b(curv)-5 b(ature)28 b(means)h Fr(m)d Fs(=)g(0\))j(then)h Fr(\013)c Fs(=)2611 2292 y Fm(m)p Fp(+3)p 2568 2306 228 4 v 2568 2353 a Fm(q)r Fp(\()p Fm(m)p Fp(+2\))2835 2325 y Fs(and)j(the)h(v)n(olume)f(will)g(ha)n(v)n(e)f(the)118 2446 y(exp)r(onen)n(t)486 2414 y Fp(1)p 486 2428 34 4 v 486 2475 a Fm(q)529 2446 y Fs(.)1364 2694 y Ft(5.)47 b(The)32 b(main)e(ideas)i(of)f(the)h(pro)s(of)331 2843 y Fs(T)-7 b(o)30 b(\014x)g(ideas)g(let)h(us)g(assume)e(that)i(w)n(e)f (w)n(ork)f(at)i(the)g(upp)r(er)f(edge)g(of)h(a)f(gap)g(i.e.)45 b(at)31 b(the)f(edge)g Fr(E)3490 2855 y Fn(\000)3547 2843 y Fs(.)45 b(A)31 b(lo)n(w)n(er)118 2943 y(edge)c(can)g(dealt)h (with)g(in)g(the)g(same)f(w)n(a)n(y)-7 b(.)36 b(Without)28 b(restriction)f(of)g(generalit)n(y)-7 b(,)26 b(w)n(e)i(will)f(also)g (assume)g Fr(E)3517 2955 y Fn(\000)3596 2943 y Fs(=)c(0)331 3042 y(First,)29 b(follo)n(wing)e(the)i(ideas)f(dev)n(elop)r(ed)g(in)h ([)p Ft(14)p Fs(],)g(w)n(e)f(sho)n(w)g(that)h(the)g(study)g(of)g(the)g (densit)n(y)f(of)h(states)f(of)h Fr(H)3850 3054 y Fm(!)118 3142 y Fs(near)24 b Fr(E)361 3154 y Fn(\000)441 3142 y Fs(can)g(b)r(e)h(reduced)f(to)g(the)h(study)f(of)h(the)f(densit)n(y)g (of)h(states)f(of)g(a)g(discrete)g(random)f(op)r(erator.)34 b(This)24 b(discrete)118 3242 y(Hamiltonian)33 b(is)g(the)g(sum)g(of)g (a)g(kinetic)g(energy)f(part)h(and)g(a)f(p)r(oten)n(tial)h(energy)f (part.)53 b(The)33 b(kinetic)g(energy)f(part)118 3341 y(is)h(made)f(of)g(the)h(Flo)r(quet)g(eigen)n(v)-5 b(alues)31 b(giving)h(rise)g(to)g(the)h(band)f(starting)g(at)h Fr(E)2786 3353 y Fn(\000)2874 3341 y Fs(for)f(the)h(bac)n(kground)e(p)r(erio)r (dic)118 3441 y(op)r(erator)37 b Fr(H)540 3411 y Fn(\000)596 3441 y Fs(.)69 b(The)38 b(p)r(oten)n(tial)g(energy)g(part)g(is)g(the)h (random)e(p)r(oten)n(tial)h(pro)5 b(jected)38 b(on)n(to)f(the)i(space)f (of)g(states)118 3541 y(generating)29 b(the)i(band)f(starting)f(at)h Fr(E)1360 3553 y Fn(\000)1417 3541 y Fs(.)44 b(This)31 b(discrete)e(Hamiltonian,)i(that)g(w)n(e)f(will)g(call)g(the)g(reduced) g(op)r(erator,)118 3640 y(can)j(b)r(e)i(realized)d(as)h(an)h(op)r (erator)e(acting)h(on)g(the)i(torus)e Fq(T)2078 3610 y Fn(\003)2115 3640 y Fs(.)55 b(T)-7 b(o)34 b(p)r(erform)f(this)h (reduction,)h(one)e(needs)h(to)g(kno)n(w)118 3740 y(that)d(the)h(edge)f Fr(E)702 3752 y Fn(\000)789 3740 y Fs(of)g(\006)g(corresp)r(onds)e(to)i (an)g(edge)g(of)g(a)f(single)h(p)r(erio)r(dic)g(realization)e(\(namely) i Fr(H)3390 3710 y Fn(\000)3446 3740 y Fs(\).)48 b(Therefore)118 3839 y(w)n(e)29 b(\014rst)f(need)h(the)h(p)r(ositivit)n(y)e(assumption) h(of)g Fr(V)47 b Fs(and,)30 b(second,)e(w)n(e)h(need)g(to)g(kno)n(w)f (that,)h(when)g(switc)n(hing)g(on)f(the)118 3939 y(random)23 b(p)r(oten)n(tial)h(\(remem)n(b)r(er)f(that)h Fr(H)1434 3909 y Fn(\000)1513 3939 y Fs(is)g(no)n(w)f(the)h(unp)r(erturb)r(ed)g (Hamiltonian\),)h(none)e(of)h(the)g(sp)r(ectrum)g(b)r(elo)n(w)118 4039 y Fr(E)179 4051 y Fn(\000)269 4039 y Fs(passes)33 b(ab)r(o)n(v)n(e)f Fr(E)828 4051 y Fn(\000)884 4039 y Fs(.)55 b(In)34 b(the)g(presen)n(t)f(exp)r(osition,)h(this)g(is)g (ensured)f(b)n(y)g(the)h(fact)g(that)g(the)g(gap)f(b)r(elo)n(w)g Fr(E)3752 4051 y Fn(\000)3842 4039 y Fs(is)118 4138 y(op)r(en)e(in)g (\006)g(and)g(b)n(y)g(the)g(assumption)f(that)i(the)f(random)f(v)-5 b(ariables)30 b(ha)n(v)n(e)f(connected)i(supp)r(ort.)47 b(Another)31 b(natural)118 4238 y(setting)c(where)f(there)h(cannot)f(b) r(e)h(an)n(y)f(crossing)f(is)i(when)f(one)h(assume)f(that)h Fr(H)2685 4208 y Fn(\000)2767 4238 y Fs(has)g(a)f(gap)g(and)h(that)g (the)g(random)118 4338 y(p)r(erturbation)g Fr(V)672 4307 y Fn(\000)653 4358 y Fm(!)756 4338 y Fs(is)g(smaller)g(than)h(the)g (length)f(of)h(the)g(gap.)331 4437 y(The)f(next)g(step)g(is)g(to)f(sho) n(w)g(that,)i(with)f(go)r(o)r(d)f(enough)h(precision,)f(w)n(e)g(can)h (estimate)g(the)g(in)n(tegrated)f(densit)n(y)118 4537 y(of)32 b(states)f(of)h(the)g(reduced)f(op)r(erator)f(b)r(et)n(w)n(een) h Fr(E)1740 4549 y Fp(0)1810 4537 y Fs(and)g Fr(E)2036 4549 y Fp(0)2095 4537 y Fs(+)21 b Fr(")31 b Fs(b)n(y)g(the)i (probabilit)n(y)d(that)i(the)g(reduced)f(op)r(erator)118 4636 y(\(restricted)c(to)g(some)g(large)f(cub)r(e\))i(has)f(an)g(eigen) n(v)-5 b(alue)27 b(in)g(the)h(in)n(terv)-5 b(al)27 b([)p Fr(E)2543 4648 y Fp(0)2581 4636 y Fr(;)14 b(E)2679 4648 y Fp(0)2734 4636 y Fs(+)k Fr(")p Fs(].)36 b(If)28 b(the)g(band)f(edge)g (is)g(simple,)118 4736 y(w)n(e)d(then)h(ha)n(v)n(e)e(to)h(estimate)h (the)f(probabilit)n(y)g(that,)h(for)f Fr(\016)i(>)d Fs(0)h(small,)g (there)h(exists)f Fr(u)e Fo(2)i Fr(L)3020 4706 y Fp(2)3056 4736 y Fs(\()p Fq(T)p Fs(\))h(suc)n(h)f(that)g(suc)n(h)g(that)118 4836 y(the)k(F)-7 b(ourier)27 b(co)r(e\016cien)n(ts)g(of)g Fr(u)h Fs(are)e(supp)r(orted)i(in)g([)p Fo(\000)p Fr(M)t(;)14 b(M)9 b Fs(])17 b Fo(\002)h Fs([)p Fo(\000)p Fr(M)t(;)c(M)9 b Fs(])22 b Fo(\032)h Fq(Z)2682 4806 y Fp(2)2741 4836 y Fs(and)28 b(suc)n(h)f(that)1334 4995 y Fo(h)p Fr(E)1427 5007 y Fm(n)1468 5015 y Fi(0)1524 4995 y Fo(\001)18 b Fr(u;)c(u)p Fo(i)1730 5010 y Fm(L)1776 4994 y Fi(2)1808 5010 y Fp(\()p Fj(T)1878 4994 y Fe(\003)1907 5010 y Fp(\))1955 4995 y Fs(+)19 b Fo(h)p Fr(v)2111 5007 y Fm(!)2159 4995 y Fr(u;)14 b(u)p Fo(i)2324 5010 y Fm(L)2370 4994 y Fi(2)2401 5010 y Fp(\()p Fj(T)2472 4994 y Fe(\003)2501 5010 y Fp(\))2554 4995 y Fo(\024)22 b Fr(\016)-2560 