Content-Type: multipart/mixed; boundary="-------------0203130243742" This is a multi-part message in MIME format. ---------------0203130243742 Content-Type: text/plain; name="02-120.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-120.keywords" Random Schroedinger Operator, Quantum Hall Effect ---------------0203130243742 Content-Type: application/postscript; name="spec-mix.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="spec-mix.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: spec-mix.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -N0 -f spec-mix.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.03.13:0930 %%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 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b(op)s(erators)i(and)f(therefore)i(it)e(is)g(imp)s(ortan)m(t)g(to)h (understand)265 1658 y(their)26 b(sp)s(ectral)g(prop)s(erties)f(for)h (\014nite)f(but)h(macroscopic)h(samples)f(with)f(b)s(oundaries.)37 b(This)265 1805 y(problem)f(has)h(b)s(een)f(approac)m(hed)h(recen)m (tly)h(for)f(geometries)h(where)f(only)f(one)i(b)s(oundary)265 1952 y(is)g(presen)m(t)h(and)g(the)g(op)s(erator)h(is)e(de\014ned)g(in) g(a)h(semi-in\014nite)e(region)i([MMP],)j([F)m(GW)r(],)265 2099 y([dBP].)401 2245 y(Here)31 b(w)m(e)g(will)d(tak)m(e)k(a)f (\014nite)e(system:)41 b(our)30 b(geometry)i(is)d(that)i(of)g(a)f (cylinder)f(of)h(length)265 2392 y(and)23 b(circumference)f(b)s(oth)h (equal)g(to)h Fy(L)p Fz(.)38 b(There)23 b(are)h(t)m(w)m(o)h(b)s (oundaries)20 b(at)k Fy(x)i Fz(=)f Fx(\006)3051 2356 y Fu(L)p 3051 2371 48 4 v 3057 2424 a Fw(2)3132 2392 y Fz(mo)s(delled)265 2539 y(b)m(y)31 b(t)m(w)m(o)h(smo)s(oth)e (con\014ning)f(p)s(oten)m(tials)h Fy(U)1756 2554 y Fu(`)1789 2539 y Fz(\()p Fy(x)p Fz(\))i(\()p Fy(`)f Fz(for)f(left\))h(and)f Fy(U)2620 2553 y Fu(r)2658 2539 y Fz(\()p Fy(x)p Fz(\))h(\()p Fy(r)j Fz(for)c(righ)m(t\),)h(and)265 2686 y(w)m(e)h(tak)m(e)g(p)s (erio)s(dic)d(b)s(oundary)g(conditions)h(in)g(the)h Fy(y)j Fz(direction.)42 b(These)31 b(p)s(oten)m(tials)f(v)-5 b(anish)265 2833 y(for)36 b Fx(\000)491 2797 y Fu(L)p 491 2812 V 497 2864 a Fw(2)584 2833 y Fx(\024)e Fy(x)h Fx(\024)892 2797 y Fu(L)p 892 2812 V 898 2864 a Fw(2)986 2833 y Fz(and)h(gro)m(w)h(fast)f(enough)g(for)g Fx(j)p Fy(x)p Fx(j)g(\025)2292 2797 y Fu(L)p 2292 2812 V 2298 2864 a Fw(2)2350 2833 y Fz(.)58 b(The)36 b(Hamiltonian)f(is)g(of)h(the) 265 2979 y(form)1379 3126 y Fy(H)1455 3140 y Fu(!)1531 3126 y Fz(=)24 b Fy(H)1702 3140 y Fw(0)1762 3126 y Fz(+)c Fy(V)1906 3140 y Fu(!)1976 3126 y Fz(+)g Fy(U)2129 3141 y Fu(`)2182 3126 y Fz(+)g Fy(U)2335 3140 y Fu(r)3301 3126 y Fz(\(1.1\))265 3327 y(where)k Fy(H)598 3341 y Fw(0)661 3327 y Fz(is)f(the)h(pure)f(Landau)h(Hamiltonian)e(for)i(a)h (uniform)d(\014eld)h(of)h(strength)g Fy(B)29 b Fz(and)23 b Fy(V)3437 3341 y Fu(!)265 3474 y Fz(is)29 b(a)i(suitable)e(w)m(eak)i (random)e(p)s(oten)m(tial)h(pro)s(duced)f(b)m(y)h(impurities)d(with)i (sup)14 b Fx(j)p Fy(V)3105 3488 y Fu(!)3155 3474 y Fz(\()p Fy(x;)h(y)s Fz(\))p Fx(j)27 b Fz(=)265 3620 y Fy(V)318 3634 y Fw(0)383 3620 y Fx(\034)e Fy(B)32 b Fz(\(see)d(section)g(2)f (for)g(precise)f(assumptions\).)39 b(Before)29 b(explaining)c(our)j (results)f(it)g(is)265 3767 y(useful)i(to)i(describ)s(e)e(what)h(is)f (kno)m(wn)h(ab)s(out)g(the)h(in\014nite)d(and)i(semi-in\014nite)d (cases.)401 3914 y(In)i(the)g(case)h(of)f(the)g(in\014nite)e(plane)h Fq(R)1724 3881 y Fw(2)1799 3914 y Fz(for)g(the)h(Hamiltonian)f Fy(H)2689 3928 y Fw(0)2745 3914 y Fz(+)17 b Fy(V)2886 3928 y Fu(!)2966 3914 y Fz(the)29 b(sp)s(ectrum)265 4061 y(forms)38 b(\\Landau)g(bands")g(con)m(tained)h(in)1759 3993 y Fp(S)1835 4088 y Fu(\027)t Fv(\025)p Fw(0)1983 3987 y Fp(\002)2021 4061 y Fz(\()p Fy(\027)26 b Fz(+)2228 4025 y Fw(1)p 2228 4040 36 4 v 2228 4092 a(2)2273 4061 y Fz(\))p Fy(B)f Fx(\000)20 b Fy(V)2546 4075 y Fw(0)2586 4061 y Fy(;)15 b Fz(\()p Fy(\027)26 b Fz(+)2833 4025 y Fw(1)p 2833 4040 V 2833 4092 a(2)2879 4061 y Fz(\))p Fy(B)f Fz(+)20 b Fy(V)3152 4075 y Fw(0)3191 3987 y Fp(\003)3229 4061 y Fz(.)66 b(It)38 b(is)265 4208 y(pro)m(v)m(ed)d(that)g(the)f (band)f(tails)h(ha)m(v)m(e)h(pure)e(p)s(oin)m(t)h(sp)s(ectrum)f (corresp)s(onding)f(to)j(exp)s(onen-)265 4354 y(tially)29 b(lo)s(calized)g(w)m(a)m(v)m(efunctions)h([DMP1)r(],)g([DMP2)r(],)g ([CH],)h([BCH],)g([W)-8 b(a)r(].)40 b(There)30 b(are)h(no)265 4501 y(rigorous)d(results)f(for)h(energies)g(at)h(the)g(band)e(cen)m (ters,)j(except)g(for)e(a)h(sp)s(ecial)e(mo)s(del)g(where)265 4648 y(the)33 b(impurities)d(are)j(p)s(oin)m(t)f(scatterers)i([DMP3)q (],)g([DMP4)r(].)48 b(As)33 b(\014rst)f(sho)m(wn)g(in)g([K])h(these)265 4795 y(sp)s(ectral)d(prop)s(erties)f(of)i(random)f(Sc)m(hr\177)-45 b(odinger)30 b(op)s(erators)h(imply)d(that)k(the)f(Hall)e(conduc-)265 4941 y(tivit)m(y)g({)i(giv)m(en)e(b)m(y)h(the)g(Kub)s(o)e(form)m(ula)h ({)h(considered)f(as)h(a)g(function)f(of)h(the)g(\014lling)d(factor)265 5088 y(\(ratio)34 b(of)f(electron)h(n)m(um)m(b)s(er)d(and)i(n)m(um)m(b) s(er)f(of)h(\015ux)g(quan)m(ta\))h(has)f(quan)m(tized)g(plateaux)g(at) 265 5235 y(v)-5 b(alues)27 b(equal)g(to)h Fy(\027)6 b(e)971 5202 y Fw(2)1011 5235 y Fy(=h)28 b Fz(where)f Fy(\027)33 b Fz(is)27 b(the)h(n)m(um)m(b)s(er)e(of)i(\014lled)d(Landau)i(lev)m (els.)39 b(The)27 b(presence)265 5382 y(of)h(the)g(plateaux)g(is)f(a)h (manifestation)g(of)g(Anderson)f(lo)s(calization)g(while)e(the)k(quan)m (tization)1852 5637 y Fs(2)p eop %%Page: 3 3 3 2 bop 265 100 a Fz(has)37 b(a)g(top)s(ological)f(origin.)59 b(The)36 b(latter)i(w)m(as)f(\014rst)f(disco)m(v)m(ered)h(in)f (particular)f(situations)265 247 y([TKNN],)30 b(and)f(it)g(has)g(b)s (een)g(pro)m(v)m(ed)h(for)f(more)h(general)g(mo)s(dels)e(using)g(non)h (comm)m(utativ)m(e)265 394 y(geometry)24 b([BES])f(and)e(the)i(index)e (of)i(F)-8 b(redholm)21 b(op)s(erators)i([ASS])f(\(see)i([A)m(G)q(])f (for)f(a)h(review\).)401 541 y(In)38 b(a)g(semi-in\014nite)e(system)i (where)g(the)g(particle)f(is)h(con\014ned)f(in)g(a)h(half)g(plane)f (with)265 688 y(Hamiltonian)c Fy(H)868 702 y Fw(0)930 688 y Fz(+)23 b Fy(V)1077 702 y Fu(!)1150 688 y Fz(+)g Fy(U)1306 703 y Fu(`)1373 688 y Fz(\(here)35 b(\()p Fy(x;)15 b(y)s Fz(\))36 b(b)s(elongs)d(to)i Fq(R)2361 655 y Fw(2)2406 688 y Fz(\))g(the)g(sp)s(ectrum)e(includes)g(all)265 834 y(energies)41 b(in)737 761 y Fp(\002)785 799 y Fu(B)p 785 814 57 4 v 796 866 a Fw(2)851 834 y Fy(;)15 b Fz(+)p Fx(1)1053 761 y Fp(\002)1091 834 y Fz(.)73 b(The)40 b(lo)m(w)m(er)i (edge)f(of)h(the)f(sp)s(ectrum)f(is)g(b)s(et)m(w)m(een)3016 799 y Fu(B)p 3016 814 V 3027 866 a Fw(2)3110 834 y Fx(\000)27 b Fy(V)3261 848 y Fw(0)3341 834 y Fz(and)275 945 y Fu(B)p 275 960 V 286 1012 a Fw(2)383 981 y Fz(and)42 b(in)e(its)h(vicinit)m(y) g(the)h(sp)s(ectrum)f(is)g(pure)f(p)s(oin)m(t)h(\(this)g(follo)m(ws)g (from)h(tec)m(hniques)265 1128 y(in)i([BCH)q(]\).)86 b(F)-8 b(or)46 b(energies)f(in)f(in)m(terv)-5 b(als)44 b(inside)f(the)j(gaps)g(of)f(the)h(bulk)d(Hamiltonian)265 1275 y Fy(H)341 1289 y Fw(0)407 1275 y Fz(+)27 b Fy(V)558 1289 y Fu(!)649 1275 y Fz(the)42 b(situation)d(is)h(completely)h (di\013eren)m(t.)72 b(One)40 b(can)h(sho)m(w)g(that)h(the)f(a)m(v)m (erage)265 1421 y(v)m(elo)s(cit)m(y)36 b(\()p Fy( )s(;)15 b(v)787 1435 y Fu(y)829 1421 y Fy( )s Fz(\))36 b(in)e(the)h Fy(y)j Fz(direction)c(of)h(an)g(assumed)f(eigenstate)i Fy( )j Fz(do)s(es)c(not)g(v)-5 b(anish,)265 1568 y(but)30 b(since)g(the)h(v)m(elo)s(cit)m(y)h Fy(v)1193 1582 y Fu(y)1265 1568 y Fz(is)e(the)h(comm)m(utator)i(b)s(et)m(w)m(een)e Fy(y)j Fz(and)c(the)h(Hamiltonian,)f(this)265 1715 y(implies)23 b(that)j(the)g(eigenstate)i(cannot)e(exist,)h(and)e(that)i(therefore)f (the)g(sp)s(ectrum)e(is)h(purely)265 1862 y(con)m(tin)m(uous)k([MMP)q (],)h([F)q(].)40 b(In)29 b(fact)h(Mourre)g(theory)g(has)f(b)s(een)g (suitably)e(applied)g(to)k(pro)m(v)m(e)265 2009 y(that)39 b(the)g(sp)s(ectrum)e(is)h(purely)f(absolutely)g(con)m(tin)m(uous)i([F) m(GW)r(],)i([dBP].)65 b(These)39 b(w)m(orks)265 2155 y(put)33 b(on)h(a)g(rigorous)f(basis)f(the)i(exp)s(ectation)g(that,)i (b)s(ecause)e(of)f(the)h(c)m(hiral)f(nature)g(of)h(the)265 2302 y(b)s(oundary)42 b(curren)m(ts,)48 b(the)d(states)g(remain)f (extended)g(in)f(the)i Fy(y)i Fz(direction)c(ev)m(en)i(in)e(the)265 2449 y(presence)34 b(of)g(disorder)d([H)q(].)51 b(The)33 b(same)h(sort)g(of)g(analysis)e(sho)m(ws)i(that)g(if)e(the)i Fy(y)j Fz(direction)265 2596 y(is)32 b(made)g(p)s(erio)s(dic)e(of)j (length)f Fy(L)p Fz(,)i(the)e(same)h(energy)g(in)m(terv)-5 