This is a multi-part message in MIME format. ---------------0110080658492 Content-Type: text/plain; name="01-358.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-358.keywords" Random operators, localization, weak disorder, Lifshitz tails ---------------0110080658492 Content-Type: application/postscript; name="weak-dis-cont.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="weak-dis-cont.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: weak-dis-cont.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips weak-dis-cont.dvi -o weak-dis-cont.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.10.08:1338 %%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 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y(norm.)44 b(In)e Fs(\(0.4\))o Ff(,)35 b(the)g(supr)-5 b(emum)34 b(is)h(taken)f(over)h(b)-5 b(ounde)g(d,)34 b(Bor)-5 b(el)34 b(me)-5 b(asur)g(able)34 b(functions)g(on)h Fj(R)5 b Ff(.)-118 3204 y Fs(Theorem)31 b(0.1)f(is)f(a)h(consequence)k(of)c(the)h(b)s(eha)m(vior)f(of)g(the)g (in)m(tegrated)h(densit)m(y)g(of)f(states)h(near)g(the)f(in\014m)m(um)f (of)-118 3320 y(the)h(sp)s(ectrum.)42 b(More)30 b(precisely)-8 b(,)30 b(let)f(\003)1379 3335 y Fp(L)1460 3320 y Fs(b)s(e)h(the)f(cub)s (e)h(of)f(cen)m(ter)i(0)e(and)g(side)h(length)e Fq(L)i Fs(in)f Fj(R)3400 3284 y Fp(d)3446 3320 y Fs(,)h(and)f(let)g Fq(H)3908 3335 y Fp(!)r(;\025)p Fm(j)p Fi(\003)4084 3346 y Fk(L)-118 3436 y Fs(b)s(e)36 b(the)h(Hamiltonian)32 b Fq(H)835 3451 y Fp(!)r(;\025)982 3436 y Fs(restricted)37 b(to)e(the)i(cub)s(e)g(\003)2016 3451 y Fp(L)2104 3436 y Fs(with)e(Diric)m(hlet)f(b)s(oundary)j(conditions.)53 b(De\014ne)36 b(the)-118 3552 y(in)m(tegrated)c(densit)m(y)i(of)e (states)h(of)f Fq(H)1262 3567 y Fp(!)r(;\025)1406 3552 y Fs(b)m(y)984 3786 y Fo(N)1066 3801 y Fp(\025)1111 3786 y Fs(\()p Fq(E)6 b Fs(\))28 b(=)111 b(lim)1396 3849 y Fi(#\003)p Fm(!)p Fi(+)p Fm(1)1727 3715 y Fs(#)p Fo(f)p Fs(eigen)m(v)-5 b(alues)32 b(of)g Fq(H)2554 3731 y Fp(!)r(;\025)p Fm(j)p Fi(\003)2762 3715 y Fo(\024)c Fq(E)6 b Fo(g)p 1727 3763 1268 4 v 2299 3855 a(j)p Fs(\003)p Fo(j)3005 3786 y Fq(:)-3150 b Fs(\(0.5\))-118 4015 y(The)33 b(limit)c(in)j (\(0.5\))g(exists)i Fq(!)t Fs(-a.e.;)d(it)h(is)g(non-random)f(and)i (non-decreasing)g(\([3,)f(27]\).)43 b(W)-8 b(e)33 b(pro)m(v)m(e)-118 4195 y Fw(Theorem)k(0.2.)49 b Ff(Fix)26 b Fq(\021)31 b Fo(2)d Fs(\(0)p Fq(;)17 b(d=)p Fs(\(4)p Fq(d)t Fs(+)t(4\)\))p Ff(.)40 b(Then,)27 b(ther)-5 b(e)27 b(exists)f Fq(")h(>)h Fs(0)e Ff(and)g Fq(\025)2826 4210 y Fp(\021)2896 4195 y Fq(>)h Fs(0)g Ff(such)f(that,)j(for)d Fq(\025)i Fo(2)g Fs(\(0)p Fq(;)17 b(\025)4025 4210 y Fp(\021)4066 4195 y Fs(\))p Ff(,)-118 4312 y(one)34 b(has)1420 4480 y Fo(N)1502 4495 y Fp(\025)1547 4480 y Fs(\()p 1585 4400 79 4 v Fq(E)6 b Fs(\()p Fq(\025)p 1758 4425 65 4 v(!)s Fs(\))22 b Fo(\000)h Fq(\025)2039 4439 y Fi(1+)p Fp(\021)2171 4480 y Fs(\))28 b Fo(\024)g Fq(e)2387 4439 y Fm(\000)p Fp(\025)2483 4415 y Fd(\000)p Fk(")2569 4480 y Fq(:)-2714 b Fs(\(0.6\))-118 4661 y(So,)38 b(the)g(densit)m(y)h(of)e(states)h(in)e(the)i(in)m(terv) -5 b(al)36 b Fq(I)1654 4676 y Fp(\021)r(;\025)1794 4661 y Fs(is)h(small.)56 b(This)38 b(in)m(terv)-5 b(al)36 b(is)h(in)f(the)i(\015uctuational)e(region)g(of)-118 4777 y(the)f(sp)s(ectrum)g(\(see)g([26]\).)49 b(That)34 b(in)g(\015uctuation)g(regions,)h(the)f(sp)s(ectrum)h(is)f(scarce)i (and)e(th)m(us)i(the)e(states)i(are)-118 4893 y(lo)s(calized)27 b(is)i(a)h(basic)f(mec)m(hanism)g(for)g(lo)s(calization)d(and)j(has)h (b)s(een)h(kno)m(wn)f(for)g(a)f(long)f(time)g(\(see)j(e.g.)f([25,)f (26]\).)-118 5010 y(It)h(also)f(is)g(the)h(only)g(mec)m(hanism)f(that)g (has)h(b)s(een)h(understo)s(o)s(d)f(mathematically)d(in)i(dimensions)f (larger)h(than)h(1;)-118 5126 y(this)i(mec)m(hanism)g(is)g(also)g (basic)g(in)g(the)h(study)h(of)e(large)f(disorder)i(lo)s(calization.)94 5242 y(The)g(asymptotics)e(\(0.6\))h(is)f(directly)g(related)g(to)g (the)h(celebrated)h(\\Lifshitz)d(tail")f(b)s(eha)m(vior)i(\(see)i(e.g.) f([18])-118 5358 y(for)h(a)h(recen)m(t)i(review\).)48 b(The)35 b(main)d(di\013erence)j(with)f(Lifshitz)f(tails)f(is)i(that)f (Lifshitz)g(tails)g(are)h(asymptotics)g(in)-118 5475 y(the)f(limit)28 b(when)34 b(one)e(approac)m(hes)i(the)e(band)h(edge,)g (and)f(\(0.6\))g(holds)g(on)g(an)g(in)m(terv)-5 b(al.)43 b(This)32 b(is)g(a)g(consequence)-118 5591 y(of)g(the)h(w)m(eak)h (disorder)e(limit.)1989 5690 y Fg(2)p eop %%Page: 3 3 3 2 bop 94 241 a Fs(T)-8 b(ec)m(hnically)42 b(sp)s(eaking)f(the)h (passage)h(from)d(Theorem)i(0.2)f(to)h(Theorem)g(0.1)f(mak)m(es)h(use)h (of)e(m)m(ultiscale)-118 357 y(analysis)24 b(and)h(W)-8 b(egner)25 b(estimates)g(\(see)g([30,)g(8,)f(11]\).)41 b(These)26 b(are)f(w)m(ell)f(kno)m(wn)i(to)s(ols)e(in)g(the)h(pro)s(of) f(of)g(lo)s(calization)-118 473 y(for)32 b(random)g(Sc)m(hr\177)-49 b(odinger)32 b(op)s(erators.)44 b(In)33 b(section)f(3,)h(w)m(e)g (explain)f(ho)m(w)h(w)m(e)h(use)f(these)h(to)s(ols.)94 589 y(As)i(the)e(main)f(result)i(of)f(this)g(pap)s(er,)h(w)m(e)g(sho)m (w)h(that)e(the)h(b)s(eha)m(vior)f(\(0.6\),)h(found)f(at)g(the)h(b)s (ottom)e(of)h(the)-118 706 y(sp)s(ectrum)g(also)g(happ)s(ens)h(at)f (simple)f(non-degenerate)i(sp)s(ectral)f(edges.)49 b(Hence,)36 b(at)e(suc)m(h)i(sp)s(ectral)e(edges,)i(one)-118 822 y(also)c(has)g(lo)s(calization.)40 b(This)33 b(is)f(the)h(con)m(ten)m (t)h(of)e(Theorems)h(1.1)f(and)h(1.3.)94 938 y(T)-8 b(o)40 b(complete)g(this)f(section,)j(let)e(us)g(note)g(that)g(results)h (similar)36 b(to)k(Theorems)h(0.1)e(and)h(0.2)g(ha)m(v)m(e)h(b)s(een) -118 1054 y(obtained)32 b(for)g(discrete)h(random)f(op)s(erators)g(in)g ([22].)1640 1420 y(1.)55 b Fr(The)38 b(resul)-7 b(ts)-118 1594 y Fs(Let)44 b Fq(W)58 b Fs(b)s(e)45 b(a)e(b)s(ounded)i Fj(Z)934 1558 y Fp(d)972 1594 y Fs(-p)s(erio)s(dic)d(p)s(oten)m(tial)h (and)h(consider)h(the)f(p)s(erio)s(dic)f(Sc)m(hr\177)-49 b(odinger)44 b(op)s(erator)g Fq(H)55 b Fs(=)-118 1710 y Fo(\000)p Fs(\001)23 b(+)f Fq(W)46 b Fs(acting)32 b(on)g Fq(L)793 1674 y Fi(2)833 1710 y Fs(\()p Fj(R)937 1674 y Fp(d)983 1710 y Fs(\).)44 b(It)32 b(is)g(self-adjoin)m(t)f(on)i Fq(H)2022 1674 y Fi(2)2061 1710 y Fs(\()p Fj(R)2165 1674 y Fp(d)2211 1710 y Fs(\);)g(let)e(\006)2519 1725 y Fi(0)2592 1710 y Fs(b)s(e)i(its)f(sp)s(ectrum.)-118 1827 y(Consider)h(the)g(con)m (tin)m(uous)g(Anderson)h(mo)s(del)d(i.e.)43 b(the)33 b(random)f(Sc)m(hr\177)-49 b(odinger)32 b(op)s(erator)g(de\014ned)i(b)m (y)1236 2058 y Fq(H)1317 2073 y Fp(!)r(;\025)1456 2058 y Fs(=)27 b Fq(H)j Fs(+)22 b Fq(\025V)1882 2073 y Fp(!)1960 2058 y Fs(=)27 b Fq(H)j Fs(+)22 b Fq(\025)2357 1964 y Fn(X)2346 2183 y Fp(\015)t Fm(2)p Fl(Z)2483 2164 y Fk(d)2530 2058 y Fq(!)2591 2073 y Fp(\015)2635 2058 y Fq(V)2692 2073 y Fp(\015)2736 2058 y Fq(;)-2881 b Fs(\(1.1\))-118 2396 y(where)48 b Fq(V)235 2411 y Fp(\015)279 2396 y Fs(\()p Fo(\001)p Fs(\))k(=)g Fq(V)22 b Fs(\()p Fo(\001)31 b(\000)i Fq(\015)5 b Fs(\),)50 b Fq(V)74 b Fs(:)99 b Fj(R)1343 2360 y Fp(d)1442 2396 y Fo(!)52 b Fj(R)57 b Fs(is)47 b(a)g(p)s(oten)m(tial)e(and)i(\()p Fq(!)2650 2411 y Fp(\015)2694 2396 y Fs(\))2732 2415 y Fp(\015)t Fm(2)p Fl(Z)2870 2396 y Fk(d)i Fs(are)e(indep)s(enden)m(t)h(iden)m (tically)-118 2512 y(distributed)32 b(random)g(v)-5 b(ariables.)42 b(W)-8 b(e)33 b(assume)g(that.)-1 2678 y Fw(H0.1:)42 b Fq(V)49 b Fs(:)60 b Fj(R)562 2642 y Fp(d)636 2678 y Fo(!)28 b Fj(R)43 b Fs(is)32 b(b)s(ounded)i(and)e(deca)m(ying)h(faster) g(than)g Fo(j)p Fq(x)p Fo(j)2563 2642 y Fm(\000)p Fp(d)p Fm(\000)p Fi(1)p Fm(\000)p Fp(\017)2864 2678 y Fs(for)f(some)g Fq(\017)c(>)g Fs(0;)-1 2794 y Fw(H0.2:)42 b Fs(The)33 b(random)f(v)-5 b(ariables)31 b(\()p Fq(!)1364 2809 y Fp(\015)1408 2794 y Fs(\))1446 2814 y Fp(\015)t Fm(2)p Fl(Z)1584 2795 y Fk(d)1651 2794 y Fs(are)h(i.i.d.,)g(b)s(ounded)h(and)g (non-trivial.)-118 2960 y(Notice)f(that)g(w)m(e)i(do)f(not)f(assume)h (that)f Fq(V)54 b Fs(k)m(eeps)35 b(a)d(\014xed)i(sign.)94 3076 y(Assumption)j(\(H0\))f(ensures)i(that)e(the)h(p)s(oten)m(tial)e Fq(V)2079 3091 y Fp(!)2166 3076 y Fs(sta)m(ys)i(uniformly)e(b)s (ounded.)55 b(Hence,)39 b Fq(H)3713 3091 y Fp(!)r(;\025)3861 3076 y Fs(is)d(self-)-118 3195 y(adjoin)m(t)42 b(on)h Fq(H)458 3159 y Fi(2)497 3195 y Fs(\()p Fj(R)601 3159 y Fp(d)648 3195 y Fs(\).)75 b(By)44 b(assumption)f(\(H0.2\),)j Fq(H)1907 3210 y Fp(!)r(;\025)2061 3195 y Fs(is)d(ergo)s(dic;)48 b(hence,)f(its)c(sp)s(ectrum)h(is)f(almost)e(surely)-118 3311 y(indep)s(enden)m(t)34 b(of)e Fq(!)t Fs(.)43 b(W)-8 b(e)33 b(denote)g(it)e(b)m(y)j(\006)1466 3326 y Fp(\025)1512 3311 y Fs(.)94 3427 y(Consider)42 b(\()p Fq(E)620 3442 y Fi(+)680 3427 y Fs(\(0\))p Fq(;)17 b(E)921 3442 y Fm(\000)979 3427 y Fs(\(0\)\))41 b(a)h(gap)f(in)g(the)h(sp)s(ectrum)g(of)f Fq(H)49 b Fs(\(if)40 b Fq(E)2660 3442 y Fm(\000)2719 3427 y Fs(\(0\))i(is)f(the)h(in\014m)m(um)e(of)h(\006)3751 3442 y Fi(0)3791 3427 y Fs(,)i(w)m(e)g(set)-118 3544 y Fq(E)-46 3559 y Fi(+)13 3544 y Fs(\(0\))31 b(=)g Fo(\0001)p Fs(\).)49 b(F)-8 b(or)34 b Fq(\025)g Fs(su\016cien)m(tly)i(small,)c (let)i Fq(E)1825 3559 y Fm(\000)1885 3544 y Fs(\()p Fq(\025)p Fs(\))g(\(resp.)50 b Fq(E)2412 3559 y Fi(+)2471 3544 y Fs(\()p Fq(\025)p Fs(\)\))35 b(b)s(e)f(the)h(in\014m)m(um)e(\(resp.) 51 b(suprem)m(um\))-118 3660 y(of)31 b(\006)62 3675 y Fp(\025)139 3660 y Fs(in)f([)p Fq(E)350 3675 y Fi(0)390 3660 y Fq(;)17 b Fs(+)p Fo(1)p Fs(\))30 b(\(resp.)44 b(\()p Fo(\0001)p Fq(;)17 b(E)1291 3675 y Fi(0)1330 3660 y Fs(]\))32 b(where)g Fq(E)1779 3675 y Fi(0)1846 3660 y Fs(=)c(\()p Fq(E)2060 3675 y Fm(\000)2119 3660 y Fs(\(0\))19 b(+)g Fq(E)2430 3675 y Fi(+)2490 3660 y Fs(\(0\)\))p Fq(=)p Fs(2)30 b(,see)i(Fig.)e(1.)43 b(So,)31 b(for)g Fq(\025)g Fs(su\016cien)m(tly)-118 3776 y(small,)f(one)j(has)166 3992 y Fq(E)238 4007 y Fi(+)297 3992 y Fs(\()p Fq(\025)p Fs(\))28 b Fq(<)f(E)633 4007 y Fm(\000)692 3992 y Fs(\()p Fq(\025)p Fs(\))33 b(and)f(\006)1117 4007 y Fp(\025)1185 3992 y Fo(\\)23 b Fs([)p Fq(E)1373 4007 y Fi(+)1432 3992 y Fs(\(0\))f Fo(\000)h Fq(\016)n(;)17 b(E)1836 4007 y Fm(\000)1895 3992 y Fs(\(0\))22 b(+)g Fq(\016)t Fs(])27 b Fo(\032)i Fs([)p Fq(E)2446 4007 y Fi(+)2505 3992 y Fs(\(0\))22 b Fo(\000)g Fq(\016)n(;)17 b(E)2908 4007 y Fi(+)2968 3992 y Fs(\()p Fq(\025)p Fs(\)])22 b Fo([)g Fs([)p Fq(E)3337 4007 y Fm(\000)3397 3992 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)17 b(E)3646 4007 y Fm(\000)3705 3992 y Fs(\(0\))k(+)h Fq(\016)t Fs(])p Fq(:)-118 4213 y Fs(W)-8 b(e)30 b(study)i(the)f(sp)s (ectrum)f(of)g Fq(H)1093 4228 y Fp(!)r(;\025)1234 4213 y Fs(near)g Fq(E)1520 4228 y Fm(\000)1580 4213 y Fs(\()p Fq(\025)p Fs(\))g(for)f Fq(\025)h Fs(su\016cien)m(tly)h(small.)41 b(An)30 b(analogous)f(study)i(can)g(b)s(e)f(done)-118 4329 y(near)j Fq(E)171 4344 y Fi(+)230 4329 y Fs(\()p Fq(\025)p Fs(\).)1063 5066 y @beginspecial @setspecial tx@Dict begin STP newpath 2.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale 0.15 2.0 5. Bracket EndArrow } def [ 56.90549 56.90549 0.0 56.90549 /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 1.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 170.71646 56.90549 56.90549 56.90549 /Lineto /lineto load def false Line gsave 1.0 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 2.0 SLW 0. setgray /ArrowA { BeginArrow 1. 1. scale 0.15 2.0 5. Bracket EndArrow moveto } def /ArrowB { } def [ 227.62195 56.90549 170.71646 56.90549 /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 2.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale 0.15 2.0 5. Bracket EndArrow } def [ 85.35823 28.45274 0.0 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 1.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 128.03734 28.45274 85.35823 28.45274 /Lineto /lineto load def false Line gsave 1.0 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 2.0 SLW 0. setgray /ArrowA { BeginArrow 1. 1. scale 0.15 2.0 5. 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 4475 a tx@Dict begin { 0.0 13.23608 8.2 1.79999 0. Uput UUput } PutCoor PutBegin end 2835 4475 a 2780 4502 a Fs(\006)2850 4517 y Fi(0)2835 4475 y tx@Dict begin PutEnd end 2835 4475 a 1299 4948 a tx@Dict begin { 0.0 13.94443 8.2 1.79999 0. Uput UUput } PutCoor PutBegin end 1299 4948 a 1241 4974 a Fs(\006)1311 4989 y Fp(\025)1299 4948 y tx@Dict begin PutEnd end 1299 4948 a 1535 4475 a tx@Dict begin { 0.0 30.82277 9.0 3.0 0. Uput UUput } PutCoor PutBegin end 1535 4475 a 1407 4500 a Fq(E)1479 4515 y Fi(+)1539 4500 y Fs(\(0\))1535 4475 y tx@Dict begin PutEnd end 1535 4475 a 2480 4475 a tx@Dict begin { 0.0 30.82277 9.0 3.0 0. Uput UUput } PutCoor PutBegin end 2480 4475 a 2352 4500 a Fq(E)2424 4515 y Fm(\000)2484 4500 y Fs(\(0\))2480 4475 y tx@Dict begin PutEnd end 2480 4475 a 1748 4948 a tx@Dict begin { 0.0 31.80193 9.0 3.0 0. Uput UUput } PutCoor PutBegin end 1748 4948 a 1616 4973 a Fq(E)1688 4988 y Fi(+)1747 4973 y Fs(\()p Fq(\025)p Fs(\))1748 4948 y tx@Dict begin PutEnd end 1748 4948 a 2126 4948 a tx@Dict begin { 0.0 31.80193 9.0 3.0 0. Uput UUput } PutCoor PutBegin end 2126 4948 a 1994 4973 a Fq(E)2066 4988 y Fm(\000)2125 4973 y Fs(\()p Fq(\025)p Fs(\))2126 4948 y tx@Dict begin PutEnd end 2126 4948 a 1122 5164 a Fr(Figure)k(1.)49 b Fs(The)33 b(band)g(edges)h(for)e(\006)h(and)g(\006)2854 5179 y Fp(p)94 5475 y Fs(T)-8 b(o)45 b(describ)s(e)h(our)f(main)e (assumption,)k(w)m(e)f(need)g(to)f(recall)e(some)i(facts)g(from)e(the)j (Flo)s(quet)e(theory)h(of)-118 5591 y(p)s(erio)s(dic)31 b(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erators.)1989 5690 y Fg(3)p eop %%Page: 4 4 4 3 bop -118 241 a Fs(1.1.)56 b Fw(Flo)s(quet)47 b(theory)g(of)g(p)s (erio)s(dic)f(Sc)m(hr\177)-56 b(odinger)46 b(op)s(erators.)k Fs(The)42 b(Flo)s(quet)e(sp)s(ectrum)h(of)f Fq(H)49 b Fs(is)40 b(the)-118 357 y(sp)s(ectrum)32 b(of)f(the)h(di\013eren)m (tial)e(op)s(erator)h Fq(H)39 b Fs(acting)31 b(on)g Fq(L)2075 321 y Fp(d)2075 383 y Fi(lo)r(c)2168 357 y Fs(\()p Fj(R)2272 321 y Fp(d)2318 357 y Fs(\))h(with)f(quasi-p)s(erio)s(dic)f(b)s (oundary)i(conditions.)-118 473 y(F)-8 b(or)32 b Fq(\022)e Fo(2)f Fj(T)290 437 y Fm(\003)360 473 y Fs(=)f Fj(R)530 437 y Fp(d)576 473 y Fq(=)p Fj(Z)694 437 y Fp(d)732 473 y Fs(,)k(consider)h(the)g(follo)m(wing)d(eigen)m(v)-5 b(alue)32 b(problem)g(on)g Fq(L)2798 437 y Fp(d)2798 499 y Fi(lo)r(c)2891 473 y Fs(\()p Fj(R)2995 437 y Fp(d)3041 473 y Fs(\))1011 553 y Fn(\()1189 656 y Fq(H)8 b(')27 b Fs(=)h Fq(E)6 b(')1189 796 y(')p Fs(\()p Fq(x)22 b Fs(+)g Fq(\015)5 b Fs(\))28 b(=)g Fq(e)1737 760 y Fp(i)p Fi(2)p Fp(\031)r(\015)t(\022)1918 796 y Fq(')p Fs(\()p Fq(x)p Fs(\))p Fq(;)50 b Fo(8)p Fq(x)28 b Fo(2)g Fj(R)2488 760 y Fp(d)2535 796 y Fq(;)49 b Fo(8)p Fq(\015)33 b Fo(2)28 b Fj(Z)2913 760 y Fp(d)2951 796 y Fq(:)-118 723 y Fs(\(1.2\))-118 971 y(As)39 b Fq(H)47 b Fs(is)39 b(elliptic,)e(one)j(kno)m(ws)g(that)f (the)h(eigen)m(v)-5 b(alues)39 b(of)45 b(\(1.2\))39 b(are)g(discrete;)k (when)d(rep)s(eated)g(according)e(to)-118 1088 y(m)m(ultiplicit)m(y)-8 b(,)35 b(w)m(e)k(denote)g(them)f(b)m(y)g Fq(E)1366 1103 y Fi(0)1406 1088 y Fs(\()p Fq(\022)s Fs(\))f Fo(\024)g Fq(E)1753 1103 y Fi(1)1793 1088 y Fs(\()p Fq(\022)s Fs(\))f Fo(\024)h(\001)17 b(\001)g(\001)35 b(\024)j Fq(E)2407 1103 y Fp(n)2454 1088 y Fs(\()p Fq(\022)s Fs(\))e Fo(\024)i Fq(:)17 b(:)g(:)f Fs(.)59 b(They)39 b(are)f(called)f(the)h Ff(Flo)-5 b(quet)-118 1204 y(eigenvalues)30 b Fs(of)h Fq(H)8 b Fs(.)42 b(These)34 b(functions)d(are)g(Lipsc)m(hitz)h(con)m (tin)m(uous)g(in)e(the)i(v)-5 b(ariable)30 b Fq(\022)s Fs(;)i(when)g(of)f(m)m(ultiplicit)m(y)d(1,)-118 1320 y(the)35 b(Flo)s(quet)f(eigen)m(v)-5 b(alues)35 b(are)g(ev)m(en)i (analytic)d(in)g Fq(\022)s Fs(.)50 b(Moreo)m(v)m(er,)38 b(W)-8 b(eyl's)35 b(la)m(w)g(tells)f(us)h(that)g Fq(E)3502 1335 y Fp(n)3549 1320 y Fs(\()p Fq(\022)s Fs(\))d Fo(!)f Fs(+)p Fo(1)k Fs(as)-118 1436 y Fq(n)e Fo(!)g Fs(+)p Fo(1)j Fs(\(uniformly)d(in)i Fq(\022)s Fs(\).)54 b(In)36 b(regard)g(of)42 b(\(1.2\),)37 b(the)f(Flo)s(quet)f(eigen)m(v)-5 b(alues)36 b(are)g Fj(Z)3184 1400 y Fp(d)3222 1436 y Fs(-p)s(erio)s(dic)d(functions)j(of)-118 1553 y Fq(\022)s Fs(.)56 b(The)38 b(sp)s(ectrum)f(of)f Fq(H)44 b Fs(is)36 b(giv)m(en)h(b)m(y)h(\006)1459 1568 y Fi(0)1534 1553 y Fs(=)c Fo([)1710 1568 y Fp(n)p Fm(\025)p Fi(0)1848 1553 y Fq(E)1920 1568 y Fp(n)1967 1553 y Fs(\()p Fj(T)2068 1516 y Fm(\003)2111 1553 y Fs(\).)56 b(So)36 b(the)h(sp)s(ectrum)g(of)g Fq(H)44 b Fs(is)36 b(the)h(union)f(of)h(closed)-118 1669 y(in)m(terv)-5 b(als)29 b(called)g Ff(the)j(b)-5 b(ands)29 b Fs(of)h(the)g(sp)s(ectrum;)h(the)g(connected)g(comp)s(onen)m(ts)g(of) e Fj(R)f Fo(n)16 b Fs(\006)3178 1684 y Fi(0)3248 1669 y Fs(are)30 b(called)f Ff(the)j(gaps)d Fs(of)-118 1785 y(the)k(sp)s(ectrum)g(of)f Fq(H)8 b Fs(.)-118 1901 y(One)31 b(sa)m(ys)g(that)g(an)f(energy)h Fq(E)j Fo(2)28 b Fs(\006)1212 1916 y Fi(0)1282 1901 y Fs(is)i Ff(simple)g Fs(if)f(there)i(exists)g (exactly)g(one)g(index)f Fq(n)e Fo(\025)g Fs(1)j(suc)m(h)g(that,)g(for) f(some)-118 2017 y Fq(\022)39 b Fo(2)d Fj(T)131 1981 y Fm(\003)174 2017 y Fs(,)i(one)g(has)f Fq(E)673 2032 y Fp(n)720 2017 y Fs(\()p Fq(\022)s Fs(\))f(=)g Fq(E)6 b Fs(.)57 b(It)37 b(is)g(pro)m(v)m(ed)i(in)d([19])h(that,)i (generically)-8 b(,)37 b(the)g(band)h(edges)g(are)f(simple.)56 b(One)-118 2134 y(sa)m(ys)29 b(that)e Fq(E)6 b Fs(,)29 b(a)e(simple)f(band)i(edge,)h(is)e Ff(non-de)-5 b(gener)g(ate)26 b Fs(if)g(the)i(Flo)s(quet)f(eigen)m(v)-5 b(alue)27 b Fq(E)3177 2149 y Fp(n)3224 2134 y Fs(\()p Fo(\001)p Fs(\))g(reac)m (hing)g(this)h(band)-118 2250 y(edge)33 b(has)g(only)f(non-degenerate)i (quadratic)e(extrema)h(at)f(that)h(edge,)g(i.e.)44 b(if)31 b Fq(\022)36 b Fs(is)c(suc)m(h)i(that)f Fq(E)3504 2265 y Fp(n)3551 2250 y Fs(\()p Fq(\022)s Fs(\))28 b(=)f Fq(E)6 b Fs(,)33 b(then)-118 2366 y Fq(\022)j Fs(is)c(a)g(non-degenerate)h (quadratic)g(extrem)m(um)f(of)g Fq(E)1899 2381 y Fp(n)1947 2366 y Fs(.)