b Fs(\(5.1\))118 5155 y(Here)41 b Fr(E)389 5167 y Fm(n)430 5175 y Fi(0)494 5155 y Fo(\001)27 b Fr(u)40 b Fs(denotes)h(the)g(m)n(ultiplication)g(b) n(y)f(the)h(Flo)r(quet)g(eigen)n(v)-5 b(alue)40 b Fr(E)2724 5167 y Fm(n)2765 5175 y Fi(0)2843 5155 y Fs(and)g Fr(v)3057 5167 y Fm(!)3147 5155 y Fs(denotes)g(the)h(reduced)118 5255 y(p)r(oten)n(tial.)35 b(Notice)21 b(that)h(b)r(oth)f(terms)h(in)f (\(5.1\))g(are)f(non-negativ)n(e.)34 b(In)21 b(the)h(general)e(case,)h (the)h(situation)f(is)g(somewhat)118 5355 y(complicated)27 b(b)n(y)h(the)g(p)r(ossible)f(m)n(ultiplicities.)1991 5454 y Fp(5)p eop %%Page: 6 6 6 5 bop 331 241 a Fs(A)n(t)33 b(last,)h(w)n(e)e(estimate)h(these)g (probabilities.)51 b(Carrying)31 b(out)i(this)g(estimation)f(is)h (essen)n(tially)f(a)g(problem)g(in)118 340 y(random)22 b(F)-7 b(ourier)21 b(series;)j(indeed,)f(thanks)g(to)f(assumption)g (\(H.1'.4\),)i(for)e(our)g(purp)r(ose,)h(w)n(e)f(can)g(replace)g(the)h (random)118 440 y(v)-5 b(ariables)26 b(\()p Fr(!)546 452 y Fm(\015)589 440 y Fs(\))621 452 y Fm(\015)692 440 y Fs(b)n(y)h(i.i.d.)37 b(Bernoulli)27 b(random)g(v)-5 b(ariables.)331 540 y(Let)26 b Fr(f)34 b Fs(b)r(e)26 b(a)g(nonnegativ)n(e)e(real)h(analytic)g(function)i(on)e Fq(T)2158 510 y Fn(\003)2196 540 y Fs(,)h(let)g Fr(\016)g(>)d Fs(0)i(and)h Fr(M)31 b(<)23 b Fo(1)j Fs(b)r(e)g(parameters,)f(let)h Fr(S)k Fs(b)r(e)118 639 y(a)e(random)e(subset)i(of)g(the)g(square)f([)p Fo(\000)p Fr(M)t(;)14 b(M)9 b Fs(])18 b Fo(\002)g Fs([)p Fo(\000)p Fr(M)t(;)c(M)9 b Fs(])27 b(\(in)h Fq(Z)2211 609 y Fp(2)2243 639 y Fs(\))g(with)g(\(sa)n(y\))f(densit)n(y)2997 607 y Fp(1)p 2997 621 34 4 v 2997 668 a(2)3068 639 y Fs(and)h(let)g Fr(P)12 b Fs(\()p Fr(M)t(;)i(\016)s Fs(\))28 b(b)r(e)g(the)118 744 y(probabilit)n(y)f(that)h(there)f(is)g(a)h (function)g Fr(u)22 b Fo(2)i Fr(L)1619 714 y Fp(2)1656 744 y Fs(\()p Fq(T)p Fs(\))j(suc)n(h)g(that)1196 818 y Fk(Z)1242 1007 y Fm(T)1290 990 y Fi(2)1341 931 y Fr(f)9 b Fs(\()p Fr(\022)r Fs(\))p Fo(j)p Fr(u)p Fs(\()p Fr(\022)r Fs(\))p Fo(j)1695 897 y Fp(2)1756 931 y Fr(<)23 b(\016)s Fo(k)p Fr(u)p Fo(k)2016 897 y Fp(2)2016 952 y(2)2135 931 y Fs(and)82 b(supp)q(\()5 b(^)-47 b Fr(u)o Fs(\))24 b Fo(\032)e Fr(S:)118 1118 y Fs(Th)n(us)h(the)i(F)-7 b(ourier)22 b(co)r(e\016cien)n(ts)h(of)h Fr(u)f Fs(are)g(to)g(b)r(e)i (supp)r(orted)e(on)g(the)h(random)f(set,)i(and)e Fr(u)g Fs(is)h(to)f(b)r(e)i(mostly)e(supp)r(orted)118 1218 y(near)38 b(the)g(zero)g(set)g(of)g Fr(f)9 b Fs(.)69 b(W)-7 b(e)39 b(will)f(study)h(the)g(b)r(eha)n(vior)e(of)h(the)h(probabilit)n(y)e Fr(P)12 b Fs(\()p Fr(M)d Fs(\()p Fr(\016)s Fs(\))p Fr(;)14 b(\016)s Fs(\))39 b(as)f Fr(\016)44 b Fo(!)d