b(als)32 b(ha)m(v)m(e)i(discrete)e(eigenstates)265 2743 y(whic)m(h)40 b(carry)h(a)h(curren)m(t)e(that)i(is)e Fx(O)s Fz(\(1\))i({)g(sa)m(y)g(p)s(ositiv)m(e)e({)h(with)f(resp)s(ect)h (to)h Fy(L)f Fz([F)m(GW)r(].)265 2889 y(F)-8 b(urthermore)30 b(one)h(can)g(sho)m(w)f(that)h(the)f(eigen)m(v)-5 b(alue)31 b(spacing)e(is)h(of)g(order)g Fx(O)s Fz(\()p Fy(L)3067 2856 y Fv(\000)p Fw(1)3162 2889 y Fz(\))g([M)q(].)401 3036 y(The)j(nature)g(of)g(the)h(sp)s(ectrum)e(for)h(a)g (semi-in\014nite)e(system)i(for)g(in)m(terv)-5 b(als)32 b(inside)f(the)265 3183 y(Landau)g(bands)g(of)h(the)g(bulk)e (Hamiltonian)1854 3115 y Fp(S)1930 3210 y Fu(\027)t Fv(\025)p Fw(0)2078 3109 y Fp(\002)2116 3183 y Fz(\()p Fy(\027)c Fz(+)2323 3147 y Fw(1)p 2323 3162 36 4 v 2323 3214 a(2)2368 3183 y Fz(\))p Fy(B)g Fx(\000)19 b Fy(V)2641 3197 y Fw(0)2681 3183 y Fy(;)c Fz(\()p Fy(\027)27 b Fz(+)2928 3147 y Fw(1)p 2928 3162 V 2928 3214 a(2)2974 3183 y Fz(\))p Fy(B)e Fz(+)20 b Fy(V)3247 3197 y Fw(0)3286 3109 y Fp(\003)3356 3183 y Fz(has)265 3330 y(not)31 b(y)m(et)g(b)s(een)f(elucidated.)265 3476 y(F)-8 b(or)32 b(the)g(\014nite)f(system)h(on)f(a)i(cylinder)c (with)h(t)m(w)m(o)k(b)s(oundaries)29 b(the)j(sp)s(ectrum)e(consists)h (of)265 3623 y(\014nitely)k(degenerate)k(isolated)e(eigen)m(v)-5 b(alues.)61 b(In)37 b([FM1)q(])g(the)h(results)e(of)h([MMP)q(],)i([F)m (GW)r(])265 3770 y(for)33 b(energy)h(in)m(terv)-5 b(als)33 b(inside)f(the)h(gaps)h(of)g(the)g(bulk)e(Hamiltonian)g(are)i(extended) g(to)g(the)265 3917 y(presen)m(t)h(t)m(w)m(o)h(b)s(oundary)d(system.)55 b(The)34 b(eigen)m(v)-5 b(alues)35 b(can)g(b)s(e)f(classi\014ed)f(in)h (t)m(w)m(o)i(sets)g(dis-)265 4064 y(tinguished)29 b(b)m(y)j(the)g(sign) e(of)i(their)f(asso)s(ciated)h(curren)m(t)2224 4031 y Fw(1)2263 4064 y Fz(.)45 b(These)31 b(curren)m(ts)g(are)h(uniformly)265 4210 y(p)s(ositiv)m(e)f(or)h(uniformly)c(negativ)m(e)33 b(with)e(resp)s(ect)h(to)g Fy(L)p Fz(.)45 b(F)-8 b(or)33 b(this)d(result)h(to)i(hold)d(it)h(is)g(im-)265 4357 y(p)s(ortan)m(t)c(to)h(tak)m(e)h(the)f(circumference)e(and)h(the)g (length)g(of)g(the)h(cylinder)d(b)s(oth)h(of)i(the)f(order)265 4504 y Fy(L)p Fz(.)401 4651 y(In)h(the)h(presen)m(t)g(w)m(ork)g(w)m(e)g (study)f(the)h(curren)m(ts)f(of)h(the)g(eigenstates)h(for)e(eigen)m(v) -5 b(alues)29 b(in)265 4797 y(the)d(in)m(terv)-5 b(al)25 b(\001)817 4811 y Fu(")879 4797 y Fz(=)975 4724 y Fp(\003)1023 4762 y Fu(B)p 1023 4777 57 4 v 1034 4829 a Fw(2)1109 4797 y Fz(+)20 b Fy(";)1293 4762 y Fu(B)p 1293 4777 V 1304 4829 a Fw(2)1380 4797 y Fz(+)f Fy(V)1523 4811 y Fw(0)1563 4724 y Fp(\002)1627 4797 y Fz(where)25 b Fy(")h Fz(is)f(a)h(small)e(p)s(ositiv)m(e)h(n)m(um)m(b)s(er)g(indep)s(enden)m (t)265 4944 y(of)38 b Fy(L)p Fz(.)64 b(W)-8 b(e)39 b(limit)d(ourselv)m (es)i(to)h(the)f(\014rst)f(band)g(to)i(k)m(eep)g(the)f(discussion)e (simpler.)61 b(The)p 21 5012 1485 4 v 133 5074 a Fo(1)171 5104 y Fn(In)29 b(principle)f(the)h(ph)n(ysical)f(curren)n(t)g(is)h Fm(L)1514 5074 y Fl(\000)p Fo(1)1602 5104 y Fn(\()p Fm( )s(;)14 b(v)1768 5116 y Fk(y)1809 5104 y Fm( )s Fn(\),)29 b(but)h(here)e(w)n(e) g(will)h(call)g(curren)n(t)e(the)j(a)n(v)n(erage)25 b(v)n(elo)r(cit)n (y)21 5233 y(\()p Fm( )s(;)14 b(v)187 5245 y Fk(y)228 5233 y Fm( )s Fn(\).)1852 5637 y Fs(3)p eop %%Page: 4 4 4 3 bop 265 100 a Fz(con)m(ten)m(t)43 b(of)e(our)f(main)g(result)g (\(Theorem)h(1\))g(is)f(the)h(follo)m(wing.)71 b(Giv)m(en)41 b Fy(")p Fz(,)j(for)d Fy(L)f Fz(large)265 247 y(enough)23 b(there)h(is)f(a)h(ensem)m(ble)f(of)h(realizations)f(of)h(the)g(random) e(p)s(oten)m(tial)i(with)e(probabilit)m(y)265 394 y(1)g Fx(\000)e(O)s Fz(\()p Fy(L)595 361 y Fv(\000)p Fu(s)687 394 y Fz(\))32 b(for)g(whic)m(h)e(the)i(eigen)m(v)-5 b(alues)32 b(of)f Fy(H)1966 408 y Fu(!)2048 394 y Fz(can)h(b)s(e)f (classi\014ed)f(in)m(to)i(three)g(sets)g(that)265 541 y(w)m(e)i(call)e(\006)637 556 y Fu(`)670 541 y Fz(,)h(\006)794 555 y Fu(r)865 541 y Fz(and)f(\006)1110 556 y Fu(b)1144 541 y Fz(.)49 b(The)33 b(eigenstates)h(of)f(\006)2043 556 y Fu(`)2109 541 y Fz(\(resp.)48 b(\006)2446 555 y Fu(r)2484 541 y Fz(\))33 b(ha)m(v)m(e)h(uniformly)d(p)s(ositiv)m(e)265 688 y(\(resp.)56 b(negativ)m(e\))37 b(curren)m(ts)e(with)f(resp)s(ect)h (to)h Fy(L)p Fz(,)h(while)d(those)i(of)f(\006)2728 703 y Fu(b)2798 688 y Fz(ha)m(v)m(e)h(a)g(curren)m(t)f(of)265 834 y(the)d(order)g(of)g Fx(O)s Fz(\()p Fy(e)920 801 y Fv(\000)p Fu(\015)t(B)s Fw(\(log)15 b Fu(L)p Fw(\))1278 778 y Ft(2)1317 834 y Fz(\).)46 b(The)32 b(n)m(um)m(b)s(er)f(of)h (eigen)m(v)-5 b(alues)32 b(in)f(\006)2696 848 y Fu(\013)2777 834 y Fz(\()p Fy(\013)e Fz(=)f Fy(`;)15 b(r)s Fz(\))32 b(is)f Fx(O)s Fz(\()p Fy(L)p Fz(\))265 981 y(while)25 b(that)i(in)f(\006)861 996 y Fu(b)922 981 y Fz(is)g Fx(O)s Fz(\()p Fy(L)1182 948 y Fw(2)1221 981 y Fz(\).)40 b(This)25 b(classi\014cation)h(of)h(eigen)m(v)-5 b(alues)27 b(leads)f(to)i(a)f(w) m(ell)f(de\014ned)265 1128 y(notion)e(of)g(extended)g(edge)h(and)f(lo)s (calized)f(bulk)f(states.)40 b(The)24 b(edge)h(states)g(are)g(those)g (whic)m(h)265 1275 y(b)s(elong)35 b(to)h(\006)743 1289 y Fu(\013)827 1275 y Fz(\()p Fy(\013)f Fz(=)f Fy(`;)15 b(r)s Fz(\))35 b(and)g(are)h(extended)g(in)e(the)i(sense)f(that)h(they) g(ha)m(v)m(e)h(a)f(curren)m(t)265 1421 y(of)c(order)e Fx(O)s Fz(\(1\).)45 b(The)31 b(bulk)f(states)i(are)g(those)g(whic)m(h)e (b)s(elong)h(to)h(\006)2644 1436 y Fu(b)2709 1421 y Fz(and)f(are)h(lo)s (calized)e(in)265 1568 y(the)c(sense)g(that)h(their)e(curren)m(t)h(is)f (in\014nitesimal.)35 b(The)26 b(energy)g(lev)m(els)g(of)g(the)g (extended)g(and)265 1715 y(lo)s(calized)32 b(states)i(are)g Fj(intermixe)-5 b(d)44 b Fz(in)32 b(the)h(same)g(energy)h(in)m(terv)-5 b(al.)48 b(See)33 b(also)h([FM3)q(])f(for)g(a)265 1862 y(short)d(review)g(on)g(sp)s(ectral)g(prop)s(erties)e(of)j(systems)f (de\014ned)f(on)i(a)f(cylinder.)401 2009 y(Let)i(us)e(explain)f(the)j (mec)m(hanism)e(that)h(is)f(at)i(w)m(ork.)43 b(When)31 b(the)g(random)f(p)s(oten)m(tial)g(is)265 2155 y(remo)m(v)m(ed)i Fy(V)680 2169 y Fu(!)757 2155 y Fz(=)27 b(0)k(in)f(\(1.1\))j(the)e (eigenstates)i(with)d(energies)h(a)m(w)m(a)m(y)i(from)2881 2120 y Fu(B)p 2881 2135 57 4 v 2892 2187 a Fw(2)2979 2155 y Fz(are)f(extended)265 2302 y(in)37 b(the)h Fy(y)j Fz(direction)c(and)g(lo)s(calized)g(in)g(the)i Fy(x)f Fz(direction)e(at)j(a)g(\014nite)e(distance)h(from)f(the)265 2449 y(b)s(oundaries.)52 b(Their)34 b(energies)g(form)h(a)g(sequence)h (of)f(\\edge)h(lev)m(els")f(and)g(ha)m(v)m(e)h(a)f(spacing)265 2596 y(of)j(the)g(order)g(of)g Fx(O)s Fz(\()p Fy(L)1069 2563 y Fv(\000)p Fw(1)1163 2596 y Fz(\).)64 b(When)37 b(the)h(p)s(oten)m(tial)g(of)g(one)g(impurit)m(y)d(is)i(added)h(to)g Fy(H)3350 2610 y Fw(0)3427 2596 y Fz(it)265 2743 y(t)m(ypically)31 b(creates)k(a)e(lo)s(calized)e(b)s(ound)g(state)j(with)d(energy)i(b)s (et)m(w)m(een)g(the)g(Landau)f(lev)m(els.)265 2889 y(Supp)s(ose)i(no)m (w)j(that)f Fy(i)p Fz(\))h(a)f(coupling)f(constan)m(t)i(in)e(the)h (impurit)m(y)e(p)s(oten)m(tial)i(is)f Fj(\014ne)j(tune)-5 b(d)265 3036 y Fz(as)46 b(a)f(function)f(of)i Fy(L)f Fz(so)g(that)h(the)f(energy)h(of)f(the)h(impurit)m(y)d(lev)m(el)i(sta)m (ys)h(at)g(distance)265 3183 y(greater)c(than)e Fy(L)868 3150 y Fv(\000)p Fu(p)1003 3183 y Fz(from)g(the)h(edge)g(lev)m(els,)i Fy(ii)p Fz(\))e(the)g(p)s(osition)e(of)h(the)h(impurit)m(y)d(is)i(at)h (a)265 3330 y(distance)35 b Fy(D)k Fz(from)c(the)g(b)s(oundaries.)54 b(Then)35 b(the)h(mixing)d(b)s(et)m(w)m(een)j(the)g(lo)s(calized)f(b)s (ound)265 3476 y(state)46 b(and)f(the)g(extended)g(edge)h(states)g(is)e (con)m(trolled)g(in)g(second)h(order)g(p)s(erturbation)265 3623 y(theory)f(b)m(y)g(the)g(parameter)g Fy(L)1384 3590 y Fu(p)1424 3623 y Fy(e)1466 3590 y Fv(\000)p Fu(cB)s(D)1668 3567 y Ft(2)1706 3623 y Fz(.)81 b(Therefore)44 b(one)g(exp)s(ects)g (that)h(b)s(ound)c(states)265 3770 y(of)f(impurities)d(that)j(ha)m(v)m (e)h Fy(D)j Fx(\035)d Fz(\(log)17 b Fy(L)p Fz(\))1759 3737 y Fw(1)p Fu(=)p Fw(2)1909 3770 y Fz(are)40 b(basically)e(unp)s (erturb)s(ed)e(and)k(ha)m(v)m(e)h(an)265 3917 y(in\014nitesimal)36 b(curren)m(t.)70 b(On)39 b(the)h(other)g(hand)f(b)s(ound)f(states)k (coming)d(from)h(impurities)265 4064 y(with)26 b Fy(D)i Fx(\034)d Fz(\(log)17 b Fy(L)p Fz(\))953 4031 y Fw(1)p Fu(=)p Fw(2)1090 4064 y Fz(will)24 b(mix)i(with)g(edge)i(states.)40 b(Note)29 b(that)e Fj(even)i(for)h(impurities)h(with)265 4210 y Fy(D)k Fx(\035)e Fz(\(log)17 b Fy(L)p Fz(\))764 4177 y Fw(1)p Fu(=)p Fw(2)910 4210 y Fj(the)37 b(c)-5 b(oupling)38 