-118 2482 y(One)d(can)h(de\014ne)g Fo(N)620 2497 y Fi(0)660 2482 y Fs(\()p Fq(E)6 b Fs(\),)29 b(the)h(in)m (tegrated)f(densit)m(y)h(of)f(states)h(of)f Fq(H)37 b Fs(in)28 b(the)i(same)f(w)m(a)m(y)h(as)g(for)e Fq(H)3489 2497 y Fp(!)3569 2482 y Fs(i.e.)42 b(b)m(y)30 b(means)-118 2599 y(of)39 b(\(0.5\))o(.)44 b(One)33 b(pro)m(v)m(es)h(that)1411 2808 y Fo(N)1493 2823 y Fi(0)1532 2808 y Fs(\()p Fq(E)6 b Fs(\))28 b(=)1817 2714 y Fn(X)1828 2924 y Fp(j)t Fm(\025)p Fi(1)1978 2673 y Fn(Z)2033 2898 y Fl(T)2083 2879 y Fd(\003)2134 2808 y Fw(1)2190 2824 y Fp(E)2242 2834 y Fk(j)2275 2824 y Fi(\()p Fp(\022)r Fi(\))p Fm(\024)p Fp(E)2479 2808 y Fq(d\022)s(:)-118 3065 y Fs(The)34 b(set)g(of)e(p)s(oin)m(ts)h(of)g (increase)g(of)g Fo(N)1316 3080 y Fi(0)1388 3065 y Fs(coincides)g(with) f(the)i(sp)s(ectrum.)45 b(The)34 b(non-degeneracy)h(of)d(band)i(edges) -118 3182 y(can)g(also)f(b)s(e)h(c)m(haracterized)h(in)e(terms)g(of)h (the)g(densit)m(y)h(of)e(states.)48 b(Namely)-8 b(,)33 b Fq(E)2876 3197 y Fi(0)2916 3182 y Fs(,)h(a)g(simple)e(band)i(edge)h (of)e Fq(H)41 b Fs(is)-118 3298 y(non-degenerate)33 b(if)f(and)g(only)g (if)g Fo(jN)1252 3313 y Fi(0)1291 3298 y Fs(\()p Fq(E)6 b Fs(\))22 b Fo(\000)g(N)1648 3313 y Fi(0)1688 3298 y Fs(\()p Fq(E)1798 3313 y Fi(0)1837 3298 y Fs(\))p Fo(j)27 b(\024)i Fq(C)7 b Fo(j)p Fq(E)27 b Fo(\000)c Fq(E)2412 3313 y Fi(0)2452 3298 y Fo(j)2480 3262 y Fp(d=)p Fi(2)2623 3298 y Fs(as)33 b Fq(E)g Fo(!)28 b Fq(E)3048 3313 y Fi(0)3087 3298 y Fs(,)33 b Fq(E)h Fo(2)28 b Fs(\006)3417 3313 y Fi(0)3489 3298 y Fs(\(see)34 b([20]\).)-118 3414 y(Details)d(on)h(the)h (material)d(presen)m(ted)35 b(here)e(ma)m(y)f(b)s(e)h(found)g(in)f([28) o(,)h(24,)f(29].)-118 3589 y(1.2.)56 b Fw(The)32 b(main)g(assumptions.) 49 b Fs(Let)29 b(us)f(no)m(w)h(state)g(our)f(main)f(assumptions.)42 b(On)28 b(the)h(underlying)e(p)s(erio)s(dic)-118 3705 y(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erator,)f(w)m(e)h(assume)-1 3842 y Fw(H1:)42 b Fq(E)288 3857 y Fm(\000)347 3842 y Fs(\(0\))32 b(is)g(a)g(simple)f(non-degenerate)j(band)f(edge.)-118 3978 y(This)j(condition)e(is)h(kno)m(wn)i(to)e(hold)g(at)g(the)h(b)s (ottom)e(of)h(the)i(sp)s(ectrum)e(of)h Fq(H)43 b Fs(in)35 b(an)m(y)h(dimension,)f(see)i([1)o(,)f(28].)-118 4095 y(It)44 b(also)g(holds)g(at)g(an)m(y)h(band)g(edge)g(in)e(dimension)g (1,)48 b(see)d([31,)f(7].)79 b(As)45 b(assumption)f(\(H1\))g(is)g (stable)g(under)-118 4211 y(small)31 b(p)s(erturbations,)j(it)e(holds)i (for)f(su\016cien)m(tly)h(small)d(p)s(erturbations)i(of)h(p)s(erio)s (dic)d(op)s(erators)j(with)f(separate)-118 4327 y(v)-5 b(ariables.)42 b(F)-8 b(or)32 b(a)g(more)g(general)g(discussion)h(on)f (the)h(v)-5 b(alidit)m(y)31 b(of)h(assumption)g(\(H1\),)g(w)m(e)i (refer)f(to)f([4].)-118 4443 y(De\014ne)h(the)g(p)s(erio)s(dic)e(op)s (erator)p 1340 4537 89 4 v 1340 4617 a Fq(H)1429 4632 y Fp(\025)1502 4617 y Fs(=)c Fq(H)j Fs(+)22 b Fq(\025)1899 4522 y Fn(X)1888 4742 y Fp(\015)t Fm(2)p Fl(Z)2025 4723 y Fk(d)2072 4617 y Fq(V)2129 4632 y Fp(\015)2201 4617 y Fs(=)27 b Fq(H)j Fs(+)22 b Fq(\025)p 2570 4537 79 4 v(V)f(:)-2793 b Fs(\(1.3\))-118 4902 y(Let)p 56 4822 V 32 w Fq(E)6 b Fs(\()p Fq(\025)p Fs(\))32 b(b)s(e)g(the)g(sp)s(ectral) g(band)g(edge)h(of)p 1539 4822 89 4 v 31 w Fq(H)1628 4917 y Fp(\025)1705 4902 y Fs(closest)g(to)e Fq(E)6 b Fs(\(0\).)43 b(F)-8 b(or)31 b Fq(\025)h Fs(su\016cien)m(tly)h(small,)d (it)h(is)g(w)m(ell)g(de\014ned)-118 5018 y(\(see)j(section)e(5.1\).)43 b(W)-8 b(e)33 b(assume)g(that)-1 5155 y Fw(H2:)42 b Fs(there)33 b(exists)g Fq(C)i(>)27 b Fs(0)33 b(suc)m(h)h(that,)e(for)g Fq(\025)h Fs(su\016cien)m(tly)g(small)1177 5312 y Fo(j)p 1205 5232 79 4 v Fq(E)6 b Fs(\()p Fq(\025)p Fs(\))22 b Fo(\000)p 1537 5232 V 22 w Fq(E)6 b Fs(\(0\))p Fo(j)27 b Fs(=)h Fo(j)p 1927 5232 V Fq(E)6 b Fs(\()p Fq(\025)p Fs(\))22 b Fo(\000)g Fq(E)6 b Fs(\(0\))p Fo(j)27 b(\025)h Fq(C)7 b Fo(j)p Fq(\025)p Fo(j)p Fq(:)-2957 b Fs(\(1.4\))-118 5475 y(It)42 b(is)g(pro)m(v)m(ed)h(in)f(section)g(5.1)g(that,)i(for)e (a)g(giv)m(en)g(bac)m(kground)i(p)s(erio)s(dic)c(op)s(erator)i Fq(H)8 b Fs(,)44 b(assumption)d(\(H2\))h(is)-118 5591 y(satis\014ed)33 b(for)f(a)g(generic)h(single-site)e(p)s(erturbation)g Fq(V)22 b Fs(.)43 b(Moreo)m(v)m(er,)35 b(\(1.4\))d(implies)e(\(0.2\))o (.)1989 5690 y Fg(4)p eop %%Page: 5 5 5 4 bop -118 241 a Fs(1.3.)56 b Fw(The)37 b(results.)49 b Fs(W)-8 b(e)33 b(pro)m(v)m(e)-118 447 y Fw(Theorem)k(1.1.)49 b Ff(Assume)33 b(\(H0\),)h(\(H1\))f(and)g(\(H2\).)44 b(Fix)33 b Fq(\021)e Fo(2)d Fs(\(0)p Fq(;)17 b(d=)p Fs(\(4)p Fq(d)g Fs(+)i(4\)\))p Ff(.)44 b(Then,)32 b(ther)-5 b(e)34 b(exists)e Fq(")c(>)f Fs(0)33 b Ff(and)-118 563 y Fq(\025)-61 578 y Fp(\021)8 563 y Fq(>)28 b Fs(0)35 b Ff(such)f(that,)h(for)g Fq(\025)28 b Fo(2)g Fs(\(0)p Fq(;)17 b(\025)1169 578 y Fp(\021)1210 563 y Fs(\))p Ff(,)35 b(one)f(has)1126 793 y Fo(N)1208 808 y Fp(\025)1253 793 y Fs(\()p 1291 713 79 4 v Fq(E)6 b Fs(\()p Fq(\025)p 1464 738 65 4 v(!)s Fs(\))22 b Fo(\000)h Fq(\025)1745 752 y Fi(1+)p Fp(\021)1877 793 y Fs(\))f Fo(\000)g(N)2118 808 y Fp(\025)2163 793 y Fs(\()p Fq(E)2273 808 y Fm(\000)2333 793 y Fs(\()p Fq(\025)p Fs(\)\))27 b Fo(\024)h Fq(e)2681 752 y Fm(\000)p Fp(\025)2777 728 y Fd(\000)p Fk(")2863 793 y Fq(:)-118 1000 y Fs(This)33 b(result)f(has)h(strong)g(consequences)j(on)c(the)h (sp)s(ectral)g(b)s(eha)m(vior)f(of)g Fq(H)2698 1015 y Fp(!)2748 1000 y Fs(.)44 b(Indeed,)34 b(let)e(us)h(assume)g(that)-1 1164 y Fw(H3:)42 b Fs(the)22 b(common)f(probabilit)m(y)f(distribution)g (of)i(the)h(random)e(v)-5 b(ariables)21 b(\()p Fq(!)2861 1179 y Fp(\015)2905 1164 y Fs(\))2943 1183 y Fp(\015)t Fm(2)p Fl(Z)3080 1164 y Fk(d)k Fs(is)d(absolutely)f(con)m(tin)m(uous)99 1280 y(with)32 b(resp)s(ect)i(to)e(Leb)s(esgue)i(measure)f(and)g(its)f (densit)m(y)h(is)f(lo)s(cally)e(absolutely)i(con)m(tin)m(uous.)-118 1445 y(Under)h(this)f(additional)e(assumption,)i(in)g([11],)g(it)g(is)g (pro)m(v)m(ed)i(that)-118 1651 y Fw(Theorem)j(1.2.)49 b Ff(Assume)c(\(H0\),)j(\(H1\),)f(\(H2\))f(and)e(\(H3\).)76 b(Fix)44 b Fs(0)j Fq(<)f(\034)59 b(<)46 b Fs(1)p Ff(.)76 b(Then,)47 b(ther)-5 b(e)45 b(exists)g Fq(c)h(>)h Fs(0)-118 1768 y Ff(and)g Fq(\025)141 1783 y Fi(0)231 1768 y Fq(>)j Fs(0)d Ff(such)g(that,)k(for)c Fs(0)j Fq(<)h(\025)f(<)h(\025)1617 1783 y Fi(0)1656 1768 y Ff(,)g(the)c(inte)-5 b(gr)g(ate)g(d)47 b(density)g(of)g(states)g(of)g Fq(H)3335 1783 y Fp(!)r(;\025)3446 1768 y Ff(,)k Fo(N)3609 1783 y Fp(\025)3654 1768 y Ff(,)f(is)d(H\177) -50 b(older)-118 1892 y(c)-5 b(ontinuous)43 b(of)g(or)-5 b(der)43 b Fq(\034)55 b Ff(in)43 b(the)g(interval)g Fs([)p Fq(E)1629 1907 y Fm(\000)1689 1892 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)p 1866 1812 79 4 v 17 w(E)5 b Fs(\()p Fq(\025)p 2038 1837 65 4 v(!)s Fs(\))29 b(+)f Fq(c\025)p Fs(])43 b Ff(i.e.)70 b(ther)-5 b(e)43 b(exists)g Fq(C)3243 1907 y Fp(\025)3332 1892 y Fq(>)g Fs(0)g Ff(such)g(that,)j(for)-118 2013 y Fs(\()p Fq(E)6 b(;)17 b(E)120 1977 y Fm(0)143 2013 y Fs(\))28 b Fo(2)g Fs([)p Fq(E)402 2028 y Fm(\000)461 2013 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)p 638 1933 79 4 v 17 w(E)5 b Fs(\()p Fq(\025)p 810 1958 65 4 v(!)t Fs(\))22 b(+)g Fq(c\025)p Fs(])1159 1977 y Fi(2)1198 2013 y Ff(,)35 b(one)f(has)1289 2226 y Fo(jN)1399 2241 y Fp(\025)1444 2226 y Fs(\()p Fq(E)6 b Fs(\))22 b Fo(\000)h(N)1802 2241 y Fp(\025)1847 2226 y Fs(\()p Fq(E)1963 2185 y Fm(0)1986 2226 y Fs(\))p Fo(j)k(\024)i Fq(C)2255 2241 y Fp(\025)2300 2226 y Fo(j)p Fq(E)f Fo(\000)22 b Fq(E)2605 2185 y Fm(0)2628 2226 y Fo(j)2656 2185 y Fp(\034)2699 2226 y Fq(:)94 2438 y Fs(Theorem)38 b(1.2)f(is)g(a)g(consequence)j(of)d(the)h(W)-8 b(egner's)38 b(estimate)f(pro)m(v)m(ed)h(in)f([11];)i(a)e(w)m(eak)m(er) i(form)e(of)f(suc)m(h)-118 2555 y(an)c(estimate)g(is)g(pro)m(v)m(ed)i (in)e(Prop)s(osition)f(3.1.)94 2671 y(Using)42 b(Theorems)h(1.1)e(and)h (1.2)f(\(or)h(b)s(etter)g(said,)i(the)e(W)-8 b(egner)43 b(estimate,)g(Prop)s(osition)d(3.1\))i(together)-118 2787 y(with)32 b(the)h(m)m(ulti-scale)d(analysis)i(tec)m(hnique)i (\(see)g(e.g.)e([30,)h(8)o(]\),)g(w)m(e)h(pro)m(v)m(e)-118 2993 y Fw(Theorem)j(1.3.)49 b Ff(Assume)39 b(\(H0\),)h(\(H1\),)f (\(H2\))g(and)g(\(H3\).)56 b(Fix)39 b Fq(\021)g Fo(2)c Fs(\(0)p Fq(;)17 b(d=)p Fs(\(4)p Fq(d)24 b Fs(+)h(4\)\))p Ff(.)56 b(Ther)-5 b(e)38 b(exists)h Fq(\025)3897 3008 y Fp(\021)3974 2993 y Fq(>)c Fs(0)-118 3110 y Ff(such)g(that,)g(for)f Fq(\025)28 b Fo(2)g Fs(\(0)p Fq(;)17 b(\025)855 3125 y Fp(\021)896 3110 y Fs(\))p Ff(,)35 b(with)g(pr)-5 b(ob)g(ability)34 b(1,)h(one)f(has)-18 3282 y Fs(1.)41 b Ff(exp)-5 b(onential)43 b(lo)-5 b(c)g(alization)43 b(in)h Fq(I)1306 3297 y Fp(\021)r(;\025)1453 3282 y Fs(:=)h([)p Fq(E)1700 3297 y Fm(\000)1759 3282 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)p 1936 3202 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 2109 3227 65 4 v(!)s Fs(\))29 b Fo(\000)g Fq(\025)2403 3246 y Fi(1+)p Fp(\021)2535 3282 y Fs(])45 b Ff(\(se)-5 b(e)43 b(the)h(description)g(given)f(in)h (The)-5 b(o-)99 3399 y(r)g(em)34 b(0.1\);)h(mor)-5 b(e)g(over,)33 b(the)i(eigenfunctions)f(asso)-5 b(ciate)g(d)34 b(to)h(eigenvalues)e (in)i(this)g(interval)f(satisfy)44 b Fs(\(0.3\))o Ff(.)-18 3515 y Fs(2.)d Ff(str)-5 b(ong)29 b(Hilb)-5 b(ert-Schmidt)30 b(dynamic)-5 b(al)28 b(lo)-5 b(c)g(alization)29 b(in)g Fq(I)2191 3530 y Fp(\021)r(;\025)2324 3515 y Ff(\(se)-5 b(e)29 b(the)h(description)f(given)g(in)g(The)-5 b(or)g(em)29 b(0.1\).)94 3721 y Fs(T)-8 b(o)45 b(our)g(kno)m(wledge,)k(these)e(are)d (the)i(\014rst)f(results)h(on)f(lo)s(calization)40 b(for)45 b(sign-inde\014nite)e(p)s(oten)m(tials)h(at)-118 3837 y(in)m(ternal)f(band)h(edges.)78 b(Lo)s(calization)41 b(for)i(suc)m(h)j(p)s(oten)m(tials)c(had)i(b)s(een)h(established)f(at)f (the)i(b)s(ottom)d(of)h(the)-118 3954 y(sp)s(ectrum)49 b(assuming)e(some)i(Lifshitz)e(b)s(eha)m(vior)h(or)g(quic)m(kly)h(deca) m(ying)g(tails)e(for)h(the)g(distributions)f(of)h(the)-118 4070 y(random)42 b(v)-5 b(ariables)42 b(\()p Fq(!)764 4085 y Fp(\015)808 4070 y Fs(\))846 4089 y Fp(\015)t Fm(2)p Fl(Z)984 4070 y Fk(d)j Fs(\(see)f(e.g.)f([17,)g(2]\).)75 b(T)-8 b(o)43 b(our)g(kno)m(wledge,)k(for)42 b(single)g(site)h(p)s (oten)m(tials)f(that)h(do)-118 4186 y(not)h(ha)m(v)m(e)j(a)d (de\014nite)h(sign,)i(Lifshitz)d(tails)f(ha)m(v)m(e)j(not)f(b)s(een)g (pro)m(v)m(ed)h(to)f(hold,)i(ev)m(en)f(at)f(the)g(b)s(ottom)e(of)h(the) -118 4302 y(sp)s(ectrum.)74 b(In)43 b(the)h(presen)m(t)g(pap)s(er,)i(w) m(e)d(pro)m(v)m(e)h(a)f(kind)g(of)f(Lifshitz)f(b)s(eha)m(vior)i(at)f (in)m(ternal)g(non-degenerate)-118 4419 y(band)33 b(edges)g(without)f (assuming)g(that)g(the)h(single)e(site)h(p)s(oten)m(tial)f(has)h(a)h (de\014nite)f(sign.)43 b(This)33 b(assumption)e(has)-118 4535 y(alw)m(a)m(ys)j(pla)m(y)m(ed)g(a)g(crucial)e(role)h(in)g(the)h (analysis)f(of)g(Lifshitz)g(tails.)45 b(In)34 b(the)g(presen)m(t)h(w)m (ork,)g(it)e(is)g(replaced)h(with)-118 4651 y(assumption)27 b(\(H2\))g(and)g(the)h(assumption)f(that)g Fq(\025)p Fs(,)h(the)g(disorder,)g(is)f(small.)39 b(Note)28 b(also)e(that,)j(due) f(to)f(the)g(unique)-118 4767 y(con)m(tin)m(uation)32 b(principle,)f(assumption)h(\(H2\))g(is)g(satis\014ed)h(for)f(p)s(oten) m(tials)f(ha)m(ving)i(a)f(de\014nite)h(sign.)94 4883 y(Theorems)e(0.2)e(and)h(0.1)g(follo)m(w)e(immediately)f(from)h (Theorems)j(1.1)e(and)h(1.3.)42 b(Indeed,)32 b(for)d Fq(H)36 b Fs(=)27 b Fo(\000)p Fs(\001)k(and)-118 5000 y Fq(E)-46 5015 y Fm(\000)13 5000 y Fs(\(0\))41 b(=)g(0,)i(it)c(is)i(w) m(ell)e(kno)m(wn)j(that)f(assumption)f(\(H1\))g(is)g(ful\014lled)f(at)h (the)h(b)s(ottom)e(of)h(the)i(sp)s(ectrum,)h(see)-118 5121 y(e.g.[28,)33 b(1].)47 b(In)34 b(this)f(case,)i(one)f(pro)m(v)m (es)p 1394 5041 79 4 v 35 w Fq(E)6 b Fs(\()p Fq(\025)p 1567 5066 65 4 v(!)s Fs(\))29 b(=)h Fq(\025)p 1861 5041 79 4 v(V)44 b Fs(+)23 b Fq(o)p Fs(\()p Fq(\025)p Fs(\))33 b(\(see)h(section)g(5.1\);)g(hence,)h(assumption)e(\(H2\))g(is)-118 5242 y(ful\014lled)d(if)i(and)h(only)f(if)p 814 5162 V 31 w Fq(V)49 b Fo(6)p Fs(=)28 b(0.)94 5358 y(Analogues)35 b(of)f(Theorems)i(1.1)e(and)h(1.3)g(in)f(the)h(case)h(of)e(discrete)i (random)e(Sc)m(hr\177)-49 b(odinger)35 b(op)s(erators)f(ha)m(v)m(e)-118 5475 y(b)s(een)c(pro)m(v)m(ed)i(in)d([22)o(].)43 b(Finally)-8 b(,)28 b(let)h(us)h(sa)m(y)h(that)e(the)i(metho)s(ds)e(used)i(in)e(the) h(presen)m(t)i(pap)s(er)e(should)f(also)g(apply)-118 5591 y(to)j(other)g(random)f(mo)s(dels,)g(e.g.)43 b(to)32 b(Sc)m(hr\177)-49 b(odinger)32 b(op)s(erators)g(with)g(w)m(eak)h (random)e(magnetic)g(p)s(oten)m(tials)g(\([9]\).)1989 5690 y Fg(5)p eop %%Page: 6 6 6 5 bop -118 241 a Fs(1.4.)56 b Fw(The)49 b(outline)e(of)h(the)h(pap)s (er.)h Fs(Section)42 b(2)g(is)g(dev)m(oted)i(to)e(the)g(pro)s(of)g(of)f (Theorem)i(1.1.)72 b(This)43 b(pro)s(of)-118 357 y(is)j(made)h(in)f (four)g(steps.)88 b(In)47 b(section)g(2.1,)j(w)m(e)e(in)m(tro)s(duce)f (the)g(p)s(erio)s(dic)f(appro)m(ximations)f(that)h(enable)h(us)-118 473 y(to)35 b(estimate)f Fo(N)479 488 y Fp(\025)524 473 y Fs(.)52 b(Section)35 b(2.2)g(is)g(dev)m(oted)i(to)d(the)i (recollection)e(of)h(some)g(facts)g(from)f(the)i(Flo)s(quet)e(theory)i (of)-118 589 y(p)s(erio)s(dic)23 b(op)s(erators.)40 b(In)25 b(section)f(2.3,)i(w)m(e)f(sho)m(w)h(that)e(estimating)e Fo(N)2446 604 y Fp(\025)2516 589 y Fs(comes)i(up)h(to)f(estimating)e (the)j(probabilit)m(y)-118 706 y(that,)50 b(restricted)d(to)f(a)g (su\016cien)m(tly)h(but)g(not)f(to)s(o)g(large)f(cub)s(e,)51 b(the)c(op)s(erator)f Fq(H)3078 721 y Fp(!)r(;\025)3235 706 y Fs(has)h(an)g(eigen)m(v)-5 b(alue)46 b(in)-118 822 y Fq(I)-75 837 y Fp(\021)r(;\025)28 822 y Fs(.)j(This)35 b(probabilit)m(y)e(is)h(then)i(estimated)e(in)g(section)g(2.4)h(b)m(y)g (sho)m(wing)g(that)g(it)e(reduces)k(to)d(large)g(deviation)-118 938 y(estimates)45 b(for)g(sums)h(of)f(indep)s(enden)m(t)i(random)d(v) -5 b(ariables.)82 b(In)45 b(section)h(3,)j(w)m(e)d(pro)m(v)m(e)h (Theorem)f(1.3)f(using)-118 1054 y(Theorem)33 b(1.1)f(and)g(Prop)s (osition)f(3.1)h(that)g(is)g(pro)m(v)m(ed)h(in)f(section)g(4.)44 b(A)m(t)32 b(last,)g(in)g(the)g(app)s(endix,)h(section)g(5,)f(w)m(e) -118 1171 y(gathered)h(some)f(useful)h(results.)1270 1393 y(2.)54 b Fr(The)38 b(pr)n(oof)g(of)g(Theorem)g(1.1)94 1568 y Fs(The)k(sc)m(heme)f(of)f(this)g(pro)s(of)g(follo)m(ws)f(the)i (one)g(of)e(the)i(pro)s(of)f(of)g(Theorem)g(1.1)g(in)g([22].)67 b(W)-8 b(e)40 b(\014rst)h(state)-118 1684 y(an)31 b(appro)m(ximation)f (theorem)h(giving)f(precise)i(\014nite)g(v)m(olume)f(appro)m(ximations) e(to)j Fo(N)3124 1699 y Fp(\025)3169 1684 y Fs(;)g(suc)m(h)h(a)e (statemen)m(t)h(w)m(as)-118 1800 y(deriv)m(ed)h(in)f([20,)g(23].)43 b(Then,)34 b(w)m(e)g(study)g(the)f(\014nite)f(v)m(olume)g(appro)m (ximations.)-118 1992 y(2.1.)56 b Fw(P)m(erio)s(dic)47 b(appro)m(ximations.)i Fs(Let)43 b Fq(n)k Fo(2)g Fj(N)d Fo(n)30 b(f)p Fs(0)p Fo(g)42 b Fs(and)i(de\014ne)h(the)f(follo)m(wing)d (p)s(erio)s(dic)h(Sc)m(hr\177)-49 b(odinger)-118 2108 y(op)s(erator)744 2290 y Fq(H)833 2249 y Fp(n)825 2315 y(!)r(;\025)964 2290 y Fs(=)27 b Fq(H)j Fs(+)22 b Fq(\025)1418 2195 y Fn(X)1350 2419 y Fp(\015)t Fm(2)p Fl(Z)1487 2396 y Fk(d)1487 2440 y Fc(2)p Fk(n)p Fc(+1)1647 2290 y Fq(!)1708 2305 y Fp(\015)1894 2195 y Fn(X)1769 2415 y Fp(\014)s Fm(2)p Fi(\(2)p Fp(n)p Fi(+1\))p Fl(Z)2133 2396 y Fk(d)2180 2290 y Fq(V)f Fs(\()p Fq(x)i Fo(\000)f Fq(\015)27 b Fo(\000)c Fq(\014)6 b Fs(\))27 b(=)h Fq(H)h Fs(+)22 b Fq(\025V)3225 2249 y Fp(n)3203 2315 y(!)-118 2290 y Fs(\(2.1\))-118 2597 y(where)34 b Fj(Z)233 2561 y Fp(d)233 2622 y Fi(2)p Fp(n)p Fi(+1)430 2597 y Fs(:=)28 b Fj(Z)630 2561 y Fp(d)668 2597 y Fq(=)p Fs(\(2)p Fq(n)22 b Fs(+)g(1\))p Fj(Z)1138 2561 y Fp(d)1175 2597 y Fs(.)-118 2719 y(F)-8 b(or)30 b Fq(!)k Fs(\014xed)e(and)f Fq(n)d Fo(2)g Fj(N)818 2683 y Fm(\003)864 2719 y Fs(,)j Fq(H)1011 2683 y Fp(n)1003 2745 y(!)r(;\025)1145 2719 y Fs(is)f(a)h(\(2)p Fq(n)19 b Fs(+)f(1\))p Fj(Z)1735 2683 y Fp(d)1773 2719 y Fs(-p)s(erio)s(dic)29 b(self-adjoin)m(t)g(Sc)m(hr\177)-49 b(odinger)31 b(op)s(erator.)42 b(Let)31 b Fq(N)3901 2683 y Fp(n)3891 2745 y(!)r(;\025)4034 2719 y Fs(b)s(e)-118 2835 y(its)h(in)m(tegrated)g(densit)m(y)i(of)e (states;)h(it)f(satis\014es)1149 3048 y Fo(N)1246 3007 y Fp(n)1231 3073 y(!)r(;\025)1342 3048 y Fs(\()p Fq(E)6 b Fs(\))27 b(=)1627 2953 y Fn(X)1632 3166 y Fp(k)r Fm(2)p Fl(N)1787 2913 y Fn(Z)1843 3138 y Fm(f)p Fp(\022)r Fm(2)p Fl(T)2010 3115 y Fd(\003)2010 3159 y Fc(2)p Fk(n)p Fc(+1)2154 3138 y Fi(;)32 b Fp(E)2258 3150 y Fk(k)2296 3138 y Fi(\()p Fp(n;!)r(;\025)p Fi(;)p Fp(\022)r Fi(\))p Fm(\024)p Fp(E)t Fm(g)2741 3048 y Fq(d\022)s(:)-2985 b Fs(\(2.2\))-118 3312 y(where)40 b(\()p Fq(E)280 3327 y Fp(k)323 3312 y Fs(\()p Fq(n;)17 b(!)t(;)g(\025)p Fs(;)g Fq(\022)s Fs(\)\))797 3327 y Fp(k)r Fm(\025)p Fi(0)967 3312 y Fs(are)39 b(the)h(Flo)s(quet)e(eigen)m(v)-5 b(alues)39 b(of)f Fq(H)2396 3276 y Fp(n)2388 3338 y(!)r(;\025)2538 3312 y Fs(and)h(the)h(torus)f Fj(T)3228 3276 y Fm(\003)3228 3337 y Fp(n)3317 3312 y Fs(is)g(de\014ned)h(b)m(y)g Fj(T)3969 3276 y Fm(\003)3969 3337 y Fp(n)4058 3312 y Fs(=)-118 3442 y Fj(R)-52 3406 y Fp(d)-6 3442 y Fq(=)57 3402 y Fi(1)p 53 3419 43 4 v 53 3476 a Fp(n)106 3442 y Fj(Z)175 3406 y Fp(d)213 3442 y Fs(.)