Fs(0,)f(letting)118 1342 y Fr(M)32 b Fs(=)22 b Fr(M)9 b Fs(\()p Fr(\016)s Fs(\))26 b(go)f(to)h Fo(1)f Fs(in)h(an)g(appropriate)e(manner.)35 b(It)26 b(turns)g(out)f(that)h Fr(P)12 b Fs(\()p Fr(M)d Fs(\()p Fr(\016)s Fs(\))p Fr(;)14 b(\016)t Fs(\))26 b(b)r(eha)n(v)n(es) e(roughly)h(lik)n(e)g Fr(e)3692 1311 y Fn(\000)p Fr(\016)3783 1274 y Fe(\000)p Ff(\013)3875 1342 y Fs(.)331 1441 y(One)d(can)g(apply) g(this)h(b)r(ound)g(taking)e Fr(f)31 b Fs(to)23 b(b)r(e)g(roughly)e (the)i(Flo)r(quet)f(eigen)n(v)-5 b(alue)22 b Fr(E)2996 1453 y Fm(n)3037 1461 y Fi(0)3096 1441 y Fs(to)g(complete)h(the)g(pro)r (ofs)118 1541 y(of)28 b(Theorems)e(0.1)h(and)g(0.2,)g(in)h(a)f(manner)g (analogous)f(to)h(the)h(one)f(dimensional)g(case)g(treated)g(in)h([)p Ft(11)p Fs(].)331 1688 y(A)h(Newton)f(deca)n(y)g(exp)r(onen)n(t)h (similar)e(to)i(the)g(one)f(in)n(tro)r(duced)g(ab)r(o)n(v)n(e)g(arises) f(in)i(problems)f(concerning)f(esti-)118 1787 y(mation)f(of)f (oscillatory)f(in)n(tegrals)h(-)g(see)h([)p Ft(31)o Fs(])g(and)g([)p Ft(25)o Fs(])g(-)g(and)f(w)n(e)h(sho)n(w)f(here)g(that)h(it)g(also)f (con)n(trols)f(the)i(b)r(eha)n(vior)f(of)118 1887 y(certain)g(random)f (F)-7 b(ourier)25 b(series,)g(in)g(a)g(w)n(a)n(y)g(whic)n(h)g(to)g (some)g(exten)n(t)h(is)f(suggested)f(b)n(y)h(the)h(results)f(of)h([)p Ft(31)o Fs(])g(and)f([)p Ft(25)p Fs(].)331 2034 y(Though)j(in)i([)p Ft(13)o Fs(])f(w)n(e)g(only)g(study)g(the)h(case)e(of)h(random)f(Sc)n (hr\177)-42 b(odinger)27 b(op)r(erators,)h(the)h(metho)r(ds)h(w)n(e)f (dev)n(elop)118 2133 y(can)24 b(b)r(e)h(applied)f(to)r(o)g(man)n(y)g (other)g(families)g(of)g(random)g(op)r(erators:)33 b(acoustic)23 b(op)r(erators,)g(random)h(magnetic)g(\014elds,)118 2233 y(etc)k(\(see)f(e.g.)h([)p Ft(21)o(,)j(3)p Fs(]\).)1394 2432 y Ft(6.)47 b(Application)32 b(to)f(lo)s(calization)331 2582 y Fs(Our)j(results)h(can)g(b)r(e)h(applied)f(to)h(study)f(band)h (edge)e(lo)r(calization)h(for)g(the)g(con)n(tin)n(uous)g(Anderson)f(mo) r(del.)118 2681 y(Therefore,)39 b(in)f(addition)g(to)g(assumption)f (\(H.1\),)k(w)n(e)d(need)g(to)g(assume)f(that)h(the)g(random)f(op)r (erator)f(satis\014es)h(a)118 2781 y Fl(We)l(gner)d(estimate)f Fs(i.e)f(that)h(for)e(some)h Fr(\014)j(>)30 b Fs(0)i(and)g Fr(\027)k(>)31 b Fs(0,)i(for)f Fr(E)j Fo(2)d Fq(R)p Fs(,)39 b(for)32 b Fr(l)g Fo(\025)e Fs(1)i(and)g(0)f Fr(<)f(")h(<)f Fs(1,)j(there)f(exists)118 2881 y Fr(C)6 b Fs(\()p Fr(E)f Fs(\))24 b Fr(>)f Fs(0)k(suc)n(h)g(that)h(one)f(has)1216 3029 y Fq(P)1282 2962 y Fk(\000)1319 3029 y Fs(dist\()p Fr(\033)s Fs(\()p Fr(H)1643 2995 y Fm(\022)1636 3050 y(!)r(;C)1748 3059 y Ff(l)1776 3029 y Fr(;)14 b(E)5 b Fs(\))24 b Fo(\024)e Fr(")2061 2962 