b(c)-5 b(onstant)45 b Fz(\(equiv)-5 b(alen)m(tly)34 b(the)h(impurit)m(y)d(lev)m(el\))j Fj(has)i(to)h(b)-5 b(e)265 4357 y(\014ne)26 b(tune)-5 b(d)33 b Fz(as)23 b(a)g(function)f(of)h Fy(L)p Fz(.)38 b(Indeed,)24 b(for)f(a)g(coupling) e(constan)m(t)k(with)c(a)j(\014xed)e(v)-5 b(alue)22 b(the)265 4504 y(energy)k(of)f(the)h(impurit)m(y)d(lev)m(el)i(is)g(indep)s(enden) m(t)e(of)j Fy(L)p Fz(,)g(and)f(surely)f(for)h Fy(L)h Fz(large)f(enough)g(the)265 4651 y(energy)j(di\013erence)g(b)s(et)m(w)m (een)g(the)g(impurit)m(y)e(and)h(the)h(edge)h(lev)m(els)e(b)s(ecomes)h (m)m(uc)m(h)g(smaller)265 4797 y(than)j Fx(O)s Fz(\()p Fy(e)630 4764 y Fv(\000)p Fu(cB)s(D)832 4741 y Ft(2)871 4797 y Fz(\).)43 b(Remark)-5 b(ably)30 b(for)g(a)i(random)e(p)s(oten)m (tial)g(the)h(absence)h(of)f(resonance)g(is)265 4944 y(automatically)c(ac)m(hiev)m(ed)g(with)f(large)g(probabilit)m(y)f(and) h(no)g(\014ne)g(tuning)g(is)f(needed:)39 b(this)25 b(is)265 5091 y(wh)m(y)e(lo)s(calized)g(bulk)f(states)i(surviv)m(e.)38 b(W)-8 b(e)25 b(ha)m(v)m(e)g(analyzed)e(this)g(mec)m(hanism)g (rigorously)f(for)265 5238 y(a)28 b(mo)s(del)f(\(see)i(also)f([H]\))h (where)e(there)h(are)h(no)f(impurities)c(in)j(a)h(la)m(y)m(er)h(of)f (thic)m(kness)f(\(log)17 b Fy(L)p Fz(\))265 5385 y(along)23 b(the)g(b)s(oundary)-8 b(.)37 b(Then)22 b(the)h(edge)g(lev)m(els)g(are) g(basically)e(non)h(random)h(and)f(the)h(t)m(ypical)1852 5637 y Fs(4)p eop %%Page: 5 5 5 4 bop 265 100 a Fz(spacing)33 b(b)s(et)m(w)m(een)i(curren)m(t)f (carrying)f(eigen)m(v)-5 b(alues)34 b(is)e(easily)i(con)m(trolled.)51 b(Of)33 b(course)h(it)f(is)265 247 y(desirable)i(to)i(allo)m(w)g(for)f (impurities)e(close)j(to)g(the)g(b)s(oundary)e(but)h(then)h(the)f(edge) i(lev)m(els)265 394 y(b)s(ecome)28 b(random)f(and)g(some)h(further)f (analysis)f(is)g(needed.)40 b(Ho)m(w)m(ev)m(er)30 b(w)m(e)e(exp)s(ect)g (that)g(the)265 541 y(same)36 b(basic)f(mec)m(hanism)g(op)s(erates)h(b) s(ecause)g(the)f(t)m(ypical)g(spacing)g(b)s(et)m(w)m(een)i(edge)f(lev)m (els)265 688 y(should)25 b(still)g(b)s(e)h Fx(O)s Fz(\()p Fy(L)1016 655 y Fv(\000)p Fw(1)1110 688 y Fz(\).)40 b(In)26 b(connection)h(to)g(the)g(discussion)e(ab)s(o)m(v)m(e)j(w)m(e)f(men)m (tion)f(that)i(for)265 834 y(a)33 b(semi-in\014nite)c(system)k(the)f(b) s(ound)e(state)k(of)f(an)f(impurit)m(y)e(at)j(an)m(y)f(\014xed)g (distance)g(from)265 981 y(the)38 b(b)s(oundary)e(turns)h(in)m(to)h(a)h (resonance.)64 b(A)38 b(similar)d(situation)i(has)h(b)s(een)f(analysed) h(in)265 1128 y([GM)q(].)401 1275 y(W)-8 b(e)40 b(note)f(that)h(the)e (sp)s(ectral)g(region)h(close)f(to)2133 1239 y Fu(B)p 2133 1254 57 4 v 2144 1306 a Fw(2)2238 1275 y Fz(that)h(is)f(left)g (out)h(in)e(our)h(theorem)265 1421 y(is)h(precisely)f(the)i(one)g (where)f(resonances)h(b)s(et)m(w)m(een)h(edge)f(and)f(bulk)f(states)j (ma)m(y)g(o)s(ccur)265 1568 y(b)s(ecause)26 b(edge)g(states)h(b)s (ecome)f(v)m(ery)g(dense.)38 b(It)26 b(is)f(not)h(clear)f(what)h(is)e (the)i(connection)g(with)265 1715 y(the)i(div)m(ergence)h(of)f(the)g (lo)s(calization)g(length)f(of)h(the)h(in\014nite)d(system)i(at)h(the)f (band)f(cen)m(ter.)401 1862 y(In)32 b(the)g(presen)m(t)g(w)m(ork)g(w)m (e)g(ha)m(v)m(e)h(sho)m(wn)e(that)i(in)d(quan)m(tum)i(Hall)f(samples)g (there)h(exist)265 2009 y(w)m(ell)f(de\014ned)f(notions)h(of)h (extended)g(edge)h(states)g(\(curren)m(t)e(of)h Fx(O)s Fz(\(1\)\))i(and)d(lo)s(calized)f(bulk)265 2155 y(states)g (\(in\014nitesimal)c(curren)m(t\).)40 b(Instead)29 b(of)g(classifying)d (the)j(energy)h(lev)m(els)e(according)h(to)265 2302 y(their)36 b(curren)m(t)h(one)g(could)g(try)g(to)g(use)g(lev)m(el)g(statistics.)61 b(W)-8 b(e)38 b(exp)s(ect)g(that)f(the)h(lo)s(calized)265 2449 y(bulk)i(states)i(ha)m(v)m(e)h(P)m(oissonian)d(statistics)h (whereas)g(the)h(extended)f(edge)h(states)g(should)265 2596 y(displa)m(y)33 b(a)i(lev)m(el)f(repulsion.)51 b(In)33 b(fact)j(suc)m(h)e(a)h(strong)g(form)f(of)h(lev)m(el)f(repulsion)e(in)h (pro)m(v)m(ed)265 2743 y(in)e([M])h(for)g(energies)g(in)e(the)j(gap)f (of)g(the)g(bulk)e(Hamiltonian)g(where)i(only)f(extended)h(edge)265 2889 y(states)38 b(exist.)60 b(It)37 b(is)f(in)m(teresting)g(to)h (observ)m(e)h(that)f(in)f(the)h(presen)m(t)g(situation)e(b)s(oth)h (kind)265 3036 y(of)d(states)h(ha)m(v)m(e)g Fj(intermixe)-5 b(d)43 b Fz(energy)33 b(lev)m(els.)47 b(In)32 b(usual)f(Sc)m(hr\177)-45 b(odinger)31 b(op)s(erators)i(\(e.g.)49 b(the)265 3183 y(Anderson)31 b(mo)s(del)f(on)h(a)h(3)p Fy(D)j Fz(cubic)c(lattice\))h (it)f(is)g(accepted)i(\(but)e(not)h(pro)m(v)m(en\))g(that)g(they)265 3330 y(are)25 b(separated)g(b)m(y)g(a)g(w)m(ell)f(de\014ned)f Fj(mobilty)28 b(e)-5 b(dge)32 b Fz(\(results)24 b(in)f(this)h (direction)f(ha)m(v)m(e)j(recen)m(tly)265 3476 y(b)s(een)32 b(obtained)h([JL])g(under)e(a)j(suitable)d(h)m(yp)s(otesis\).)49 b(The)32 b(states)i(at)g(the)f(band)f(edge)i(are)265 3623 y(lo)s(calized)27 b(in)g(the)h(sense)g(that)h(the)g(sp)s(ectrum)e (is)g(dense)h(pure)f(p)s(oin)m(t)g(for)h(the)g(in\014nite)e(lattice)265 3770 y(and)g(has)g(P)m(oisson)h(statistics)f(for)h(the)g(\014nite)e (system)i([Mi].)40 b(A)m(t)27 b(the)g(band)f(cen)m(ter)h(the)g(states) 265 3917 y(are)c(b)s(eliev)m(ed)f(to)i(b)s(e)e(extended)h(in)e(the)j (sense)e(that)i(the)f(sp)s(ectrum)f(is)g(absolutely)g(con)m(tin)m(uous) 265 4064 y(for)27 b(the)f(in\014nite)f(lattice)i(and)f(has)h(the)g (statistics)f(of)h(the)g(Gaussian)f(Orthogonal)h(Ensem)m(ble)265 4210 y(for)j(the)h(\014nite)e(lattice.)401 4357 y(The)j(c)m (haracterization)h(of)f(the)g(sp)s(ectrum)f(of)39 b(\(1.1\))33 b(prop)s(osed)e(here)g(also)h(has)g(a)g(direct)265 4504 y(relev)-5 b(ance)30 b(to)h(the)f(Hall)e(conductivit)m(y)i(of)g(the)g (man)m(y)f(electron)h(\(non)g(in)m(teracting\))g(system.)265 4651 y(In)i(the)h(form)m(ulation)f(adv)m(o)s(cated)i(b)m(y)f(Halp)s (erin)d([H)q(])j(the)g(Hall)f(conductivit)m(y)g(is)g(computed)265 4797 y(as)24 b(the)g(ratio)h(of)f(the)g(net)g(equilibrium)19 b(curren)m(t)24 b(and)g(the)g(di\013erence)f(of)h(c)m(hemical)g(p)s (oten)m(tials)265 4944 y(b)s(et)m(w)m(een)d(the)g(t)m(w)m(o)h(edges.)38 b(Consider)19 b(the)i(man)m(y)f(fermion)g(state)i(\011\()p Fy(\026)2640 4959 y Fu(`)2672 4944 y Fy(;)15 b(\026)2767 4958 y Fu(r)2806 4944 y Fy(;)g(E)2913 4958 y Fu(F)2972 4944 y Fz(\))21 b(obtained)f(b)m(y)265 5091 y(\014lling)k(the)i(lev)m (els)g(of)g Fy(H)1080 5105 y Fu(!)1157 5091 y Fz(\(one)h(particle)e(p)s (er)g(state\))k(in)c(\006)2257 5106 y Fu(`)2301 5091 y Fx(\\)2374 5017 y Fp(\002)2422 5055 y Fu(B)p 2422 5070 V 2433 5122 a Fw(2)2509 5091 y Fz(+)19 b Fy(";)c(\026)2736 5106 y Fu(`)2770 5017 y Fp(\003)2808 5091 y Fz(,)27 b(\006)2926 5105 y Fu(r)2976 5091 y Fx(\\)3049 5017 y Fp(\002)3097 5055 y Fu(B)p 3097 5070 V 3108 5122 a Fw(2)3183 5091 y Fz(+)20 b Fy(";)15 b(\026)3411 5105 y Fu(r)3450 5017 y Fp(\003)265 5238 y Fz(and)33 b(\006)511 5253 y Fu(b)567 5238 y Fx(\\)650 5164 y Fp(\002)698 5202 y Fu(B)p 698 5217 V 709 5269 a Fw(2)785 5238 y Fz(+)19 b Fy(";)c(E)1024 5252 y Fu(F)1084 5164 y Fp(\003)1155 5238 y Fz(with)1375 5202 y Fu(B)p 1375 5217 V 1386 5269 a Fw(2)1464 5238 y Fz(+)22 b Fy(")31 b(<)f(\026)1786 5253 y Fu(`)1849 5238 y Fy(<)h(E)2018 5252 y Fu(F)2107 5238 y Fy(<)f(\026)2263 5252 y Fu(r)2331 5238 y Fy(<)2442 5202 y Fu(B)p 2442 5217 V 2453 5269 a Fw(2)2531 5238 y Fz(+)22 b Fy(V)2677 5252 y Fw(0)2716 5238 y Fz(.)50 b(The)33 b(total)i(curren)m(t)265 5385 y Fy(I)7 b Fz(\()p Fy(\026)402 5400 y Fu(`)435 5385 y Fy(;)15 b(\026)530 5399 y Fu(r)568 5385 y Fy(;)g(E)675 5399 y Fu(F)734 5385 y Fz(\))35 b(of)g(this)e(state)j({)f(a)f (stationary)h(state)h(of)e(the)h(man)m(y)g(particle)e(Hamiltonian)1852 5637 y Fs(5)p eop %%Page: 6 6 6 5 bop 265 100 a Fz({)30 b(is)e(giv)m(en)i(b)m(y)f(the)h(sum)f(of)g (the)h(individual)25 b(ph)m(ysical)j(curren)m(ts)h(of)h(the)g(\014lled) d(lev)m(els)i(\(giv)m(en)265 247 y(b)m(y)42 b Fy(L)465 214 y Fv(\000)p Fw(1)559 247 y Fz(\()p Fy( )s(;)15 b(v)740 261 y Fu(y)782 247 y Fy( )s Fz(\)\).)76 b(F)-8 b(rom)42 b(the)g(estimates)h(in)d(Theorem)h(1)h(the)g(leading)f(order)g(ma)m(y)h (b)s(e)265 394 y(computed)30 b(and)g(one)h(\014nds)1297 607 y Fy(I)7 b Fz(\()p Fy(\026)1434 622 y Fu(`)1467 607 y Fy(;)15 b(\026)1562 621 y Fu(r)1600 607 y Fy(;)g(E)1707 621 y Fu(F)1766 607 y Fz(\))26 b Fx(')1960 546 y Fz(1)p 1933 586 101 4 v 1933 670 a(2)p Fy(\031)2043 607 y Fz(\()p Fy(\026)2133 621 y Fu(r)2191 607 y Fx(\000)20 b Fy(\026)2337 622 y Fu(`)2370 607 y Fz(\))26 b Fy(:)845 b Fz(\(1.2\))265 820 y(In)31 