-118 3559 y(W)-8 b(e)33 b(recall)e(Theorem)i(1.2)f(from)f([21])i (namely)-118 3738 y Fw(Theorem)k(2.1)h Fs(\([21)o(]\))p Fw(.)49 b Ff(Pick)40 b Fq(\013)f(>)e Fs(0)j Ff(and)g Fq(I)46 b Fo(\032)38 b Fj(R)5 b Ff(,)48 b(a)40 b(c)-5 b(omp)g(act)40 b(interval.)60 b(Ther)-5 b(e)40 b(exists)g Fq(\027)3408 3753 y Fi(0)3486 3738 y Fq(>)d Fs(0)j Ff(and)g Fq(\032)e(>)g Fs(0)-118 3855 y Ff(such)d(that,)g(for)f Fq(\025)28 b Fo(2)g Fs([0)p Fq(;)17 b Fs(1])p Ff(,)34 b Fq(E)g Fo(2)28 b Fq(I)8 b Ff(,)35 b Fq(\027)f Fo(2)28 b Fs(\(0)p Fq(;)17 b(\027)1598 3870 y Fi(0)1637 3855 y Fs(\))35 b Ff(and)f Fq(n)28 b Fo(\025)g Fq(\027)2144 3819 y Fm(\000)p Fp(\032)2240 3855 y Ff(,)35 b(one)f(has)-118 4063 y Fs(\(2.3\))82 b Fj(E)13 b Fs(\()p Fo(N)361 4022 y Fp(n)346 4087 y(!)r(;\025)463 4063 y Fs(\()p Fq(E)28 b Fs(+)22 b Fq(\027)6 b(=)p Fs(2\)\))22 b Fo(\000)h Fj(E)12 b Fs(\()p Fo(N)1244 4022 y Fp(n)1229 4087 y(!)r(;\025)1346 4063 y Fs(\()p Fq(E)28 b Fo(\000)23 b Fq(\027)6 b(=)p Fs(2\)\))22 b Fo(\000)g Fq(e)1978 4022 y Fm(\000)p Fp(\027)2072 3998 y Fd(\000)p Fk(\013)1385 4214 y Fo(\024)29 b(N)1573 4229 y Fp(\025)1618 4214 y Fs(\()p Fq(E)f Fs(+)22 b Fq(\027)6 b Fs(\))22 b Fo(\000)h(N)2150 4229 y Fp(\025)2195 4214 y Fs(\()p Fq(E)28 b 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b(of)f(functions)h Fq(')g Fs(that)f(are)h(lo)s(cally)d(square)j(in)m(tegrable)f(\(resp.)72 b(lo)s(cally)39 b(in)i Fq(H)3026 4960 y Fi(2)3064 4996 y Fs(\))h(on)g Fj(R)3355 4960 y Fp(d)3443 4996 y Fs(and)f(that)h (satisfy)-118 5112 y Fq(')p Fs(\()p Fq(x)28 b Fs(+)h Fq(\015)5 b Fs(\))42 b(=)h Fq(e)472 5076 y Fp(i\015)t(\022)576 5112 y Fq(')p Fs(\()p Fq(x)p Fs(\),)h Fo(8)p Fq(\015)k Fo(2)c Fj(Z)1175 5076 y Fp(d)1254 5112 y Fs(is)d(denoted)i(b)m(y)f Fq(L)1949 5076 y Fi(2)1949 5138 y Fp(\022)2030 5112 y Fs(\(resp.)72 b Fq(H)2429 5076 y Fi(2)2421 5138 y Fp(\022)2468 5112 y Fs(\);)46 b(b)s(oth)41 b(spaces)i(are)f(endo)m(w)m(ed)h(with)e (their)-118 5229 y(natural)31 b(scalar)h(pro)s(duct)h(\(see)h([28)o (]\).)44 b(The)33 b(op)s(erator)f Fq(H)40 b Fs(acting)32 b(on)g(the)h Fq(H)2721 5193 y Fi(2)2713 5255 y Fp(\022)2793 5229 y Fs(is)f(denoted)i(b)m(y)f Fq(H)8 b Fs(\()p Fq(\022)s Fs(\).)-118 5345 y(Pic)m(k)30 b Fq(n)e Fo(\025)g Fs(1.)43 b(The)31 b(op)s(erator)e Fq(H)37 b Fs(is)30 b Fj(Z)1278 5309 y Fp(d)1316 5345 y Fs(-p)s(erio)s(dic;)f(hence,)j(it)c(is)i(also)f (\(2)p Fq(n)17 b Fs(+)f(1\))p Fj(Z)2837 5309 y Fp(d)2875 5345 y Fs(-p)s(erio)s(dic.)40 b(T)-8 b(o)30 b(stress)i(this)e(p)s(oin)m (t)-118 5461 y(of)38 b(view,)j(w)m(e)f(sometimes)e(call)g(it)g Fq(H)1267 5425 y Fp(n)1313 5461 y Fs(.)63 b(Let)39 b Fq(L)1650 5425 y Fi(2)1650 5487 y Fp(n;\022)1791 5461 y Fs(\(resp.)63 b Fq(H)2181 5425 y Fi(2)2173 5487 y Fp(n;\022)2274 5461 y Fs(\))39 b(b)s(e)g(the)h(set)f(of)g(functions)g Fq(')g Fs(that)f(are)h(lo)s(cally)-118 5591 y(square)28 b(in)m(tegrable)f(\(resp.)42 b(lo)s(cally)25 b(in)i Fq(H)1413 5555 y Fi(2)1452 5591 y Fs(\))g(on)g Fj(R)1713 5555 y Fp(d)1787 5591 y Fs(and)h(that)f(satisfy)g Fq(')p Fs(\()p Fq(x)12 b Fs(+)g Fq(\015)5 b Fs(\))27 b(=)h Fq(e)3004 5555 y Fp(i)p Fi(2)p Fp(\031)r(\015)t(\022)3185 5591 y Fq(')p Fs(\()p Fq(x)p Fs(\),)h Fo(8)p Fq(\015)k Fo(2)28 b Fs(\(2)p Fq(n)12 b Fs(+)g(1\))p Fj(Z)4070 5555 y Fp(d)4107 5591 y Fs(;)1989 5690 y Fg(6)p eop %%Page: 7 7 7 6 bop -118 241 a Fs(b)s(oth)28 b(spaces)i(are)e(endo)m(w)m(ed)i(with) e(their)g(natural)f(scalar)g(pro)s(duct.)43 b(The)29 b(op)s(erator)e Fq(H)36 b Fs(acting)27 b(on)i(the)f(space)i Fq(H)4041 205 y Fi(2)4033 267 y Fp(n;\022)-118 363 y Fs(is)i(denoted)i(b)m(y)f Fq(H)573 326 y Fp(n)619 363 y Fs(\()p Fq(\022)s Fs(\).)44 b(Notice)32 b(that,)h(in)e(this)i(case,)g (w)m(e)h(can)f(restrict)f(ourselv)m(es)i(to)e Fq(\022)f Fo(2)d Fj(T)3320 326 y Fm(\003)3320 387 y Fi(2)p Fp(n)p Fi(+1)3496 363 y Fs(.)94 479 y(The)j(Flo)s(quet)e(eigen)m(v)-5 b(alues)30 b(and)f(eigen)m(v)m(ectors)j(of)d Fq(H)2079 443 y Fp(n)2125 479 y Fs(\()p Fq(\022)s Fs(\))h(are)g(easily)f (computed)h(in)f(terms)g(of)h(the)g(Flo)s(quet)-118 595 y(eigen)m(v)-5 b(alues)48 b(and)h(eigen)m(v)m(ectors)h(of)e Fq(H)8 b Fs(\()p Fq(\022)s Fs(\).)91 b(Indeed,)54 b(let)48 b(\()p Fq(E)2258 610 y Fp(k)2301 595 y Fs(\()p Fq(\022)s Fs(\))p Fq(;)17 b(')2533 610 y Fp(k)2575 595 y Fs(\()p Fo(\001)p Fq(;)g(\022)s Fs(\)\))2809 610 y Fp(k)r Fm(\025)p Fi(0)2990 595 y Fs(b)s(e)49 b(the)g(Flo)s(quet)e(eigen)m(v)-5 b(alue)-118 711 y(and)38 b(eigen)m(v)m(ectors)j(for)c Fq(H)8 b Fs(\()p Fq(\022)s Fs(\))38 b(i.e.)61 b(the)39 b(solutions)f(to)g(the)h(eigen)m(v)-5 b(alue)38 b(problem)f(\(1.2\))o (.)61 b(One)39 b(c)m(hec)m(ks)j(that,)d(for)-118 828 y Fq(\015)33 b Fo(2)28 b Fj(Z)129 791 y Fp(d)129 852 y Fi(2)p Fp(n)p Fi(+1)299 828 y Fs(,)317 1165 y Fq(H)8 b(')470 1180 y Fp(k)r(;\015)572 1165 y Fs(\()p Fq(\022)s Fs(\))28 b(=)f Fq(E)899 1180 y Fp(k)r(;\015)1002 1165 y Fs(\()p Fq(\022)s Fs(\))p Fq(')1190 1180 y Fp(k)r(;\015)1292 1165 y Fs(\()p Fq(\022)s Fs(\))33 b(where)1747 961 y Fn(8)1747 1050 y(<)1747 1230 y(:)1836 1056 y Fq(E)1908 1071 y Fp(k)r(;\015)2011 1056 y Fs(\()p Fq(\022)s Fs(\))27 b(=)h Fq(E)2338 1071 y Fp(k)2381 1056 y Fs(\()p Fq(\022)d Fs(+)d Fq(\015)5 b(=)p Fs(\(2)p Fq(n)22 b Fs(+)g(1\)\))p Fq(;)1836 1229 y(')1900 1244 y Fp(k)r(;\015)2002 1229 y Fs(\()p Fq(\022)s Fs(\))28 b(=)2475 1162 y(1)p 2268 1206 463 4 v 2268 1297 a(\(2)p Fq(n)22 b Fs(+)g(1\))2620 1269 y Fp(d=)p Fi(2)2740 1229 y Fq(')2804 1244 y Fp(k)2847 1229 y Fs(\()p Fq(\022)j Fs(+)d Fq(\015)5 b(=)p Fs(\(2)p Fq(n)22 b Fs(+)g(1\)\))p Fq(:)-118 1503 y Fs(F)-8 b(or)32 b Fq(\015)g Fs(=)c(\()p Fq(\015)333 1518 y Fi(1)372 1503 y Fq(;)17 b(:)g(:)g(:)f(;)h(\015)642 1518 y Fp(d)681 1503 y Fs(\))28 b Fo(2)g Fj(Z)910 1467 y Fp(d)981 1503 y Fs(and)k Fq(l)1201 1467 y Fm(0)1252 1503 y Fo(\025)d Fs(0,)j(de\014ne)356 1710 y Fq(C)426 1725 y Fp(\015)t(;l)508 1706 y Fd(0)561 1710 y Fs(=)c Fo(f)p Fq(x)g Fs(=)f(\()p Fq(x)994 1725 y Fi(1)1034 1710 y Fq(;)17 b(:)g(:)g(:)f(;)h(x)1308 1725 y Fp(d)1349 1710 y Fs(\);)49 b Fo(8)p Fs(1)28 b Fo(\024)g Fq(i)g Fo(\024)g Fq(d;)49 b Fo(\000)p Fq(l)2101 1669 y Fm(0)2147 1710 y Fo(\000)23 b Fs(1)p Fq(=)p Fs(2)k Fo(\024)h Fq(x)2581 1725 y Fp(i)2631 1710 y Fo(\000)23 b Fs(\(2)p Fq(l)2849 1669 y Fm(0)2894 1710 y Fs(+)g(1\))p Fq(\015)3131 1725 y Fp(i)3186 1710 y Fq(<)k(l)3320 1669 y Fm(0)3366 1710 y Fs(+)22 b(1)p Fq(=)p Fs(2)p Fo(g)-3779 b Fs(\(2.4\))-118 1927 y(The)33 b(v)m(ectors)i(\()p Fq(')516 1942 y Fp(k)r(;\015)618 1927 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\)\))852 1942 y Fp(k)r(;\015)986 1927 y Fs(form)32 b(an)g(orthonormal)e(family)g(in)i Fq(L)2388 1891 y Fi(2)2428 1927 y Fs(\()p Fq(C)2536 1942 y Fi(0)p Fp(;n)2638 1927 y Fs(\))g(as)190 2053 y Fn(Z)246 2279 y Fp(C)296 2288 y Fc(0)p Fk(;n)409 2189 y Fq(')473 2204 y Fp(k)r(;\015)575 2189 y Fs(\()p Fq(x;)17 b(\022)s Fs(\))p 798 2102 435 4 v Fq(')862 2204 y Fp(k)901 2185 y Fd(0)923 2204 y Fp(;\015)983 2185 y Fd(0)1009 2189 y Fs(\()p Fq(x;)g(\022)s Fs(\))q Fq(dx)27 b Fs(=)1653 2094 y Fn(X)1470 2314 y Fp(\014)s Fm(2)p Fl(Z)1610 2295 y Fk(d)1640 2314 y Fp(=)p Fi(\(2)p Fp(n)p Fi(+1\))p Fl(Z)1949 2295 y Fk(d)1996 2053 y Fn(Z)2051 2279 y Fp(\014)s Fi(+)p Fp(C)2199 2288 y Fc(0)p Fk(;)p Fc(0)2304 2189 y Fq(')2368 2204 y Fp(k)r(;\015)2471 2189 y Fs(\()p Fq(x;)17 b(\022)s Fs(\))p 2694 2102 412 4 v Fq(')2758 2204 y Fp(k)2797 2185 y Fd(0)2819 2204 y Fp(;\015)2883 2189 y Fs(\()p Fq(x;)g(\022)s Fs(\))p Fq(dk)201 2582 y Fs(=)487 2515 y(1)p 315 2559 393 4 v 315 2651 a(\(2)p Fq(n)22 b Fs(+)g(1\))667 2622 y Fp(d)734 2382 y Fn(0)734 2561 y(@)891 2488 y(X)821 2711 y Fp(\014)s Fm(2)p Fl(Z)961 2688 y Fk(d)961 2732 y Fc(2)p Fk(n)p Fc(+1)1122 2582 y Fq(e)1167 2541 y Fi(2)p Fp(\031)r(\014)s Fi(\()p Fp(\015)t Fm(\000)p Fp(\015)1450 2518 y Fd(0)1474 2541 y Fi(\))p Fp(=)p Fi(\(2)p Fp(n)p Fi(+1\))1764 2382 y Fn(1)1764 2561 y(A)1867 2447 y(Z)1923 2672 y Fp(C)1973 2681 y Fc(0)p Fk(;)p Fc(0)2078 2582 y Fq(')2142 2597 y Fp(k)2184 2582 y Fs(\()p Fq(x;)17 b(\022)26 b Fs(+)c Fq(\015)5 b(=)p Fs(\(2)p Fq(n)22 b Fs(+)g(1\)\))p 2985 2495 953 4 v Fq(')3049 2597 y Fp(k)3088 2578 y Fd(0)3113 2582 y Fs(\()p Fq(x;)17 b(\022)26 b Fs(+)c Fq(\015)3475 2553 y Fm(0)3498 2582 y Fq(=)p Fs(\(2)p Fq(n)g Fs(+)g(1\)\))o Fq(dx)201 2856 y Fs(=)28 b Fq(\016)348 2871 y Fp(k)r(k)426 2852 y Fd(0)452 2856 y Fq(\016)495 2871 y Fp(\015)t(\015)575 2852 y Fd(0)602 2856 y Fq(:)-118 2499 y Fs(\(2.5\))-118 3076 y(The)33 b(family)d(\()p Fq(')482 3091 y Fp(k)r(;\015)585 3076 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\)\))819 3091 y Fp(k)r(;\015)953 3076 y Fs(is)32 b(complete)g(as)h(the)g(family)d(\() p Fq(')2151 3091 y Fp(k)2194 3076 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\)\))2428 3091 y Fp(k)r Fm(\025)p Fi(0)2592 3076 y Fs(is.)43 b(Indeed,)34 b(for)e Fq(u)c Fo(2)g Fq(L)3465 3040 y Fi(2)3505 3076 y Fs(\()p Fj(R)3608 3040 y Fp(d)3655 3076 y Fs(\),)k(one)h(has)-118 3454 y Fq(u)27 b Fs(=)69 3360 y Fn(X)76 3572 y Fp(k)r Fm(\025)p Fi(0)229 3319 y Fn(Z)284 3544 y Fl(T)334 3525 y Fd(\003)392 3454 y Fs(^)-56 b Fq(u)441 3469 y Fp(n)488 3454 y Fs(\()p Fq(\022)s Fs(\))p Fq(')676 3469 y Fp(n)723 3454 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))p Fq(d\022)30 b Fs(=)1148 3360 y Fn(X)1156 3572 y Fp(k)r Fm(\025)p Fi(0)1378 3360 y Fn(X)1309 3583 y Fp(\015)t Fm(2)p Fl(Z)1446 3560 y Fk(d)1446 3604 y Fc(2)p Fk(n)p Fc(+1)1607 3319 y Fn(Z)1662 3544 y Fl(T)1712 3521 y Fd(\003)1712 3565 y Fc(2)p Fk(n)p Fc(+1)1883 3454 y Fs(^)-55 b Fq(u)1933 3469 y Fp(\015)t(;n)2040 3454 y Fs(\()p Fq(\022)s Fs(\))p Fq(')2228 3469 y Fp(\015)t(;n)2335 3454 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))p Fq(d\022)34 b Fs(where)40 b(^)-55 b Fq(u)2999 3469 y Fp(\015)t(;n)3106 3454 y Fs(\()p Fq(\022)s Fs(\))27 b(=)34 b(^)-55 b Fq(u)3417 3469 y Fp(n)3463 3454 y Fs(\()p Fq(\022)25 b Fs(+)e Fq(\015)5 b(=)p Fs(\(2)p Fq(n)21 b Fs(+)h(1\)\))p Fq(:)-118 3802 y Fs(So,)34 b(the)h(pairs)e(\()p Fq(E)566 3817 y Fp(k)r(;\015)669 3802 y Fs(\()p Fq(\022)s Fs(\))p Fq(;)17 b(')901 3817 y Fp(k)r(;\015)1003 3802 y Fs(\()p Fo(\001)p Fq(;)g(\022)s Fs(\)\))1237 3825 y Fp(k)r Fm(\025)p Fi(0)p Fp(;\015)t Fm(2)p Fl(Z)1522 3803 y Fk(d)1522 3847 y Fc(2)p Fk(n)p Fc(+1)1704 3802 y Fs(form)33 b(a)g(family)f(of)h(\(2)p Fq(n)23 b Fs(+)g(1\))p Fj(Z)2853 3766 y Fp(d)2891 3802 y Fs(-p)s(erio)s(dic)32 b(Flo)s(quet)h(eigen)m(v)-5 b(alues)-118 3931 y(and)33 b(eigen)m(v)m(ectors)h(of)e Fq(H)8 b Fs(.)94 4047 y(W)-8 b(e)40 b(also)d(mak)m(e)i(use)h(of)e(sp)s(ectral)g(pro)5 b(jectors.)63 b(W)-8 b(e)39 b(de\014ne)h(\005)2445 4062 y Fp()1032 1514 y(<)1032 1693 y(>)1032 1723 y(:)1121 1629 y Fq(!)t Fs(;)1308 1463 y Fo(9)p Fq(\022)j Fo(2)d Fj(R)1599 1422 y Fp(d)1645 1463 y Fq(;)1278 1632 y Fo(9)p Fq(E)34 b Fo(2)28 b Fs([)p Fq(E)1632 1647 y Fm(\000)1691 1632 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)p 1868 1552 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 2041 1577 65 4 v(!)s Fs(\))22 b Fo(\000)h Fq(\025)2322 1591 y Fi(1+)p Fp(\021)2449 1567 y Fd(0)2476 1632 y Fs(])p Fq(;)1292 1792 y Fo(9)p Fq(')28 b Fo(2)g Fq(H)1622 1751 y Fi(2)1614 1817 y Fp(n;\022)1715 1792 y Fq(;)49 b Fo(k)p Fq(')p Fo(k)28 b Fs(=)f(1)p Fq(;)2579 1629 y Fs(s.t.)44 b Fq(H)2842 1587 y Fp(n)2834 1653 y(!)r(;\025)2945 1629 y Fs(\()p Fq(\022)s Fs(\))p Fq(')28 b Fs(=)g Fq(E)6 b(')3407 1394 y Fn(9)3407 1484 y(>)3407 1514 y(=)3407 1693 y(>)3407 1723 y(;)3512 1629 y Fq(:)-118 1962 y Fs(Flo)s(quet)32 b(theory)h(giv)m(es)g(us)g(another)g(c)m(haracterization)e(of)h(\012\() p Fq(n;)17 b(\025;)g(\021)2458 1925 y Fm(0)2481 1962 y Fs(\),)33 b(namely)-8 b(,)777 2157 y(\012\()p Fq(n;)17 b(\025;)g(\021)1140 2116 y Fm(0)1163 2157 y Fs(\))28 b(=)1332 2046 y Fn(n)1399 2157 y Fq(!)t Fs(;)48 b Fq(\033)t Fs(\()p Fq(H)1725 2116 y Fp(n)1717 2181 y(!)r(;\025)1828 2157 y Fs(\))22 b Fo(\\)h Fs([)p Fq(E)2076 2172 y Fm(\000)2135 2157 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)p 2312 2077 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 2485 2102 65 4 v(!)s Fs(\))22 b Fo(\000)h Fq(\025)2766 2116 y Fi(1+)p Fp(\021)2893 2092 y Fd(0)2920 2157 y Fs(])28 b Fo(6)p Fs(=)f Fo(;)3128 2046 y Fn(o)3211 2157 y Fq(:)-3356 b Fs(\(2.7\))-118 2352 y(By)34 b(\(2.6\))e(and)h(Theorem)g(2.1,)f(Theorem)h(1.1)f(is)g (an)h(immediate)d(consequence)35 b(of)-118 2530 y Fw(Prop)s(osition)g (2.1.)50 b Ff(Fix)38 b Fq(\021)j Fo(2)36 b Fs(\(0)p Fq(;)17 b(d=)p Fs(\(4)p Fq(d)24 b Fs(+)i(4\)\))39 b Ff(and)g Fq(\032)e(>)f(d)p Ff(.)58 b(Then,)40 b(ther)-5 b(e)40 b(exists)f Fq(\025)3136 2545 y Fp(\021)r(;\032)3270 2530 y Fq(>)d Fs(0)j Ff(and)g Fq(")d(>)h Fs(0)i Ff(such)-118 2646 y(that,)c(for)g Fq(\025)27 b Fo(2)h Fs(\(0)p Fq(;)17 b(\025)633 2661 y Fp(\021)r(;\032)730 2646 y Fs(\))p Ff(,)35 b(one)f(has)1557 2826 y Fj(P)p Fs([\012\()p Fq(n;)17 b(\025;)g(\021)t Fs(\)])29 b Fo(\024)f Fq(e)2250 2785 y Fm(\000)p Fp(\025)2346 2762 y Fd(\000)p Fk(")2432 2826 y Fq(:)-2577 b Fs(\(2.8\))-118 2990 y Ff(wher)-5 b(e)701 3154 y Fs(2)p Fq(n)23 b Fs(+)f(1)27 b(=)h([)p Fq(\025)1193 3113 y Fm(\000)p Fi(1)p Fp(=)p Fi(2+2)p Fp(\021)1480 3089 y Fd(0)1508 3154 y Fs(])1535 3169 y Fp(o)1595 3154 y Fo(\001)22 b Fs([)p Fq(\025)1729 3113 y Fm(\000)p Fp(\021)1821 3089 y Fd(0)1848 3154 y Fs(])1875 3169 y Fp(o)1936 3154 y Fo(\001)g Fs([)p Fq(\025)2070 3113 y Fm(\000)p Fp(\032)2165 3154 y Fs(])2192 3169 y Fp(o)2265 3154 y Ff(and)34 b Fq(\021)e(<)27 b(\021)2689 3113 y Fm(0)2740 3154 y Fq(<)g(d=)p Fs(\(4)p Fq(d)21 b Fs(+)h(4\))p Fq(:)-118 3323 y Ff(Her)-5 b(e,)35 b Fs([)p 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Fq(L)1056 3930 y Fi(2)1096 3966 y Fs(\()p Fj(R)1200 3930 y Fp(d)1247 3966 y Fs(\))32 b(in)m(to)g Fq(L)1581 3930 y Fi(2)1648 3966 y Fs(=)c(\005)1825 3981 y Fm(\024)p Fp(E)1932 3990 y Fc(0)1970 3966 y Fq(L)2036 3930 y Fi(2)2098 3966 y Fs(+)22 b(\005)2269 3981 y Fp(>E)2376 3990 y Fc(0)2415 3966 y Fq(L)2481 3930 y Fi(2)2553 3966 y Fs(and)33 b(write)705 4193 y Fq(H)794 4152 y Fp(n)786 4218 y(!)r(;\025)919 4193 y Fo(\000)23 b Fq(E)34 b Fs(=)1228 4053 y Fn(\022)1301 4129 y Fs(\005)1374 4144 y Fm(\024)p Fp(E)1481 4153 y Fc(0)1520 4129 y Fs(\()p Fq(H)1647 4093 y Fp(n)1639 4155 y(!)r(;\025)1772 4129 y Fo(\000)23 b Fq(E)6 b Fs(\)\005)2061 4144 y Fm(\024)p Fp(E)2168 4153 y Fc(0)2432 4129 y Fq(\025)p Fs(\005)2562 4144 y Fm(\024)p Fp(E)2669 4153 y Fc(0)2707 4129 y Fq(V)2786 4093 y Fp(n)2764 4154 y(!)2833 4129 y Fs(\005)2906 4144 y Fp(>E)3013 4153 y Fc(0)1444 4249 y Fq(\025)p Fs(\005)1574 4264 y Fp(>E)1681 4273 y Fc(0)1719 4249 y Fq(V)1798 4213 y Fp(n)1776 4274 y(!)1845 4249 y Fs(\005)1918 4264 y Fm(\024)p Fp(E)2025 4273 y Fc(0)2289 4249 y Fs(\005)2362 4264 y Fp(>E)2469 4273 y Fc(0)2508 4249 y Fs(\()p Fq(H)2635 4213 y Fp(n)2627 4275 y(!)r(;\025)2760 4249 y Fo(\000)22 b Fq(E)6 b Fs(\)\005)3048 4264 y Fp(>E)3155 4273 y Fc(0)3194 4053 y Fn(\023)3284 4193 y Fq(:)-3429 b Fs(\(2.9\))-118 4419 y(F)-8 b(or)43 b Fq(E)53 b Fo(2)47 b Fq(I)349 4434 y Fp(\021)r(;\025)452 4419 y Fs(,)f(the)f(op)s(erator)e(\005)1182 4434 y Fm(\024)p Fp(E)1289 4443 y Fc(0)1327 4419 y Fs(\()p Fq(H)1454 4383 y Fp(n)1446 4445 y(!)r(;\025)1587 4419 y Fo(\000)30 b Fq(E)6 b Fs(\)\005)1883 4434 y Fm(\024)p Fp(E)1990 4443 y Fc(0)2072 4419 y Fs(is)44 b(in)m(v)m(ertible)f(and)h(its)f(in)m(v)m (erse)i(b)s(ounded)g(b)m(y)f(a)g(con-)-118 4535 y(stan)m(t)c(indep)s (enden)m(t)h(of)e Fq(\025)g Fs(and)h Fq(!)t Fs(.)64 b(This)39 b(follo)m(ws)g(from)f(the)i(fact)f(that)h Fq(V)2707 4550 y Fp(!)2797 4535 y Fs(is)f(uniformly)e(b)s(ounded)j(and)g(that)-118 4651 y(\005)-45 4666 y Fm(\024)p Fp(E)62 4675 y Fc(0)100 4651 y Fq(H)189 4615 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Fs(\)\005)2678 4978 y Fm(\024)p Fp(E)2785 4987 y Fc(0)2823 4882 y Fn(\003)2865 4905 y Fm(\000)p Fi(1)2975 4963 y Fs(\005)3048 4978 y Fm(\024)p Fp(E)3155 4987 y Fc(0)3194 4963 y Fq(V)3272 4922 y Fp(n)3251 4988 y(!)3319 4963 y Fs(\005)3392 4978 y Fp(>E)3499 4987 y Fc(0)3538 4963 y Fq(:)-3683 b Fs(\(2.10\))-118 5137 y(As)39 b(the)g(second)g(term)f(in)g(\(2.10\))f(is)h(b)s(ounded)h(b)m (y)h Fq(C)7 b(\025)1956 5101 y Fi(2)1995 5137 y Fs(,)40 b(w)m(e)f(see)h(that)e Fq(E)44 b Fo(2)38 b Fq(I)2855 5152 y Fp(\021)r(;\025)2996 5137 y Fs(is)g(in)f(the)i(sp)s(ectrum)g(of) f Fq(H)4031 5101 y Fp(n)4023 5163 y(!)r(;\025)-118 5272 y Fs(implies)27 b(that)i(the)h(op)s(erator)e(\005)1045 5287 y Fp(>E)1152 5296 y Fc(0)1191 5272 y Fq(H)1280 5236 y Fp(n)1272 5298 y(!)r(;\025)1383 5272 y Fs(\005)1456 5287 y Fp(>E)1563 5296 y Fc(0)1630 5272 y Fs(has)i(sp)s(ectrum)g(b)s (elo)m(w)f(the)h(energy)p 2970 5192 79 4 v 30 w Fq(E)6 b Fs(\()p Fq(\025)p 3143 5217 65 4 v(!)s Fs(\))15 b Fo(\000)g Fq(\025)3409 5236 y Fi(1+)p Fp(\021)3558 5272 y Fs(+)g Fq(C)7 b(\025)3783 5236 y Fi(2)3823 5272 y Fs(.)42 b(Th)m(us,)-118 5388 y(w)m(e)34 b(ha)m(v)m(e)f(pro)m(v)m(ed)360 5552 y(\012\()p Fq(n;)17 b(\025;)g(\021)t Fs(\))28 b Fo(\032)894 5471 y Fn(\010)952 5552 y Fq(!)t Fs(;)48 b Fo(9)p Fq(')28 b Fo(2)g Fs(\005)1406 5567 y Fp(>E)1513 5576 y Fc(0)1552 5552 y Fq(H)1641 5511 y Fi(2)1680 5552 y Fq(;)49 b Fo(k)p Fq(')p Fo(k)27 b Fs(=)h(1)p Fq(;)49 b Fs(s.t.)44 b Fo(h)p Fq(H)2478 5511 y Fp(n)2470 5576 y(!)r(;\025)2580 5552 y Fq(';)17 b(')p Fo(i)27 b(\024)p 2924 5472 79 4 v 29 w Fq(E)6 b Fs(\()p Fq(\025)p 3097 5497 65 4 v(!)s Fs(\))22 b Fo(\000)g Fq(\025)3377 5511 y Fi(1+)p Fp(\021)3532 5552 y Fs(+)g Fq(C)7 b(\025)3764 5511 y Fi(2)3803 5471 y Fn(\011)3878 5552 y Fq(:)-4023 b Fs(\(2.11\))1989 5690 y Fg(8)p eop %%Page: 9 9 9 8 bop -118 241 a Fs(Let)32 b(us)i(estimate)d(the)i(probabilit)m(y)d (of)i(this)g(last)g(ev)m(en)m(t.)45 b(As)33 b Fq(V)2242 205 y Fp(n)2220 265 y(!)2321 241 y Fs(is)f(uniformly)e(b)s(ounded,)k(p) s(erturbation)d(theory)-118 357 y(tells)g(us)j(that,)e(for)g(some)g Fq(C)j(>)28 b Fs(0)k(and)h Fq(\025)f Fs(su\016cien)m(tly)h(small,)1071 537 y Fo(j)p Fq(E)1171 552 y Fm(\000)1230 537 y Fs(\()p Fq(\025)p Fs(\))21 b Fo(\000)i Fq(E)1556 552 y Fm(\000)1615 537 y Fs(\(0\))p Fo(j)f Fs(+)g Fo(j)p 1916 457 79 4 v Fq(E)6 b Fs(\()p Fq(\025)p 2089 483 65 4 v(!)s Fs(\))22 b Fo(\000)g Fq(E)2384 552 y Fm(\000)2444 537 y Fs(\(0\))p Fo(j)27 b(\024)h Fq(C)7 b Fo(j)p Fq(\025)p Fo(j)p Fq(:)-118 724 y Fs(Hence,)34 b(for)e Fq(')g Fs(as)h(in)f(\(2.11\))o(,)h(one)g (has)943 904 y