y Fk(\001)2122 3029 y Fo(\024)h Fr(C)6 b Fs(\()p Fr(E)f Fs(\)V)-7 b(ol)q(\()p Fr(C)2617 3041 y Fm(l)2643 3029 y Fs(\))2675 2995 y Fm(\014)2720 3029 y Fr(")2759 2995 y Fm(\027)118 3029 y Fs(\(6.1\))118 3188 y(where)27 b(the)g(constan)n(t)g Fr(C)6 b Fs(\()p Fr(E)f Fs(\))28 b(is)f(lo)r(cally)g(uniform)g(in)g(energy)-7 b(.)36 b(Here)27 b Fr(l)h Fs(is)f(an)g(in)n(teger)g(and)g Fr(H)3061 3157 y Fm(\022)3054 3210 y(!)r(;C)3166 3219 y Ff(l)3220 3188 y Fs(is)h(the)f(op)r(erator)f Fr(H)3850 3200 y Fm(!)118 3301 y Fs(restricted)h(to)g(the)h(cub)r(e)g Fr(C)987 3313 y Fm(l)1041 3301 y Fs(with)g(b)r(oundary)f(conditions:)36 b Fr(')p Fs(\()p Fr(x)20 b Fs(+)e Fr(\015)5 b Fs(\))23 b(=)f Fr(e)2498 3270 y Fm(i\022)r(\015)2598 3301 y Fr(')p Fs(\()p Fr(x)p Fs(\))29 b(for)e(an)n(y)f Fr(\015)i Fo(2)c Fs(2)p Fr(l)r Fq(Z)3354 3270 y Fp(2)3385 3301 y Fs(.)118 3400 y(W)-7 b(e)28 b(pro)n(v)n(e)331 3556 y Fg(Theorem)k Fs(6.1)p Fg(.)40 b Fl(L)l(et)27 b Fr(H)1120 3568 y Fm(!)1196 3556 y Fl(b)l(e)g(a)h(r)l(andom)h(op)l(er)l(ator)f(describ)l(e)l(d)h (by)36 b Fs(\(1.1\))27 b Fl(such)h(that)g(\(H.1\))g(and)g(estimate)34 b Fs(\(6.1\))118 3655 y Fl(ar)l(e)26 b(satis\014e)l(d.)38 b(Then,)28 b(for)f(every)f(b)l(and)h(e)l(dge)f(of)h Fs(\006)p Fl(,)g(the)f(almost)g(sur)l(e)f(sp)l(e)l(ctrum)h(of)g Fr(H)2839 3667 y Fm(!)2887 3655 y Fl(,)h(ther)l(e)f(exists)g(a)g (neighb)l(orho)l(o)l(d)118 3755 y(of)34 b(that)f(b)l(and)g(e)l(dge)g (wher)l(e)h(the)f(sp)l(e)l(ctrum)f(of)i Fr(H)1676 3767 y Fm(!)1757 3755 y Fl(is)f(pur)l(ely)g(punctual.)47 b(Mor)l(e)l(over)35 b(the)e(eigenfunctions)g(asso)l(ciate)l(d)118 3854 y(to)d(ener)l(gies)g (in)g(that)g(neighb)l(orho)l(o)l(d)i(ar)l(e)e(exp)l(onential)t(ly)h(lo) l(c)l(alize)l(d.)331 4010 y Fs(This)i(result)h(is)f(a)h(direct)f (application)g(of)h(Theorem)f(3.1)g(and)g(the)h(m)n(ulti-scale)f (analysis)g(dev)n(elop)r(ed)g(in)h([)p Ft(10)p Fs(])118 4109 y(\(see)28 b(also)e([)p Ft(1,)31 b(9)p Fs(]\).)1784 4309 y Ft(References)153 4441 y Fv([1])k(J.)27 b(M.)g(Barbaroux,)h(J.)f (M.)f(Com)n(b)r(es,)i(and)g(P)-6 b(.)27 b(D.)f(Hislop.)h(Lo)r (calization)h(near)g(band)g(edges)g(for)f(random)f(Sc)n(hr\177)-35 b(odinger)28 b(op)r(erators.)263 4524 y Fd(Helv.)i(Phys.)g(A)l(cta)p Fv(,)g(70\(1-2\):16{43,)h(1997.)f(P)n(ap)r(ers)e(honouring)h(the)h (60th)f(birthda)n(y)h(of)e(Klaus)g(Hepp)h(and)g(of)f(W)-6 b(alter)30 b(Hunzik)n(er,)263 4607 y(P)n(art)24 b(I)r(I)g(\(Z)r(\177) -37 b(uric)n(h,)24 b(1995\).)153 4690 y([2])35 b(R.)23 b(Carmona)g(and)h(J.)g(Lacroix.)f Fd(Sp)l(e)l(ctr)l(al)k(The)l(ory)f (of)g(R)l(andom)i(Schr\177)-36 b(odinger)26 b(Op)l(er)l(ators)p Fv(.)f(Birkh\177)-35 b(auser,)22 b(Basel,)i(1990.)153 