b(\(1.2\))i(the)e(Hall)f(conductance)j(is)d(equal)h(to)h (one)f(\(this)g(is)f(b)s(ecause)h(w)m(e)h(ha)m(v)m(e)h(considered)265 967 y(only)j(the)g(\014rst)g(band\).)58 b(When)36 b Fy(\026)1469 982 y Fu(`)1538 967 y Fz(and)g Fy(\026)1776 981 y Fu(r)1850 967 y Fz(v)-5 b(ary)36 b(the)h(densit)m(y)e(of)i(particles)e(in)h(the)g (state)265 1114 y(\011\()p Fy(\026)426 1129 y Fu(`)459 1114 y Fy(;)15 b(\026)554 1128 y Fu(r)592 1114 y Fy(;)g(E)699 1128 y Fu(F)758 1114 y Fz(\))38 b(do)s(es)g(not)g(c)m(hange)h(since)e (the)h(n)m(um)m(b)s(er)e(of)i(lev)m(els)g(in)e(\006)2794 1128 y Fu(\013)2881 1114 y Fz(\()p Fy(\013)i Fz(=)g Fy(`;)15 b(r)s Fz(\))38 b(is)e(of)265 1261 y(order)c Fx(O)s Fz(\()p Fy(L)p Fz(\).)47 b(Ho)m(w)m(ev)m(er)34 b(if)e Fy(E)1305 1275 y Fu(F)1396 1261 y Fz(is)f(increased)h(the)g(particle)g(densit)m (y)g(\(and)g(th)m(us)g(the)g(\014lling)265 1408 y(factor\))i(increases) e(since)g(the)g(n)m(um)m(b)s(er)f(of)i(lev)m(els)f(in)f(\006)2186 1423 y Fu(b)2252 1408 y Fz(is)h(of)g(order)g Fx(O)s Fz(\()p Fy(L)2863 1375 y Fw(2)2903 1408 y Fz(\),)h(but)f(the)g(Hall)265 1554 y(conductance)j(do)s(es)g(not)g(c)m(hange)g(and)f(hence)h(has)g(a) g(plateau.)53 b(In)34 b(other)h(w)m(ords)f(the)h(edge)265 1701 y(states)g(con)m(tribute)e(to)h(the)g(Hall)e(conductance)i(but)f (not)h(to)g(the)g(densit)m(y)e(of)i(states)h(of)e(the)265 1848 y(sample)k(in)f(the)i(thermo)s(dynamic)e(limit.)61 b(An)37 b(in)m(teresting)g(problem)f(is)h(the)h(relationship)265 1995 y(b)s(et)m(w)m(een)43 b(the)f(presen)m(t)g(picture)f(and)g(the)h (one)h(using)d(the)i(Kub)s(o)f(form)m(ula,)j(for)e(this)f(see)265 2142 y([SKR1])30 b(and)g([SKR2].)401 2288 y(The)c(precise)g (de\014nition)e(of)j(the)f(mo)s(del)f(and)h(the)h(statemen)m(t)h(of)e (the)h(main)e(result)g(\(The-)265 2435 y(orem)31 b(1\))g(are)f(the)h (sub)5 b(ject)30 b(of)h(the)f(next)h(section.)265 2806 y Fr(2)161 b(The)53 b(Structure)f(of)i(the)f(Sp)t(ectrum)265 3042 y Fz(W)-8 b(e)30 b(consider)e(the)h(family)f(of)h(random)f (Hamiltonians)f(\(1.1\))k(acting)e(on)g(the)g(Hilb)s(ert)e(space)265 3189 y Fy(L)327 3156 y Fw(2)366 3189 y Fz(\()p Fq(R)i Fx(\002)19 b Fz([)p Fx(\000)683 3153 y Fu(L)p 683 3168 48 4 v 689 3221 a Fw(2)741 3189 y Fy(;)791 3153 y Fu(L)p 791 3168 V 797 3221 a Fw(2)849 3189 y Fz(]\))31 b(with)d(p)s(erio)s 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4741 a Fw(2)1168 4710 y Fx(j)1193 4672 y Fu(m)1255 4681 y Ft(1)1320 4710 y Fx(\024)k Fy(U)1478 4725 y Fu(`)1511 4710 y Fz(\()p Fy(x)p Fz(\))h Fx(\024)f Fy(c)1794 4724 y Fw(2)1834 4710 y Fx(j)p Fy(x)20 b Fz(+)2032 4674 y Fu(L)p 2032 4689 V 2038 4741 a Fw(2)2090 4710 y Fx(j)2115 4672 y Fu(m)2177 4681 y Ft(2)2473 4710 y Fz(for)30 b Fy(x)c Fx(\024)f(\000)2867 4674 y Fu(L)p 2867 4689 V 2873 4741 a Fw(2)3301 4710 y Fz(\(2.2\))828 4881 y Fy(c)867 4895 y Fw(1)907 4881 y Fx(j)p Fy(x)c Fx(\000)1105 4846 y Fu(L)p 1105 4861 V 1111 4913 a Fw(2)1163 4881 y Fx(j)1188 4844 y Fu(m)1250 4853 y Ft(1)1315 4881 y Fx(\024)k Fy(U)1473 4895 y Fu(r)1511 4881 y Fz(\()p Fy(x)p Fz(\))h Fx(\024)f Fy(c)1794 4895 y Fw(2)1834 4881 y Fx(j)p Fy(x)20 b Fx(\000)2032 4846 y Fu(L)p 2032 4861 V 2038 4913 a Fw(2)2090 4881 y Fx(j)2115 4844 y Fu(m)2177 4853 y Ft(2)2473 4881 y Fz(for)30 b Fy(x)c Fx(\025)2796 4846 y Fu(L)p 2796 4861 V 2802 4913 a Fw(2)3301 4881 y Fz(\(2.3\))265 5095 y(for)k(some)h(constan)m(ts)h(0) 25 b Fy(<)h(c)1241 5109 y Fw(1)1306 5095 y Fy(<)f(c)1441 5109 y Fw(2)1511 5095 y Fz(and)30 b(2)c Fx(\024)f Fy(m)1935 5109 y Fw(1)1999 5095 y Fy(<)g(m)2175 5109 y Fw(2)2240 5095 y Fy(<)g Fx(1)p Fz(.)41 b(Recall)30 b(that)h Fy(U)3026 5110 y Fu(`)3059 5095 y Fz(\()p Fy(x)p Fz(\))26 b(=)f(0)31 b(for)265 5241 y Fy(x)e Fx(\025)g(\000)527 5206 y Fu(L)p 527 5221 V 533 5273 a Fw(2)618 5241 y Fz(and)j Fy(U)859 5255 y Fu(r)897 5241 y Fz(\()p Fy(x)p Fz(\))e(=)f(0)k(for)g Fy(x)c Fx(\024)1560 5206 y Fu(L)p 1560 5221 V 1566 5273 a Fw(2)1618 5241 y Fz(.)48 b(W)-8 b(e)34 b(could)e(allo)m(w)g(steep)s (er)h(con\014nemen)m(ts)g(but)f(the)265 5388 y(presen)m(t)e(p)s (olynomial)e(conditions)h(turn)g(out)i(to)g(b)s(e)f(tec)m(hnically)f (con)m(v)m(enien)m(t.)1852 5637 y Fs(6)p eop %%Page: 7 7 7 6 bop 401 100 a Fz(W)-8 b(e)46 b(assume)e(that)h(eac)m(h)h(impurit)m (y)c(is)h(the)i(source)f(of)h(a)g(lo)s(cal)e(p)s(oten)m(tial)h Fy(V)69 b Fx(2)48 b Fy(C)3423 67 y Fw(2)3462 100 y Fz(,)265 247 y(0)26 b Fx(\024)f(j)p Fy(V)20 b Fz(\()p Fy(x;)15 b(y)s Fz(\))p Fx(j)27 b(\024)e Fy(V)941 261 y Fw(0)1005 247 y Fy(<)g Fx(1)p Fz(,)31 b(supp)13 b Fy(V)46 b Fx(\032)25 b Fq(B)1715 174 y Fp(\000)1762 247 y Fz(0)p Fy(;)1858 211 y Fw(1)p 1858 226 36 4 v 1858 279 a(4)1904 174 y Fp(\001)1945 247 y Fz(,)31 b(and)f(that)h(they)f(are)h(lo)s(cated)g(at) g(the)g(sites)265 394 y(of)47 b(a)f(\014nite)g(lattice)g(\003)53 b(=)1256 320 y Fp(\010)1309 394 y Fz(\()p Fy(n;)15 b(m)p Fz(\))26 b Fx(2)f Fq(Z)1731 361 y Fw(2)1766 394 y Fz(;)15 b Fy(n)25 b Fx(2)g Fz([)p Fx(\000)2078 358 y Fu(L)p 2078 373 48 4 v 2084 425 a Fw(2)2156 394 y Fz(+)20 b(log)d Fy(L;)2492 358 y Fu(L)p 2492 373 V 2498 425 a Fw(2)2570 394 y Fx(\000)j Fz(log)c Fy(L)p Fz(])p Fy(;)f(m)26 b Fx(2)f Fz([)p Fx(\000)3218 358 y Fu(L)p 3218 373 V 3224 425 a Fw(2)3276 394 y Fy(;)3326 358 y Fu(L)p 3326 373 V 3332 425 a Fw(2)3384 394 y Fz(])3409 320 y Fp(\011)3462 394 y Fz(.)265 541 y(The)30 b(random)f(p)s(oten)m(tial)h Fy(V)1222 555 y Fu(!)1303 541 y Fz(has)g(the)h(form)1025 760 y Fy(V)1078 774 y Fu(!)1129 760 y Fz(\()p Fy(x;)15 b(y)s Fz(\))26 b(=)1533 674 y Fp(X)1461 875 y Fw(\()p Fu(n;m)p Fw(\))p Fv(2)p Fw(\003)1752 760 y Fy(X)1827 774 y Fu(n;m)1956 760 y Fz(\()p Fy(!)s Fz(\))p Fy(V)21 b Fz(\()p Fy(x)f Fx(\000)g Fy(n;)15 b(y)23 b Fx(\000)d Fy(m)p Fz(\))574 b(\(2.4\))265 1045 y(where)45 b(the)h(coupling)e (constan)m(ts)i Fy(X)1586 1059 y Fu(n;m)1761 1045 y Fz(are)g(i.i.d.)85 b(random)45 b(v)-5 b(ariables)44 b(with)g(common)265 1192 y(densit)m(y)37 b Fy(h)g Fx(2)g Fy(C)845 1159 y Fw(2)884 1192 y Fz(\([)p Fx(\000)p Fz(1)p Fy(;)15 b Fz(1]\))39 b(that)f(satis\014es)f Fx(k)p Fy(h)p Fx(k)1932 1206 y Fv(1)2034 1192 y Fy(<)25 b Fx(1)o Fz(,)40 b(supp)13 b Fy(h)38 b Fz(=)f([)p Fx(\000)p Fz(1)p Fy(;)15 b Fz(1].)63 b(W)-8 b(e)39 b(will)34 b(de-)265 1339 y(note)42 b(b)m(y)f Fq(P)670 1353 y Fw(\003)765 1339 y Fz(the)h(pro)s(duct)e(measure)h (de\014ned)g(on)g(the)h(set)f(of)h(all)e(p)s(ossible)f(realizations)265 1486 y Fy(!)28 b Fx(2)d 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b(non)h(random)f(Hamiltonians)1206 2732 y Fy(H)1282 2746 y Fu(\013)1357 2732 y Fz(=)c Fy(H)1529 2746 y Fw(0)1588 2732 y Fz(+)20 b Fy(U)1741 2746 y Fu(\013)1790 2732 y Fy(;)379 b(\013)26 b Fz(=)f Fy(`;)15 b(r)28 b(:)755 b Fz(\(2.6\))265 2951 y(All)34 b(the)h(Hamiltonians)e(considered)h(ab)s (o)m(v)m(e)i(ha)m(v)m(e)g(p)s(erio)s(dic)c(b)s(oundary)h(conditions)g (along)265 3098 y(the)44 b Fy(y)i Fz(direction)c(and)h(are)h(essen)m (tially)e(self-adjoin)m(t)h(on)g Fy(C)2413 3065 y Fv(1)2406 3123 y Fw(0)2487 3098 y Fz(\()p Fq(R)30 b Fx(\002)19 b Fz([)p Fx(\000)2805 3062 y Fu(L)p 2805 3077 V 2811 3130 a Fw(2)2863 3098 y Fy(;)2914 3062 y Fu(L)p 2914 3077 V 2920 3130 a Fw(2)2972 3098 y Fz(]\).)80 b(F)-8 b(or)44 b(eac)m(h)265 3245 y(realization)33 b Fy(!)k Fz(and)c(size)g Fy(L)h Fz(the)g(sp)s(ectrum)e Fy(\033)s Fz(\()p Fy(H)1984 3259 y Fu(!)2035 3245 y Fz(\))i(of)41 b(\(1.1\))35 b(\(it)f(dep)s(ends)e(on)h Fy(L)p Fz(\))h(consists)265 3392 y(of)42 b(isolated)g(eigen)m(v)-5 b(alues)42 b(of)h(\014nite)e(m)m (ultiplicit)m(y)-8 b(.)74 b(In)41 b(order)h(to)h(state)h(our)d(main)h (result)265 3539 y(c)m(haracterizing)30 b(these)h(eigen)m(v)-5 b(alues)30 b(w)m(e)g(\014rst)f(ha)m(v)m(e)i(to)g(describ)s(e)d(the)i (sp)s(ectra)g(of)37 b(\(2.5\))32 b(and)265 3685 y(\(2.6\))r(.)401 3832 y(Let)f(us)e(b)s(egin)f(with)h(the)h(edge)g(Hamiltonians)e (\(2.6\))r(.)40 b(Here)31 b(w)m(e)f(state)h(their)e(prop)s(erties)265 3979 y(without)c(pro)s(ofs)g(and)g(refer)h(the)g(reader)g(to)h([MMP)q (],)g([F])g(for)e(more)h(details.)39 b(Since)25 b(the)h(edge)265 4126 y(Hamiltonians)j Fy(H)900 4140 y Fu(\013)979 4126 y Fz(comm)m(ute)i(with)e Fy(p)1623 4140 y Fu(y)1665 4126 y Fz(,)h(they)h(are)f(decomp)s(osable)g(in)m(to)h(a)f(direct)g(sum)813 4345 y Fy(H)889 4359 y Fu(\013)963 4345 y Fz(=)1074 4259 y Fp(X)1206 4276 y Fv(\010)1031 4470 y Fu(k)r Fv(2)1127 4443 y Ft(2)p Fi(\031)p 1126 4455 69 3 v 1139 4496 a(L)1205 4470 y Fh(Z)1280 4345 y Fy(H)1356 4359 y Fu(\013)1405 4345 y Fz(\()p Fy(k)s Fz(\))c(=)1662 4259 y Fp(X)1793 4276 y Fv(\010)1619 4470 y Fu(k)r Fv(2)1715 4443 y Ft(2)p