Fo(h)p Fq(H)c Fo(\000)23 b Fq(E)1264 919 y Fm(\000)1323 904 y Fs(\(0\))p Fq(';)17 b(')p Fo(i)27 b(\024)h(j)p Fq(E)g Fo(\000)22 b Fq(E)2090 919 y Fm(\000)2150 904 y Fs(\(0\))p Fo(j)f Fs(+)h Fq(\025)p Fo(k)p Fq(V)2607 863 y Fp(n)2586 929 y(!)2654 904 y Fo(k)2704 919 y Fm(1)2807 904 y Fo(\024)28 b Fq(C)7 b(\025:)-3191 b Fs(\(2.12\))-118 1090 y(Rewrite)32 b(the)h(condition)e(in)h(\(2.11\))g(as)549 1278 y Fo(h)p Fs(\()p 626 1198 89 4 v Fq(H)714 1216 y Fp(n)714 1303 y(\025)p 755 1265 47 3 v(!)828 1278 y Fo(\000)p 928 1198 79 4 v 23 w Fq(E)6 b Fs(\()p Fq(\025)p 1101 1223 65 4 v(!)s Fs(\)\))p Fq(';)17 b(')p Fo(i)27 b(\024)h Fs(\()p Fo(\000)p Fq(\025)1756 1237 y Fi(1+)p Fp(\021)1910 1278 y Fs(+)22 b Fq(C)7 b(\025)2142 1237 y Fi(2)2182 1278 y Fs(\))p Fo(k)p Fq(')p Fo(k)2384 1237 y Fi(2)2445 1278 y Fs(+)22 b Fq(\025)p Fo(h)p Fs(\()p Fj(E)12 b Fs(\()p Fq(V)2859 1237 y Fp(n)2832 1303 y(!)2906 1278 y Fs(\))22 b Fo(\000)h Fq(V)3144 1237 y Fp(n)3123 1303 y(!)3191 1278 y Fs(\))p Fq(';)17 b(')p Fo(i)p Fq(:)-3585 b Fs(\(2.13\))-118 1480 y(Note)37 b(that)g Fj(E)12 b Fs(\()p Fq(V)493 1495 y Fp(!)549 1480 y Fs(\))35 b(=)g Fj(E)12 b Fs(\()p Fq(V)916 1444 y Fp(n)888 1504 y(!)963 1480 y Fs(\))35 b(=)p 1147 1425 V 35 w Fq(!)p 1211 1400 79 4 v 3 w(V)21 b Fs(.)57 b(By)37 b(the)g(de\014nition)f(of)p 2256 1400 V 37 w Fq(E)6 b Fs(\()p Fq(\025)p 2429 1425 65 4 v(!)s Fs(\),)38 b(for)e Fq(\025)h Fs(su\016cien)m(tly)g(small,)p 3623 1400 89 4 v 36 w Fq(H)3712 1418 y Fp(n)3712 1504 y(\025)p 3753 1467 47 3 v(!)3803 1480 y Fs(,)h(hence,)-118 1603 y(\005)-45 1618 y Fp(>E)62 1627 y Fc(0)p 100 1523 89 4 v 100 1603 a Fq(H)189 1541 y Fp(n)189 1628 y(\025)p 230 1590 47 3 v(!)280 1603 y Fs(\005)353 1618 y Fp(>E)460 1627 y Fc(0)532 1603 y Fs(has)c(no)g(sp)s(ectrum)g(in)e(the)i(in)m (terv)-5 b(al)33 b([)p Fq(E)2007 1618 y Fi(0)2046 1603 y Fq(;)p 2090 1523 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 2263 1548 65 4 v(!)s Fs(\)\);)34 b(hence,)i(equation)d(\(2.13\))g (implies)e(that,)i(for)-118 1719 y(some)f Fq(')c Fo(2)g Fs(\005)385 1734 y Fp(>E)492 1743 y Fc(0)530 1719 y Fq(H)619 1683 y Fi(2)658 1719 y Fs(,)1308 1900 y Fo(h)p Fs(\()p Fj(E)12 b Fs(\()p Fq(V)1568 1859 y Fp(n)1540 1925 y(!)1615 1900 y Fs(\))22 b Fo(\000)g Fq(V)1853 1859 y Fp(n)1831 1925 y(!)1900 1900 y Fs(\))p Fq(';)17 b(')p Fo(i)27 b(\025)h Fq(\025)2338 1859 y Fp(\021)2380 1900 y Fq(=)p Fs(2)p Fo(k)p Fq(')p Fo(k)2642 1859 y Fi(2)2680 1900 y Fq(:)-2825 b Fs(\(2.14\))-118 2090 y(Indeed,)34 b(if)d(this)i(is)f(not)g(the)h (case,)h(then)f(\(2.13\))f(implies)e(that,)i(for)g(some)h Fq(')27 b Fo(2)h Fs(\005)2904 2105 y Fp(>E)3011 2114 y Fc(0)3050 2090 y Fq(H)3139 2054 y Fi(2)3178 2090 y Fs(,)1009 2278 y Fo(h)p Fs(\()p 1086 2198 89 4 v Fq(H)1174 2216 y Fp(n)1174 2303 y(\025)p 1215 2265 47 3 v(!)1288 2278 y Fo(\000)p 1387 2198 79 4 v 22 w Fq(E)6 b Fs(\()p Fq(\025)p 1560 2223 65 4 v(!)t Fs(\)\))p Fq(';)17 b(')p Fo(i)26 b(\024)j Fs(\()p Fo(\000)p Fq(\025)2216 2237 y Fi(1+)p Fp(\021)2348 2278 y Fq(=)p Fs(2)22 b(+)g Fq(C)7 b(\025)2700 2237 y Fi(2)2739 2278 y Fs(\))p Fo(k)p Fq(')p Fo(k)2941 2237 y Fi(2)2980 2278 y Fq(:)-118 2480 y Fs(whic)m(h)32 b(means)g(that,)f(for)g Fq(\025)h Fs(su\016cien)m(tly)g(small,)e(\005) 1774 2495 y Fp(>E)1881 2504 y Fc(0)p 1919 2400 89 4 v 1919 2480 a Fq(H)2008 2418 y Fp(n)2008 2505 y(\025)p 2049 2467 47 3 v(!)2099 2480 y Fs(\005)2172 2495 y Fp(>E)2279 2504 y Fc(0)2349 2480 y Fs(has)i(sp)s(ectrum)g(in)f(the)h(in)m(terv)-5 b(al)31 b([)p Fq(E)3678 2495 y Fi(0)3717 2480 y Fq(;)p 3761 2400 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 3934 2425 65 4 v(!)s Fs(\))20 b Fo(\000)-118 2596 y Fq(\025)-61 2560 y Fi(1+)p Fp(\021)71 2596 y Fq(=)p Fs(4\).)43 b(So,)32 b(taking)i(\(2.11\))d(in)m(to)h(accoun)m(t,)i(w)m(e)f(pro)m(v)m(ed)h (that)1529 2777 y(\012\()p Fq(n;)17 b(\025;)g(\021)t Fs(\))27 b Fo(\032)i Fs(\012)2133 2735 y Fm(0)2156 2777 y Fs(\()p Fq(n;)17 b(\025;)g(\021)t Fs(\))-118 2957 y(where)389 3138 y(\012)459 3096 y Fm(0)483 3138 y Fs(\()p Fq(n;)g(\025;)g(\021)t Fs(\))26 b(=)944 3057 y Fn(\010)1002 3138 y Fq(!)t Fs(;)49 b Fo(9)p Fq(')28 b Fo(2)g Fs(\005)1457 3153 y Fp(>E)1564 3162 y Fc(0)1602 3138 y Fq(H)1691 3096 y Fi(2)1730 3138 y Fq(;)49 b Fo(k)p Fq(')p Fo(k)27 b Fs(=)h(1)p Fq(;)49 b Fs(s.t.)44 b(equations)33 b(\(2.12\))f(and)g(\(2.14\))g(hold)3774 3057 y Fn(\011)3849 3138 y Fq(:)-118 3328 y Fs(Let)g(us)g(no)m(w)h (estimate)e(the)h(probabilit)m(y)e(of)h(\012)1618 3292 y Fm(0)1642 3328 y Fs(\()p Fq(n;)17 b(\025;)g(\021)t Fs(\).)42 b(W)-8 b(rite)32 b Fq(')27 b Fo(2)h Fs(\005)2571 3343 y Fp(>E)2678 3352 y Fc(0)2717 3328 y Fq(H)2806 3292 y Fi(2)2876 3328 y Fs(using)k(the)g(Flo)s(quet)f(decomp)s(osi-)-118 3444 y(tion)g(in)m(tro)s(duced)i(in)f(section)h(2.2)1419 3652 y Fq(')27 b Fs(=)1622 3558 y Fn(X)1614 3768 y Fp(p)p Fm(\025)p Fp(p)1741 3777 y Fc(0)1791 3517 y Fn(Z)1846 3742 y Fl(T)1896 3723 y Fd(\003)1947 3652 y Fq(\037)2008 3667 y Fp(p)2048 3652 y Fs(\()p Fq(\022)s Fs(\))p Fq(')2236 3667 y Fp(p)2276 3652 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))p Fq(d\022)s(:)-2716 b Fs(\(2.15\))-118 3938 y(The)33 b(index)g Fq(p)394 3953 y Fi(0)466 3938 y Fs(is)f(the)h(largest)f (index)h Fq(p)f Fs(so)h(that,)g(for)f(all)e Fq(\022)h Fo(2)d Fj(T)2267 3902 y Fm(\003)2310 3938 y Fs(,)33 b(one)f(has)h Fq(E)2794 3953 y Fp(p)p Fm(\000)p Fi(1)2924 3938 y Fs(\()p Fq(\022)s Fs(\))28 b Fo(\024)g Fq(E)3253 3953 y Fi(0)3293 3938 y Fs(.)-118 4054 y(By)34 b(assumption)e(\(H.2\),)i Fq(E)910 4069 y Fm(\000)969 4054 y Fs(\(0\))f(is)f(a)h(simple)f (degenerate)i(Flo)s(quet)f(eigen)m(v)-5 b(alue)33 b(of)f Fq(H)41 b Fs(i.e.,)33 b(hence,)i(there)f(exists)-118 4171 y Fq(C)h(>)27 b Fs(0)32 b(suc)m(h)i(that)8 4319 y Fo(\017)41 b Fs(for)32 b Fq(p)c Fo(6)p Fs(=)f Fq(p)477 4334 y Fi(0)517 4319 y Fs(,)32 b Fo(8)p Fq(\022)f Fo(2)d Fj(T)864 4283 y Fm(\003)907 4319 y Fs(,)1560 4499 y Fo(j)p Fq(E)1660 4514 y Fp(p)1722 4499 y Fo(\000)22 b Fq(E)1893 4514 y Fm(\000)1952 4499 y Fs(\(0\))p Fo(j)27 b(\025)i Fs(1)p Fq(=C)7 b Fs(;)-2558 b(\(2.16\))8 4693 y Fo(\017)41 b Fs(there)33 b(exists)h Fq(Z)g Fs(=)28 b Fo(f)p Fq(\022)918 4708 y Fp(j)954 4693 y Fs(;)50 b(1)27 b Fo(\024)h Fq(j)34 b Fo(\024)28 b Fq(n)1449 4708 y Fp(z)1489 4693 y Fo(g)k Fs(suc)m(h)i(that)f Fq(E)2075 4708 y Fp(p)2111 4717 y Fc(0)2149 4693 y Fs(\()p Fq(\022)2232 4708 y Fp(j)2269 4693 y Fs(\))28 b(=)f Fq(E)2510 4708 y Fm(\000)2570 4693 y Fs(\(0\))32 b(and)g(for)h Fq(\022)d Fo(2)e Fj(R)3301 4657 y Fp(d)3348 4693 y Fs(,)k(one)h(has)1233 4877 y Fo(j)p Fq(E)1333 4892 y Fp(p)1369 4901 y Fc(0)1429 4877 y Fo(\000)23 b Fq(E)1601 4892 y Fm(\000)1660 4877 y Fs(\(0\))p Fo(j)k(\025)h Fs(1)p Fq(=C)92 b Fs(inf)2136 4938 y Fi(1)p Fm(\024)p Fp(j)t Fm(\024)p Fp(n)2357 4946 y Fk(z)2409 4877 y Fo(j)p Fq(\022)25 b Fo(\000)e Fq(\022)2652 4892 y Fp(j)2689 4877 y Fo(j)2717 4836 y Fi(2)2756 4877 y Fq(:)-2901 b Fs(\(2.17\))-118 5103 y(W)-8 b(e)33 b(refer)g(to)f (section)h(5.1)f(for)g(more)g(details)f(on)i(these)h(prop)s(erties.) -118 5219 y(Equation)e(\(2.12\))g(implies)e(that)1119 5357 y Fn(X)1111 5568 y Fp(p)p Fm(\025)p Fp(p)1238 5577 y Fc(0)1288 5317 y Fn(Z)1344 5542 y Fl(T)1394 5523 y Fd(\003)1445 5452 y Fo(j)p Fq(E)1545 5467 y Fp(p)1584 5452 y Fs(\()p Fq(\022)s Fs(\))22 b Fo(\000)h Fq(E)1902 5467 y Fm(\000)1961 5452 y Fs(\(0\))p Fo(j)2114 5411 y Fi(2)2153 5452 y Fo(j)p Fq(\037)2242 5467 y Fp(p)2281 5452 y Fs(\()p Fq(\022)s Fs(\))p Fo(j)2433 5411 y Fi(2)2473 5452 y Fq(d\022)30 b Fo(\024)e Fq(C)7 b(\025)2838 5411 y Fi(2)2878 5452 y Fq(:)-3023 b Fs(\(2.18\))1989 5690 y Fg(9)p eop %%Page: 10 10 10 9 bop -118 245 a Fs(Fix)43 b(2)p Fq(l)31 b Fs(+)f(1)46 b(=)g([)p Fq(\025)584 209 y Fm(\000)p Fi(1)p Fp(=)p Fi(2+2)p Fp(\021)871 185 y Fd(0)899 245 y Fs(])926 260 y Fp(o)994 245 y Fo(\001)29 b Fs([)p Fq(\025)1135 209 y Fm(\000)p Fp(\021)1227 185 y Fd(0)1255 245 y Fs(])1282 260 y Fp(o)1364 245 y Fs(and)43 b(2)p Fq(k)33 b Fs(+)c(1)46 b(=)g([)p Fq(\025)2103 209 y Fm(\000)p Fp(\032)2199 245 y Fs(])2226 260 y Fp(o)2307 245 y Fs(where)f Fq(\021)50 b(<)c(\021)2872 209 y Fm(0)2942 245 y Fq(<)g(d=)p Fs(\(4)p Fq(d)28 b Fs(+)i(4\))43 b(is)g(\014xed)i(as)f(in)-118 361 y(Prop)s(osition)29 b(2.1.)42 b(Note)31 b(that)g(2)p Fq(n)18 b Fs(+)h(1)27 b(=)h(\(2)p Fq(l)20 b Fs(+)e(1\)\(2)p Fq(k)j Fs(+)e(1\).)42 b(Equations)31 b(\(2.18\))o(,)g(\(2.16\))f(and)h(\(2.17\))f(imply)f (that)623 476 y Fn(X)615 686 y Fp(p>p)742 695 y Fc(0)792 435 y Fn(Z)847 661 y Fl(T)897 642 y Fd(\003)948 571 y Fo(j)p Fq(\037)1037 586 y Fp(p)1077 571 y Fs(\()p Fq(\022)s Fs(\))p Fo(j)1229 530 y Fi(2)1268 571 y Fq(d\022)c Fs(+)1543 476 y Fn(X)1487 687 y Fi(1)p Fm(\024)p Fp(j)t Fm(\024)p Fp(n)1708 695 y Fk(z)1760 435 y Fn(Z)1815 661 y Fm(j)p Fp(\022)r Fm(\000)p Fp(\022)1958 671 y Fk(j)1990 661 y Fm(j)p Fp(>)p Fi(1)p Fp(=l)2178 571 y Fo(j)p Fq(\037)2267 586 y Fp(p)2303 595 y Fc(0)2341 571 y Fs(\()p Fq(\022)s Fs(\))p Fo(j)2493 530 y Fi(2)2532 571 y Fq(d\022)30 b Fo(\024)e Fq(C)7 b(\025)2897 530 y Fi(2)2937 571 y Fq(l)2968 530 y Fi(2)3035 571 y Fo(\024)28 b Fq(C)7 b(\025)3274 530 y Fi(2)p Fp(\021)3346 507 y Fd(0)3374 571 y Fq(:)-118 827 y Fs(Hence,)34 b(w)m(e)g(write)395 1032 y Fq(')28 b Fs(=)646 937 y Fn(X)590 1147 y Fi(1)p Fm(\024)p Fp(j)t Fm(\024)p Fp(n)811 1155 y Fk(z)863 1032 y Fq(')927 1047 y Fp(j)985 1032 y Fs(+)22 b Fq(')1147 1047 y Fp(e)1217 1032 y Fs(where)34 b Fq(')1563 1047 y Fp(j)1627 1032 y Fs(=)1730 896 y Fn(Z)1786 1121 y Fm(j)p Fp(\022)r Fm(\000)p Fp(\022)1929 1131 y Fk(j)1960 1121 y Fm(j\024)p Fi(1)p Fp(=l)2148 1032 y Fq(\037)2209 1047 y Fp(p)2245 1056 y Fc(0)2283 1032 y Fs(\()p Fq(\022)s Fs(\))p Fq(')2471 1047 y Fp(p)2507 1056 y Fc(0)2546 1032 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))p Fq(d\022)34 b Fs(and)f Fo(k)p Fq(')3176 1047 y Fp(e)3213 1032 y Fo(k)27 b(\024)h Fq(C)7 b(\025)3529 990 y Fp(\021)3566 967 y Fd(0)3594 1032 y Fq(:)-3739 b Fs(\(2.19\))-118 1289 y(W)-8 b(e)33 b(note)g(that)1182 1351 y Fn(X)1126 1562 y Fi(1)p Fm(\024)p Fp(j)t Fm(\024)p Fp(n)1347 1570 y Fk(z)1399 1446 y Fo(k)p Fq(')1513 1461 y Fp(j)1549 1446 y Fo(k)1599 1405 y Fi(2)1666 1446 y Fs(=)27 b Fo(k)p Fq(')p Fo(k)1933 1405 y Fi(2)1994 1446 y Fo(\000)c Fq(C)7 b(\025)2228 1405 y Fi(2)p Fp(\021)2300 1381 y Fd(0)2355 1446 y Fs(=)28 b(1)22 b Fo(\000)g Fq(C)7 b(\025)2763 1405 y Fi(2)p Fp(\021)2835 1381 y Fd(0)2863 1446 y Fq(:)-3008 b Fs(\(2.20\))-118 1709 y(Pluging)32 b(\(2.19\))g(in)m(to)g(\(2.14\))o(,)h(for)f Fq(\025)g Fs(su\016cien)m(tly)i(small,)c(w)m(e)k(get)1290 1793 y Fn(X)1199 2006 y Fi(1)p Fm(\024)p Fp(j;j)1371 1987 y Fd(0)1392 2006 y Fm(\024)p Fp(n)1490 2014 y Fk(z)1525 1888 y Fo(h)p Fs(\()p Fj(E)12 b Fs(\()p Fq(V)1785 1846 y Fp(n)1757 1912 y(!)1832 1888 y Fs(\))22 b Fo(\000)h Fq(V)2070 1846 y Fp(n)2049 1912 y(!)2117 1888 y Fs(\))p Fq(')2219 1903 y Fp(j)2255 1888 y Fq(;)17 b(')2363 1903 y Fp(j)2396 1884 y Fd(0)2422 1888 y Fo(i)27 b(\025)h Fq(\025)2650 1846 y Fp(\021)2692 1888 y Fq(=)p Fs(4)p Fq(:)-118 2153 y Fs(By)34 b(\(2.20\),)e(this)g(implies)f(that,)h(for)g (some)g(1)c Fo(\024)g Fq(j;)17 b(j)1810 2116 y Fm(0)1861 2153 y Fo(\024)28 b Fq(n)2024 2168 y Fp(z)2064 2153 y Fs(,)33 b(one)g(has)1255 2311 y Fo(h)p Fs(\()p Fj(E)12 b Fs(\()p Fq(V)1515 2270 y Fp(n)1487 2335 y(!)1562 2311 y Fs(\))22 b Fo(\000)h Fq(V)1800 2270 y Fp(n)1779 2335 y(!)1847 2311 y Fs(\))p Fq(')1949 2326 y Fp(j)1985 2311 y Fq(;)17 b(')2093 2326 y Fp(j)2126 2307 y Fd(0)2152 2311 y Fo(i)27 b(\025)i Fq(\025)2381 2270 y Fp(\021)2422 2311 y Fq(=)p Fs(\(2)p Fq(n)2616 2326 y Fp(z)2656 2311 y Fs(\))2694 2270 y Fi(2)2733 2311 y Fq(:)-2878 b Fs(\(2.21\))-118 2474 y(Let)33 b(us)g(study)h(the)f(functions)f(\()p Fq(')1141 2489 y Fp(j)1178 2474 y Fs(\))1216 2489 y Fi(1)p Fm(\024)p Fp(j)t Fm(\024)p Fp(n)1437 2497 y Fk(z)1508 2474 y Fs(in)g(more)g (detail.)42 b(W)-8 b(e)33 b(pro)m(v)m(e)-118 2651 y Fw(Lemma)k(2.1.)49 b Ff(Fix)34 b Fs(1)28 b Fo(\024)g Fq(j)34 b Fo(\024)28 b Fq(n)1094 2666 y Fp(z)1134 2651 y Ff(.)44 b(F)-7 b(or)34 b Fs(1)28 b Fo(\024)g Fq(l)1603 2614 y Fm(0)1654 2651 y Fo(\024)g Fq(l)r Ff(,)35 b(ther)-5 b(e)35 b(exists)50 b Fs(~)-64 b Fq(')2429 2666 y Fp(j)2493 2651 y Fo(2)28 b Fq(L)2653 2614 y Fi(2)2693 2651 y Fs(\()p Fj(R)2797 2614 y Fp(d)2843 2651 y Fs(\))35 b Ff(such)g(that)-18 2787 y Fs(1.)57 b(~)-65 b Fq(')163 2802 y Fp(j)234 2787 y Ff(is)35 b(c)-5 b(onstant)34 b(on)h(e)-5 b(ach)34 b(cub)-5 b(e)478 2945 y Fq(C)548 2960 y Fp(\015)t(;l)630 2941 y Fd(0)684 2945 y Fs(=)27 b Fo(f)p Fq(x)h Fs(=)g(\()p Fq(x)1117 2960 y Fi(1)1157 2945 y Fq(;)17 b(:)g(:)g(:)e(;)i(x)1430 2960 y Fp(d)1471 2945 y Fs(\);)51 b Fo(8)p Fs(1)28 b Fo(\024)g Fq(i)g 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Fs(])308 3865 y Fp(o)384 3850 y Fs(and)f(2)p Fq(k)682 3814 y Fm(0)731 3850 y Fs(+)26 b(1)37 b(=)f([)p Fq(\025)1115 3814 y Fm(\000)p Fp(\032)1210 3850 y Fs(])1237 3865 y Fp(o)1302 3850 y Fo(\001)25 b Fs([)p Fq(\025)1439 3814 y Fm(\000)p Fp(\021)1531 3791 y Fd(0)1558 3850 y Fs(])1585 3865 y Fp(o)1661 3850 y Fs(where)40 b Fq(\021)g(<)d(\021)2202 3814 y Fm(0)2262 3850 y Fq(<)f(d=)p Fs(\(4)p Fq(d)25 b Fs(+)g(4\))38 b(is)g(\014xed)h(as)f(in)f(Prop)s(osition)f(2.1.)-118 3966 y(Note)d(that)f(2)p Fq(n)22 b Fs(+)g(1)28 b(=)f(\(2)p Fq(l)854 3930 y Fm(0)899 3966 y Fs(+)c(1\)\(2)p Fq(k)1226 3930 y Fm(0)1270 3966 y Fs(+)f(1\).)44 b(Using)32 b(p)s(oin)m(t)g (\(1\))g(of)g(Lemma)f(2.1,)h(w)m(e)i(de\014ne)504 4145 y(\011)580 4160 y Fp(j)616 4145 y Fs(\()p Fq(x)p Fs(\))28 b(=)g Fq( )942 4160 y Fp(p)978 4169 y Fc(0)1016 4145 y Fs(\()p Fq(x;)17 b(\022)1198 4160 y Fp(j)1235 4145 y Fs(\))f(~)-65 b Fq(')1337 4160 y Fp(j)1374 4145 y Fs(\()p Fq(x)p Fs(\))28 b(=)f Fq( )1699 4160 y Fp(p)1735 4169 y Fc(0)1774 4145 y Fs(\()p Fq(x;)17 b(\022)1956 4160 y Fp(j)1993 4145 y Fs(\))2061 4051 y Fn(X)2048 4270 y Fp(\014)s Fm(2)p Fl(Z)2188 4251 y Fk(d)2218 4145 y Fs(\(2)p Fq(l)2336 4104 y Fm(0)2381 4145 y Fs(+)22 b(1\))2566 4104 y Fm(\000)p Fp(d=)p Fi(2)2732 4145 y Fq(a)2783 4160 y Fp(j)2820 4145 y Fs(\()p Fq(\014)6 b Fs(\))p Fw(1)3013 4161 y Fi(\(2)p Fp(l)3097 4142 y Fd(0)3119 4161 y Fi(+1\))p Fp(\014)s Fi(+)p Fp(C)3384 4180 y Fc(0)p Fk(;l)3453 4166 y Fd(0)3485 4145 y Fq(:)-118 4425 y Fs(As)33 b Fq( )89 4440 y Fp(p)125 4449 y Fc(0)164 4425 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)319 4440 y Fp(j)355 4425 y Fs(\))f(~)-65 b Fq(')457 4440 y Fp(j)493 4425 y Fs(\()p Fo(\001)p Fs(\))28 b Fo(2)g Fq(L)785 4388 y Fi(2)824 4425 y Fs(\()p Fj(R)928 4388 y Fp(d)975 4425 y Fs(\),)k(using)h(the)g(p)s(erio)s(dicit)m(y)d (of)i Fq( )2157 4440 y Fp(p)2193 4449 y Fc(0)2232 4425 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)2387 4440 y Fp(j)2423 4425 y Fs(\),)33 b(w)m(e)h(compute)501 4637 y Fo(k)p Fq( )614 4652 y Fp(p)650 4661 y Fc(0)688 4637 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)843 4652 y Fp(j)880 4637 y Fs(\))f(~)-65 b Fq(')982 4652 y Fp(j)1018 4637 y Fo(k)1068 4596 y Fi(2)1068 4666 y Fp(L)1116 4647 y Fc(2)1151 4666 y Fi(\()p Fl(R)1226 4647 y Fk(d)1262 4666 y Fi(\))1321 4637 y Fs(=)1438 4543 y Fn(X)1425 4762 y Fp(\014)s Fm(2)p Fl(Z)1565 4743 y Fk(d)1595 4637 y Fs(\(2)p Fq(l)1713 4596 y Fm(0)1759 4637 y Fs(+)22 b(1\))1944 4596 y Fm(\000)p Fp(d)2039 4637 y Fo(j)p Fq(a)2118 4652 y Fp(j)2154 4637 y Fs(\()p Fq(\014)6 b Fs(\))p Fo(j)2319 4596 y Fi(2)2374 4502 y Fn(Z)2430 4727 y Fi(\(2)p Fp(l)2514 4708 y Fd(0)2537 4727 y Fi(+1\))p Fp(\014)s Fi(+)p Fp(C)2802 4746 y Fc(0)p Fk(;l)2871 4732 y Fd(0)2919 4637 y Fo(j)p Fq( )3010 4652 y Fp(p)3046 4661 y Fc(0)3085 4637 y Fs(\()p Fq(x;)17 b(\022)3267 4652 y Fp(j)3304 4637 y Fs(\))p Fo(j)3370 4596 y Fi(2)3409 4637 y Fq(dx)1321 4958 y Fs(=)1438 4864 y Fn(X)1425 5083 y Fp(\014)s Fm(2)p Fl(Z)1565 5064 y Fk(d)1612 4958 y Fo(j)p Fq(a)1691 4973 y Fp(j)1727 4958 y Fs(\()p Fq(\014)6 b Fs(\))p Fo(j)1892 4917 y Fi(2)1948 4823 y Fn(Z)2003 5048 y Fp(C)2053 5057 y Fc(0)p Fk(;)p Fc(0)2158 4958 y Fo(j)p Fq( )2249 4973 y Fp(p)2285 4982 y Fc(0)2323 4958 y Fs(\()p Fq(x;)17 b(\022)2505 4973 y Fp(j)2542 4958 y Fs(\))p Fo(j)2608 4917 y Fi(2)2647 4958 y Fq(dx:)-118 5236 y Fs(Hence,)34 b(as)319 5155 y Fn(R)366 5270 y Fp(C)416 5279 y Fc(0)p Fk(;)p Fc(0)521 5236 y Fo(j)p Fq(')613 5251 y Fp(p)649 5260 y Fc(0)687 5236 y Fs(\()p Fq(x;)17 b(\022)869 5251 y Fp(j)906 5236 y Fs(\))p Fo(j)972 5199 y Fi(2)1011 5236 y Fq(dx)27 b Fs(=)1248 5155 y Fn(R)1295 5270 y Fp(C)1345 5279 y Fc(0)p Fk(;)p Fc(0)1450 5236 y Fo(j)p Fq( )1541 5251 y Fp(p)1577 5260 y Fc(0)1615 5236 y Fs(\()p Fq(x;)17 b(\022)1797 5251 y Fp(j)1834 5236 y Fs(\))p Fo(j)1900 5199 y Fi(2)1939 5236 y Fq(dx)33 b Fs(is)f(p)s(ositiv)m(e,)g(b)m(y)j(\(2.22\))o(,)e(w)m (e)h(get)1283 5348 y Fn(X)1270 5568 y Fp(\014)s Fm(2)p Fl(Z)1410 5549 y Fk(d)1457 5443 y Fo(j)p Fq(a)1536 5458 y Fp(j)1573 5443 y Fs(\()p Fq(\014)6 b Fs(\))p Fo(j)1738 5402 y Fi(2)1804 5443 y Fo(\024)28 b Fq(C)7 b Fo(k)p Fq(')2100 5458 y Fp(j)2136 5443 y Fo(k)2186 5402 y Fi(2)2186 5471 y Fp(L)2234 5452 y Fc(2)2269 5471 y Fi(\()p Fl(R)2344 5452 y Fk(d)2380 5471 y Fi(\))2439 5443 y Fq(<)28 b Fs(+)p Fo(1)p Fq(:)-2864 b Fs(\(2.23\))1969 5690 y Fg(10)p eop %%Page: 11 11 11 10 bop -118 241 a Fs(By)34 b(\(2.22\))e(and)h(the)g(de\014nitions)f (of)g Fq(l)j Fs(and)d Fq(l)1543 205 y Fm(0)1567 241 y Fs(,)h(the)g(lo)m(w)m(er)f(b)s(ound)h(\(2.21\))f(implies)e(that)1083 448 y Fo(h)p Fs(\()p Fj(E)13 b Fs(\()p Fq(V)1343 407 y Fp(n)1316 472 y(!)1390 448 y Fs(\))22 b Fo(\000)h Fq(V)1628 407 y Fp(n)1607 472 y(!)1675 448 y Fs(\)\011)1789 463 y Fp(j)1826 448 y Fq(;)17 b Fs(\011)1946 463 y Fp(j)1979 444 y Fd(0)2004 448 y Fo(i)27 b(\025)i Fq(\025)2233 407 y Fp(\021)2274 448 y Fq(=)p Fs(\(2)p Fq(n)2468 463 y Fp(z)2508 448 y Fs(\))2546 407 y Fi(2)2607 448 y Fo(\000)23 b Fq(C)7 b(\025)2841 407 y Fp(\021)2878 383 y Fd(0)2905 448 y Fq(:)-3050 b Fs(\(2.24\))-118 