4773 y([3])35 b(F.)21 b(Ghribi.)f(Asymptotique)i(de)f(Lifshitz)g(p)r (our)g(des)h(op)n(\023)-33 b(erateurs)22 b(de)g(Sc)n(hr\177)-35 b(odinger)22 b(\022)-35 b(a)21 b(c)n(hamp)g(magn)n(\023)-33 b(etique)22 b(al)n(\023)-33 b(eatoire.)21 b(Thesis)g(Univ)n(er-)263 4856 y(sit)n(\023)-33 b(e)24 b(P)n(aris)f(13,)g(2000.)i(in)e (preparation.)153 4939 y([4])35 b(Bernard)30 b(Hel\013er)g(and)g(Ab)r (deremane)g(Mohamed.)f(Asymptotic)h(of)f(the)i(densit)n(y)g(of)e (states)i(for)f(the)g(Sc)n(hr\177)-35 b(odinger)31 b(op)r(erator)f (with)263 5023 y(p)r(erio)r(dic)24 b(electric)g(p)r(oten)n(tial.)g Fd(Duke)i(Math.)f(J.)p Fv(,)f(92\(1\):1{60,)h(1998.)153 5106 y([5])35 b(W.)30 b(Kirsc)n(h.)f(Random)g(Sc)n(hr\177)-35 b(odinger)31 b(op)r(erators.)f(In)g(A.)f(Jensen)i(H.)e(Holden,)j (editor,)f Fd(Schr\177)-36 b(odinger)32 b(Op)l(er)l(ators)p Fv(,)g(n)n(um)n(b)r(er)d(345)i(in)263 5189 y(Lecture)25 b(Notes)f(in)g(Ph)n(ysics,)f(Berlin,)f(1989.)j(Springer)e(V)-6 b(erlag.)23 b(Pro)r(ceedings,)h(Sonderb)r(org,)g(Denmark)f(1988.)153 5272 y([6])35 b(W.)23 b(Kirsc)n(h)g(and)h(F.)e(Martinelli.)g(On)h(the)h (sp)r(ectrum)f(of)g(Sc)n(hr\177)-35 b(odinger)24 b(op)r(erators)f(with) h(a)f(random)f(p)r(oten)n(tial.)j Fd(Communic)l(ations)i(in)263 5355 y(Mathematic)l(al)g(Physics)p Fv(,)c(85:329{350,)j(1982.)1991 5454 y Fp(6)p eop %%Page: 7 7 7 6 bop 153 241 a Fv([7])35 b(W.)25 b(Kirsc)n(h)g(and)h(F.)f (Martinelli.)f(Large)h(deviations)i(and)f(Lifshitz)f(singularities)f (of)h(the)i(in)n(tegrated)g(densit)n(y)f(of)f(states)h(of)f(random)263 324 y(hamiltonians.)e Fd(Communic)l(ations)k(in)f(Mathematic)l(al)g (Physics)p Fv(,)e(89:27{40,)h(1983.)153 407 y([8])35 b(W.)24 b(Kirsc)n(h)f(and)h(B.)f(Simon.)f(Lifshitz)h(tails)h(for)f(the) h(Anderson)g(mo)r(del.)e Fd(Journal)28 b(of)e(Statistic)l(al)f(Physics) p Fv(,)f(38:65{76,)h(1985.)153 490 y([9])35 b(W)-6 b(erner)23 b(Kirsc)n(h,)e(P)n(eter)i(Stollmann,)f(and)h(G)r(\177)-37 b(un)n(ter)23 b(Stolz.)g(Anderson)f(lo)r(calization)h(for)f(random)f (Sc)n(hr\177)-35 b(odinger)23 b(op)r(erators)g(with)f(long)263 573 y(range)i(in)n(teractions.)h Fd(Comm.)h(Math.)g(Phys.)p Fv(,)d(195\(3\):495{507,)k(1998.)118 656 y([10])35 b(W)-6 b(erner)32 b(Kirsc)n(h,)g(P)n(eter)g(Stollmann,)h(and)f(G)r(\177)-37 b(un)n(ter)33 b(Stolz.)f(Lo)r(calization)g(for)f(random)g(p)r (erturbations)h(of)f(p)r(erio)r(dic)h(Sc)n(hr\177)-35 b(odinger)263 739 y(op)r(erators.)24 b Fd(R)l(andom)j(Op)l(er.)f(Sto)l (chastic)g(Equations)p Fv(,)f(6\(3\):241{268,)h(1998.)118 822 y([11])35 b(F.)22 b(Klopp.)g(Band)h(edge)g(b)r(eha)n(viour)g(for)f (the)h(in)n(tegrated)h(densit)n(y)f(of)f(states)h(of)f(random)g(Jacobi) h(matrices)e(in)h(dimension)g(1.)g