Fi(\031)p 1714 4455 V 1727 4496 a(L)1793 4470 y Fh(Z)1868 4272 y Fp(\002)1915 4310 y Fw(1)p 1915 4325 36 4 v 1915 4377 a(2)1961 4345 y Fy(p)2007 4308 y Fw(2)2007 4368 y Fu(x)2070 4345 y Fz(+)2171 4310 y Fw(1)p 2171 4325 V 2171 4377 a(2)2217 4345 y Fz(\()p Fy(k)d Fx(\000)d Fy(B)5 b(x)p Fz(\))2574 4308 y Fw(2)2634 4345 y Fz(+)20 b Fy(U)2787 4359 y Fu(\013)2836 4272 y Fp(\003)2914 4345 y Fy(:)362 b Fz(\(2.7\))265 4654 y(F)-8 b(or)22 b(eac)m(h)g Fy(k)j Fz(the)c(one)g(dimensional)e(Hamiltonian)g Fy(H)2071 4668 y Fu(\013)2120 4654 y Fz(\()p Fy(k)s Fz(\))j(has)f(a)h(compact)g (resolv)m(en)m(t,)i(th)m(us)d(it)265 4801 y(has)h(discrete)f(eigen)m(v) -5 b(alues)22 b(and)f(b)m(y)g(standard)g(argumen)m(ts)h(one)g(can)g (sho)m(w)g(that)g(they)g(are)g(not)265 4948 y(degenerate.)42 b(If)28 b(the)g Fy(y)j Fz(direction)d(w)m(ould)f(b)s(e)h(in\014nitely)d (extended,)k Fy(k)j Fz(w)m(ould)27 b(v)-5 b(ary)29 b(o)m(v)m(er)g(the) 265 5095 y(real)g(axis)f(and)g(the)i(eigen)m(v)-5 b(alues)28 b(of)h Fy(H)1601 5109 y Fu(\013)1651 5095 y Fz(\()p Fy(k)s Fz(\))g(w)m(ould)f(form)h(sp)s(ectral)f(branc)m(hes)g Fy(")3035 5062 y Fu(\013)3035 5117 y(\027)3085 5095 y Fz(\()3122 5071 y(^)3120 5095 y Fy(k)t Fz(\),)3263 5071 y(^)3261 5095 y Fy(k)g Fx(2)d Fq(R)265 5241 y Fz(lab)s(elled)k(b)m(y)i (the)h(Landau)e(lev)m(el)h(index)f Fy(\027)6 b Fz(.)43 b(These)31 b(sp)s(ectral)g(branc)m(hes)g(are)h(strictly)e(mono-)265 5388 y(tone,)43 b(en)m(tire)c(functions)g(with)f(the)i(prop)s(erties)e Fy(")2031 5355 y Fu(`)2031 5411 y(\027)2074 5388 y Fz(\()p Fx(\0001)p Fz(\))j(=)g(+)p Fx(1)p Fz(,)g Fy(")2729 5355 y Fu(`)2729 5411 y(\027)2773 5388 y Fz(\(+)p Fx(1)p Fz(\))25 b(=)g(\()p Fy(\027)i Fz(+)3333 5352 y Fw(1)p 3333 5367 V 3333 5419 a(2)3379 5388 y Fz(\))p Fy(B)1852 5637 y Fs(7)p eop %%Page: 8 8 8 7 bop 265 100 a Fz(and)28 b Fy(")482 67 y Fu(r)482 123 y(\027)526 100 y Fz(\()p Fx(\0001)p Fz(\))e(=)e(\()p Fy(\027)g Fz(+)1081 65 y Fw(1)p 1081 80 36 4 v 1081 132 a(2)1126 100 y Fz(\))p Fy(B)5 b Fz(,)30 b Fy(")1332 67 y Fu(r)1332 123 y(\027)1375 100 y Fz(\(+)p Fx(1)p Fz(\))c(=)f(+)p Fx(1)p Fz(.)39 b(Here)30 b(b)s(ecause)f(of)g(the)g(p)s(erio)s(dic)e(b)s (oundary)265 247 y(conditions)i(the)h(set)h(of)g Fy(k)i Fz(v)-5 b(alues)30 b(is)g(discrete)g(so)g(that)h(the)g(sp)s(ectrum)e (of)i Fy(H)2921 261 y Fu(\013)1249 485 y Fy(\033)s Fz(\()p Fy(H)1415 499 y Fu(\013)1464 485 y Fz(\))26 b(=)1621 412 y Fp(\010)1674 485 y Fy(E)1746 448 y Fu(\013)1741 508 y(\027)t(k)1823 485 y Fz(;)15 b Fy(\027)31 b Fx(2)25 b Fq(N)7 b Fy(;)15 b(k)34 b Fx(2)2302 450 y Fw(2)p Fu(\031)p 2302 465 79 4 v 2317 517 a(L)2390 485 y Fq(Z)2455 412 y Fp(\011)3301 485 y Fz(\(2.8\))265 723 y(consists)h(of)g(isolated)g(p) s(oin)m(ts)g(on)g(the)g(sp)s(ectral)g(branc)m(hes)g Fy(E)2424 690 y Fu(\013)2419 751 y(\027)t(k)2535 723 y Fz(=)e Fy(")2681 690 y Fu(\013)2681 746 y(\027)2731 723 y Fz(\()p Fy(k)s Fz(\),)38 b Fy(k)e Fx(2)3102 688 y Fw(2)p Fu(\031)p 3102 703 V 3117 755 a(L)3190 723 y Fq(Z)p Fz(.)51 b(The)265 870 y(corresp)s(onding)28 b(eigenfunctions)h Fy( )1505 837 y Fu(\013)1502 898 y(\027)t(k)1615 870 y Fz(ha)m(v)m(e)i(the)g (form)1356 1118 y Fy( )1418 1080 y Fu(\013)1415 1141 y(\027)t(k)1497 1118 y Fz(\()p Fy(x;)15 b(y)s Fz(\))26 b(=)1885 1056 y(1)p 1839 1097 138 4 v 1839 1115 a Fx(p)p 1915 1115 62 4 v 77 x Fy(L)1987 1118 y(e)2029 1080 y Fu(ik)r(y)2133 1118 y Fy(')2192 1080 y Fu(\013)2192 1141 y(\027)t(k)2274 1118 y Fz(\()p Fy(x)p Fz(\))905 b(\(2.9\))265 1365 y(with)55 b Fy(')557 1332 y Fu(\013)557 1393 y(\027)t(k)696 1365 y Fz(the)h(normalized)f(eigenfunctions)h(of)g(the)h (one-dimensional)d(Hamiltonian)265 1512 y Fy(H)341 1526 y Fu(\013)390 1512 y Fz(\()p Fy(k)s Fz(\).)77 b(By)43 b(de\014nition,)g(the)g(curren)m(t)f(of)g(the)h(state)g Fy( )2288 1479 y Fu(\013)2285 1539 y(\027)t(k)2410 1512 y Fz(in)e(the)h Fy(y)j Fz(direction)c(is)h(giv)m(en)265 1658 y(b)m(y)30 b(the)h(exp)s(ectation)g(v)-5 b(alue)30 b(of)g(the)h(v)m(elo)s(cit)m(y)g Fy(v)1913 1672 y Fu(y)1979 1658 y Fz(=)25 b Fy(p)2121 1672 y Fu(y)2183 1658 y Fx(\000)20 b Fy(B)5 b(x)p Fz(,)504 1908 y Fy(J)563 1871 y Fu(\013)554 1932 y(\027)t(k)661 1908 y Fz(=)25 b(\()p Fy( )854 1871 y Fu(\013)851 1932 y(\027)t(k)933 1908 y Fy(;)15 b(v)1017 1922 y Fu(y)1059 1908 y Fy( )1121 1871 y Fu(\013)1118 1932 y(\027)t(k)1200 1908 y Fz(\))26 b(=)1357 1785 y Fp(Z)1407 1991 y Fh(R)1474 1908 y Fx(j)p Fy(')1558 1871 y Fu(\013)1558 1932 y(\027)t(k)1641 1908 y Fz(\()p Fy(x)p Fz(\))p Fx(j)1788 1871 y Fw(2)1828 1908 y Fz(\()p Fy(k)e Fx(\000)c Fy(B)5 b(x)p Fz(\))15 b(d)o Fy(x)26 b Fz(=)e Fy(@)2474 1923 y Fw(^)2472 1941 y Fu(k)2515 1908 y Fy(")2557 1871 y Fu(\013)2557 1931 y(\027)2607 1908 y Fz(\()2644 1884 y(^)2642 1908 y Fy(k)t Fz(\))2728 1804 y Fp(\014)2728 1858 y(\014)2728 1913 y(\014)2760 1954 y Fw(^)2758 1972 y Fu(k)r Fw(=)2862 1936 y(2)p Fu(\031)r(m)p 2862 1951 141 4 v 2908 2003 a(L)3255 1908 y Fz(\(2.10\))265 2171 y(where)30 b(the)h(last)f(equalit)m(y)g(follo)m(ws)f(from)h(the)h(F)-8 b(eynman-Hellman)30 b(theorem.)41 b(F)-8 b(rom)31 b(\(2.10\))265 2318 y(w)m(e)24 b(notice)g(that)g(for)f(an)m(y)h Fy(")i(>)f Fz(0,)h(one)d(can)h(\014nd)e Fy(j)5 b Fz(\()p Fy(")p Fz(\))27 b Fy(>)e Fz(0)f(and)f Fy(L)p Fz(\()p Fy(")p Fz(\))i(suc)m(h)e(that)h(for)g Fy(L)h(>)g(L)p Fz(\()p Fy(")p Fz(\))265 2464 y(the)31 b(states)g(of)g(the)f(t)m(w)m(o)i(branc) m(hes)e Fy(\027)h Fz(=)25 b(0,)31 b Fy(\013)26 b Fz(=)f Fy(`;)15 b(r)33 b Fz(with)c(energies)h Fy(E)2717 2431 y Fu(\013)2712 2492 y Fw(0)p Fu(k)2815 2464 y Fx(\025)2921 2429 y Fw(1)p 2921 2444 36 4 v 2921 2496 a(2)2966 2464 y Fy(B)25 b Fz(+)20 b Fy(")31 b Fz(satisfy)1107 2702 y Fy(J)1166 2665 y Fu(`)1157 2726 y Fw(0)p Fu(k)1261 2702 y Fx(\024)25 b(\000)p Fy(j)5 b Fz(\()p Fy(")p Fz(\))27 b Fy(<)e Fz(0)273 b Fy(J)2082 2665 y Fu(r)2073 2726 y Fw(0)p Fu(k)2176 2702 y Fx(\025)25 b Fy(j)5 b Fz(\()p Fy(")p Fz(\))27 b Fy(>)e Fz(0)h Fy(:)610 b Fz(\(2.11\))265 2941 y(In)44 b(other)h(w)m(ords)f(the)g(eigenstates)i(of)f(the)g(edge)g (Hamiltonians)e(carry)h(an)h(appreciable)265 3087 y(curren)m(t.)40 b(The)30 b(spacing)g(of)g(t)m(w)m(o)h(consecutiv)m(e)g(eigen)m(v)-5 b(alues)30 b(greater)i(than)2882 3052 y Fw(1)p 2882 3067 V 2882 3119 a(2)2927 3087 y Fy(B)24 b Fz(+)c Fy(")30 b Fz(satis\014es)1063 3218 y Fp(\014)1063 3273 y(\014)1063 3327 y(\014)1063 3382 y(\014)1094 3350 y Fy(E)1166 3313 y Fu(\013)1161 3398 y Fw(0)1206 3364 y Ft(2)p Fi(\031)r Ft(\()p Fi(m)p Ft(+1\))p 1206 3383 248 3 v 1309 3424 a Fi(L)1488 3350 y Fx(\000)20 b Fy(E)1651 3313 y Fu(\013)1646 3386 y Fw(0)1691 3359 y Ft(2)p Fi(\031)r(m)p 1691 3371 123 3 v 1732 3412 a(L)1828 3218 y Fp(\014)1828 3273 y(\014)1828 3327 y(\014)1828 3382 y(\014)1884 3350 y Fy(>)1990 3289 y(j)5 b Fz(\()p Fy(")p Fz(\))p 1990 3329 156 4 v 2037 3412 a Fy(L)2337 3350 y(\013)26 b Fz(=)f Fy(`;)15 b(r)28 b(:)566 b Fz(\(2.12\))265 3604 y(Note)33 b(that)f(these)g(observ)-5 b(ations)31 b(extend)h(to)g(other)g(branc)m(hes)f(but)g Fy(j)5 b Fz(\()p Fy(")p Fz(\))33 b(and)e Fy(L)p Fz(\()p Fy(")p Fz(\))h(are)g(not)265 3751 y(uniform)25 b(with)h(resp)s(ect)i (to)g(the)f(index)f Fy(\027)6 b Fz(.)40 b(In)26 b(the)i(rest)f(of)h (the)f(pap)s(er)g(w)m(e)h(limit)d(ourselv)m(es)i(to)265 3898 y Fy(\027)k Fz(=)25 b(0)i(for)f(simplicit)m(y)-8 b(.)37 b(On)25 b(the)i(other)f(hand)g(the)g(spacing)g(b)s(et)m(w)m(een) h(the)f(energies)h(of)f Fy(\033)s Fz(\()p Fy(H)3419 3913 y Fu(`)3452 3898 y Fz(\))265 4045 y(and)33 b Fy(\033)s Fz(\()p Fy(H)611 4059 y Fu(r)649 4045 y Fz(\))g(is)f(a)i(priori)c (arbitrary)-8 b(.)49 b(W)-8 b(e)34 b(assume)f(that)h(the)f(con\014ning) f(p)s(oten)m(tials)g Fy(U)3275 4060 y Fu(`)3341 4045 y Fz(and)265 4191 y Fy(U)327 4205 y Fu(r)395 4191 y Fz(are)f(suc)m(h)f (that)h(the)g(follo)m(wing)e(h)m(yp)s(othesis)g(is)g(ful\014lled.)265 4413 y FA(Hyp)s(othesis)34 b(1.)41 b Fj(Fix)32 b(any)h Fy(")25 b(>)g Fz(0)33 b Fj(and)f(let)g Fz(\001)1883 4427 y Fu(")1945 4413 y Fz(=)2041 4339 y Fp(\002)2089 4377 y Fw(1)p 2089 4392 36 4 v 2089 4444 a(2)2134 4413 y Fy(B)25 b Fz(+)20 b Fy(";)2411 4377 y Fw(1)p 2411 4392 V 2411 4444 a(2)2457 4413 y Fy(B)k Fz(+)c Fy(V)2694 4427 y Fw(0)2734 4339 y Fp(\003)2772 4413 y Fj(.)41 b(Ther)-5 b(e)33 b(exist)f Fy(L)p Fz(\()p Fy(")p Fz(\))265 4560 y Fj(and)i Fy(d)p Fz(\()p Fy(")p Fz(\))26 b Fy(>)f Fz(0)33 b Fj(such)g(that)h(for)f(al)5 b(l)33 b Fy(L)25 b(>)g(L)p Fz(\()p Fy(")p Fz(\))1113 4812 y(dist)14 b(\()q Fy(\033)s Fz(\()p Fy(H)1477 4827 y Fu(`)1510 4812 y Fz(\))20 b Fx(\\)g Fz(\001)1722 4826 y Fu(")1759 4812 