651 y(Let)33 b(us)g(compute)f Fo(h)p Fs(\()p Fj(E)13 b Fs(\()p Fq(V)837 615 y Fp(n)810 676 y(!)884 651 y Fs(\))22 b Fo(\000)h Fq(V)1122 615 y Fp(n)1101 676 y(!)1169 651 y Fs(\)\011)1283 666 y Fp(j)1320 651 y Fq(;)17 b Fs(\011)1440 666 y Fp(j)1473 647 y Fd(0)1498 651 y Fo(i)p Fs(.)43 b(One)33 b(has)211 874 y Fo(h)p Fs(\()p Fj(E)12 b Fs(\()p Fq(V)471 833 y Fp(n)443 898 y(!)518 874 y Fs(\))22 b Fo(\000)h Fq(V)756 833 y Fp(n)735 898 y(!)803 874 y Fs(\)\011)917 889 y Fp(j)953 874 y Fq(;)17 b Fs(\011)1073 889 y Fp(j)1106 870 y Fd(0)1132 874 y Fo(i)27 b Fs(=)1315 779 y Fn(X)1302 999 y Fp(\014)s Fm(2)p Fl(Z)1442 980 y Fk(d)1558 779 y Fn(X)1489 1003 y Fp(\015)t Fm(2)p Fl(Z)1627 980 y Fk(d)1627 1024 y Fc(2)p Fk(n)p Fc(+1)1771 874 y Fs(\()p 1809 819 65 4 v Fq(!)e Fo(\000)e Fq(!)2056 889 y Fp(\015)2100 874 y Fs(\)\(2)p Fq(l)2256 833 y Fm(0)2301 874 y Fs(+)f(1\))2486 833 y Fm(\000)p Fp(d)2581 874 y Fq(a)2632 889 y Fp(j)2669 874 y Fs(\()p Fq(\014)6 b Fs(\))p 2806 787 247 4 v Fq(a)2857 889 y Fp(j)2890 870 y Fd(0)2915 874 y Fs(\()p Fq(\014)g Fs(\))1754 1078 y Fn(Z)1809 1304 y Fi(\(2)p Fp(l)1893 1285 y Fd(0)1916 1304 y Fi(+1\))p Fp(\014)s Fi(+)p Fp(C)2181 1323 y Fc(0)p Fk(;l)2250 1309 y Fd(0)2298 1214 y Fq(V)2377 1173 y Fp(n)2424 1214 y Fs(\()p Fq(x)22 b Fo(\000)h Fq(\015)5 b Fs(\))p Fq( )2796 1229 y Fp(p)2832 1238 y Fc(0)2871 1214 y Fs(\()p Fq(x;)17 b(\022)3053 1229 y Fp(j)3090 1214 y Fs(\))p 3128 1127 418 4 v Fq( )3191 1229 y Fp(p)3227 1238 y Fc(0)3265 1214 y Fs(\()p Fq(x;)g(\022)3447 1229 y Fp(j)3480 1210 y Fd(0)3507 1214 y Fs(\))p Fq(dx)584 1474 y Fs(=)721 1380 y Fn(X)687 1599 y Fp(\014)730 1580 y Fd(00)771 1599 y Fm(2)p Fl(Z)868 1580 y Fk(d)1005 1380 y Fn(X)915 1603 y Fp(\014)958 1584 y Fd(0)980 1603 y Fm(2)p Fl(Z)1077 1580 y Fk(d)1077 1634 y Fc(2)p Fk(k)1141 1620 y Fd(0)1158 1634 y Fc(+1)1325 1380 y Fn(X)1256 1603 y Fp(\015)t Fm(2)p Fl(Z)1394 1580 y Fk(d)1394 1624 y Fc(2)p Fk(n)p Fc(+1)1538 1474 y Fs(\()p 1576 1420 65 4 v Fq(!)25 b Fo(\000)e Fq(!)1823 1489 y Fp(\015)1867 1474 y Fs(\)\(2)p Fq(l)2023 1433 y Fm(0)2068 1474 y Fs(+)f(1\))2253 1433 y Fm(\000)p Fp(d)2348 1474 y Fq(a)2399 1489 y Fp(j)2436 1474 y Fs(\()p Fq(\014)2535 1433 y Fm(0)2580 1474 y Fs(+)g(\(2)p Fq(k)2819 1433 y Fm(0)2864 1474 y Fs(+)g(1\))p Fq(\014)3110 1433 y Fm(0)o(0)3152 1474 y Fs(\))p 3190 1388 864 4 v Fq(a)3241 1489 y Fp(j)3274 1470 y Fd(0)3300 1474 y Fs(\()p Fq(\014)3399 1446 y Fm(0)3444 1474 y Fs(+)g(\(2)p Fq(k)3683 1446 y Fm(0)3728 1474 y Fs(+)g(1\))p Fq(\014)3974 1446 y Fm(0)o(0)4016 1474 y Fs(\))1754 1689 y Fn(Z)1809 1914 y Fp(C)1859 1933 y Fc(0)p Fk(;l)1928 1919 y Fd(0)1976 1824 y Fq(V)2054 1783 y Fp(n)2101 1824 y Fs(\()p Fq(x)h Fo(\000)f Fq(\015)28 b Fs(+)22 b(\(2)p Fq(l)2611 1783 y Fm(0)2656 1824 y Fs(+)g(1\))p Fq(\014)2902 1783 y Fm(0)2925 1824 y Fs(\))p Fq( )3026 1839 y Fp(p)3062 1848 y Fc(0)3100 1824 y Fs(\()p Fq(x;)17 b(\022)3282 1839 y Fp(j)3319 1824 y Fs(\))p 3357 1737 418 4 v Fq( )3420 1839 y Fp(p)3456 1848 y Fc(0)3495 1824 y Fs(\()p Fq(x;)g(\022)3677 1839 y Fp(j)3710 1820 y Fd(0)3736 1824 y Fs(\))p Fq(dx:)-118 2115 y Fs(as)33 b Fq(V)80 2079 y Fp(n)160 2115 y Fs(is)f(\(2)p Fq(n)22 b Fs(+)g(1\))p Fj(Z)679 2079 y Fp(d)716 2115 y Fs(-p)s(erio)s(dic)31 b(and)i(\(2)p Fq(l)1431 2079 y Fm(0)1476 2115 y Fs(+)22 b(1\)\(2)p Fq(k)1802 2079 y Fm(0)1847 2115 y Fs(+)g(1\))27 b(=)h(\(2)p Fq(n)22 b Fs(+)g(1\).)-118 2231 y(Then,)34 b(setting)829 2445 y Fq(c)871 2460 y Fp(j)t(j)937 2441 y Fd(0)962 2445 y Fs(\()p Fq(\014)1061 2404 y Fm(0)1084 2445 y Fs(\))28 b(=)1287 2350 y Fn(X)1253 2570 y Fp(\014)1296 2551 y Fd(0)q(0)1337 2570 y Fm(2)p Fl(Z)1434 2551 y Fk(d)1481 2445 y Fq(a)1532 2460 y Fp(j)1569 2445 y Fs(\()p Fq(\014)1668 2404 y Fm(0)1713 2445 y Fs(+)22 b(\(2)p Fq(k)1952 2404 y Fm(0)1997 2445 y Fs(+)g(1\))p Fq(\014)2243 2404 y Fm(0)o(0)2285 2445 y Fs(\))p 2323 2358 864 4 v Fq(a)2374 2460 y Fp(j)2407 2441 y Fd(0)2433 2445 y Fs(\()p Fq(\014)2532 2416 y Fm(0)2577 2445 y Fs(+)g(\(2)p Fq(k)2816 2416 y Fm(0)2861 2445 y Fs(+)g(1\))p Fq(\014)3107 2416 y Fm(0)o(0)3149 2445 y Fs(\))-118 2752 y(w)m(e)34 b(obtain)238 3076 y Fo(h)p Fs(\()p Fj(E)12 b Fs(\()p Fq(V)497 3035 y Fp(n)470 3101 y(!)544 3076 y Fs(\))22 b Fo(\000)h Fq(V)783 3035 y Fp(n)761 3101 y(!)830 3076 y Fs(\)\011)944 3091 y Fp(j)980 3076 y Fq(;)17 b Fs(\011)1100 3091 y Fp(j)1133 3072 y Fd(0)1158 3076 y Fo(i)28 b Fs(=)f(\(2)p Fq(l)1446 3035 y Fm(0)1492 3076 y Fs(+)22 b(1\))1677 3035 y Fm(\000)p Fp(d)1857 2981 y Fn(X)1788 3205 y Fp(\015)t Fm(2)p Fl(Z)1926 3182 y Fk(d)1926 3226 y Fc(2)p Fk(n)p Fc(+1)2070 3076 y Fs(\()p 2108 3021 65 4 v Fq(!)j Fo(\000)e Fq(!)2355 3091 y Fp(\015)2399 3076 y Fs(\))2454 2846 y Fn(0)2454 3021 y(B)2454 3085 y(@)2631 2981 y(X)2541 3205 y Fp(\014)2584 3186 y Fd(0)2606 3205 y Fm(2)p Fl(Z)2704 3182 y Fk(d)2704 3236 y Fc(2)p Fk(k)2768 3222 y Fd(0)2784 3236 y Fc(+1)2882 3076 y Fq(c)2924 3091 y Fp(j)t(j)2990 3072 y Fd(0)3015 3076 y Fs(\()p Fq(\014)3114 3035 y Fm(0)3137 3076 y Fs(\))1696 3336 y Fn(Z)1751 3562 y Fp(C)1801 3581 y Fc(0)p Fk(;l)1870 3567 y Fd(0)1918 3472 y Fq(V)1997 3431 y Fp(n)2044 3472 y Fs(\()p Fq(x)f Fo(\000)h Fq(\015)k Fs(+)22 b(\(2)p Fq(l)2553 3431 y Fm(0)2599 3472 y Fs(+)g(1\))p Fq(\014)2845 3431 y Fm(0)2867 3472 y Fs(\))p Fq( )2968 3487 y Fp(p)3004 3496 y Fc(0)3043 3472 y Fs(\()p Fq(x;)17 b(\022)3225 3487 y Fp(j)3262 3472 y Fs(\))p 3300 3385 418 4 v Fq( )3363 3487 y Fp(p)3399 3496 y Fc(0)3437 3472 y Fs(\()p Fq(x;)g(\022)3619 3487 y Fp(j)3652 3468 y Fd(0)3679 3472 y Fs(\))p Fq(dx)3823 3302 y Fn(!)1225 3868 y Fs(=)27 b(\(2)p Fq(l)1446 3827 y Fm(0)1492 3868 y Fs(+)22 b(1\))1677 3827 y Fm(\000)p Fp(d)1859 3773 y Fn(X)1788 3997 y Fp(\015)t Fm(2)p Fl(Z)1926 3974 y Fk(d)1926 4028 y Fc(2)p Fk(l)1976 4014 y Fd(0)1993 4028 y Fc(+1)2091 3638 y Fn(0)2091 3813 y(B)2091 3877 y(@)2267 3773 y(X)2178 3997 y Fp(\015)2218 3978 y Fd(0)2240 3997 y Fm(2)p Fl(Z)2338 3974 y Fk(d)2338 4028 y Fc(2)p Fk(k)2402 4014 y Fd(0)2418 4028 y Fc(+1)2500 3868 y Fs(\()p 2538 3813 65 4 v Fq(!)j Fo(\000)e Fq(!)2785 3884 y Fp(\015)t Fi(+\(2)p Fp(l)2964 3865 y Fd(0)2987 3884 y Fi(+1\))p Fp(\015)3144 3865 y Fd(0)3171 3868 y Fs(\))3226 3638 y Fn(2)3226 3813 y(6)3226 3877 y(4)3382 3773 y(X)3292 3997 y Fp(\014)3335 3978 y Fd(0)3357 3997 y Fm(2)p Fl(Z)3455 3974 y Fk(d)3455 4028 y Fc(2)p Fk(k)3519 4014 y Fd(0)3535 4028 y Fc(+1)3633 3868 y Fq(c)3675 3883 y Fp(j)t(j)3741 3864 y Fd(0)3766 3868 y Fs(\()p Fq(\014)3865 3827 y Fm(0)3888 3868 y Fs(\))1470 4128 y Fn(Z)1525 4354 y Fp(C)1575 4373 y Fc(0)p Fk(;l)1644 4359 y Fd(0)1692 4264 y Fq(V)1771 4223 y Fp(n)1818 4264 y Fs(\()p Fq(x)f Fo(\000)h Fq(\015)k Fs(+)22 b(\(2)p Fq(l)2327 4223 y Fm(0)2372 4264 y Fs(+)g(1\)\()p Fq(\014)2656 4223 y Fm(0)2701 4264 y Fo(\000)h Fq(\015)2857 4223 y Fm(0)2880 4264 y Fs(\)\))p Fq( )3019 4279 y Fp(p)3055 4288 y Fc(0)3094 4264 y Fs(\()p Fq(x;)17 b(\022)3276 4279 y Fp(j)3313 4264 y Fs(\))p 3351 4177 418 4 v Fq( )3414 4279 y Fp(p)3450 4288 y Fc(0)3488 4264 y Fs(\()p Fq(x;)g(\022)3670 4279 y Fp(j)3703 4260 y Fd(0)3730 4264 y Fs(\))p Fq(dx)3874 4094 y Fn(#!)1225 4531 y Fs(=)27 b(\(2)p Fq(l)1446 4489 y Fm(0)1492 4531 y Fs(+)22 b(1\))1677 4489 y Fm(\000)p Fp(d)1859 4436 y Fn(X)1788 4659 y Fp(\015)t Fm(2)p Fl(Z)1926 4637 y Fk(d)1926 4690 y Fc(2)p Fk(l)1976 4676 y Fd(0)1993 4690 y Fc(+1)2091 4531 y Fq(X)2172 4546 y Fp(j)t(j)2238 4527 y Fd(0)2263 4531 y Fs(\()p Fq(\015)5 b Fs(\))-118 3800 y(\(2.25\))-118 4866 y(where)1043 5064 y Fq(X)1124 5079 y Fp(j)t(j)1190 5060 y Fd(0)1215 5064 y Fs(\()p Fq(\015)g Fs(\))28 b(=)1567 4970 y Fn(X)1478 5193 y Fp(\015)1518 5174 y Fd(0)1541 5193 y Fm(2)p Fl(Z)1638 5170 y Fk(d)1638 5224 y Fc(2)p Fk(k)1702 5210 y Fd(0)1719 5224 y Fc(+1)1800 5064 y Fs(\()p 1838 5009 65 4 v Fq(!)e Fo(\000)c Fq(!)2085 5080 y Fp(\015)t Fi(+\(2)p Fp(l)2264 5061 y Fd(0)2287 5080 y Fi(+1\))p Fp(\015)2444 5061 y Fd(0)2472 5064 y Fs(\))p Fq(A)2583 5079 y Fp(j)t(j)2649 5060 y Fd(0)2674 5064 y Fs(\()p Fq(\015)5 b(;)17 b(\015)2868 5023 y Fm(0)2891 5064 y Fs(\))p Fq(;)192 5414 y(A)265 5429 y Fp(j)t(j)331 5410 y Fd(0)356 5414 y Fs(\()p Fq(\015)5 b(;)17 b(\015)550 5373 y Fm(0)573 5414 y Fs(\))28 b(=)833 5319 y Fn(X)742 5543 y Fp(\014)785 5524 y Fd(0)808 5543 y Fm(2)p Fl(Z)905 5520 y Fk(d)905 5574 y Fc(2)p Fk(k)969 5560 y Fd(0)986 5574 y Fc(+1)1083 5414 y Fq(c)1125 5429 y Fp(j)t(j)1191 5410 y Fd(0)1217 5414 y Fs(\()p Fq(\014)1316 5373 y Fm(0)1339 5414 y Fs(\))1394 5278 y Fn(Z)1448 5504 y Fp(C)1498 5523 y Fc(0)p Fk(;l)1567 5509 y Fd(0)1615 5414 y Fq(V)1694 5373 y Fp(n)1741 5414 y Fs(\()p Fq(x)22 b Fo(\000)h Fq(\015)k Fs(+)22 b(\(2)p Fq(l)2250 5373 y Fm(0)2296 5414 y Fs(+)g(1\)\()p Fq(\014)2580 5373 y Fm(0)2624 5414 y Fo(\000)h Fq(\015)2780 5373 y Fm(0)2803 5414 y Fs(\)\))p Fq( )2942 5429 y Fp(p)2978 5438 y Fc(0)3017 5414 y Fs(\()p Fq(x;)17 b(\022)3199 5429 y Fp(j)3236 5414 y Fs(\))p 3274 5327 418 4 v Fq( )3337 5429 y Fp(p)3373 5438 y Fc(0)3411 5414 y Fs(\()p Fq(x;)g(\022)3593 5429 y Fp(j)3626 5410 y Fd(0)3653 5414 y Fs(\))p Fq(dx:)1969 5690 y Fg(11)p eop %%Page: 12 12 12 11 bop -118 241 a Fs(The)31 b(random)d(v)-5 b(ariables)29 b(\()p Fq(X)955 256 y Fp(j)t(j)1021 237 y Fd(0)1046 241 y Fs(\()p Fq(\015)5 b Fs(\)\))1216 264 y Fp(\015)t Fm(2)p Fl(Z)1353 241 y Fk(d)1353 295 y Fc(2)p Fk(l)1403 281 y Fd(0)1420 295 y Fc(+1)1535 241 y Fs(are)30 b(indep)s(enden)m(t)h(and) e(cen)m(tered)j(\(i.e.)42 b Fj(E)12 b Fs(\()p Fq(X)3200 256 y Fp(j)t(j)3266 237 y Fd(0)3297 241 y Fs(\()p Fq(\015)5 b Fs(\)\))28 b(=)f(0\).)43 b(They)31 b(are)-118 379 y(b)s(ounded;)i (indeed,)g(one)g(computes)35 671 y Fo(j)p Fq(X)144 686 y Fp(j)t(j)210 667 y Fd(0)235 671 y Fs(\()p Fq(\015)5 b Fs(\))p Fo(j)27 b(\024)h Fq(C)710 577 y Fn(X)621 800 y Fp(\015)661 781 y Fd(0)683 800 y Fm(2)p Fl(Z)781 777 y Fk(d)781 831 y Fc(2)p Fk(k)845 817 y Fd(0)861 831 y Fc(+1)959 671 y Fo(j)p Fq(A)1060 686 y Fp(j)t(j)1126 667 y Fd(0)1151 671 y Fs(\()p Fq(\015)5 b(;)17 b(\015)1345 630 y Fm(0)1368 671 y Fs(\))p Fo(j)422 990 y(\024)28 b Fq(C)711 896 y Fn(X)621 1119 y Fp(\014)664 1100 y Fd(0)686 1119 y Fm(2)p Fl(Z)783 1096 y Fk(d)783 1150 y Fc(2)p Fk(k)847 1136 y Fd(0)864 1150 y Fc(+1)1051 896 y Fn(X)962 1119 y Fp(\015)1002 1100 y Fd(0)1025 1119 y Fm(2)p Fl(Z)1122 1096 y Fk(d)1122 1150 y Fc(2)p Fk(k)1186 1136 y Fd(0)1203 1150 y Fc(+1)1300 990 y Fo(j)p Fq(c)1370 1005 y Fp(j)t(j)1436 986 y Fd(0)1461 990 y Fs(\()p Fq(\014)1560 949 y Fm(0)1583 990 y Fs(\))p Fo(j)1120 1204 y Fn(Z)1175 1430 y Fp(C)1225 1449 y Fc(0)p Fk(;l)1294 1435 y Fd(0)1342 1340 y Fo(j)p Fq(V)1448 1299 y Fp(n)1495 1340 y Fs(\()p Fq(x)23 b Fo(\000)f Fq(\015)27 b Fs(+)22 b(\(2)p Fq(l)2004 1299 y Fm(0)2050 1340 y Fs(+)g(1\)\()p Fq(\014)2334 1299 y Fm(0)2378 1340 y Fo(\000)h Fq(\015)2534 1299 y Fm(0)2557 1340 y Fs(\)\))p Fo(jj)p Fq( )2752 1355 y Fp(p)2788 1364 y Fc(0)2826 1340 y Fs(\()p Fq(x;)17 b(\022)3008 1355 y Fp(j)3045 1340 y Fs(\))p Fo(jj)p Fq( )3202 1355 y Fp(p)3238 1364 y Fc(0)3276 1340 y Fs(\()p Fq(x;)g(\022)3458 1355 y Fp(j)3491 1336 y Fd(0)3518 1340 y Fs(\))p Fo(j)p Fq(dx)422 1632 y Fo(\024)28 b Fq(C)711 1537 y Fn(X)621 1761 y Fp(\014)664 1742 y Fd(0)686 1761 y Fm(2)p Fl(Z)783 1738 y Fk(d)783 1792 y Fc(2)p Fk(k)847 1778 y Fd(0)864 1792 y Fc(+1)962 1632 y Fo(j)p Fq(c)1032 1647 y Fp(j)t(j)1098 1628 y Fd(0)1123 1632 y Fs(\()p Fq(\014)1222 1591 y Fm(0)1245 1632 y Fs(\))p Fo(j)1416 1537 y Fn(X)1328 1761 y Fp(\015)1368 1742 y Fd(0)1390 1761 y Fm(2)p Fl(Z)1487 1738 y Fk(d)1487 1792 y Fc(2)p Fk(k)1551 1778 y Fd(0)1568 1792 y Fc(+1)1665 1496 y Fn(Z)1721 1722 y Fp(C)1771 1741 y Fc(0)p Fk(;l)1840 1727 y Fd(0)1888 1632 y Fo(j)p Fq(V)1994 1591 y Fp(n)2041 1632 y Fs(\()p Fq(x)22 b Fo(\000)h Fq(\015)k Fs(+)22 b(\(2)p Fq(l)2550 1591 y Fm(0)2596 1632 y Fs(+)g(1\))p Fq(\015)2837 1591 y Fm(0)2860 1632 y Fs(\))p Fo(jj)p Fq( )3017 1647 y Fp(p)3053 1656 y Fc(0)3091 1632 y Fs(\()p Fq(x;)17 b(\022)3273 1647 y Fp(j)3310 1632 y Fs(\))p Fo(jj)p Fq( )3467 1647 y Fp(p)3503 1656 y Fc(0)3541 1632 y Fs(\()p Fq(x;)g(\022)3723 1647 y Fp(j)3756 1628 y Fd(0)3782 1632 y Fs(\))p Fo(j)p Fq(dx:)-118 1943 y Fs(Using)38 b(the)i(fact)f(that)f Fq(V)832 1906 y Fp(n)918 1943 y Fs(is)g(\(2)p Fq(n)27 b Fs(+)f(1\))p Fj(Z)1452 1906 y Fp(d)1490 1943 y Fs(-p)s(erio)s(dic,)38 b(the)i(relation)d(\(2)p Fq(n)26 b Fs(+)g(1\))39 b(=)f(\(2)p Fq(l)3101 1906 y Fm(0)3151 1943 y Fs(+)26 b(1\)\(2)p Fq(k)3481 1906 y Fm(0)3530 1943 y Fs(+)h(1\))38 b(and)h(esti-)-118 2059 y(mate)32 b(\(2.23\))o(,)h(w)m(e)g(obtain)375 2271 y Fo(j)p Fq(X)484 2286 y Fp(j)t(j)550 2267 y Fd(0)575 2271 y Fs(\()p Fq(\015)5 b Fs(\))p Fo(j)27 b(\024)h Fq(C)974 2177 y Fn(X)961 2396 y Fp(\014)s Fm(2)p Fl(Z)1101 2377 y Fk(d)1148 2271 y Fo(j)p Fq(a)1227 2286 y Fp(j)1263 2271 y Fs(\()p Fq(\014)6 b Fs(\))p Fo(j)21 b(\001)h(j)p Fq(a)1578 2286 y Fp(j)1611 2267 y Fd(0)1637 2271 y Fs(\()p Fq(\014)6 b Fs(\))p Fo(j)1898 2177 y Fn(X)1819 2400 y Fp(\015)1859 2381 y Fd(0)1880 2400 y Fm(2)p Fl(Z)1978 2377 y Fk(d)1978 2421 y Fc(2)p Fk(n)p Fc(+1)2138 2136 y Fn(Z)2194 2361 y Fp(C)2244 2370 y Fc(0)p Fk(;)p Fc(0)2349 2271 y Fo(j)p Fq(V)2455 2230 y Fp(n)2502 2271 y Fs(\()p Fq(x)22 b Fo(\000)h Fq(\015)2773 2230 y Fm(0)2796 2271 y Fs(\))p Fo(jj)p Fq( )2953 2286 y Fp(p)2989 2295 y Fc(0)3027 2271 y Fs(\()p Fq(x;)17 b(\022)3209 2286 y Fp(j)3246 2271 y Fs(\))p Fo(jj)p Fq( )3403 2286 y Fp(p)3439 2295 y Fc(0)3477 2271 y Fs(\()p Fq(x;)g(\022)3659 2286 y Fp(j)3692 2267 y Fd(0)3719 2271 y Fs(\))p Fo(j)p Fq(dx)762 2611 y Fo(\024)28 b Fq(C)7 b Fo(k)p Fq(')1058 2626 y Fp(j)1094 2611 y Fo(kk)p Fq(')1258 2626 y Fp(j)1291 2607 y Fd(0)1317 2611 y Fo(k)1406 2516 y Fn(X)1384 2736 y Fp(\015)1424 2717 y Fd(0)1446 2736 y Fm(2)p Fl(Z)1543 2717 y Fk(d)1590 2476 y Fn(Z)1645 2701 y Fp(C)1695 2710 y Fc(0)p Fk(;)p Fc(0)1800 2611 y Fo(j)p Fq(V)21 b Fs(\()p Fq(x)i Fo(\000)f Fq(\015)2177 2570 y Fm(0)2201 2611 y Fs(\))p Fo(jj)p Fq( )2358 2626 y Fp(p)2394 2635 y Fc(0)2432 2611 y Fs(\()p Fq(x;)17 b(\022)2614 2626 y Fp(j)2651 2611 y Fs(\))p Fo(jj)p Fq( )2808 2626 y Fp(p)2844 2635 y Fc(0)2882 2611 y Fs(\()p Fq(x;)g(\022)3064 2626 y Fp(j)3097 2607 y Fd(0)3123 2611 y Fs(\))p Fo(j)p Fq(dx)762 2871 y Fo(\024)28 b Fq(C)7 b Fo(k)p Fq(')1058 2886 y Fp(j)1094 2871 y Fo(kk)p Fq(')1258 2886 y Fp(j)1291 2867 y Fd(0)1317 2871 y Fo(k)27 b Fq(<)h Fs(+)p Fo(1)-118 3037 y Fs(By)34 b(\(2.24\))e(and)g(\(2.25\))o(,)h(to)f (estimate)f(the)i(probabilit)m(y)d(of)i(\012\()p Fq(n;)17 b(\025;)g(\021)t Fs(\),)32 b(it)f(su\016ces)k(to)d(estimate)f(the)i (probabilit)m(y)-118 3153 y(that)759 3313 y Fq(\025)816 3271 y Fp(\021)857 3313 y Fq(=)p Fs(\(4)p Fq(n)1051 3328 y Fp(z)1091 3313 y Fs(\))1129 3271 y Fi(2)1196 3313 y Fo(\024)28 b Fq(\025)1358 3271 y Fp(\021)1400 3313 y Fq(=)p Fs(\(2)p Fq(n)1594 3328 y Fp(z)1634 3313 y Fs(\))1672 3271 y Fi(2)1733 3313 y Fo(\000)23 b Fq(C)7 b(\025)1967 3271 y Fp(\021)2004 3248 y Fd(0)2059 3313 y Fo(\024)28 b Fs(\(2)p Fq(l)2282 3271 y Fm(0)2327 3313 y Fs(+)22 b(1\))2512 3271 y Fm(\000)p Fp(d)2695 3218 y Fn(X)2624 3441 y Fp(\015)t Fm(2)p Fl(Z)2761 3419 y Fk(d)2761 3473 y Fc(2)p Fk(l)2811 3459 y Fd(0)2828 3473 y Fc(+1)2926 3313 y Fq(X)3007 3328 y Fp(j)t(j)3073 3309 y Fd(0)3098 3313 y Fs(\()p Fq(\015)5 b Fs(\))p Fq(:)-118 3610 y Fs(This)36 b(is)g(a)g(large)g(deviation)f(estimate;)i(it)f(is)f(b)s(ounded)j(ab)s (o)m(v)m(e)f(in)e(the)i(usual)f(w)m(a)m(y)i(using)e(exp)s(onen)m(tial)f (Mark)m(o)m(v)-118 3727 y(inequalities)c(\(see)i(e.g.)g([6,)g(5)o(]\))g (and)g(yields)476 4038 y Fj(P)552 3807 y Fn(2)552 3983 y(6)552 4047 y(4)620 4038 y Fq(\025)677 3996 y Fp(\021)719 4038 y Fq(=)p Fs(\(4)p Fq(n)913 4053 y Fp(z)953 4038 y Fs(\))991 3996 y Fi(2)1058 4038 y Fo(\024)28 b Fs(\(2)p Fq(l)1281 3996 y Fm(0)1326 4038 y Fs(+)22 b(1\))1511 3996 y Fm(\000)p Fp(d)1694 3943 y Fn(X)1623 4166 y Fp(\015)t Fm(2)p Fl(Z)1760 4144 y Fk(d)1760 4197 y Fc(2)p Fk(l)1810 4183 y Fd(0)1827 4197 y Fc(+1)1925 4038 y Fq(X)2006 4053 y Fp(j)t(j)2072 4034 y Fd(0)2097 4038 y Fs(\()p Fq(\015)5 b Fs(\))2229 3807 y Fn(3)2229 3983 y(7)2229 4047 y(5)2323 4038 y Fo(\024)28 b Fq(e)2473 3996 y Fm(\000)p Fp(c)p Fi(\()p Fp(l)2608 3973 y Fd(0)2631 3996 y Fi(\))2658 3973 y Fk(d)2694 3996 y Fp(\025)2735 3973 y Fc(2)p Fk(\021)2836 4038 y Fo(\024)g Fq(e)2986 3996 y Fm(\000)p Fp(c\025)3113 3973 y Fd(\000)p Fk(d=)p Fc(2+2)p Fk(d\021)3398 3952 y Fd(0)3420 3973 y Fc(+2)p Fk(\021)-118 4348 y Fs(recalling)h(the)i (de\014nition)f(of)h Fq(l)1010 4312 y Fm(0)1033 4348 y Fs(.)43 b(As)32 b Fq(\021)f(<)d(\021)1481 4312 y Fm(0)1531 4348 y Fq(<)g(d=)p Fs(\(4)p Fq(d)18 b Fs(+)h(4\),)31 b(one)g(has)g Fo(\000)p Fq(d=)p Fs(2)19 b(+)g(2)p Fq(d\021)2972 4312 y Fm(0)3013 4348 y Fs(+)g(2)p Fq(\021)31 b(<)d Fs(0.)43 b(Hence,)32 b(for)f(some)-118 4465 y Fq(")c(>)h Fs(0)k(and)h Fq(\025)f Fs(su\016cien)m(tly)h(small,)e(one)i(has)905 4770 y Fj(P)981 4540 y Fn(2)981 4715 y(6)981 4779 y(4)1049 4770 y Fq(\025)1106 4729 y Fp(\021)1148 4770 y Fq(=)p Fs(\(4)p Fq(n)1342 4785 y Fp(z)1381 4770 y Fs(\))1419 4729 y Fi(2)1486 4770 y Fo(\024)28 b Fs(\(2)p Fq(l)1709 4729 y Fm(0)1755 4770 y Fs(+)22 b(1\))1940 4729 y Fm(\000)p Fp(d)2122 4675 y Fn(X)2051 4899 y Fp(\015)t Fm(2)p Fl(Z)2189 4876 y Fk(d)2189 4930 y Fc(2)p Fk(l)2239 4916 y Fd(0)2256 4930 y Fc(+1)2354 4770 y Fq(X)2435 4785 y Fp(j)t(j)2501 4766 y Fd(0)2526 4770 y Fs(\()p Fq(\015)5 b Fs(\))2658 4540 y Fn(3)2658 4715 y(7)2658 4779 y(5)2752 4770 y Fo(\024)28 b Fq(e)2902 4729 y Fm(\000)p Fp(\025)2998 4705 y Fd(\000)p Fk(")3084 4770 y Fq(:)-118 5081 y Fs(On)d(the)g(other)g(hand,)i(the)e (probabilit)m(y)e(of)h(\012\()p Fq(n;)17 b(\025;)g(\021)t Fs(\))25 b(is)f(b)s(ounded)i(b)m(y)f(the)h(sum)e(o)m(v)m(er)i Fq(j)31 b Fs(and)25 b Fq(j)3356 5045 y Fm(0)3404 5081 y Fs(of)g(the)g(probabilit)m(y)-118 5197 y(estimate)32 b(ab)s(o)m(v)m(e.)44 b(This)33 b(yields)f(\(2.8\))g(and)h(completes)f (the)h(pro)s(of)f(of)g(Prop)s(osition)f(2.1.)p 4063 5197 4 66 v 4067 5135 59 4 v 4067 5197 V 4125 5197 4 66 v -118 5313 a Fw(Pro)s(of)37 b(of)h(Lemma)f(2.1.)44 b Fs(Recall)30 b(that)1454 5501 y Fq(')1518 5516 y Fp(j)1582 5501 y Fs(=)1686 5365 y Fn(Z)1741 5591 y Fl(T)1791 5572 y Fd(\003)1854 5501 y Fs(~)-61 b Fq(\037)1903 5516 y Fp(p)1939 5525 y Fc(0)1978 5501 y Fs(\()p Fq(\022)s Fs(\))p Fq(')2166 5516 y Fp(p)2202 5525 y Fc(0)2240 5501 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))p Fq(d\022)s(:)1969 5690 y Fg(12)p eop %%Page: 13 13 13 12 bop -118 241 a Fs(where)45 b(~)-60 b Fq(\037)225 256 y Fp(p)261 265 y Fc(0)327 241 y Fs(=)27 b Fw(1)486 256 y Fm(j)p Fp(\022)r Fm(\000)p Fp(\022)629 266 y Fk(j)661 256 y Fm(j\024)p Fi(1)p Fp(=l)854 241 y Fo(\001)22 b Fq(\037)965 256 y Fp(p)1001 265 y Fc(0)1072 241 y Fs(\(see)34 b(\(2.19\))o(\).)43 b(Using)33 b(the)g(p)s(erio)s(dic)e(comp)s(onen)m (ts,)i(w)m(e)g(write)442 480 y Fq(')506 495 y Fp(j)570 480 y Fs(=)674 339 y Fn(\022)747 344 y(Z)803 570 y Fl(T)853 551 y Fd(\003)915 480 y Fs(~)-60 b Fq(\037)965 495 y Fp(p)1001 504 y Fc(0)1039 480 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)1208 438 y Fp(i)p Fm(\001\001)p Fp(\022)1311 480 y Fq(d\022)1410 339 y Fn(\023)1499 480 y Fq( )1562 495 y Fp(p)1598 504 y Fc(0)1637 480 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)1792 495 y Fp(j)1828 480 y Fs(\))22 b(+)1986 344 y Fn(Z)2042 570 y Fl(T)2092 551 y Fd(\003)2154 480 y Fs(~)-60 b Fq(\037)2204 495 y Fp(p)2240 504 y Fc(0)2278 480 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)2447 438 y Fp(i)p Fm(\001\001)p Fp(\022)2550 480 y Fs(\()p Fq( )2651 495 y Fp(p)2687 504 y Fc(0)2725 480 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))22 b Fo(\000)h Fq( )3106 495 y Fp(p)3142 504 y Fc(0)3180 480 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)3335 495 y Fp(j)3372 480 y Fs(\)\))p Fq(d\022)s(:)-118 708 y Fs(As)34 b(discussed)h(in)e(section)g(5.1,)g(under)h(our)g (assumptions,)f(for)g Fq(\022)k Fs(close)c(to)g Fq(\022)2743 723 y Fp(j)2780 708 y Fs(,)g(the)h(function)f(\()p Fq(x;)17 b(\022)s Fs(\))29 b Fo(7!)g Fq( )3836 723 y Fp(p)3872 732 y Fc(0)3911 708 y Fs(\()p Fq(x;)17 b(\022)s Fs(\))-118 826 y(is)28 b(analytic)f(in)h Fq(\022)j Fs(v)-5 b(alued)28 b(in)g(the)h Fj(Z)1174 790 y Fp(d)1212 826 y Fs(-p)s(erio)s(dic,)e(lo)s (cally)f(square)k(in)m(tegrable)d(functions)h(in)g Fq(x)p Fs(;)i(hence,)h(w)m(e)f(compute)589 921 y Fn(\015)589 981 y(\015)589 1041 y(\015)589 1101 y(\015)644 930 y(Z)700 1156 y Fl(T)750 1137 y Fd(\003)812 1066 y Fs(~)-60 b Fq(\037)862 1081 y Fp(p)898 1090 y Fc(0)936 1066 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)1105 1025 y Fp(i)p Fm(\001\001)p Fp(\022)1207 1066 y Fs(\()p Fq( )1308 1081 y Fp(p)1344 1090 y Fc(0)1383 1066 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))22 b Fo(\000)g Fq( )1763 1081 y Fp(p)1799 1090 y Fc(0)1838 1066 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)1993 1081 y Fp(j)2029 1066 y Fs(\)\))p Fq(d\022)2204 921 y Fn(\015)2204 981 y(\015)2204 1041 y(\015)2204 1101 y(\015)2259 948 y Fi(2)2259 1165 y Fp(L)2307 1146 y Fc(2)2342 1165 y Fi(\()p Fl(R)2417 1146 y Fk(d)2454 1165 y Fi(\))600 1380 y Fs(=)715 1285 y Fn(X)704 1504 y Fp(\015)t Fm(2)p Fl(Z)841 1486 y Fk(d)888 1244 y Fn(Z)943 1469 y Fp(C)993 1478 y Fk(\015)s(;)p Fc(0)1103 1235 y Fn(\014)1103 1295 y(\014)1103 1355 y(\014)1103 1414 y(\014)1137 1244 y(Z)1192 1469 y Fl(T)1242 1451 y Fd(\003)1305 1380 y Fs(~)-61 b Fq(\037)1354 1395 y Fp(p)1390 1404 y Fc(0)1428 1380 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)1597 1338 y Fp(ix)p Fm(\001)p Fp(\022)1720 1380 y Fs(\()p Fq( )1821 1395 y Fp(p)1857 1404 y Fc(0)1896 1380 y Fs(\()p Fq(x;)17 b(\022)s Fs(\))22 b Fo(\000)h Fq( )2304 1395 y Fp(p)2340 1404 y Fc(0)2378 1380 y Fs(\()p Fq(x;)17 b(\022)2560 1395 y Fp(j)2597 1380 y Fs(\)\))p Fq(d\022)2772 1235 y Fn(\014)2772 1295 y(\014)2772 1355 y(\014)2772 1414 y(\014)2805 1261 y Fi(2)2861 1380 y Fq(dx)600 1726 y Fs(=)715 1631 y Fn(X)704 1851 y Fp(\015)t Fm(2)p Fl(Z)841 1832 y Fk(d)888 1590 y Fn(Z)943 1816 y Fp(C)993 1825 y Fc(0)p Fk(;)p Fc(0)1098 1581 y Fn(\014)1098 1641 y(\014)1098 1701 y(\014)1098 1761 y(\014)1131 1590 y(Z)1187 1816 y Fl(T)1237 1797 y Fd(\003)1299 1726 y Fs(~)-60 b Fq(\037)1349 1741 y Fp(p)1385 1750 y Fc(0)1423 1726 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)1592 1685 y Fp(ix\022)1695 1726 y Fq(e)1740 1685 y Fp(i\015)t(\022)1844 1726 y Fs(\()p Fq( )1945 1741 y Fp(p)1981 1750 y Fc(0)2019 1726 y Fs(\()p Fq(x)23 b Fs(+)f Fq(\015)5 b(;)17 b(\022)s Fs(\))22 b Fo(\000)g Fq( )2603 1741 y Fp(p)2639 1750 y Fc(0)2678 1726 y Fs(\()p Fq(x)h Fs(+)f Fq(\015)5 b(;)17 b(\022)3037 1741 y Fp(j)3073 1726 y Fs(\)\))p Fq(d\022)3248 1581 y Fn(\014)3248 1641 y(\014)3248 1701 y(\014)3248 1761 y(\014)3281 1607 y Fi(2)3337 1726 y Fq(dx)-118 2015 y Fs(In)37 b(the)g(\014rst)g(step,)h (w)m(e)g(rewrote)f(the)g Fq(x)p Fs(-in)m(tegral)e(o)m(v)m(er)j Fj(R)2025 1979 y Fp(d)2108 2015 y Fs(as)f(the)g(sum)f(of)g(the)h(in)m (tegrals)f(o)m(v)m(er)h(cub)s(es)h(co)m(v)m(ering)-118 2131 y Fj(R)-52 2095 y Fp(d)-6 2131 y Fs(;)33 b(then,)g(w)m(e)h (shifted)e(the)h(argumen)m(t)g Fq(x)g Fs(so)f(as)h(to)f(cen)m(ter)i (the)f(cub)s(e)h(at)e(0.)-118 2247 y(In)23 b(the)g(next)g(step,)i(w)m (e)f(use)f(the)g(p)s(erio)s(dicit)m(y)e(of)h(the)h(p)s(erio)s(dic)d (comp)s(onen)m(ts,)25 b(and)e(P)m(arsev)-5 b(al's)23 b(form)m(ula)e(to)h(compute)-118 2363 y(the)33 b Fq(\022)s Fs(-in)m(tegral.)678 2458 y Fn(\015)678 2518 y(\015)678 2578 y(\015)678 2638 y(\015)733 2467 y(Z)789 2693 y Fl(T)839 2674 y Fd(\003)901 2603 y Fs(~)-60 b Fq(\037)951 2618 y Fp(p)987 2627 y Fc(0)1025 2603 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)1194 2562 y Fp(i)p Fm(\001\001)p Fp(\022)1296 2603 y Fs(\()p Fq( )1397 2618 y Fp(p)1433 2627 y Fc(0)1472 2603 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\))22 b Fo(\000)g Fq( )1852 2618 y Fp(p)1888 2627 y Fc(0)1927 2603 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)2082 2618 y Fp(j)2118 2603 y Fs(\)\))p Fq(d\022)2293 2458 y Fn(\015)2293 2518 y(\015)2293 2578 y(\015)2293 2638 y(\015)2348 2485 y Fi(2)2348 2702 y Fp(L)2396 2683 y Fc(2)2431 2702 y Fi(\()p Fl(R)2506 2683 y Fk(d)2542 2702 y Fi(\))689 2960 y Fs(=)793 2824 y Fn(Z)848 3050 y Fp(C)898 3059 y Fc(0)p Fk(;)p Fc(0)1003 2760 y Fn(0)1003 2939 y(@)1102 2865 y(X)1090 3085 y Fp(\015)t Fm(2)p Fl(Z)1228 3066 y Fk(d)1274 2815 y Fn(\014)1274 2875 y(\014)1274 2935 y(\014)1274 2995 y(\014)1308 2824 y(Z)1363 3050 y Fl(T)1413 3031 y Fd(\003)1476 2960 y Fs(~)-61 b Fq(\037)1525 2975 y Fp(p)1561 2984 y Fc(0)1599 2960 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)1768 2919 y Fp(ix\022)1871 2960 y Fq(e)1916 2919 y Fp(i\015)t(\022)2020 2960 y Fs(\()p Fq( )2121 2975 y Fp(p)2157 2984 y Fc(0)2196 2960 y Fs(\()p Fq(x;)17 b(\022)s Fs(\))22 b Fo(\000)g Fq( )2603 2975 y Fp(p)2639 2984 y Fc(0)2678 2960 y Fs(\()p Fq(x;)17 b(\022)2860 2975 y Fp(j)2897 2960 y Fs(\)\))p Fq(d\022)3072 2815 y Fn(\014)3072 2875 y(\014)3072 2935 y(\014)3072 2995 y(\014)3105 2842 y Fi(2)3145 2760 y Fn(1)3145 2939 y(A)3248 2960 y Fq(dx)689 3287 y Fs(=)793 3152 y Fn(Z)848 3377 y Fp(C)898 3386 y Fc(0)p Fk(;)p Fc(0)1003 3152 y Fn(Z)1058 3377 y Fl(T)1108 3358 y Fd(\003)1159 3287 y Fo(j)11 b Fs(~)-60 b Fq(\037)1248 3302 y Fp(p)1284 3311 y Fc(0)1322 3287 y Fs(\()p Fq(\022)s Fs(\)\()p Fq( )1547 3302 y Fp(p)1583 3311 y Fc(0)1622 3287 y Fs(\()p Fq(x;)17 b(\022)s Fs(\))22 b Fo(\000)h Fq( )2030 3302 y Fp(p)2066 3311 y Fc(0)2105 3287 y Fs(\()p Fq(x;)17 b(\022)2287 3302 y Fp(j)2324 3287 y Fs(\)\))p Fo(j)2427 3239 y Fi(2)2483 3287 y Fq(d\022)s(dx)689 3608 y Fs(=)793 3472 y Fn(Z)848 3698 y Fl(T)898 3679 y Fd(\003)949 3608 y Fo(j)11 b Fs(~)-60 b Fq(\037)1038 3623 y Fp(p)1074 3632 y Fc(0)1112 3608 y Fs(\()p Fq(\022)s Fs(\))p Fo(j)1264 3559 y Fi(2)1320 3437 y Fn( )1399 3472 y(Z)1454 3698 y Fp(C)1504 3707 y Fc(0)p Fk(;)p Fc(0)1609 3608 y Fo(j)p Fq( )1700 3623 y Fp(p)1736 3632 y Fc(0)1774 3608 y Fs(\()p Fq(x;)17 b(\022)s Fs(\))23 b Fo(\000)f Fq( )2182 3623 y Fp(p)2218 3632 y Fc(0)2257 3608 y Fs(\()p Fq(x;)17 b(\022)2439 3623 y Fp(j)2476 3608 y Fs(\))p Fo(j)2542 3567 y Fi(2)2581 3608 y Fq(dx)2687 3437 y Fn(!)2783 3608 y Fq(d\022)30 b Fo(\024)3024 3540 y Fq(C)p 3024 3585 77 4 v 3027 3676 a(l)3058 3647 y Fi(2)3111 3608 y Fq(:)-118 3866 y Fs(T)-8 b(o)33 b(conclude,)g(w)m(e)g(used)h(the)f(analyticit)m(y)e(of)h Fq(\022)f Fo(7!)c Fq( )1850 3881 y Fp(p)1886 3890 y Fc(0)1925 3866 y Fs(\()p Fo(\001)p Fq(;)17 b(\022)s Fs(\).)-118 3982 y(So)32 b(w)m(e)i(no)m(w)f(are)g(left)f(with)g(pro)m(ving)g(Lemma) f(2.1)h(when)i Fq(')2096 3997 y Fp(j)2165 3982 y Fs(is)e(replaced)h (with)f(the)h(function)1269 4214 y Fq(x)28 b Fo(7!)1479 4073 y Fn(\022)1553 4078 y(Z)1608 4304 y Fl(T)1658 4285 y Fd(\003)1721 4214 y Fs(~)-61 b Fq(\037)1770 4229 y Fp(p)1806 4238 y Fc(0)1845 4214 y Fs(\()p Fq(\022)s Fs(\))p Fq(e)2014 4172 y Fp(ix)p Fm(\001)p Fp(\022)2136 4214 y Fq(d\022)2235 4073 y Fn(\023)2325 4214 y Fq( )2388 4229 y Fp(p)2424 4238 y Fc(0)2463 4214 y Fs(\()p Fq(x;)17 b(\022)2645 4229 y Fp(j)2682 4214 y Fs(\))p Fq(:)-118 4442 y Fs(This)29 b(is)g(an)g(immediate)d(consequence)32 b(of)d(Lemma)f(5.1)g(when)j(one)e(pic)m(ks)h Fq(")d Fs(=)h(1)p Fq(=l)j Fs(and)e Fq(\021)i Fs(=)d(\(2)p Fq(l)3444 4406 y Fm(0)3482 4442 y Fs(+)15 b(1\))p Fq(=)p Fs(\()p Fq(\031)t(l)r Fs(\).)42 b(This)-118 4558 y(completes)32 b(the)h(pro)s(of)f(of)g (Lemma)f(2.1.)p 4063 4558 4 66 v 4067 4496 59 4 v 4067 4558 V 4125 4558 4 66 v 1270 4785 a(3.)54 b Fr(The)38 b(pr)n(oof)g(of)g(Theorem)g(1.3)94 4959 y Fs(Theorem)31 b(1.3)e(is)h(deriv)m(ed)h(from)e(Theorem)h(1.1)g(using)f(the)i(m)m (ultiscale)d(analysis)h(done)i(in)e([30,)h(8].)42 b(In)31 b(order)-118 5075 y(to)h(apply)g(the)h(results)g(of)f([30,)h(8],)f(w)m (e)i(ha)m(v)m(e)g(to)e(c)m(hec)m(k)j(t)m(w)m(o)e(assumptions:)8 5217 y Fo(\017)41 b Fs(the)33 b(W)-8 b(egner)33 b(estimate;)8 5333 y Fo(\017)41 b Fs(the)33 b(initial)c(length)j(scale)g(estimate.) -118 5475 y(W)-8 b(e)38 b(\014rst)h(state)f(the)h(W)-8 b(egner)39 b(estimate.)59 b(This)38 b(estimate)f(is)h(an)g(estimate)f (of)h(the)g(probabilit)m(y)e(of)i(presence)i(of)-118 5591 y(eigen)m(v)-5 b(alues)32 b(in)g(a)h(giv)m(en)f(in)m(terv)-5 b(al.)42 b(More)33 b(precisely)-8 b(,)33 b(w)m(e)h(pro)m(v)m(e)1969 5690 y Fg(13)p eop %%Page: 14 14 14 13 bop -118 241 a Fw(Prop)s(osition)35 b(3.1.)50 b Ff(Fix)38 b Fq(\034)46 b Fo(2)36 b Fs(\(0)p Fq(;)17 b Fs(1\))p Ff(.)56 b(Then,)39 b(ther)-5 b(e)38 b(exists)h Fq(\025)2245 256 y Fi(0)2319 241 y Fq(>)c Fs(0)k Ff(and)f Fq(c)2753 256 y Fi(0)2828 241 y Fq(>)d Fs(0)j Ff(such)h(that,)h(for)f Fq(\025)c Fo(2)g Fs(\(0)p Fq(;)17 b(\025)4027 256 y Fi(0)4066 241 y Fs(\))p Ff(,)-118 362 y(one)34 b(has,)g(for)h Fq(n)28 b Fo(\025)g Fs(1)35 b Ff(and)f Fq(\022)d Fo(2)d Fj(T)1123 326 y Fm(\003)1123 387 y Fi(2)p Fp(n)p Fi(+1)1299 362 y Ff(,)35 b(for)f Fq(I)8 b Ff(,)35 b(an)f(interval)h(in)f Fs([)p Fq(E)2350 377 y Fm(\000)2410 362 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)p 2587 282 79 4 v 17 w(E)5 b Fs(\()p Fq(\025)p 2759 307 65 4 v(!)s Fs(\))22 b(+)g Fq(c)3023 377 y Fi(0)3063 362 y Fq(\025)p Fs(])p Ff(,)35 b(one)f(has)168 547 y Fj(P)244 466 y Fn(\000)291 547 y Fo(f)p Fq(H)430 506 y Fp(n)422 572 y(!)r(;\025)533 547 y Fs(\()p Fq(\022)s Fs(\))h Ff(has)f(an)h(eigenvalue)e(in)i Fq(I)8 b Fo(g)1697 466 y Fn(\001)1770 547 y Fo(\024)28 b Fj(E)1953 466 y Fn(\000)2004 547 y Fs(#)p Fo(f)p Ff(eigenvalues)34 b(of)g Fq(H)2849 506 y Fp(n)2841 572 y(!)r(;\025)2952 547 y Fs(\()p Fq(\022)s Fs(\))h Ff(in)g Fq(I)8 b Fo(g)3332 466 y Fn(\001)3405 547 y Fo(\024)28 b Fq(C)7 b(\025)3644 506 y Fm(\000)p Fi(1)3738 547 y Fq(n)3796 506 y Fi(2)p Fp(d)3872 547 y Fo(j)p Fq(I)h Fo(j)3979 506 y Fp(\034)4021 547 y Fq(:)-4166 b Fs(\(3.1\))94 731 y(Prop)s(osition)30 b(3.1)g(is)g(pro)m(v)m(ed)j(in)d(section)h(4.)43 b(It)31 b(do)s(es)g(not)g(imply)e(Theorem)i(1.2.)42 b(The)32 b(v)m(olume)e(dep)s(endence)-118 848 y(in)j(\(3.1\))g(is)h(not)f (optimal.)45 b(But)34 b(it)e(is)i(su\016cien)m(t)h(for)e(our)h(purp)s (ose.)48 b(In)34 b([11],)g(the)g(optimal)d(v)m(olume)i(dep)s(endence) -118 964 y(w)m(as)38 b(obtained)f(but)g(for)g(an)g(appro)m(ximation)e (sc)m(heme)j(di\013eren)m(t)f(from)f(the)i(p)s(erio)s(dic)d(one)j(used) g(in)f(the)g(presen)m(t)-118 1080 y(pap)s(er.)94 1196 y(W)-8 b(e)33 b(no)m(w)h(state)f(the)g(initial)28 b(length)33 b(scale)f(estimate;)g(it)g(is)g(a)g(consequence)k(of)c(Theorem)h(1.1.) -118 1380 y Fw(Prop)s(osition)i(3.2.)50 b Ff(Fix)40 b Fq(\021)j Fo(2)d Fs(\(0)p Fq(;)17 b(d=)p Fs(\(4)p Fq(d)25 b Fs(+)i(4\)\))41 b Ff(and)g Fq(\032)e(>)h(d)h Ff(su\016ciently)g(lar) -5 b(ge.)63 b(Then,)42 b(ther)-5 b(e)41 b(exists)g Fq(a)f(>)f Fs(0)p Ff(,)-118 1497 y Fq(")-72 1512 y Fi(0)3 1497 y Fq(>)c Fs(0)40 b Ff(and)e Fq(\025)453 1512 y Fi(0)528 1497 y Fq(>)e Fs(0)j Ff(such)g(that,)i(for)e Fs(0)c Fq(<)h(\025)g(<)g (\025)1807 1512 y Fi(0)1885 1497 y Ff(and)j Fs(0)c Fq(<)h(")g(<)f(") 2514 1512 y Fi(0)2553 1497 y Ff(,)41 b(for)e Fq(\025)2841 1461 y Fm(\000)p Fp(\032)2972 1497 y Fq(<)c(n)h(<)g(e)3334 1461 y Fp(\025)3375 1437 y Fk(")3413 1497 y Ff(,)k(with)f(pr)-5 b(ob)g(ability)-118 1616 y Fs(1)22 b Fo(\000)g Fq(e)97 1580 y Fm(\000)p Fp(\015)192 1556 y Fd(\000)p Fk(")278 1616 y Ff(,)35 b(for)g Fq(E)e Fo(2)c Fq(I)742 1631 y Fp(\021)r(;\025)879 1616 y Ff(and)34 b Fq(\022)d Fo(2)d Fj(T)1301 1580 y Fm(\003)1301 1640 y Fi(2)p Fp(n)p Fi(+1)1477 1616 y Ff(,)35 b(one)f(has)911 1829 y Fo(k)p Fw(1)1017 1844 y Fp(C)1067 1853 y Fc(0)p Fk(;)p Fc(2)p Fk(n)p Fc(+1)1266 1844 y Fm(n)p Fp(C)1351 1853 y Fc(0)p Fk(;)p Fc(2)p Fk(n)p Fd(\000)p Fc(1)1557 1829 y Fs(\()p Fq(E)28 b Fo(\000)23 b Fq(H)1884 1788 y Fp(n)1876 1854 y(!)r(;\025)1987 1829 y Fs(\()p Fq(\022)s Fs(\)\))2149 1788 y Fm(\000)p Fi(1)2243 1829 y Fw(1)2299 1844 y Fp(C)2349 1853 y Fc(0)p Fk(;n)2445 1829 y Fo(k)28 b(\024)g Fq(e)2673 1788 y Fp(a)2710 1722 y Fo(p)p 2794 1722 237 4 v 2794 1788 a Fm(j)p Fp(E)t Fm(\000)p Fp(\025)p 2966 1750 47 3 v(!)q Fm(j)p Fp(n)3078 1829 y Fq(:)-3223 b Fs(\(3.2\))-118 2013 y(Using)29 b(Prop)s(ositions)e (3.1)i(and)g(3.2)g(for)f Fq(\022)j Fs(=)c(0,)j(Theorem)f(1.2)g(is)f(an) h(immediate)e(consequence)32 b(of)c(Theorem)h(12.2)-118 2130 y(and)k(Prop)s(osition)d(13.1)i(of)g([30],)h(and)g(Theorem)f(3.8)h (of)f([8].)-118 2338 y(3.1.)56 b Fw(The)34 b(pro)s(of)g(of)g(the)f (initial)e(length)i(scale)g(estimate.)48 b Fs(W)-8 b(e)29 b(no)m(w)h(pro)m(v)m(e)h(Prop)s(osition)c(3.2.)42 b(Fix)29 b Fq(\021)t(;)17 b(\021)3995 2302 y Fm(0)4047 2338 y Fs(as)-118 2455 y(in)32 b(Prop)s(osition)f(3.2)h(i.e.)44 b Fq(\021)31 b(<)d(\021)1082 2418 y Fm(0)1133 2455 y Fq(<)g(d=)p Fs(\(4)p Fq(d)21 b Fs(+)h(4\))32 b(and)h(\014x)h Fq(\032)28 b(>)g(d)p Fs(.)43 b(De\014ne)34 b(the)f(ev)m(en)m(t)h(\012) 3143 2470 y Fp(\015)t(;\021)3240 2451 y Fd(0)3264 2470 y Fp(;n)3358 2455 y Fs(=)28 b Fo(f)p Fs(there)33 b(exists)h(an)-118 2579 y(eigen)m(v)-5 b(alue)30 b(of)h Fq(H)545 2543 y Fp(n)537 2605 y(!)r(;\025)679 2579 y Fs(in)f([)p Fq(E)890 2594 y Fm(\000)949 2579 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)p 1126 2499 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 1299 2524 65 4 v(!)s Fs(\))19 b Fo(\000)g Fq(\025)1573 2543 y Fi(1+)p Fp(\021)1700 2519 y Fd(0)1727 2579 y Fs(])p Fo(g)p Fs(.)43 b(Using)31 b(Theorem)g(2.1)f(in)h(conjunction)f(with)h(Theorem)g(1.1,) -118 2723 y(w)m(e)36 b(obtain)e(that,)h(there)h(exists)f Fq(\015)1149 2738 y Fp(\021)1223 2723 y Fq(>)c Fs(0)k(and)g Fq(")c(>)h Fs(0)i(suc)m(h)j(that,)e(for)f(0)e Fq(<)f(\025)h(<)f(\025) 2930 2738 y Fp(\021)3007 2723 y Fs(and)k Fq(\025)3256 2687 y Fm(\000)p Fp(\032)3383 2723 y Fo(\024)d Fq(n)g Fo(\024)g Fq(e)3736 2687 y Fp(\025)3777 2663 y Fd(\000)p Fk("=)p Fc(2)3925 2723 y Fs(,)k(one)-118 2839 y(has)1238 3013 y Fj(P)p Fs(\(\012)1405 3028 y Fp(\025;\021)1503 3009 y Fd(0)1529 3028 y Fp(;n)1595 3013 y Fs(\))28 b Fo(\024)g Fq(C)7 b(e)1888 2972 y Fp(d\025)1965 2949 y Fd(\000)p Fk("=)p Fc(2)2113 3013 y Fq(e)2158 2972 y Fm(\000)p Fp(\025)2254 2949 y Fd(\000)p Fk(")2368 3013 y Fo(\024)28 b Fq(e)2518 2972 y Fm(\000)2583 2945 y Fc(1)p 2583 2957 31 3 v 2583 2998 a(2)2623 2972 y Fp(\025)2664 2949 y Fd(\000)p Fk(")2750 3013 y Fq(:)-118 3193 y Fs(By)38 b(a)g(Com)m(b)s(es-Thomas)f(estimate)g(\(see)i(e.g.)f([10])f(and)h (references)i(therein\),)f(for)e Fq(E)43 b Fo(2)37 b Fq(I)3310 3208 y Fp(\021)r(;\025)3450 3193 y Fs(and)h Fq(!)i Fo(62)d Fs(\012)3919 3208 y Fp(\025;\021)4017 3189 y Fd(0)4040 3208 y Fp(;n)4107 3193 y Fs(,)-118 3309 y(one)c(has)622 3382 y Fn(\015)622 3442 y(\015)622 3501 y(\015)678 3496 y Fw(1)734 3511 y Fp(C)784 3520 y Fk(\015)s(;)p Fc(0)877 3496 y Fs(\()p Fq(E)28 b Fo(\000)23 b Fq(H)1204 3455 y Fp(n)1196 3521 y(!)r(;\025)1307 3496 y Fs(\()p Fq(\022)s Fs(\)\))1469 3455 y Fm(\000)p Fi(1)1563 3496 y Fw(1)1619 3511 y Fp(C)1669 3530 y Fk(\015)1704 3516 y Fd(0)1728 3530 y Fk(;)p Fc(0)1785 3382 y Fn(\015)1785 3442 y(\015)1785 3501 y(\015)1868 3496 y Fo(\024)29 b Fq(C)7 b(\025)2108 3455 y Fm(\000)p Fi(1)p Fm(\000)p Fp(\021)2294 3496 y Fq(e)2339 3455 y Fm(\000)2394 3383 y Fo(p)p 2478 3383 581 4 v 2478 3455 a Fm(j)p Fp(E)t Fm(\000)p 2609 3401 56 3 v Fp(E)r Fi(\()p Fp(\025)p 2731 3418 47 3 v(!)s Fi(\)+)p Fp(\025)2901 3436 y Fk(\021)2935 3422 y Fd(0)2958 3436 y Fc(+1)3039 3455 y Fm(j)o(j)p Fp(\015)t Fm(\000)p Fp(\015)3213 3432 y Fd(0)3236 3455 y Fm(j)p Fp(=C)3350 3496 y Fq(;)-3495 b Fs(\(3.3\))-118 3757 y(for)26 b(some)h(constan)m(t)h Fq(C)34 b Fs(indep)s(enden)m(t)29 b(of)d Fq(\025)h Fs(and)g Fq(\021)t Fs(.)42 b(De\014ne)27 b Fq(\016)t Fs(\()p Fq(E)6 b Fs(\))28 b(:=)2454 3635 y Fn(q)p 2553 3635 862 4 v 2553 3757 a Fo(j)p Fq(E)g Fo(\000)p 2781 3677 79 4 v 23 w Fq(E)6 b Fs(\()p Fq(\025)p 