Fd(Journal)263 905 y(of)k(Statistic)l(al)g(Physics)p Fv(,)d(90\(3-4\):927{947,)j(1998.)118 988 y([12])35 b(F.)23 b(Klopp)h(and)g(J.)g(Ralston.)f(Endp)r(oin)n(ts)i (of)e(the)i(sp)r(ectrum)e(of)g(p)r(erio)r(dic)h(op)r(erators)g(are)g (generically)g(simple.)d(Preprin)n(t,)i(2000.)118 1071 y([13])35 b(F.)23 b(Klopp)h(and)g(T.)f(W)-6 b(ol\013.)24 b(Lifshitz)f(tails)h(for)e(2-dimensional)h(random)g(Sc)n(hr\177)-35 b(odinger)24 b(op)r(erators.)g(in)f(preparation,)h(2000.)118 1154 y([14])35 b(F)-6 b(r)n(\023)-33 b(ed)n(\023)g(eric)32 b(Klopp.)f(In)n(ternal)i(Lifshits)e(tails)h(for)f(random)g(p)r (erturbations)i(of)e(p)r(erio)r(dic)h(Sc)n(hr\177)-35 b(odinger)33 b(op)r(erators.)f Fd(Duke)g(Math.)i(J.)p Fv(,)263 1237 y(98\(2\):335{396,)27 b(1999.)118 1320 y([15])35 b(P)-6 b(.)23 b(Kuc)n(hmen)n(t.)h Fd(Flo)l(quet)i(the)l(ory)g (for)f(p)l(artial)i(di\013er)l(ential)f(e)l(quations)p Fv(,)d(v)n(olume)g(60)h(of)f Fd(Op)l(er)l(ator)k(The)l(ory:)33 b(A)l(dvanc)l(es)26 b(and)h(Applic)l(a-)263 1403 y(tions)p Fv(.)d(Birkh\177)-35 b(auser,)22 b(Basel,)i(1993.)118 1486 y([16])35 b(I.)23 b(M.)e(Lifshitz.)h(Structure)i(of)e(the)h (energy)h(sp)r(ectrum)e(of)g(impurit)n(y)f(bands)i(in)f(disordered)h (solid)f(solutions.)g Fd(Soviet)i(Physics)h(JETP)p Fv(,)263 1569 y(17:1159{1170,)h(1963.)118 1652 y([17])35 b(I.)19 b(M.)e(Lifshitz.)h(Energy)h(sp)r(ectrum)f(structure)i(and)f(quan)n(tum) g(states)h(of)e(disordered)h(condensed)h(systems.)e Fd(Soviet)i (Physics)h(Usp)l(ekhi)p Fv(,)263 1735 y(7:549{573,)k(1965.)118 1818 y([18])35 b(I.M.)29 b(Lifshitz,)h(S.A.)e(Gredeskul,)j(and)f(L.A.)f (P)n(astur.)g Fd(Intr)l(o)l(duction)k(to)e(the)g(the)l(ory)g(of)g (disor)l(der)l(e)l(d)j(systems)p Fv(.)29 b(Wiley)-6 b(,)30 b(New-Y)-6 b(ork,)263 1901 y(1988.)118 1984 y([19])35 b(G.)f(Mezincescu.)g(Lifshitz)f(singularities)g(for)f(p)r(erio)r(dic)i (op)r(erators)g(plus)f(random)g(p)r(oten)n(tials.)h Fd(Journal)j(of)e (Statistic)l(al)g(Physics)p Fv(,)263 2067 y(49:1081{1090,)26 b(1987.)118 2150 y([20])35 b(G.)27 b(Mezincescu.)h(In)n(ternal)f (Lifshits)f(singularities)h(for)f(one)h(dimensional)f(Sc)n(hr\177)-35 b(odinger)28 b(op)r(erators.)f Fd(Communic)l(ations)j(in)f(Mathe-)263 2233 y(matic)l(al)e(Physics)p Fv(,)c(158:315{325,)j(1993.)118 2316 y([21])35 b(H.)19 b(Na)t(jar.)g(Asymptotique)h(de)h(la)e(densit)n (\023)-33 b(e)21 b(d')n(\023)-33 b(etats)21 b(in)n(t)n(\023)-33 b(egr)n(\023)g(ee)21 b(des)f(mo)r(d)n(\022)-33 b(eles)19 b(al)n(\023)-33 b(eatoires)20 b(con)n(tin)n(us.)h(Thesis)e(Univ)n (ersit)n(\023)-33 b(e)20 b(P)n(aris)f(13,)i(2000.)263 2399 y(in)j(pr)n(\023)-33 b(eparation.)118 2482 y([22])35 b(S.)26 