y Fy(;)15 b(\033)s Fz(\()p Fy(H)1965 4826 y Fu(r)2003 4812 y Fz(\))21 b Fx(\\)e Fz(\001)2215 4826 y Fu(")2252 4812 y Fz(\))26 b Fx(\025)2419 4750 y Fy(d)p Fz(\()p Fy(")p Fz(\))p 2419 4791 161 4 v 2468 4874 a Fy(L)2614 4812 y(:)616 b Fz(\(2.13\))401 5050 y(This)23 b(h)m(yp)s(othesis)g(is)h(imp)s(ortan)m(t)g(b)s(ecause)g(a)h (minimal)d(amoun)m(t)j(of)f(non-degeneracy)i(b)s(e-)265 5197 y(t)m(w)m(een)31 b(the)f(sp)s(ectra)g(of)g(the)g(t)m(w)m(o)h(edge) f(systems)g(is)f(needed)g(in)f(order)i(to)g(con)m(trol)g(bac)m(kscat-) 265 5343 y(tering)43 b(e\013ects)i(induced)c(b)m(y)i(the)h(random)f(p)s (oten)m(tial.)79 b(Indeed)43 b(in)f(a)h(system)h(with)e(t)m(w)m(o)1852 5637 y Fs(8)p eop %%Page: 9 9 9 8 bop 265 100 a Fz(b)s(oundaries)30 b(bac)m(kscattering)35 b(fa)m(v)m(ors)e(lo)s(calization)f(and)g(has)h(a)g(tendency)g(to)g (destro)m(y)h(cur-)265 247 y(ren)m(ts.)53 b(This)32 b(h)m(yp)s(othesis) h(can)i(easily)e(b)s(e)h(realized)f(b)m(y)h(taking)h(non-symmetric)e (con\014ning)265 394 y(p)s(oten)m(tials)28 b Fy(U)745 409 y Fu(`)806 394 y Fz(and)g Fy(U)1043 408 y Fu(r)1081 394 y Fz(.)40 b(In)28 b(a)g(more)h(realistic)e(mo)s(del)g(with)g (impurities)f(close)i(to)h(the)g(edges)265 541 y(one)i(exp)s(ects)f (that)h(it)f(is)g(automatically)g(satis\014ed)f(with)h(a)g(large)h (probabilit)m(y)-8 b(.)401 688 y(No)m(w)25 b(w)m(e)g(describ)s(e)e(the) i(sp)s(ectral)e(prop)s(erties)g(of)i(the)f(bulk)f(random)h(Hamiltonian) e(\(2.5\))r(.)265 834 y(F)-8 b(rom)39 b(the)g(b)s(ound)d(\(A.5\))k(on)e (the)h(k)m(ernel)f(of)h Fy(R)1969 848 y Fw(0)2008 834 y Fz(\()p Fy(z)t Fz(\))g(and)f(the)h(fact)g(that)g Fy(V)2958 848 y Fu(!)3047 834 y Fz(is)f(b)s(ounded)265 981 y(with)d(compact)j (supp)s(ort)c(w)m(e)j(can)g(see)g(that)g Fy(V)1912 995 y Fu(!)1999 981 y Fz(is)e(relativ)m(ely)h(compact)i(w.r.t.)59 b Fy(H)3216 995 y Fw(0)3255 981 y Fz(,)38 b(th)m(us)265 1128 y Fy(\033)317 1142 y Fu(ess)419 1128 y Fz(\()p Fy(H)530 1143 y Fu(b)564 1128 y Fz(\))26 b(=)721 1054 y Fp(\010)774 1128 y Fz(\()p Fy(\027)g Fz(+)981 1092 y Fw(1)p 981 1107 36 4 v 981 1159 a(2)1026 1128 y Fz(\))p Fy(B)5 b Fz(;)15 b Fy(\027)32 b Fx(2)24 b Fq(N)1397 1054 y Fp(\011)1456 1128 y Fz(.)57 b(Since)35 b Fx(j)p Fy(V)1859 1142 y Fu(!)1910 1128 y Fz(\()p Fy(x;)15 b(y)s Fz(\))p Fx(j)35 b(\024)f Fy(V)2338 1142 y Fw(0)2412 1128 y Fy(<)g(B)41 b Fz(the)36 b(eigen)m(v)-5 b(alues)35 b Fy(E)3336 1095 y Fu(b)3331 1156 y(\014)3414 1128 y Fz(of)265 1275 y Fy(H)341 1290 y Fu(b)403 1275 y Fz(are)29 b(con)m(tained)g(in)e(Landau)h(bands)1656 1206 y Fp(S)1732 1302 y Fu(\027)t Fv(\025)p Fw(0)1881 1201 y Fp(\002)1918 1275 y Fz(\()p Fy(\027)f Fz(+)2126 1239 y Fw(1)p 2126 1254 V 2126 1306 a(2)2171 1275 y Fz(\))p Fy(B)e Fx(\000)20 b Fy(V)2444 1289 y Fw(0)2483 1275 y Fy(;)15 b Fz(\()p Fy(\027)27 b Fz(+)2731 1239 y Fw(1)p 2731 1254 V 2731 1306 a(2)2776 1275 y Fz(\))p Fy(B)e Fz(+)20 b Fy(V)3049 1289 y Fw(0)3089 1201 y Fp(\003)3126 1275 y Fz(.)40 b(W)-8 b(e)30 b(will)265 1421 y(assume)265 1623 y FA(Hyp)s(othesis)39 b(2.)k Fj(Fix)36 b(any)g Fy(")31 b(>)f Fz(0)p Fj(.)51 b(Ther)-5 b(e)36 b(exist)f Fy(\026)p Fz(\()p Fy(")p Fz(\))i Fj(a)f(strictly)g(p)-5 b(ositive)36 b(c)-5 b(onstant)38 b(and)265 1769 y Fy(L)p Fz(\()p Fy(")p Fz(\))g Fj(such)e(that)i(for)f(al)5 b(l)38 b Fy(L)32 b(>)g(L)p Fz(\()p Fy(")p Fz(\))38 b Fj(one)f(c)-5 b(an)37 b(\014nd)g(a)g(set)g(of)g(r)-5 b(e)g(alizations)39 b(of)e(the)g(r)-5 b(andom)265 1916 y(p)g(otential)36 b Fz(\012)704 1860 y Fg(0)704 1943 y Fw(\003)790 1916 y Fj(with)e Fq(P)1043 1930 y Fw(\003)1097 1916 y Fz(\(\012)1198 1860 y Fg(0)1198 1943 y Fw(\003)1251 1916 y Fz(\))27 b Fx(\025)f Fz(1)21 b Fx(\000)g Fy(L)1630 1883 y Fv(\000)p Fu(\022)1724 1916 y Fj(,)33 b Fy(\022)c(>)d Fz(0)p Fj(,)34 b(with)g(the)g(pr)-5 b(op)g(erty)36 b(that)f(if)e Fy(!)d Fx(2)c Fz(\012)3283 1860 y Fg(0)3283 1943 y Fw(\003)3369 1916 y Fj(the)265 2063 y(eigenstates)33 b(c)-5 b(orr)g(esp)g(onding)36 b(to)d Fy(E)1480 2030 y Fu(b)1475 2091 y(\014)1548 2063 y Fx(2)24 b Fy(\033)s Fz(\()p Fy(H)1799 2078 y Fu(b)1834 2063 y Fz(\))c Fx(\\)g Fz(\001)2046 2077 y Fu(")2115 2063 y Fj(satisfy)813 2277 y Fx(j)p Fy( )900 2239 y Fu(b)897 2300 y(\014)945 2277 y Fz(\()p Fy(x;)i Fz(\026)-52 b Fy(y)1117 2292 y Fu(\014)1164 2277 y Fz(\))p Fx(j)26 b(\024)f Fy(e)1388 2239 y Fv(\000)p Fu(\026)p Fw(\()p Fu(")p Fw(\))p Fu(L)1811 2277 y Fy(;)202 b Fx(j)p Fy(@)2111 2291 y Fu(y)2152 2277 y Fy( )2214 2239 y Fu(b)2211 2300 y(\014)2259 2277 y Fz(\()p Fy(x;)22 b Fz(\026)-52 b Fy(y)2431 2292 y Fu(\014)2478 2277 y Fz(\))p Fx(j)26 b(\024)f Fy(e)2702 2239 y Fv(\000)p Fu(\026)p Fw(\()p Fu(")p Fw(\))p Fu(L)3255 2277 y Fz(\(2.14\))265 2490 y Fj(for)33 b(some)40 b Fz(\026)-51 b Fy(y)688 2505 y Fu(\014)767 2490 y Fj(dep)-5 b(ending)34 b(on)f Fy(!)i Fj(and)f Fy(L)p Fj(.)401 2692 y Fz(Since)27 b Fy(V)689 2706 y Fu(!)767 2692 y Fz(is)f(random)h(w)m(e)h(exp)s(ect)g (that)g(w)m(a)m(v)m(efunctions)f(with)g(energies)g(in)f(\001)3101 2706 y Fu(")3165 2692 y Fz(\(not)i(to)s(o)265 2838 y(close)23 b(to)h(the)f(Landau)f(lev)m(els)h(where)f(the)h(lo)s(calization)f (length)h(div)m(erges\))g(are)g(exp)s(onen)m(tially)265 2985 y(lo)s(calized)f(on)g(a)h(scale)g Fx(O)s Fz(\(1\))h(with)e(resp)s (ect)g(to)i Fy(L)p Fz(.)38 b(Inequalities)21 b(\(2.14\))k(are)e(a)g(w)m (eak)m(er)h(v)m(ersion)265 3132 y(of)35 b(this)e(statemen)m(t,)k(and)d (ha)m(v)m(e)i(b)s(een)d(c)m(hec)m(k)m(ed)k(for)d(the)g(sp)s(ecial)f (case)j(where)e(the)g(random)265 3279 y(p)s(oten)m(tial)22 b(is)g(a)h(sum)f(of)g(rank)h(one)g(p)s(erturbations)d([FM2)q(])j(using) e(the)i(metho)s(ds)f(of)h(Aizenman)265 3426 y(and)28 b(Molc)m(hano)m(v)j([AM])e(\(see)h(for)f(example)g([DMP4)q(])g(where)g (the)g(case)h(of)f(p)s(oin)m(t)f(impurities)265 3572 y(is)38 b(treated)i(b)m(y)f(these)h(metho)s(ds\).)66 b(Presumably)37 b(one)j(could)e(adapt)h(existing)f(tec)m(hniques)265 3719 y(for)f(m)m(ultiplicativ)m(e)e(p)s(oten)m(tials)h(to)i(our)f (geometry)-8 b(,)41 b(to)d(pro)m(v)m(e)f(h)m(yp)s(othesis)f(\()p Fy(H)7 b Fz(2\))38 b(at)g(least)265 3866 y(for)g(energies)g(close)h(to) g(the)g(band)e(tail)1689 3830 y Fu(B)p 1689 3845 57 4 v 1700 3897 a Fw(2)1781 3866 y Fz(+)25 b Fy(V)1930 3880 y Fw(0)1970 3866 y Fz(.)64 b(One)38 b(also)h(exp)s(ects)f(that)h Fy(\026)p Fz(\()p Fy(")p Fz(\))h Fx(!)e Fz(0)h(as)265 4013 y Fy(")d Fx(!)g Fz(0.)60 b(The)37 b(main)e(ph)m(ysical)h (consequence)h(of)g(\()p Fy(H)7 b Fz(2\))38 b(\(as)f(sho)m(wn)f(in)g (section)h(5\))g(is)f(that)265 4159 y(a)f(state)h(satisfying)e (\(2.14\))j(do)s(es)d(not)g(carry)h(an)m(y)f(appreciable)f(curren)m(t)i (\(con)m(trary)g(to)g(the)265 4306 y(eigenstates)d(of)e Fy(H)905 4320 y Fu(\013)954 4306 y Fz(\))h(in)e(the)h(sense)h(that)g Fy(J)1772 4273 y Fu(b)1763 4334 y(\014)1836 4306 y Fz(=)25 b(\()p Fy( )2029 4273 y Fu(b)2026 4334 y(\014)2073 4306 y Fy(;)15 b(v)2157 4320 y Fu(y)2199 4306 y Fy( )2261 4273 y Fu(b)2258 4334 y(\014)2306 4306 y Fz(\))25 b(=)g Fx(O)s Fz(\()p Fy(e)2614 4273 y Fv(\000)p Fu(\026)p Fw(\()p Fu(")p Fw(\))p Fu(L)2852 4306 y Fz(\).)401 4453 y(W)-8 b(e)32 b(no)m(w)e(state)i(our)e(main)f(result.)265 4654 y FA(Theorem)42 b(1.)k Fj(Fix)39 b Fy(")f(>)f Fz(0)i Fj(and)i(assume)e(that)i Fz(\()p Fy(H)7 b Fz(1\))40 b Fj(and)g Fz(\()p Fy(H)7 b Fz(2\))40 b Fj(ar)-5 b(e)41 b(ful\014l)5 b(le)-5 b(d.)62 b(Assume)265 4801 y Fy(B)30 b(>)25 b Fz(4)p Fy(V)558 4815 y Fw(0)598 4801 y Fj(.)40 b(L)-5 b(et)30 b Fy(p)25 b Fx(\025)g Fz(7)k Fj(and)h Fy(s)25 b Fz(=)g(min)o(\()p Fy(\022)s(;)15 b(p)e Fx(\000)g Fz(6\))p Fj(.)40 b(Then)29 b(ther)-5 b(e)30 b(exists)g(a)f(numeric)-5 b(al)31 b(c)-5 b(onstant)265 4948 y Fy(\015)30 b(>)25 b Fz(0)31 b Fj(and)g(an)g Fy(L)p Fz(\()p Fy(";)15 b(p;)g(B)5 b(;)15 b(V)1248 4962 y Fw(0)1289 4948 y Fz(\))30 b Fj(such)g(that)i (for)f(al)5 b(l)31 b(for)f(al)5 b(l)31 b Fy(L)25 b(>)g(L)p Fz(\()p Fy(";)15 b(p;)g(B)5 b(;)15 b(V)2898 4962 y Fw(0)2939 4948 y Fz(\))30 b Fj(one)h(c)-5 b(an)31 b(\014nd)265 5095 y(a)e(set)489 5072 y Fz(^)478 5095 y(\012)544 5109 y Fw(\003)626 5095 y Fj(of)g(r)-5 b(e)g(alizations)32 b(of)d(the)h(r)-5 b(andom)31 b(p)-5 b(otential)31 b(with)f Fq(P)2398 5109 y Fw(\003)2452 5095 y Fz(\()2497 5072 y(^)2487 5095 y(\012)2553 5109 y Fw(\003)2606 5095 y Fz(\))25 b Fx(\025)g Fz(1)12 b Fx(\000)g Fz(3)p Fy(L)3009 5062 y Fv(\000)p Fu(s)3131 5095 y Fj(such)29 b(that)265 5241 y(for)f(any)h Fy(!)f Fx(2)756 5218 y Fz(^)746 5241 y(\012)812 5255 y Fw(\003)865 5241 y Fj(,)g Fy(\033)s Fz(\()p Fy(H)1087 5255 y Fu(!)1138 5241 y Fz(\))10 b Fx(\\)g Fz(\001)1330 5255 