2954 3702 65 4 v(!)s Fs(\))22 b(+)g Fq(\025)3233 3728 y Fi(1+)p Fp(\021)3360 3709 y Fd(0)3387 3757 y Fo(j)p Fs(.)42 b(W)-8 b(e)27 b(notice)g(that,)-118 3907 y(for)32 b Fq(E)i Fo(2)28 b Fq(I)274 3922 y Fp(\021)r(;\025)376 3907 y Fs(,)33 b Fq(E)28 b Fo(\000)p 636 3827 79 4 v 23 w Fq(E)6 b Fs(\()p Fq(\025)p 809 3852 65 4 v(!)s Fs(\))28 b Fo(\025)g Fq(\025)1101 3871 y Fi(1+)p Fp(\021)1233 3907 y Fs(;)33 b(as)f Fq(\021)g(<)27 b(\021)1647 3871 y Fm(0)1670 3907 y Fs(,)33 b(for)f Fq(\025)g Fs(su\016cien)m(tly)i (small,)c(one)j(has)1478 4136 y Fq(\016)t Fs(\()p Fq(E)6 b Fs(\))28 b Fo(\025)1822 4068 y Fs(1)p 1822 4113 49 4 v 1822 4204 a(2)1881 4009 y Fn(q)p 1980 4009 531 4 v 1980 4136 a Fo(j)p Fq(E)g Fo(\000)p 2208 4056 79 4 v 23 w Fq(E)6 b Fs(\()p Fq(\025)p 2381 4081 65 4 v(!)s Fs(\))p Fo(j)o Fq(:)-118 4345 y Fs(Pic)m(k)39 b(0)e Fq(<)g(")351 4360 y Fi(0)428 4345 y Fq(<)g(")p Fs(.)61 b(Summing)37 b(\(3.3\))h(o)m(v)m(er)i Fq(\015)i Fo(2)c Fq(C)1836 4360 y Fi(0)p Fp(;)p Fi(2)p Fp(n)p Fi(+1)2089 4345 y Fo(n)26 b Fq(C)2235 4360 y Fi(0)p Fp(;)p Fi(2)p Fp(n)p Fm(\000)p Fi(1)2501 4345 y Fs(and)38 b Fq(\015)2752 4309 y Fm(0)2813 4345 y Fo(2)g Fq(C)2987 4360 y Fi(0)p Fp(;n)3127 4345 y Fs(and)g(using)g(the)h(fact)f(that)-118 4472 y Fq(\025)-61 4436 y Fm(\000)p Fp(\032)63 4472 y Fq(<)29 b(n)g(<)g(e)405 4436 y Fp(\025)446 4413 y Fd(\000)p Fk(")523 4428 y Fc(0)567 4472 y Fs(,)k(w)m(e)i(get)e(that,)g(for)g Fq(!)g Fo(62)c Fs(\012)1583 4487 y Fp(\015)t(;\021)1680 4468 y Fd(0)1703 4487 y Fp(;n)1803 4472 y Fs(and)34 b Fq(E)h Fo(2)29 b Fq(I)2239 4487 y Fp(\021)r(;\025)2342 4472 y Fs(,)34 b(estimate)e(\(3.2\))h(holds.)46 b(This)33 b(completes)g(the)-118 4589 y(pro)s(of)f(of)g(Theorem)h(3.2.)1145 4835 y(4.)55 b Fr(Pr)n(oof)39 b(of)f(the)f(Wegner)g(estima)-7 b(te)94 5010 y Fs(The)40 b(pro)s(of)d(of)h(Prop)s(osition)f(3.1)h(follo)m(ws)f (the)i(philosoph)m(y)f(in)m(tro)s(duced)h(in)e([17])h(and)h(is)f(quite) g(similar)d(to)-118 5126 y(the)30 b(pro)s(of)f(of)h(Theorem)g(1.2)f(in) g([11].)43 b(F)-8 b(or)29 b(the)h(reader's)h(con)m(v)m(enience,)i(w)m (e)e(nev)m(ertheless)h(repro)s(duce)f(the)f(details)-118 5242 y(here.)94 5358 y(As)45 b(ab)s(o)m(v)m(e,)i(w)m(e)e(\014x)f(a)g (band)g(edge)g Fq(E)1524 5373 y Fm(\000)1584 5358 y Fs(\(0\))f(and)h Fq(\022)50 b Fo(2)d Fj(T)2224 5322 y Fm(\003)2224 5383 y Fi(2)p Fp(n)p Fi(+1)2400 5358 y Fs(.)77 b(All)42 b(the)i(op)s (erators)g(w)m(e)g(no)m(w)h(discuss)g(are)-118 5475 y(considered)e (with)f Fq(\022)s Fs(-quasi-p)s(erio)s(dic)e(b)s(oundary)j(condition)d (in)i(the)h(sense)h(explained)e(in)f(section)i(2.2;)j(in)c(our)-118 5591 y(notations,)d(w)m(e)h(forget)e(ab)s(out)g(the)h(parameter)f Fq(\022)k Fs(i.e.)61 b(w)m(e)39 b(write)g Fq(E)2461 5606 y Fp(k)2542 5591 y Fs(instead)f(of)h Fq(E)3074 5606 y Fp(k)3116 5591 y Fs(\()p Fq(\022)s Fs(\),)i Fq(H)46 b Fs(instead)38 b(of)g Fq(H)8 b Fs(\()p Fq(\022)s Fs(\),)1969 5690 y Fg(14)p eop %%Page: 15 15 15 14 bop -118 241 a Fs(etc.)60 b(Our)38 b(statemen)m(t)g(are)g (uniform)e(in)h(the)i(parameter)e Fq(\022)s Fs(.)60 b(Recall)36 b(that)i(the)g(Flo)s(quet)f(eigen)m(v)-5 b(alues)38 b(of)f Fq(H)46 b Fs(are)-118 357 y(denoted)33 b(b)m(y)h(\()p Fq(E)496 372 y Fp(k)539 357 y Fs(\))577 372 y Fp(k)r Fm(\025)p Fi(1)742 357 y Fs(and)f(ordered)g(increasingly)-8 b(.)42 b(De\014ne)33 b(\005)2223 372 y Fi(0)2290 357 y Fs(=)28 b(\005)2467 321 y Fp(n)2467 387 y()27 b Fs(0)32 b(suc)m(h)i(that,)f(for)f Fq(\025)g Fs(su\016cien)m(tly)i (small,)p 1474 2118 79 4 v 1474 2198 a Fq(E)6 b Fs(\()p Fq(\025)p 1647 2143 65 4 v(!)s Fs(\))22 b Fo(\000)p 1871 2118 79 4 v 23 w Fq(E)6 b Fs(\()p Fq(\025!)2109 2157 y Fi(+)2167 2198 y Fs(\))28 b Fo(\025)g Fq(\025=C)r(:)-2661 b Fs(\(4.2\))-118 2403 y(F)-8 b(or)42 b(more)g(details,)i(w)m(e)g (refer)g(to)e(section)h(5.1.)74 b(W)-8 b(e)43 b(rewrite)g Fq(H)2351 2418 y Fp(!)r(;\025)2507 2403 y Fs(=)p 2628 2323 89 4 v 45 w Fq(H)2717 2419 y Fp(\025!)2804 2400 y Fc(+)2888 2403 y Fs(+)29 b Fq(\025V)3112 2418 y Fi(~)-40 b Fp(!)3200 2403 y Fs(where)44 b(\()8 b(~)-57 b Fq(!)3591 2418 y Fp(\015)3636 2403 y Fs(\))3674 2418 y Fp(\015)3763 2403 y Fs(=)45 b(\()p Fq(!)3983 2418 y Fp(\015)4056 2403 y Fo(\000)-118 2527 y Fq(!)-53 2491 y Fi(+)5 2527 y Fs(\))43 2542 y Fp(\015)88 2527 y Fs(.)75 b(So,)45 b(for)e Fq(\025)g Fs(su\016cien)m(tly)h(small,)f(the)h(op)s(erator)p 2001 2447 V 42 w Fq(H)2090 2543 y Fp(\025!)2177 2524 y Fc(+)2275 2527 y Fs(has)g(no)f(sp)s(ectrum)g(in)f(the)i(in)m(terv)-5 b(al)42 b([\()p Fq(E)3845 2542 y Fm(\000)3904 2527 y Fs(\(0\))29 b(+)-118 2653 y(2)p Fq(E)3 2668 y Fi(+)62 2653 y Fs(\(0\)\))p Fq(=)p Fs(3)p Fq(;)p 367 2573 79 4 v 17 w(E)5 b Fs(\()p Fq(\025)p 539 2598 65 4 v(!)s Fs(\))32 b(+)g Fq(C)7 b(\025)p Fs(])47 b(\(for)g(some)g Fq(C)59 b(>)53 b Fs(0\).)87 b(Hence,)52 b(for)47 b(an)m(y)h Fq(n)k Fo(\025)h Fs(1,)e(the)d(op)s(erator)p 3564 2573 89 4 v 46 w Fq(H)3653 2592 y Fp(n)3653 2678 y(\025!)3740 2659 y Fc(+)3842 2653 y Fs(has)g(no)-118 2774 y(sp)s(ectrum)37 b(in)e([\()p Fq(E)566 2789 y Fm(\000)626 2774 y Fs(\(0\))24 b(+)h(2)p Fq(E)997 2789 y Fi(+)1056 2774 y Fs(\(0\)\))p Fq(=)p Fs(3)p Fq(;)p 1361 2694 79 4 v 17 w(E)5 b Fs(\()p Fq(\025)p 1533 2720 65 4 v(!)s Fs(\))25 b(+)f Fq(C)7 b(\025)p Fs(].)55 b(W)-8 b(e)37 b(decomp)s(ose)g(\005)2743 2789 y Fi(0)2817 2774 y Fs(=)d(\005)3000 2733 y Fi(+)3000 2799 y(0)3084 2774 y Fs(+)25 b(\005)3258 2733 y Fm(\000)3258 2799 y Fi(0)3353 2774 y Fs(where)38 b(\005)3712 2733 y Fm(\000)3712 2799 y Fi(0)3805 2774 y Fs(=)c(\005)3988 2738 y Fp(n)3988 2800 y()g Fs(0)33 b(suc)m(h)i(that)d([)p Fq(E)2143 5490 y Fi(0)2206 5475 y Fo(\000)23 b Fq(";)17 b(E)2468 5490 y Fi(0)2529 5475 y Fs(+)22 b Fq(")p Fs(])29 b Fo(\032)f Fs([)p Fq(E)2933 5490 y Fi(0)2973 5475 y Fq(;)p 3017 5395 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 3190 5420 65 4 v(!)s Fs(\))22 b(+)h Fq(C)7 b(\025=)p Fs(2],)32 b(if)g Fq(E)39 b Fs(is)33 b(an)-118 5591 y(eigen)m(v)-5 b(alue)23 b(in)f([)p Fq(E)542 5606 y Fi(0)585 5591 y Fo(\000)s Fq(";)17 b(E)827 5606 y Fi(0)869 5591 y Fs(+)s Fq(")p Fs(],)25 b(then,)h(b)m(y)f(\(4.5\),)g Fq(G)p Fs(\()p Fq(E)1882 5606 y Fi(0)1922 5591 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(has)j(an)f(eigen)m(v)-5 b(alue)23 b(in)f([)p Fo(\000)p Fs(1)s Fo(\000)s Fq(C)7 b("\025)3459 5555 y Fm(\000)p Fi(1)3554 5591 y Fq(;)17 b Fo(\000)p Fs(1)s(+)s Fq(C)7 b("\025)3986 5555 y Fm(\000)p Fi(1)4080 5591 y Fs(].)1969 5690 y Fg(15)p eop %%Page: 16 16 16 15 bop -118 241 a Fs(This)33 b(yields)-35 431 y(#)p Fo(f)p Fs(eigen)m(v)-5 b(alues)33 b(of)f Fq(H)801 390 y Fp(n)793 455 y(!)r(;\025)936 431 y Fs(in)g([)p Fq(E)1149 446 y Fi(0)1211 431 y Fo(\000)23 b Fq(";)17 b(E)1473 446 y Fi(0)1534 431 y Fs(+)22 b Fq(")p Fs(])p Fo(g)1346 591 y(\024)28 b Fs(#)p Fo(f)p Fs(eigen)m(v)-5 b(alues)33 b(of)f Fq(G)p Fs(\()p Fq(E)2385 606 y Fi(0)2424 591 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))32 b(in)f([)p Fo(\000)p Fs(1)23 b Fo(\000)f Fq(C)7 b("\025)3272 550 y Fm(\000)p Fi(1)3366 591 y Fq(;)17 b Fo(\000)p Fs(1)22 b(+)g Fq(C)7 b("\025)3836 550 y Fm(\000)p Fi(1)3930 591 y Fs(])p Fo(g)p Fq(;)-118 752 y Fs(the)33 b(eigen)m(v)-5 b(alues)32 b(b)s(eing)g(coun)m (ted)i(with)e(m)m(ultiplicit)m(y)-8 b(.)-118 868 y(Hence,)34 b(one)f(has)-118 1058 y(\(4.6\))82 b Fj(E)13 b Fs(\(#)p Fo(f)p Fs(eigen)m(v)-5 b(alues)38 b(of)33 b Fq(H)1106 1017 y Fp(n)1098 1083 y(!)r(;\025)1241 1058 y Fs(in)f([)p Fq(E)1454 1073 y Fi(0)1516 1058 y Fo(\000)22 b Fq(";)17 b(E)1777 1073 y Fi(0)1839 1058 y Fs(+)22 b Fq(")p Fs(])p Fo(g)p Fs(\))1220 1219 y Fo(\024)28 b Fj(E)13 b Fs(\(#)p Fo(f)p Fs(eigen)m(v)-5 b(alues)38 b(of)33 b Fq(G)p Fs(\()p Fq(E)2364 1234 y Fi(0)2403 1219 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))31 b(in)h([)p Fo(\000)p Fs(1)23 b Fo(\000)f Fq(C)7 b("\025)3251 1178 y Fm(\000)p Fi(1)3345 1219 y Fq(;)17 b Fo(\000)p Fs(1)22 b(+)g Fq(C)7 b("\025)3815 1178 y Fm(\000)p Fi(1)3909 1219 y Fs(])p Fo(g)p Fs(\))p Fq(:)-118 1385 y Fs(T)-8 b(o)29 b(estimate)g(the)h(last)f(exp)s(ectation,)h(w)m (e)h(use)f(the)g(computations)e(done)i(in)f([11].)42 b(De\014ne)30 b Fq(I)3286 1400 y Fp(")3351 1385 y Fs(=)d([)p Fo(\000)p Fs(1)16 b Fo(\000)g Fq(\024;)h Fo(\000)p Fs(1)f(+)g Fq(\024)p Fs(])-118 1502 y(where)25 b Fq(\024)j Fs(=)g Fq(C)7 b("\025)523 1465 y Fm(\000)p Fi(1)617 1502 y Fs(.)40 b(Let)25 b Fq(\032)g Fs(b)s(e)f(a)g(nonnegativ)m(e,)j(smo)s(oth,)e (monotone)e(decreasing)i(function)f(suc)m(h)i(that)e Fq(\032)p Fs(\()p Fq(x)p Fs(\))k(=)g(1,)-118 1618 y(for)45 b Fq(x)50 b(<)f Fo(\000)p Fq(\024=)p Fs(2,)g(and)d Fq(\032)p Fs(\()p Fq(x)p Fs(\))k(=)f(0,)g(for)c Fq(x)50 b Fo(\025)g Fq(\024=)p Fs(2.)81 b(W)-8 b(e)46 b(can)g(assume)g(that)f Fq(\032)h Fs(has)g(compact)f(supp)s(ort)g(since)-118 1734 y Fq(G)p Fs(\()p Fq(E)69 1749 y Fi(0)108 1734 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))31 b(is)i(lo)m(w)m(er)f(semib)s(ounded)h (indep)s(enden)m(tly)g(of)f Fq(n)p Fs(.)44 b(Then,)34 b(one)f(has)612 1887 y Fj(E)13 b Fs(\(#)p Fo(f)p Fs(eigen)m(v)-5 b(alues)38 b(of)33 b Fq(G)p Fs(\()p Fq(E)1651 1902 y Fi(0)1690 1887 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))31 b(in)h([)p Fo(\000)p Fs(1)22 b Fo(\000)h Fq(\024;)17 b Fo(\000)p Fs(1)22 b(+)g Fq(\024)p Fs(])p Fo(g)p Fs(\))640 2038 y Fo(\024)28 b Fj(E)13 b Fo(f)p Fs(tr)5 b([)p Fq(\032)p Fs(\()p Fq(\025G)p Fs(\()p Fq(E)1296 2053 y Fi(0)1336 2038 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)g Fs(3)p Fq(\024=)p Fs(2\))g Fo(\000)h Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)2511 2053 y Fi(0)2550 2038 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g(+)g(3)p Fq(\024=)p Fs(2\)])p Fo(g)640 2288 y(\024)28 b Fj(E)823 2118 y Fn(\()909 2288 y Fs(tr)1001 2118 y Fn(")1059 2153 y(Z)1159 2179 y Fi(3)p Fp(\024=)p Fi(2)1114 2378 y Fm(\000)p Fi(3)p Fp(\024=)p Fi(2)1364 2221 y Fq(d)p 1347 2266 86 4 v 1347 2357 a(dt)1442 2288 y(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)1717 2303 y Fi(0)1757 2288 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)h Fq(t)p Fs(\))k Fq(dt)2481 2118 y Fn(#\))2636 2288 y Fq(:)-118 2137 y Fs(\(4.7\))-118 2541 y(In)33 b(order)g(to)f(ev)-5 b(aluate)32 b(the)h Fq(\032)978 2505 y Fm(0)1034 2541 y Fs(term)f(in)g(\(4.7\))o(,)h(de\014ne)h(the)f (v)m(ector)g(\014eld)1636 2723 y Fo(V)j Fs(=)1906 2629 y Fn(X)1837 2852 y Fp(\015)t Fm(2)p Fl(Z)1974 2829 y Fk(d)1974 2873 y Fc(2)p Fk(n)p Fc(+1)2143 2723 y Fs(~)-57 b Fq(!)2196 2738 y Fp(\015)2240 2723 y Fq(@)2291 2738 y Fp(!)2335 2746 y Fk(\015)-118 3012 y Fs(and)33 b(compute)1090 3173 y Fo(V)8 b Fq(G)p Fs(\()p Fq(E)1346 3188 y Fi(0)1386 3173 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))26 b(=)h Fq(G)p Fs(\()p Fq(E)1950 3188 y Fi(0)1990 3173 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h Fq(\025G)2491 3188 y Fi(1)2530 3173 y Fs(\()p Fq(E)2640 3188 y Fi(0)2679 3173 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))-118 3353 y(where)29 b Fq(G)236 3368 y Fi(1)275 3353 y Fs(\()p Fq(E)385 3368 y Fi(0)425 3353 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))26 b(is)h(b)s(ounded)h(uniformly)d(in)i Fq(!)t Fs(,)h Fq(\025)g Fs(and)f Fq(E)2194 3368 y Fi(0)2262 3353 y Fo(2)h Fs([)p Fq(E)2455 3368 y Fi(0)2494 3353 y Fq(;)p 2538 3273 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 2711 3298 65 4 v(!)s Fs(\))12 b(+)g Fq(C)7 b(\025=)p Fs(2].)41 b(This)28 b(follo)m(ws)e(from)g(the)-118 3477 y(fact)36 b(that)g(\005)366 3435 y Fi(+)366 3501 y(0)p 425 3397 89 4 v 425 3477 a Fq(H)514 3415 y Fp(n)514 3501 y(\025!)601 3482 y Fc(+)656 3477 y Fs(\005)729 3435 y Fi(+)729 3501 y(0)824 3477 y Fs(do)s(es)h(not)f(dep)s(end)h(on)g Fq(!)1768 3492 y Fp(\015)1812 3477 y Fs(,)g(that)f Fo(V)8 b Fq(V)2239 3440 y Fp(n)2222 3503 y Fi(~)-40 b Fp(!)r(;\025)2362 3477 y Fs(=)34 b Fq(V)2550 3440 y Fp(n)2534 3503 y Fi(~)-40 b Fp(!)r(;\025)2640 3477 y Fs(,)37 b(that)f Fo(V)8 b Fs(\000)3049 3440 y Fi(0)3089 3477 y Fs(\()p Fq(!)t(;)17 b(E)3308 3492 y Fi(0)3346 3477 y Fs(\))36 b(and)h Fo(V)8 b Fs(\000)3744 3440 y Fi(0+)3838 3477 y Fs(\()p Fq(!)t(;)17 b(E)4057 3492 y Fi(0)4096 3477 y Fs(\))-118 3612 y(sta)m(y)33 b(b)s(ounded,)h(and)e(that)h(for)f Fq(E)1136 3627 y Fi(0)1203 3612 y Fo(2)c Fs([)p Fq(E)1396 3627 y Fi(0)1436 3612 y Fq(;)p 1480 3532 79 4 v 17 w(E)6 b Fs(\()p Fq(\025)p 1653 3557 65 4 v(!)s Fs(\))22 b(+)g Fq(C)7 b(\025=)p Fs(2],)32 b(dist\()p Fq(E)6 b(;)17 b(\033)t Fs(\()p 2608 3532 89 4 v Fq(H)2696 3627 y Fp(\025)p 2737 3589 47 3 v(!)2788 3612 y Fs(\)\))27 b Fo(\025)h Fq(\025=C)39 b Fs(for)32 b(some)h Fq(C)i(>)27 b Fs(1.)-118 3728 y(W)-8 b(e)33 b(no)m(w)g(write)f(the)h Fq(\032)720 3692 y Fm(0)777 3728 y Fs(term)f(in)f(\(4.7\))h(as)531 3884 y Fo(V)8 b Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)875 3899 y Fi(0)915 3884 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)h Fq(t)p Fs(\))k(=)h Fq(\032)1707 3843 y Fm(0)1730 3884 y Fs(\()p Fq(G)p Fs(\()p Fq(E)1955 3899 y Fi(0)1995 3884 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)g Fq(t)p Fs(\))p Fo(V)8 b Fq(G)p Fs(\()p Fq(E)2861 3899 y Fi(0)2901 3884 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))1553 4038 y(=)28 b Fq(\032)1707 3997 y Fm(0)1730 4038 y Fs(\()p Fq(G)p Fs(\()p Fq(E)1955 4053 y Fi(0)1995 4038 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)g Fq(t)p Fs(\)\()p Fq(G)p Fs(\()p Fq(E)2830 4053 y Fi(0)2870 4038 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h Fq(O)s Fs(\()p Fq(\025)p Fs(\)\))-118 4205 y(W)-8 b(e)30 b(no)m(w)h(note)f(that)g Fq(\032)721 4169 y Fm(0)772 4205 y Fo(\024)e Fs(0,)j(and)f(that)f(on)h (supp)p Fq(\032)1762 4169 y Fm(0)1787 4205 y Fs(,)h(one)f(has)g Fq(G)p Fs(\()p Fq(E)2379 4220 y Fi(0)2419 4205 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))26 b Fo(\024)i Fs(\()p Fo(\000)p Fs(1)17 b(+)g(2)p Fq(\024)p Fs(\).)42 b(So,)31 b(for)e Fq(\025)h Fs(su\016cien)m(tly)-118 4321 y(small,)g(w)m(e)k(obtain)425 4530 y Fo(\000)p Fq(\032)552 4489 y Fm(0)576 4530 y Fs(\()p Fq(G)p Fs(\()p Fq(E)801 4545 y Fi(0)840 4530 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)h Fq(t)p Fs(\))k Fo(\024)i(\000)1847 4463 y Fs(1)p 1671 4507 401 4 v 1671 4598 a(2\(1)22 b Fo(\000)g Fs(2)p Fq(\024)p Fs(\))2170 4435 y Fn(X)2098 4659 y Fp(!)r Fm(2)p Fl(Z)2241 4636 y Fk(d)2241 4680 y Fc(2)p Fk(n)p Fc(+1)2410 4530 y Fs(~)-57 b Fq(!)2463 4545 y Fp(\015)2544 4463 y Fq(@)5 b(\032)p 2517 4507 163 4 v 2517 4598 a(@)g(!)2634 4613 y Fp(\015)2689 4530 y Fs(\()p Fq(G)p Fs(\()p Fq(E)2914 4545 y Fi(0)2953 4530 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)h Fq(t)p Fs(\))p Fq(:)-3709 b Fs(\(4.8\))-118 4825 y(With)37 b(this)h(estimate,)h(and)f(the)g(fact)g(that)g Fq(d\032)p Fs(\()p Fq(x)26 b Fs(+)g(1)g Fo(\000)g Fq(t)p Fs(\))p Fq(=dt)37 b Fs(=)g Fo(\000)p Fq(\032)2528 4789 y Fm(0)2552 4825 y Fs(\()p Fq(x)26 b Fs(+)g(1)f Fo(\000)i Fq(t)p Fs(\),)39 b(the)g(righ)m(t)e(side)h(of)45 b(\(4.7\))37 b(is)-118 4941 y(b)s(ounded)c(ab)s(o)m(v)m(e)h(b)m(y)652 5172 y Fo(\000)915 5105 y Fs(1)p 739 5150 401 4 v 739 5241 a(2\(1)22 b Fo(\000)h Fs(2)p Fq(\024)p Fs(\))1235 5078 y Fn(X)1166 5301 y Fp(\015)t Fm(2)p Fl(Z)1304 5278 y Fk(d)1304 5322 y Fc(2)p Fk(n)p Fc(+1)1465 5037 y Fn(Z)1564 5063 y Fi(3)p Fp(\024=)p Fi(2)1520 5262 y Fm(\000)p Fi(3)p Fp(\024=)p Fi(2)1742 5172 y Fj(E)12 b Fo(f)c Fs(~)-57 b Fq(!)1913 5187 y Fp(\015)2026 5105 y Fq(@)p 1973 5150 163 4 v 1973 5241 a(@)5 b(!)2090 5256 y Fp(\015)2145 5172 y Fs(tr[)p Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)2523 5187 y Fi(0)2563 5172 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)g Fq(t)p Fs(\)])p Fo(g)p Fq(dt:)-3481 b Fs(\(4.9\))-118 5475 y(In)38 b(order)g(to)f(ev)-5 b(aluate)37 b(the)h(exp)s(ectation,)h (w)m(e)g(select)f(one)g(random)f(v)-5 b(ariable,)37 b(sa)m(y)i Fq(!)3113 5490 y Fp(\015)3195 5475 y Fs(\()p Fq(\015)i Fo(2)c Fj(Z)3497 5438 y Fp(d)3497 5499 y Fi(2)p Fp(n)p Fi(+1)3667 5475 y Fs(\),)i(and)e(\014rst)-118 5591 y(in)m(tegrate)d (with)g(resp)s(ect)h(to)f(this)g(v)-5 b(ariable)32 b(using)i(h)m(yp)s (othesis)i(\(H3\).)48 b(Let)35 b Fq(h)2778 5606 y Fi(0)2851 5591 y Fs(b)s(e)g(the)f(common)f(densit)m(y)j(of)d(the)1969 5690 y Fg(16)p eop %%Page: 17 17 17 16 bop -118 244 a Fs(random)29 b(v)-5 b(ariables)29 b(\()p Fq(!)738 259 y Fp(\015)782 244 y Fs(\))820 264 y Fp(\015)t Fm(2)p Fl(Z)957 245 y Fk(d)992 244 y Fs(;)i(b)m(y)g (\(H3\),)f(there)h(is)e(a)h(decomp)s(osition)e(supp\()p Fq(!)2798 259 y Fi(0)2839 244 y Fs(\))f(=)h Fo([)3074 203 y Fp(N)7 b Fm(\000)p Fi(1)3074 272 y Fp(l)q Fi(=0)3232 244 y Fs(\()p Fq(M)3364 259 y Fp(l)3390 244 y Fq(;)17 b(M)3528 259 y Fp(l)q Fi(+1)3644 244 y Fs(\))30 b(so)g(that)g Fq(h)4094 259 y Fi(0)-118 377 y Fs(is)e(absolutely)f(con)m(tin)m(uous)i (on)f(eac)m(h)h(subin)m(terv)-5 b(al.)41 b(Let)1972 351 y(~)1971 377 y Fq(h)2027 392 y Fi(0)2094 377 y Fs(b)s(e)29 b(the)f(function)2765 351 y(~)2764 377 y Fq(h)2820 392 y Fi(0)2860 377 y Fs(\()p Fq(x)p Fs(\))g(:=)f Fq(xh)3260 392 y Fi(0)3300 377 y Fs(\()p Fq(x)p Fs(\).)42 b(As)3641 351 y(~)3640 377 y Fq(h)3696 392 y Fi(0)3764 377 y Fs(is)27 b(lo)s(cally)-118 493 y(absolutely)32 b(con)m(tin)m(uous,)h(w)m(e)h (can)f(in)m(tegrate)f(b)m(y)h(parts)g(and)g(obtain)223 587 y Fn(\014)223 646 y(\014)223 706 y(\014)223 766 y(\014)257 595 y(Z)312 821 y Fl(R)381 731 y Fq(d!)493 746 y Fp(\015)538 705 y Fs(~)537 731 y Fq(h)593 746 y Fi(0)632 731 y Fs(\()p Fq(!)731 746 y Fp(\015)775 731 y Fs(\))876 664 y Fq(@)p 823 708 163 4 v 823 799 a(@)5 b(!)940 814 y Fp(\015)995 731 y Fs(tr)p Fo(f)p Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)1396 746 y Fi(0)1436 731 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))j(+)i(1)g Fo(\000)h Fq(t)p Fs(\))f Fo(\000)h Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)2443 746 y Fi(0)2483 731 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))2731 690 