b(Nak)l(ao.)h(On)g(the)g(sp)r(ectral)g(distribution)g(of)f(the) h(Sc)n(hr\177)-35 b(odinger)27 b(op)r(erator)h(with)e(random)g(p)r (oten)n(tial.)h Fd(Jap)l(an)j(Journal)g(of)f(Mathe-)263 2565 y(matics)p Fv(,)24 b(3:117{139,)h(1977.)118 2648 y([23])35 b(L.)24 b(P)n(astur.)g(Beha)n(viour)g(of)g(some)g(Wiener)g (in)n(tegrals)g(as)g Fc(t)d Fb(!)f Fv(+)p Fb(1)k Fv(and)g(the)h(densit) n(y)g(of)f(states)h(of)f(the)h(Sc)n(hr\177)-35 b(odinger)25 b(equation)g(with)263 2731 y(a)f(random)f(p)r(oten)n(tial.)h Fd(T)-5 b(e)l(or.-Mat.-Fiz)p Fv(,)23 b(32:88{95,)i(1977.)f(\(in)g (russian\).)118 2814 y([24])35 b(L.)23 b(P)n(astur)h(and)g(A.)f (Figotin.)h Fd(Sp)l(e)l(ctr)l(a)j(of)f(R)l(andom)h(and)g(A)n (lmost-Perio)l(dic)f(Op)l(er)l(ators)p Fv(.)f(Springer)e(V)-6 b(erlag,)24 b(Berlin,)e(1992.)118 2897 y([25])35 b(D.)21 b(H.)g(Phong)i(and)f(E.)f(M.)g(Stein.)h(The)g(Newton)h(p)r(olyhedron)g (and)f(oscillatory)g(in)n(tegral)g(op)r(erators.)g Fd(A)l(cta)i(Math.)p Fv(,)d(179\(1\):105{152,)263 2980 y(1997.)118 3064 y([26])35 b(M.)27 b(Reed)h(and)g(B.)e(Simon.)g Fd(Metho)l(ds)k(of)f(Mo)l(dern)h (Mathematic)l(al)h(Physics,)f(V)-5 b(ol)29 b(IV:)g(Analysis)g(of)g(Op)l (er)l(ators)p Fv(.)f(Academic)g(Press,)263 3147 y(New-Y)-6 b(ork,)23 b(1978.)118 3230 y([27])35 b(J.)g(Sj\177)-35 b(ostrand.)35 b(Microlo)r(cal)g(analysis)g(for)f(p)r(erio)r(dic)h (magnetic)g(Sc)n(hr\177)-35 b(odinger)36 b(equation)h(and)e(related)h (questions.)g(In)f Fd(Micr)l(olo)l(c)l(al)263 3313 y(analysis)27 b(and)f(applic)l(ations)p Fv(,)f(v)n(olume)e(1495)i(of)f Fd(L)l(e)l(ctur)l(e)h(Notes)h(in)f(Mathematics)p Fv(,)f(Berlin,)e (1991.)i(Springer)g(V)-6 b(erlag.)118 3396 y([28])35 b(M.)26 b(M.)g(Skrigano)n(v.)i(Pro)r(of)e(of)h(the)g(Bethe-Sommerfeld)f (conjecture)i(in)f(dimension)f(2.)h Fd(Dokl.)h(A)n(kad.)h(Nauk)g(SSSR)p Fv(,)e(248\(1\):39{42,)263 3479 y(1979.)118 3562 y([29])35 b(P)n(eter)24 b(Stollmann.)f(Lifshitz)g(asymptotics)h(via)g(linear)f (coupling)h(of)g(disorder.)e Fd(Math.)k(Phys.)g(A)n(nal.)g(Ge)l(om.)p Fv(,)e(2\(3\):279{289,)i(1999.)118 3645 y([30])35 b(Alain-Sol)23 b(Sznitman.)g Fd(Br)l(ownian)k(motion,)g(obstacles)f(and)g(r)l(andom)i (me)l(dia)p Fv(.)c(Springer-V)-6 b(erlag,)23 b(Berlin,)f(1998.)118 3728 y([31])35 b(V.)29 b(V)-6 b(arc)n(henk)n(o.)30 b(Newton)g(p)r (olyhedra)g(and)f(estimations)g(of)g(oscillatory)g(in)n(tegrals.)g Fd(F)-5 b(unctional)31 b(A)n(nalysis)h(and)f(its)f(Applic)l(ations)p Fv(,)263 3811 y(8:175{196,)25 b(1976.)331 3966 y(\(F)-6 b(r)n(\023)-33 b(ed)n(\023)g(eric)34 b(Klopp\))f Fw(D)943 3960 y(\023)943 3966 y(ep)l(ar)l(tement)h(de)h(Ma)l(th)1663 3960 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