y Fu(")1393 5241 y Fj(is)28 b(the)g(union)g(of)g(thr)-5 b(e)g(e)30 b(sets)e Fz(\006)2445 5256 y Fu(`)2487 5241 y Fx([)10 b Fz(\006)2624 5256 y Fu(b)2667 5241 y Fx([)g Fz(\006)2804 5255 y Fu(r)2841 5241 y Fj(,)29 b(e)-5 b(ach)28 b(dep)-5 b(ending)265 5388 y(on)33 b Fy(!)i Fj(and)f Fy(L)p Fj(,)e(and)i(char)-5 b(acterize)g(d)35 b(by)e(the)g(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)1852 5637 y Fs(9)p eop %%Page: 10 10 10 9 bop 347 100 a Fj(a\))50 b Fy(E)552 67 y Fu(\013)547 128 y(k)627 100 y Fx(2)25 b Fz(\006)779 114 y Fu(\013)860 100 y Fz(\()p Fy(\013)h Fz(=)f Fy(`;)15 b(r)s Fz(\))33 b Fj(ar)-5 b(e)33 b(a)g(smal)5 b(l)34 b(p)-5 b(erturb)g(ation)35 b(of)e Fy(E)2430 67 y Fu(\013)2425 128 y Fw(0)p Fu(k)2528 100 y Fx(2)25 b Fy(\033)s Fz(\()p Fy(H)2780 114 y Fu(\013)2830 100 y Fz(\))20 b Fx(\\)g Fz(\001)3042 114 y Fu(")3111 100 y Fj(with)1107 339 y Fx(j)p Fy(E)1204 301 y Fu(\013)1199 362 y(k)1275 339 y Fx(\000)g Fy(E)1438 301 y Fu(\013)1433 362 y Fw(0)p Fu(k)1511 339 y Fx(j)25 b(\024)g Fy(e)1699 301 y Fv(\000)p Fu(\015)t(B)s Fw(\(log)14 b Fu(L)p Fw(\))2056 278 y Ft(2)2096 339 y Fy(;)387 b(\013)26 b Fz(=)e Fy(`;)15 b(r)29 b(:)395 b Fz(\(2.15\))352 597 y Fj(b\))49 b(F)-7 b(or)33 b Fy(E)721 564 y Fu(\013)716 625 y(k)796 597 y Fx(2)25 b Fz(\006)948 611 y Fu(\013)1030 597 y Fj(the)33 b(curr)-5 b(ent)33 b Fy(J)1558 564 y Fu(\013)1549 625 y(k)1640 597 y Fj(of)g(the)g(asso)-5 b(ciate)g(d)35 b(eigenstate)e (satis\014es)1122 835 y Fx(j)q Fy(J)1207 798 y Fu(\013)1198 859 y(k)1276 835 y Fx(\000)20 b Fy(J)1426 798 y Fu(\013)1417 859 y Fw(0)p Fu(k)1496 835 y Fx(j)25 b(\024)g Fy(e)1684 798 y Fv(\000)p Fu(\015)t(B)s Fw(\(log)15 b Fu(L)p Fw(\))2042 775 y Ft(2)2081 835 y Fy(;)387 b(\013)26 b Fz(=)f Fy(`;)15 b(r)28 b(:)410 b Fz(\(2.16\))352 1094 y Fj(c\))49 b Fz(\006)546 1109 y Fu(b)612 1094 y Fj(c)-5 b(ontains)34 b(the)f(same)h(numb)-5 b(er)33 b(of)g(ener)-5 b(gy)32 b(levels)h(as)g Fy(\033)s Fz(\()p Fy(H)2597 1109 y Fu(b)2632 1094 y Fz(\))20 b Fx(\\)g Fz(\001)2844 1108 y Fu(")2913 1094 y Fj(and)1181 1332 y Fz(dist)o(\(\006)1429 1347 y Fu(b)1463 1332 y Fy(;)15 b Fz(\006)1569 1346 y Fu(\013)1619 1332 y Fz(\))25 b Fx(\025)g Fy(L)1837 1295 y Fv(\000)p Fu(p)p Fw(+1)2022 1332 y Fy(;)387 b(\013)26 b Fz(=)f Fy(`;)15 b(r)28 b(:)469 b Fz(\(2.17\))347 1591 y Fj(d\))50 b(The)32 b(curr)-5 b(ent)34 b(asso)-5 b(ciate)g(d)35 b(to)e(e)-5 b(ach)34 b(level)e Fy(E)1994 1606 y Fu(\014)2067 1591 y Fx(2)24 b Fz(\006)2218 1606 y Fu(b)2285 1591 y Fj(satis\014es)1604 1829 y Fx(j)p Fy(J)1679 1844 y Fu(\014)1727 1829 y Fx(j)i(\024)f Fy(e)1916 1792 y Fv(\000)p Fu(\015)t(B)s Fw(\(log)14 b Fu(L)p Fw(\))2273 1768 y Ft(2)2338 1829 y Fy(:)892 b Fz(\(2.18\))401 2088 y(The)39 b(pro)s(of)g(of)h(the)g(theorem)g(is)e (organized)i(as)g(follo)m(ws.)67 b(In)39 b(section)h(3)f(w)m(e)i(set)f (up)e(a)265 2235 y(decoupling)e(sc)m(heme)i(b)m(y)g(whic)m(h)e(w)m(e)j (express)e(the)h(resolv)m(en)m(t)g(of)g Fy(H)2652 2249 y Fu(!)2740 2235 y Fz(as)g(an)f(appro)m(ximate)265 2382 y(sum)h(of)g(those)h(of)g(the)f(edge)h(and)f(bulk)f(systems.)65 b(P)m(arts)39 b Fy(a)p Fz(\))g(and)e Fy(c)p Fz(\))j(of)e(Theorem)g(1)h (are)265 2529 y(pro)m(v)m(en)27 b(in)f(section)h(4.)40 b(First)26 b(w)m(e)i(compute)f(appro)m(ximations)f(for)h(the)g(sp)s (ectral)f(pro)5 b(jections)265 2675 y(of)32 b Fy(H)446 2689 y Fu(!)527 2675 y Fz(in)f(terms)g(of)h(the)g(pro)5 b(jectors)32 b Fy(P)13 b Fz(\()p Fy(E)1759 2642 y Fu(\013)1754 2703 y Fw(0)p Fu(k)1832 2675 y Fz(\))32 b(of)g Fy(H)2080 2689 y Fu(\013)2161 2675 y Fz(and)f Fy(P)2397 2690 y Fu(b)2431 2675 y Fz(\()2482 2652 y(\026)2466 2675 y(\001\))h(of)g Fy(H)2790 2690 y Fu(b)2856 2675 y Fz(\(Prop)s(osition)e(1\).)265 2822 y(This)39 b(is)g(done)h(for)g(realizations)f(of)i(the)f(disorder)f (suc)m(h)h(that)h(the)f(lev)m(els)g(of)g Fy(H)3120 2837 y Fu(b)3195 2822 y Fz(are)g(not)265 2969 y(\\to)s(o)28 b(close")f(to)g(those)g(of)g Fy(H)1242 2983 y Fu(\013)1291 2969 y Fz(.)39 b(W)-8 b(e)28 b(then)e(sho)m(w)g(that)h(these)g (realizations)f(are)h(t)m(ypical)f(\(ha)m(v)m(e)265 3116 y(large)34 b(probabilit)m(y\))d(thanks)j(to)g(a)g(W)-8 b(egner)35 b(estimate)g(\(Prop)s(osition)d(2\).)51 b(P)m(arts)35 b Fy(b)p Fz(\))f(and)f Fy(d)p Fz(\))265 3263 y(are)42 b(pro)m(v)m(en)h(in)d(section)i(5)h(b)m(y)e(estimating)h(curren)m(ts)f (in)g(term)h(of)g(norms)f(of)h(di\013erences)265 3409 y(b)s(et)m(w)m(een)31 b(pro)5 b(jectors.)41 b(The)30 b(app)s(endices)f(con)m(tain)h(some)h(tec)m(hnical)f(estimates.)265 3784 y Fr(3)161 b(Decoupling)53 b(of)g(the)g(Bulk)h(and)f(the)g(Edge)f (Sys-)265 4021 y(tems)265 4258 y Fz(The)30 b(resolv)m(en)m(t)g Fy(R)q Fz(\()p Fy(z)t Fz(\))c(=)f(\()p Fy(z)g Fx(\000)19 b Fy(H)1409 4272 y Fu(!)1459 4258 y Fz(\))1494 4225 y Fv(\000)p Fw(1)1619 4258 y Fz(can)30 b(b)s(e)f(expressed,)h(up)f(to)i (a)g(small)d(term,)j(as)f(a)g(sum)265 4405 y(of)e(the)f(resolv)m(en)m (ts)i(of)e(the)h(bulk)e(system)i Fy(R)1759 4420 y Fu(b)1793 4405 y Fz(\()p Fy(z)t Fz(\))e(=)f(\()p Fy(z)19 b Fx(\000)c Fy(H)2289 4420 y Fu(b)2322 4405 y Fz(\))2357 4372 y Fv(\000)p Fw(1)2480 4405 y Fz(and)27 b(the)g(t)m(w)m(o)i(edge)g(systems)265 4552 y Fy(R)334 4566 y Fu(\013)384 4552 y Fz(\()p Fy(z)t Fz(\))d(=)f(\()p Fy(z)17 b Fx(\000)c Fy(H)876 4566 y Fu(\013)924 4552 y Fz(\))959 4519 y Fv(\000)p Fw(1)1081 4552 y Fz(\()p Fy(\013)26 b Fz(=)f Fy(`;)15 b(r)s Fz(\).)39 b(Here)27 b(this)f(will)e(b)s(e)i(ac)m(hiev)m(ed)h(b)m(y)f(a)h Fj(de)-5 b(c)g(oupling)31 b(formula)265 4698 y Fz(dev)m(elop)s(ed)36 b(in)g(other)i(con)m(texts)g([BCD)q(],)h([BG)q(].)61 b(W)-8 b(e)38 b(set)g Fy(D)h Fz(=)d(log)17 b Fy(L)37 b Fz(and)f(in)m(tro)s(duce)g(the)265 4845 y(c)m(haracteristic)31 b(functions)767 5060 y(~)745 5083 y Fy(J)795 5098 y Fu(`)829 5083 y Fz(\()p Fy(x)p Fz(\))83 b(=)g Fy(\037)1245 5102 y Fw(])p Fv(\0001)p Fu(;)p Fg(\000)1468 5082 y Ff(L)p 1468 5090 40 3 v 1474 5123 a Fe(2)1531 5102 y Ft(+)1602 5082 y Ff(D)p 1602 5090 48 3 v 1612 5123 a Fe(2)1660 5102 y Fw(])1683 5083 y Fz(\()p Fy(x)p Fz(\))2010 5060 y(~)1988 5083 y Fy(J)2038 5098 y Fu(b)2073 5083 y Fz(\()p Fy(x)p Fz(\))26 b(=)e Fy(\037)2373 5102 y Fw([)p Fg(\000)2451 5082 y Ff(L)p 2451 5090 40 3 v 2457 5123 a Fe(2)2515 5102 y Ft(+)2585 5082 y Ff(D)p 2585 5090 48 3 v 2595 5123 a Fe(2)2643 5102 y Fu(;)2673 5082 y Ff(L)p 2672 5090 40 3 v 2678 5123 a Fe(2)2736 5102 y Fg(\000)2808 5082 y Ff(D)p 2808 5090 48 3 v 2818 5123 a Fe(2)2866 5102 y Fw(])2889 5083 y Fz(\()p Fy(x)p Fz(\))762 5232 y(~)740 5255 y Fy(J)790 5269 y Fu(r)829 5255 y Fz(\()p Fy(x)p Fz(\))83 b(=)g Fy(\037)1245 5273 y Fw([)1275 5254 y Ff(L)p 1275 5262 40 3 v 1281 5295 a Fe(2)1338 5273 y Fg(\000)1411 5254 y Ff(D)p 1411 5262 48 3 v 1420 5295 a Fe(2)1468 5273 y Fu(;)p Fw(+)p Fv(1)p Fw([)1637 5255 y Fz(\()p Fy(x)p Fz(\))26 b Fy(:)1491 b Fz(\(3.1\))1828 5637 y Fs(10)p eop %%Page: 11 11 11 10 bop 265 100 a Fz(W)-8 b(e)39 b(will)34 b(also)k(use)f(three)g(b)s (ounded)f Fy(C)1651 67 y Fv(1)1725 100 y Fz(\()p Fq(R)s Fz(\))43 b(functions)36 b Fx(j)p Fy(J)2372 114 y Fu(i)2401 100 y Fz(\()p Fy(x)p Fz(\))p Fx(j)i(\024)f Fz(1,)j Fy(i)d Fx(2)f(I)44 b(\021)36 b(f)p Fy(`;)15 b(b;)g(r)s Fx(g)p Fz(,)265 247 y(with)32 b(b)s(ounded)f(\014rst)i(and)g(second)g(deriv)-5 b(ativ)m(es)33 b(sup)2109 269 y Fu(x)2168 247 y Fx(j)p Fy(@)2246 214 y Fu(n)2241 270 y(x)2294 247 y Fy(J)2344 261 y Fu(i)2372 247 y Fz(\()p Fy(x)p Fz(\))p Fx(j)e(\024)g Fz(2,)j Fy(n)c Fz(=)g(1)p Fy(;)15 b Fz(2,)36 b(and)d(suc)m(h)265 394 y(that)319 673 y Fy(J)369 688 y Fu(`)402 673 y Fz(\()p Fy(x)p Fz(\))84 b(=)762 487 y Fp(8)762 569 y(<)762 732 y(:)842 598 y Fz(1)92 b(if)29 b Fy(x)c Fx(\024)g(\000)1316 562 y Fu(L)p 1316 577 48 4 v 1322 629 a Fw(2)1394 598 y Fz(+)1495 562 y Fw(3)p Fu(D)p 1495 577 96 4 v 1525 629 a Fw(4)842 774 y Fz(0)92 b(if)29 b Fy(x)c Fx(\025)g(\000)1316 738 y Fu(L)p 1316 753 48 4 v 1322 805 a Fw(2)1394 774 y Fz(+)1495 738 y Fw(3)p Fu(D)p 1495 753 96 4 v 1525 805 a Fw(4)1620 774 y Fz(+)20 b(1)2055 673 y Fy(J)2105 688 y Fu(b)2139 673 y Fz(\()p Fy(x)p Fz(\))26 b(=)2383 487 y Fp(8)2383 569 y(<)2383 732 y(:)2464 598 y Fz(1)91 b(if)30 b Fx(j)p Fy(x)p Fx(j)25 b(\024)2917 562 y Fu(L)p 2917 577 48 4 v 2923 629 a Fw(2)2995 598 y Fx(\000)3096 562 y Fu(D)p 3096 577 60 4 v 3109 629 a Fw(4)2464 774 y Fz(0)91 b(if)30 b Fx(j)p Fy(x)p Fx(j)25 b(\025)2917 738 y Fu(L)p 2917 753 48 4 v 2923 805 a Fw(2)2995 774 y Fx(\000)3096 738 y Fu(D)p 3096 753 60 4 v 3109 805 a Fw(4)3186 774 y Fz(+)20 b(1)314 1059 y Fy(J)364 1073 y Fu(r)402 1059 y Fz(\()p Fy(x)p Fz(\))84 