y Fp(M)2799 699 y Fc(0)2832 690 y Fp(;\015)2918 731 y Fs(+)22 b(1)g Fo(\000)g Fq(t)p Fs(\))p Fo(g)3309 587 y Fn(\014)3309 646 y(\014)3309 706 y(\014)3309 766 y(\014)235 1045 y Fs(=)338 870 y Fn(\014)338 930 y(\014)338 990 y(\014)338 1050 y(\014)338 1109 y(\014)371 920 y Fp(N)7 b Fm(\000)p Fi(1)376 950 y Fn(X)392 1162 y Fp(l)q Fi(=0)541 909 y Fn(Z)641 935 y Fp(M)709 947 y Fk(l)p Fc(+1)597 1134 y Fp(M)665 1146 y Fk(l)831 1045 y Fq(d!)943 1060 y Fp(\015)987 1018 y Fs(~)986 1045 y Fq(h)1042 1060 y Fi(0)1082 1045 y Fs(\()p Fq(!)1181 1060 y Fp(\015)1225 1045 y Fs(\))1326 977 y Fq(@)p 1273 1022 V 1273 1113 a(@)e(!)1390 1128 y Fp(\015)1445 1045 y Fs(tr)p Fo(f)p Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)1846 1060 y Fi(0)1885 1045 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)h Fq(t)p Fs(\))f Fo(\000)h Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)2893 1060 y Fi(0)2932 1045 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))3180 1003 y Fp(M)3248 1012 y Fc(0)3281 1003 y Fp(;\015)3368 1045 y Fs(+)22 b(1)f Fo(\000)i Fq(t)p Fs(\))p Fo(g)3759 870 y Fn(\014)3759 930 y(\014)3759 990 y(\014)3759 1050 y(\014)3759 1109 y(\014)235 1302 y Fo(\024)28 b(k)391 1275 y Fs(~)390 1302 y Fq(h)446 1317 y Fi(0)485 1302 y Fo(k)535 1317 y Fm(1)610 1302 y Fo(j)p Fs(tr)o Fo(f)p Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)1038 1317 y Fi(0)1078 1302 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))1326 1261 y Fp(M)1394 1272 y Fk(N)1450 1261 y Fp(;\015)1536 1302 y Fs(+)22 b(1)g Fo(\000)h Fq(t)p Fs(\))f Fo(\000)g Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)2274 1317 y Fi(0)2314 1302 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))2562 1261 y Fp(M)2630 1270 y Fc(0)2663 1261 y Fp(;\015)2749 1302 y Fs(+)22 b(1)g Fo(\000)h Fq(t)p Fs(\))p Fo(gj)229 1471 y Fs(+)f Fo(k)378 1444 y Fs(~)377 1471 y Fq(h)433 1429 y Fm(0)433 1495 y Fi(0)472 1471 y Fo(k)522 1486 y Fm(1)679 1471 y Fs(sup)614 1551 y Fp(x)p Fm(2)p Fi(supp)5 b(~)-40 b Fp(!)909 1471 y Fo(j)p Fs(tr)o Fo(f)p Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)1337 1486 y Fi(0)1377 1471 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))1625 1429 y Fp(x;\015)1749 1471 y Fs(+)22 b(1)g Fo(\000)h Fq(t)p Fs(\))f Fo(\000)h Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)2488 1486 y Fi(0)2528 1471 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))2776 1429 y Fp(M)2844 1438 y Fc(0)2876 1429 y Fp(;\015)2963 1471 y Fs(+)22 b(1)g Fo(\000)g Fq(t)p Fs(\))p Fo(gj)p Fq(;)-118 1730 y Fs(where)38 b Fq(G)p Fs(\()p Fq(E)355 1745 y Fi(0)394 1730 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))642 1694 y Fp(x;\015)781 1730 y Fs(is)36 b(the)h(op)s(erator)f Fq(G)p Fs(\()p Fq(E)1639 1745 y Fi(0)1679 1730 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))35 b(with)h(the)h(coupling)e(constan)m(t)j Fq(!)3213 1745 y Fp(\015)3293 1730 y Fs(at)e(the)h Fq(\015)3644 1694 y Fp(th)3715 1730 y Fs(-site)f(\014xed)-118 1846 y(at)i(the)h(v)-5 b(alue)39 b Fq(!)501 1861 y Fp(\015)583 1846 y Fs(=)f Fq(x)p Fs(.)62 b(Similarly)-8 b(,)36 b(the)k(v)-5 b(alue)38 b(0)g(or)h Fq(M)49 b Fs(denotes)40 b(the)f(coupling)e (constan)m(t)j Fq(!)3461 1861 y Fp(\015)3544 1846 y Fs(\014xed)g(at)e (those)-118 1965 y(v)-5 b(alues.)43 b(No)m(w,)34 b(as)e Fq(G)p Fs(\()p Fq(E)766 1980 y Fi(0)806 1965 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))31 b(is)h(of)g(rank)h(at)f(most)g Fq(C)7 b(n)2012 1929 y Fp(d)2053 1965 y Fs(,)33 b(w)m(e)g(get)g(that,)f (for)g(some)h Fq(C)h(>)28 b Fs(0,)k(one)h(has)405 2064 y Fn(\014)405 2123 y(\014)405 2183 y(\014)405 2243 y(\014)438 2072 y(Z)494 2298 y Fl(R)563 2208 y Fq(d!)675 2223 y Fp(\015)719 2182 y Fs(~)718 2208 y Fq(h)774 2223 y Fi(0)814 2208 y Fs(\()p Fq(!)913 2223 y Fp(\015)957 2208 y Fs(\))1058 2141 y Fq(@)p 1005 2185 V 1005 2276 a(@)5 b(!)1122 2291 y Fp(\015)1177 2208 y Fs(tr)p Fo(f)p Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)1578 2223 y Fi(0)1617 2208 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))k(+)h(1)g Fo(\000)h Fq(t)p Fs(\))f Fo(\000)h Fq(\032)p Fs(\()p Fq(G)p Fs(\()p Fq(E)2625 2223 y Fi(0)2664 2208 y Fq(;)17 b(!)t(;)g(\025)p Fs(\))2912 2167 y Fp(M)2980 2176 y Fc(0)3013 2167 y Fp(;\015)3100 2208 y Fs(+)22 b(1)f Fo(\000)i Fq(t)p Fs(\))p Fo(g)3491 2064 y Fn(\014)3491 2123 y(\014)3491 2183 y(\014)3491 2243 y(\014)3552 2208 y Fo(\024)28 b Fq(C)7 b(n)3792 2167 y 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b(is)h(non)g(degenerate)h(in)e(a)h(neigh)m(b)s(orho)s (o)s(d)e(of)i Fq(\022)4097 5606 y Fp(j)1969 5690 y Fg(17)p eop %%Page: 18 18 18 17 bop -118 241 a Fs(for)33 b Fq(\025)g Fs(small.)45 b(This)33 b(is)g(a)h(consequence)i(of)e(p)s(erturbation)e(theory)i(as)g (it)f(holds)g(for)g Fq(\025)c Fs(=)h(0)j(b)m(y)i(assumption.)45 b(One)-118 357 y(computes)356 556 y Fq(@)407 571 y Fp(\025)453 556 y Fq(E)525 571 y Fp(n)572 556 y Fs(\()618 529 y(~)610 556 y Fq(\022)655 571 y Fp(j)692 556 y Fq(;)17 b(\025)p Fs(\))831 571 y Fb(\026)p Fp(\025)p Fi(=0)1023 556 y Fs(=)1127 445 y Fn(\020)1186 556 y Fo(h)p Fq(V)22 b(')1368 571 y Fp(n)1415 556 y Fs(\()p Fq(x;)1559 529 y Fs(~)1552 556 y Fq(\022)1597 571 y Fp(j)1634 556 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)17 b(\025)p Fs(\))p Fq(;)g(')2014 571 y Fp(n)2059 556 y Fs(\()p Fq(x;)2204 529 y Fs(~)2196 556 y Fq(\022)2241 571 y Fp(j)2279 556 y Fs(\()p Fq(\025)p Fs(\))p Fq(;)g(\025)p Fs(\))p Fo(i)k Fs(+)h Fo(r)2792 571 y Fp(\022)2831 556 y Fq(E)2903 571 y Fp(n)2950 556 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Fs(\))p Fo(j)2596 2025 y Fi(2)2657 2066 y Fs(+)22 b Fq(O)s Fs(\()p Fq(\025)2928 2025 y Fi(2)2967 2066 y Fs(\))-3123 b(\(5.3\))94 2290 y(On)24 b(the)g(other)f(hand,)i (using)e(the)h(c)m(haracterization)f(of)f(\006)2195 2305 y Fp(\025)2241 2290 y Fs(,)j(the)f(almost)e(sure)i(sp)s(ectrum)f(of)g Fq(H)3554 2305 y Fp(!)r(;\025)3689 2290 y Fs(in)f(terms)h(of)-118 2406 y(admissible)28 b(p)s(erio)s(dic)h(sp)s(ectra)i(\(see)g([14,)f (27]\),)h(w)m(e)g(kno)m(w)g(that,)g(for)f Fq(\025)g Fs(su\016cien)m (tly)h(small)d(and)j Fq(t)c Fo(2)p Fs(ess-supp\()p Fq(!)4027 2421 y Fi(0)4069 2406 y Fs(\),)-118 2522 y(w)m(e)34 b(ha)m(v)m(e)1228 2651 y(1)p 1228 2696 49 4 v 1228 2787 a(2)1287 2719 y(\()p Fq(E)1397 2734 y Fm(\000)1456 2719 y Fs(\(0\))22 b(+)g Fq(E)1773 2734 y Fi(+)1832 2719 y Fs(\(0\)\))27 b Fo(\024)i Fq(E)2200 2734 y Fm(\000)2259 2719 y Fs(\()p Fq(\025)p Fs(\))e Fo(\024)p 2524 2638 79 4 v 28 w Fq(E)6 b Fs(\()p Fq(\025t)p Fs(\))p Fq(:)-118 2926 y Fs(Let)26 b Fq(!)111 2941 y Fi(+)195 2926 y Fs(and)f Fq(!)438 2941 y Fm(\000)523 2926 y Fs(resp)s(ectiv)m(ely)h(b)s(e)g(the)g(essen)m(tial)g(suprem)m (um)f(and)h(in\014m)m(um)e(of)h(the)h(random)f(v)-5 b(ariables)24 b(\()p Fq(!)3853 2941 y Fp(\015)3897 2926 y Fs(\))3935 2945 y Fp(\015)t Fm(2)p Fl(Z)4072 2926 y Fk(d)s Fs(;)-118 3042 y(as)i(the)h(random)f(v)-5 b(ariables)25 b(are)h(not)g(trivial,)f (using)j(\(5.2\))e(and)g(the)h(computation)e(\(5.1\))o(,)j(one)f(sees)h (that,)f(for)f(some)-118 3159 y Fq(C)35 b(>)27 b Fs(0)32 b(and)h(for)f Fq(\025)g Fs(su\016cien)m(tly)i(small)732 3331 y Fo(j)p 760 3251 V Fq(E)6 b Fs(\()p Fq(\025)p 933 3276 65 4 v(!)s Fs(\))22 b Fo(\000)p 1157 3251 79 4 v 23 w Fq(E)6 b Fs(\()p Fq(\025!)1391 3346 y Fi(+)1450 3331 y Fs(\))p Fo(j)27 b(\025)h(j)p Fq(\025)p Fo(j)p Fq(=C)38 b Fs(and)33 b Fo(j)p 2136 3251 V Fq(E)6 b Fs(\()p Fq(\025)p 2309 3276 65 4 v(!)s Fs(\))22 b Fo(\000)p 2533 3251 79 4 v 23 w Fq(E)6 b Fs(\()p Fq(\025!)2767 3346 y Fm(\000)2825 3331 y Fs(\))p Fo(j)28 b(\025)g(j)p Fq(\025)p Fo(j)p Fq(=C)r(:)-118 3510 y Fs(This)33 b(implies)d(that)i(\(0.2\))g (holds)g(for)g(some)h Fq(C)h(>)28 b Fs(0)k(and)h Fq(\025)f Fs(su\016cien)m(tly)i(small.)-118 3715 y(5.2.)56 b Fw(A)37 b(useful)g(lemma.)48 b Fs(W)-8 b(e)33 b(pro)m(v)m(e)-118 3899 y Fw(Lemma)k(5.1.)49 b Ff(Pick)37 b Fq(")31 b(>)h Fs(0)37 b Ff(and)f Fq(u)31 b Fo(2)i Fq(L)1439 3862 y Fi(2)1478 3899 y Fs(\()p Fj(R)1582 3862 y Fp(d)1629 3899 y Fs(\))k Ff(such)g(that)g(supp)p Fs(\()p Fq(u)p Fs(\))31 b Fo(\032)h(fj)p Fq(x)p Fo(j)g(\024)g Fq(")p Fo(g)p Ff(.)51 b(Then,)36 b(for)h(any)g Fq(\021)f Fo(2)c Fs(\(0)p Fq(;)17 b Fs(1\))p Ff(,)-118 4015 y(ther)-5 b(e)35 b(exists)40 b Fs(~)-55 b Fq(u)27 b Fo(2)i Fq(L)635 3979 y Fi(2)674 4015 y Fs(\()p Fj(R)778 3979 y Fp(d)825 4015 y Fs(\))34 b Ff(such)h(that)8 4159 y Fo(\017)41 b(k)6 b Fs(~)-55 b Fq(u)o Fo(k)254 4176 y Fp(L)302 4157 y Fc(2)368 4159 y Fs(=)28 b Fo(k)p Fq(u)p Fo(k)628 4176 y Fp(L)676 4157 y Fc(2)714 4159 y Ff(,)8 4275 y Fo(\017)41 b(k)p Fq(u)21 b Fo(\000)29 b Fs(~)-55 b Fq(u)p Fo(k)432 4292 y Fp(L)480 4273 y Fc(2)546 4275 y Fo(\024)28 b Fq(C)7 b(\021)t Fo(k)p Fq(u)p Fo(k)936 4292 y Fp(L)984 4273 y Fc(2)1021 4275 y Ff(,)8 4392 y Fo(\017)47 b Fs(^)-55 b Fq(u)155 4407 y Fp(\021)231 4392 y Ff(is)35 b(c)-5 b(onstant)34 b(on)h(e)-5 b(ach)34 b(cub)-5 b(e)35 b Fq(C)1370 4410 y Fp(\015)t(;)1452 4379 y Fk(\021)p 1439 4394 60 3 v 1439 4436 a Fc(2)p Fk(")1513 4392 y Ff(,)g Fq(\015)e Fo(2)28 b Fj(Z)1825 4355 y Fp(d)1897 4392 y Ff(wher)-5 b(e)924 4575 y Fq(C)994 4590 y Fp(\015)t(;r)1120 4575 y Fs(=)27 b Fo(f)p Fq(x)h Fs(=)g(\()p Fq(x)1553 4590 y Fi(1)1592 4575 y Fq(;)17 b(:)g(:)g(:)f(;)h(x)1866 4590 y Fp(d)1907 4575 y Fs(\);)51 b Fo(\000)p Fq(\031)t(r)31 b Fo(\024)d Fq(x)2394 4590 y Fp(i)2445 4575 y Fo(\000)23 b Fs(2)p Fq(\031)t(r)s(\015)2751 4590 y Fp(i)2805 4575 y Fq(<)28 b(\031)t(r)s Fo(g)p Fq(:)-118 4759 y Fs(Lemma)34 b(5.1)g(is)h(a)g(v)m(ery)i(simple)c(quan)m(titativ)m (e)i(v)m(ersion)h(of)f(the)g(Uncertain)m(t)m(y)i(Principle.)50 b(It)35 b(is)g(the)g(analogue)f(of)-118 4875 y(Lemma)d(6.2)h(in)g([22]) g(dev)m(elop)s(ed)i(for)e(the)h(discrete)g(setting.)-118 4991 y Fw(Pro)s(of.)56 b Fs(Pic)m(k)37 b Fq(")p Fs(,)g Fq(\021)k Fs(and)36 b Fq(u)h Fs(as)g(in)f(Lemma)f(5.1.)55 b(Consider)38 b Fq(u)e Fs(as)h(a)f(\(2)p Fq("=\021)t Fs(\))p Fj(Z)2803 4955 y Fp(d)2840 4991 y Fs(-p)s(erio)s(dic)e (function)j(\(con)m(tin)m(ue)g(it)-118 5107 y(b)m(y)c(0)e(on)h(the)g (fundamen)m(tal)f(cell)g(of)g(\(2)p Fq("=\021)t Fs(\))p Fj(Z)1586 5071 y Fp(d)1655 5107 y Fs(and)h(p)s(erio)s(dically)c(to)k (the)g(rest)h(of)e Fj(R)3023 5071 y Fp(d)3069 5107 y Fs(\).)44 b(Expand)33 b(it)d(in)h(a)h(F)-8 b(ourier)-118 5223 y(series)33 b(to)f(get)776 5443 y Fq(u)p Fs(\()p Fq(x)p Fs(\))c(=)1106 5348 y Fn(X)1094 5568 y Fp(\015)t Fm(2)p Fl(Z)1232 5549 y Fk(d)1284 5443 y Fs(^)-55 b Fq(u)1334 5458 y Fp(\015)1378 5443 y Fq(e)1423 5402 y Fp(i)1457 5371 y Fk(\031)r(\021)p 1458 5387 72 3 v 1479 5428 a(")1539 5402 y Fp(\015)t(x)1656 5443 y Fs(where)40 b(^)-55 b Fq(u)1994 5458 y Fp(\015)2065 5443 y Fs(=)2169 5332 y Fn(\020)2260 5376 y Fq(\021)p 2238 5420 95 4 v 2238 5511 a Fs(2)p Fq(")2343 5332 y Fn(\021)2402 5355 y Fp(d)2459 5307 y Fn(Z)2515 5533 y Fl(R)2563 5514 y Fk(d)2620 5443 y Fq(u)p Fs(\()p Fq(x)p Fs(\))p Fq(e)2852 5402 y Fm(\000)p Fp(i)2941 5371 y Fk(\031)r(\021)p 2941 5387 72 3 v 2962 5428 a(")3022 5402 y Fp(\015)t(x)3107 5443 y Fq(dx:)1969 5690 y Fg(18)p eop %%Page: 19 19 19 18 bop -118 241 a Fs(P)m(arsev)-5 b(al's)33 b(iden)m(tit)m(y)g(then) g(reads)1459 476 y Fo(k)p Fq(u)p Fo(k)1615 435 y Fi(2)1615 502 y Fp(L)1663 483 y Fc(2)1729 476 y Fs(=)1844 381 y Fn(X)1832 601 y Fp(\015)t Fm(2)p Fl(Z)1970 582 y Fk(d)2017 335 y Fn(\022)2100 408 y Fs(2)p Fq(")p 2100 453 95 4 v 2121 544 a(\021)2204 335 y Fn(\023)2278 358 y Fp(d)2335 476 y Fo(j)6 b Fs(^)-55 b Fq(u)2419 491 y Fp(\015)2462 476 y Fo(j)2490 435 y Fi(2)2529 476 y Fq(:)-2674 b Fs(\(5.4\))-118 757 y(De\014ne)33 b(the)g(function)f Fq(v)f Fs(:)61 b Fj(R)966 721 y Fp(d)1040 757 y Fo(!)27 b Fj(R)1233 721 y Fp(d)1312 757 y Fs(b)m(y)1170 992 y Fq(v)t Fs(\()p Fq(\030)5 b Fs(\))27 b(=)1475 852 y Fn(\022)1559 925 y Fs(2)p Fq(")p 1559 969 V 1580 1061 a(\021)1663 852 y Fn(\023)1736 874 y Fp(d)1800 992 y Fs(^)-55 b Fq(u)1850 1007 y Fp(\015)1926 992 y Fs(for)32 b Fq(\030)g Fo(2)c Fq(C)2314 1010 y Fp(\015)t(;)2397 979 y Fk(\021)p 2384 995 60 3 v 2384 1036 a Fc(2)p Fk(")2458 992 y Fq(;)49 b(\015)33 b Fo(2)28 b Fj(Z)2781 951 y Fp(d)2818 992 y Fq(:)-118 1220 y Fs(By)40 b(\(5.4\))o(,)g Fq(v)g Fo(2)d Fq(L)565 1184 y Fi(2)605 1220 y Fs(\()p Fj(R)709 1184 y Fp(d)755 1220 y Fs(\).)60 b(Let)44 b(~)-55 b Fq(u)37 b Fs(b)s(e)i(the)f(in)m(v)m(erse)h(F)-8 b(ourier)37 b(transform)g(of)g Fq(v)t Fs(.)60 b(Chec)m(k)40 b(that)d(it)g(satis\014es)i(the)f(the)-118 1337 y(prop)s(erties)32 b(stated)i(in)d(Lemma)h(5.1.)43 b(First,)31 b(\(5.4\))h(yields)760 1579 y Fo(k)6 b Fs(~)-55 b Fq(u)o Fo(k)915 1538 y Fi(2)915 1605 y Fp(L)963 1586 y Fc(2)1029 1579 y Fs(=)1231 1512 y(1)p 1143 1556 225 4 v 1143 1648 a(\(2)p Fq(\031)t Fs(\))1327 1619 y Fp(d)1377 1579 y Fo(k)p Fq(v)t Fo(k)1528 1538 y Fi(2)1528 1605 y Fp(L)1576 1586 y Fc(2)1642 1579 y Fs(=)1745 1439 y Fn(\022)1837 1512 y Fs(2)p Fq(")1932 1476 y Fi(2)p 1829 1556 151 4 v 1829 1648 a Fq(\031)t(\021)1940 1619 y Fi(2)1989 1439 y Fn(\023)2062 1459 y Fp(d)2131 1485 y Fn(X)2119 1704 y Fp(\015)t Fm(2)p Fl(Z)2257 1685 y Fk(d)2303 1444 y Fn(Z)2359 1669 y Fp(C)2409 1693 y Fc(0)p Fk(;)2481 1668 y(\021)p 2468 1683 60 3 v 2468 1717 a Fc(2)p Fk(")2563 1579 y Fo(j)6 b Fs(^)-55 b Fq(u)2647 1594 y Fp(\015)2690 1579 y Fo(j)2718 1538 y Fi(2)2758 1579 y Fq(d\030)31 b Fs(=)d Fo(k)p Fq(u)p Fo(k)3143 1538 y Fi(2)3143 1605 y Fp(L)3191 1586 y Fc(2)3229 1579 y Fq(:)-118 1847 y Fs(Let)39 b(^)-55 b Fq(u)32 b Fs(b)s(e)g(the)h(F)-8 b(ourier)32 b(transform)f(of)i Fq(u)p Fs(;)f(w)m(e)h(compute)670 1973 y Fn(Z)725 2199 y Fl(R)773 2180 y Fk(d)830 2109 y Fo(j)6 b Fs(^)-55 b Fq(u)o Fs(\()p Fq(\030)5 b Fs(\))22 b Fo(\000)g Fq(v)t Fs(\()p Fq(\030)5 b Fs(\))p Fo(j)1361 2068 y Fi(2)1400 2109 y Fq(d\030)31 b Fs(=)1641 2014 y Fn(X)1629 2234 y Fp(\015)t Fm(2)p Fl(Z)1767 2215 y Fk(d)1813 1973 y Fn(Z)1869 2199 y Fp(C)1919 2223 y Fc(0)p Fk(;)1991 2198 y(\021)p 1978 2213 V 1978 2247 a Fc(2)p Fk(")2073 1935 y Fn(\014)2073 1994 y(\014)2073 2054 y(\014)2073 2114 y(\014)2073 2174 y(\014)2112 2109 y Fs(^)-55 b Fq(u)p Fs(\()p Fq(\030)26 b Fs(+)c Fq(\015)2433 2042 y(\031)t(\021)p 2433 2086 111 4 v 2465 2177 a(")2554 2109 y Fs(\))g Fo(\000)2713 1969 y Fn(\022)2797 2042 y Fs(2)p Fq(")p 2797 2086 95 4 v 2818 2177 a(\021)2901 1969 y Fn(\023)2974 1991 y Fp(d)3037 2109 y Fs(^)-55 b Fq(u)3087 2124 y Fp(\015)3131 1935 y Fn(\014)3131 1994 y(\014)3131 2054 y(\014)3131 2114 y(\014)3131 2174 y(\014)3165 1961 y Fi(2)3221 2109 y Fq(d\030)5 b(:)-3465 b Fs(\(5.5\))-118 2377 y(On)32 b(the)h(other)g(hand,)g(w)m(e)h(note)849 2612 y(^)-55 b Fq(u)o Fs(\()p Fq(\030)27 b Fs(+)22 b Fq(\015)1170 2545 y(\031)t(\021)p 1170 2589 111 4 v 1202 2681 a(")1290 2612 y Fs(\))h Fo(\000)1450 2472 y Fn(\022)1533 2545 y Fs(2)p Fq(")p 1533 2589 95 4 v 1555 2681 a(\021)1638 2472 y Fn(\023)1711 2494 y Fp(d)1774 2612 y Fs(^)-55 b Fq(u)1824 2627 y Fp(\015)1896 2612 y Fs(=)1999 2477 y Fn(Z)2055 2702 y Fl(R)2103 2683 y Fk(d)2160 2612 y Fq(u)p Fs(\()p Fq(x)p Fs(\)\()p Fq(e)2430 2571 y Fp(ix\030)2553 2612 y Fo(\000)23 b Fs(1\))p Fq(e)2785 2571 y Fm(\000)p Fp(i)2874 2540 y Fk(\031)r(\021)p 2874 2556 72 3 v 2895 2597 a(")2956 2571 y Fp(\015)t(x)3040 2612 y Fq(dx:)-118 2841 y Fs(As)40 b(w)m(e)g(did)e(for)h Fq(u)p Fs(,)h(w)m(e)g(consider)g Fq(x)f Fo(7!)f Fq(u)p Fs(\()p Fq(x)p Fs(\)\()p Fq(e)1675 2805 y Fp(ix\030)1803 2841 y Fo(\000)27 b Fs(1\))39 b(as)g(a)g(\(2)p Fq("=\021)t Fs(\))p Fj(Z)2588 2805 y Fp(d)2625 2841 y Fs(-p)s(erio)s(dic)e(function;)42 b(then,)f(P)m(arsev)-5 b(al's)-118 2957 y(iden)m(tit)m(y)32 b(yields)626 3124 y Fn(X)614 3343 y Fp(\015)t Fm(2)p Fl(Z)751 3324 y Fk(d)798 3044 y Fn(\014)798 3104 y(\014)798 3164 y(\014)798 3223 y(\014)798 3283 y(\014)837 3218 y Fs(^)-55 b Fq(u)p Fs(\()p Fq(\030)26 b Fs(+)c Fq(\015)1158 3151 y(\031)t(\021)p 1158 3196 111 4 v 1191 3287 a(")1279 3218 y Fs(\))g Fo(\000)1439 3078 y Fn(\022)1522 3151 y Fs(2)p Fq(")p 1522 3196 95 4 v 1543 3287 a(\021)1626 3078 y Fn(\023)1700 3100 y Fp(d)1763 3218 y Fs(^)-55 b Fq(u)1813 3233 y Fp(\015)1857 3044 y Fn(\014)1857 3104 y(\014)1857 3164 y(\014)1857 3223 y(\014)1857 3283 y(\014)1890 3070 y Fi(2)1957 3218 y Fs(=)2061 3078 y Fn(\022)2144 3151 y Fs(2)p Fq(")p 2144 3196 V 2165 3287 a(\021)2248 3078 y Fn(\023)2322 3100 y Fp(d)2379 3083 y Fn(Z)2434 3308 y Fm(j)p Fp(x)p Fm(j\024)p Fp(")2621 3218 y Fo(j)p Fq(u)p Fs(\()p Fq(x)p Fs(\)\()p Fq(e)2919 3177 y Fp(ix\030)3042 3218 y Fo(\000)23 b Fs(1\))p Fo(j)3257 3177 y Fi(2)3296 3218 y Fq(dx)-118 3492 y Fs(Substituting)32 b(in)f(\(5.5\),)i(w)m(e)g(obtain)574 3607 y Fn(Z)629 3833 y Fl(R)677 3814 y Fk(d)734 3743 y Fo(j)6 b Fs(^)-55 b Fq(u)o Fs(\()p Fq(\030)5 b Fs(\))22 b Fo(\000)g Fq(v)t Fs(\()p Fq(\030)5 b Fs(\))p Fo(j)1265 3702 y Fi(2)1304 3743 y Fq(d\030)31 b Fs(=)1533 3607 y Fn(Z)1588 3833 y Fm(j)p Fp(x)p Fm(j\024)p Fp(")1775 3743 y Fo(j)p Fq(u)p Fs(\()p Fq(x)p Fs(\))p Fo(j)2018 3702 y Fi(2)2074 3573 y Fn( )2152 3602 y(\022)2236 3676 y Fs(2)p Fq(")p 2236 3720 V 2257 3811 a(\021)2340 3602 y Fn(\023)2413 3625 y Fp(d)2471 3607 y Fn(Z)2526 3833 y Fp(C)2576 3857 y Fc(0)p Fk(;)2648 3831 y(\021)p 2636 3846 60 3 v 2636 3881 a Fc(2)p Fk(")2730 3743 y Fo(j)p Fq(e)2803 3702 y Fp(ix\030)2927 3743 y Fo(\000)22 b Fs(1)p Fo(j)3103 3702 y Fi(2)3142 3743 y Fq(d\030)3241 3573 y Fn(!)3336 3743 y Fq(dx)-118 4001 y Fs(that)32 b(is)836 4052 y Fn(Z)891 4278 y Fl(R)939 4259 y Fk(d)996 4188 y Fo(j)6 b Fs(^)-55 b Fq(u)o Fs(\()p Fq(\030)5 b Fs(\))22 b Fo(\000)g Fq(v)t Fs(\()p Fq(\030)5 b Fs(\))p Fo(j)1527 4147 y Fi(2)1566 4188 y Fq(d\030)31 b Fo(\024)e Fq(C)7 b(\021)1926 4147 y Fi(2)1981 4052 y Fn(Z)2037 4278 y Fm(j)p Fp(x)p Fm(j\024)p Fp(")2224 4188 y Fo(j)p Fq(u)p Fs(\()p Fq(x)p Fs(\))p Fo(j)2467 4147 y Fi(2)2505 4188 y Fq(dx)28 b Fs(=)g Fq(C)7 b(\021)2872 4147 y Fi(2)2911 4188 y Fo(k)p Fq(u)p 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