b(=)762 873 y Fp(8)762 954 y(<)762 1118 y(:)842 983 y Fz(1)92 b(if)29 b Fy(x)c Fx(\025)1245 948 y Fu(L)p 1245 963 48 4 v 1251 1015 a Fw(2)1323 983 y Fx(\000)1424 948 y Fw(3)p Fu(D)p 1424 963 96 4 v 1454 1015 a Fw(4)842 1159 y Fz(0)92 b(if)29 b Fy(x)c Fx(\024)1245 1124 y Fu(L)p 1245 1139 48 4 v 1251 1191 a Fw(2)1323 1159 y Fx(\000)1424 1124 y Fw(3)p Fu(D)p 1424 1139 96 4 v 1454 1191 a Fw(4)1550 1159 y Fx(\000)20 b Fz(1)1827 1059 y Fy(:)1449 b Fz(\(3.2\))265 1356 y(F)-8 b(or)31 b Fy(i)26 b Fx(2)f(I)36 b Fz(w)m(e)31 b(ha)m(v)m(e)h Fy(H)1077 1370 y Fu(!)1127 1356 y Fy(J)1177 1370 y Fu(i)1230 1356 y Fz(=)25 b Fy(H)1402 1370 y Fu(i)1430 1356 y Fy(J)1480 1370 y Fu(i)1539 1356 y Fz(th)m(us)722 1575 y(\()p Fy(z)g Fx(\000)20 b Fy(H)991 1589 y Fu(!)1041 1575 y Fz(\))1091 1489 y Fp(X)1100 1686 y Fu(i)p Fv(2I)1238 1575 y Fy(J)1288 1589 y Fu(i)1317 1575 y Fy(R)1386 1589 y Fu(i)1414 1575 y Fz(\()p Fy(z)t Fz(\))1553 1552 y(~)1530 1575 y Fy(J)1580 1589 y Fu(i)1635 1575 y Fz(=)1731 1489 y Fp(X)1739 1686 y Fu(i)p Fv(2I)1862 1575 y Fz(\()p Fy(z)25 b Fx(\000)20 b Fy(H)2131 1589 y Fu(i)2158 1575 y Fz(\))p Fy(J)2243 1589 y Fu(i)2272 1575 y Fy(R)2341 1589 y Fu(i)2369 1575 y Fz(\()p Fy(z)t Fz(\))2508 1552 y(~)2485 1575 y Fy(J)2535 1589 y Fu(i)2590 1575 y Fz(=)25 b(1)c Fx(\000)f(K)q Fz(\()p Fy(z)t Fz(\))272 b(\(3.3\))265 1843 y(where)1075 1989 y Fx(K)q Fz(\()p Fy(z)t Fz(\))27 b(=)1384 1903 y Fp(X)1392 2100 y Fu(i)p Fv(2I)1530 1989 y Fy(K)1607 2003 y Fu(i)1635 1989 y Fz(\()p Fy(z)t Fz(\))g(=)1874 1903 y Fp(X)1882 2100 y Fu(i)p Fv(2I)2030 1953 y Fw(1)p 2030 1968 36 4 v 2030 2021 a(2)2075 1989 y Fz([)p Fy(p)2146 1952 y Fw(2)2146 2012 y Fu(x)2190 1989 y Fy(;)15 b(J)2280 2003 y Fu(i)2309 1989 y Fz(])p Fy(R)2403 2003 y Fu(i)2432 1989 y Fz(\()p Fy(z)t Fz(\))2571 1966 y(~)2548 1989 y Fy(J)2598 2003 y Fu(i)2652 1989 y Fy(:)624 b Fz(\(3.4\))265 2234 y(T)-8 b(o)35 b(obtain)g(the)g(second)g(equalit)m(y)f(one)i(comm)m (utes)f(\()p Fy(z)28 b Fx(\000)23 b Fy(H)2378 2248 y Fu(i)2406 2234 y Fz(\))35 b(and)f Fy(J)2707 2248 y Fu(i)2771 2234 y Fz(and)g(then)h(uses)f(the)265 2381 y(iden)m(tit)m(y)601 2313 y Fp(P)697 2408 y Fu(i)p Fv(2I)831 2381 y Fy(J)881 2395 y Fu(i)931 2358 y Fz(~)909 2381 y Fy(J)959 2395 y Fu(i)1013 2381 y Fz(=)1109 2313 y Fp(P)1205 2408 y Fu(i)p Fv(2I)1361 2358 y Fz(~)1339 2381 y Fy(J)1389 2395 y Fu(i)1443 2381 y Fz(=)25 b(1.)41 b(F)-8 b(rom)31 b(\(3.3\))h(w)m(e)f (deduce)f(the)g(decoupling)f(form)m(ula)1096 2652 y Fy(R)q Fz(\()p Fy(z)t Fz(\))d(=)1404 2496 y Fp( )1476 2565 y(X)1485 2762 y Fu(i)p Fv(2I)1623 2652 y Fy(J)1673 2666 y Fu(i)1701 2652 y Fy(R)1770 2666 y Fu(i)1799 2652 y Fz(\()p Fy(z)t Fz(\))1938 2629 y(~)1915 2652 y Fy(J)1965 2666 y Fu(i)1994 2496 y Fp(!)2081 2652 y Fz(\()q(1)20 b Fx(\000)g(K)q Fz(\()p Fy(z)t Fz(\)\))2496 2610 y Fv(\000)p Fw(1)2631 2652 y Fy(:)645 b Fz(\(3.5\))401 2924 y(The)35 b(main)g(result)f(of)i (this)e(section)i(is)e(an)i(estimate)g(of)g(the)g(op)s(erator)f(norm)g (of)h Fx(K)q Fz(\()p Fy(z)t Fz(\).)265 3071 y(In)30 b(particular)e(it)i (will)e(assure)i Fx(kK)q Fz(\()p Fy(z)t Fz(\))p Fx(k)e Fy(<)d Fz(1.)265 3276 y FA(Lemma)h(1.)38 b Fj(L)-5 b(et)27 b Fx(R)p Fy(e)15 b(z)30 b Fx(2)25 b Fz(\001)1267 3290 y Fu(")1331 3276 y Fj(such)j(that)h Fz(dist)o(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)2148 3291 y Fu(`)2182 3276 y Fz(\))9 b Fx([)g Fy(\033)s Fz(\()p Fy(H)2462 3290 y Fu(r)2500 3276 y Fz(\))g Fx([)g Fy(\033)s Fz(\()p Fy(H)2780 3291 y Fu(b)2815 3276 y Fz(\)\))26 b Fx(\025)f Fy(e)3049 3241 y Fv(\000)3135 3213 y Fi(B)p 3114 3225 92 3 v 3114 3267 a Ft(512)3215 3241 y Fw(\(log)14 b Fu(L)p Fw(\))3421 3217 y Ft(2)3460 3276 y Fj(.)265 3423 y(Then)38 b(for)g Fy(L)g Fj(lar)-5 b(ge)39 b(enough)f(ther)-5 b(e)38 b(exists)g(a)h(c)-5 b(onstant)39 b Fy(C)7 b Fz(\()p Fy(B)e(;)15 b(V)2507 3437 y Fw(0)2546 3423 y Fz(\))35 b Fy(>)f Fz(0)k Fj(indep)-5 b(endent)40 b(of)e Fy(L)265 3570 y Fj(such)33 b(that)1144 3717 y Fx(kK)q Fz(\()p Fy(z)t Fz(\))p Fx(k)28 b(\024)d Fy(")1586 3679 y Fv(\000)p Fw(1)1680 3717 y Fy(C)7 b Fz(\()p Fy(B)e(;)15 b(V)1954 3731 y Fw(0)1993 3717 y Fz(\))p Fy(Le)2132 3676 y Fv(\000)2223 3640 y Fu(B)p 2197 3655 106 4 v 2197 3707 a Fw(512)2314 3676 y(\(log)e Fu(L)p Fw(\))2519 3652 y Ft(2)2583 3717 y Fy(:)693 b Fz(\(3.6\))265 3922 y Fj(Pr)-5 b(o)g(of.)50 b Fz(Computing)38 b(the)i(comm)m(utator)h(in)e(the)h(de\014nition)d(of)j Fy(K)2610 3936 y Fu(i)2638 3922 y Fz(\()p Fy(z)t Fz(\))h(and)e (applying)e(the)265 4069 y(second)30 b(resolv)m(en)m(t)i(form)m(ula)d (w)m(e)i(ha)m(v)m(e)789 4288 y Fy(K)866 4302 y Fu(i)895 4288 y Fz(\()p Fy(z)t Fz(\))84 b(=)f Fx(\000)1330 4252 y Fw(1)p 1330 4267 36 4 v 1330 4320 a(2)1375 4288 y Fz(\()p Fy(@)1463 4251 y Fw(2)1458 4311 y Fu(x)1503 4288 y Fy(J)1553 4302 y Fu(i)1581 4288 y Fz(\))p Fy(R)1685 4302 y Fu(i)1714 4288 y Fz(\()p Fy(z)t Fz(\))1853 4265 y(~)1830 4288 y Fy(J)1880 4302 y Fu(i)1930 4288 y Fx(\000)19 b Fz(\()p Fy(@)2103 4302 y Fu(x)2148 4288 y Fy(J)2198 4302 y Fu(i)2227 4288 y Fz(\))p Fy(@)2310 4302 y Fu(x)2354 4288 y Fy(R)2423 4302 y Fu(i)2451 4288 y Fz(\()p Fy(z)t Fz(\))2590 4265 y(~)2567 4288 y Fy(J)2617 4302 y Fu(i)1095 4460 y Fz(=)83 b Fx(\000)1330 4424 y Fw(1)p 1330 4439 V 1330 4491 a(2)1375 4460 y Fz(\()p Fy(@)1463 4422 y Fw(2)1458 4482 y Fu(x)1503 4460 y Fy(J)1553 4474 y Fu(i)1581 4460 y Fz(\))p Fy(R)1685 4474 y Fw(0)1725 4460 y Fz(\()p Fy(z)t Fz(\))1864 4437 y(~)1841 4460 y Fy(J)1891 4474 y Fu(i)1941 4460 y Fx(\000)2042 4424 y Fw(1)p 2042 4439 V 2042 4491 a(2)2087 4460 y Fz(\()p Fy(@)2175 4422 y Fw(2)2170 4482 y Fu(x)2215 4460 y Fy(J)2265 4474 y Fu(i)2294 4460 y Fz(\))p Fy(R)2398 4474 y Fw(0)2437 4460 y Fz(\()p Fy(z)t Fz(\))p Fy(W)2639 4474 y Fu(i)2668 4460 y Fy(R)2737 4474 y Fu(i)2766 4460 y Fz(\()p Fy(z)t Fz(\))2905 4437 y(~)2882 4460 y Fy(J)2932 4474 y Fu(i)1095 4632 y Fx(\000)g Fz(\()p Fy(@)1332 4646 y Fu(x)1376 4632 y Fy(J)1426 4646 y Fu(i)1455 4632 y Fz(\))p Fy(@)1538 4646 y Fu(x)1582 4632 y Fy(R)1651 4646 y Fw(0)1691 4632 y Fz(\()p Fy(z)t Fz(\))1830 4609 y(~)1807 4632 y Fy(J)1857 4646 y Fu(i)1906 4632 y Fx(\000)20 b Fz(\()p Fy(@)2080 4646 y Fu(x)2125 4632 y Fy(J)2175 4646 y Fu(i)2203 4632 y Fz(\))p Fy(@)2286 4646 y Fu(x)2331 4632 y Fy(R)2400 4646 y Fw(0)2439 4632 y Fz(\()p Fy(z)t Fz(\))p Fy(W)2641 4646 y Fu(i)2670 4632 y Fy(R)2739 4646 y Fu(i)2768 4632 y Fz(\()p Fy(z)t Fz(\))2907 4609 y(~)2884 4632 y Fy(J)2934 4646 y Fu(i)3301 4632 y Fz(\(3.7\))265 4851 y(where)30 b(w)m(e)h(ha)m(v)m(e)g(set)g Fy(W)1099 4866 y Fu(`)1157 4851 y Fz(=)25 b Fy(U)1315 4866 y Fu(`)1349 4851 y Fz(,)30 b Fy(W)1490 4866 y Fu(b)1550 4851 y Fz(=)25 b Fy(V)1699 4865 y Fu(!)1779 4851 y Fz(and)30 b Fy(W)2042 4865 y Fu(r)2105 4851 y Fz(=)25 b Fy(U)2263 4865 y Fu(r)2301 4851 y Fz(.)41 b(F)-8 b(rom)31 b(the)g(triangle)e(inequalit)m(y)265 4997 y(and)h Fx(k)p Fy(R)556 5011 y Fu(i)584 4997 y Fz(\()p Fy(z)t Fz(\))p Fx(k)d Fz(=)e(dist)o(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)1302 5011 y Fu(i)1331 4997 y Fz(\)\))1401 4964 y Fv(\000)p Fw(1)1527 4997 y Fz(w)m(e)31 b(obtain)445 5216 y Fx(k)p Fy(K)567 5230 y Fu(i)596 5216 y Fz(\()p Fy(z)t Fz(\))p Fx(k)84 b(\024)1005 5181 y Fw(1)p 1005 5196 V 1005 5248 a(2)1050 5216 y Fx(k)p Fz(\()p Fy(@)1183 5179 y Fw(2)1178 5239 y Fu(x)1224 5216 y Fy(J)1274 5230 y Fu(i)1302 5216 y Fz(\))p Fy(R)1406 5230 y Fw(0)1446 5216 y Fz(\()p Fy(z)t Fz(\))1585 5193 y(~)1562 5216 y Fy(J)1612 5230 y Fu(i)1642 5216 y Fx(k)20 b Fz(+)1808 5181 y Fw(1)p 1808 5196 V 1808 5248 a(2)1853 5216 y Fx(k)p Fz(\()p Fy(@)1986 5179 y Fw(2)1981 5239 y Fu(x)2027 5216 y Fy(J)2077 5230 y Fu(i)2106 5216 y Fz(\))p Fy(R)2210 5230 y Fw(0)2249 5216 y Fz(\()p Fy(z)t Fz(\))p Fy(W)2451 5230 y Fu(i)2480 5216 y Fx(k)31 b Fz(dist)o(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)2990 5230 y Fu(i)3020 5216 y Fz(\)\))3090 5179 y Fv(\000)p Fw(1)841 5388 y Fz(+)83 b Fx(k)p Fz(\()p Fy(@)1123 5402 y Fu(x)1168 5388 y Fy(J)1218 5402 y Fu(i)1246 5388 y Fz(\))p Fy(@)1329 5402 y Fu(x)1374 5388 y Fy(R)1443 5402 y Fw(0)1482 5388 y Fz(\()p Fy(z)t Fz(\))1621 5365 y(~)1598 5388 y Fy(J)1648 5402 y Fu(i)1678 5388 y Fx(k)21 b Fz(+)f Fx(k)p Fz(\()p Fy(@)1963 5402 y Fu(x)2008 5388 y Fy(J)2058 5402 y Fu(i)2086 5388 y Fz(\))p Fy(@)2169 5402 y Fu(x)2214 5388 y Fy(R)2283 5402 y Fw(0)2322 5388 y Fz(\()p Fy(z)t Fz(\))p Fy(W)2524 5402 y Fu(i)2553 5388 y Fx(k)31 b Fz(dist)o(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)3063 5402 y Fu(i)3092 5388 y Fz(\)\))3162 5351 y Fv(\000)p Fw(1)3282 5388 y Fy(:)-6 b Fz(\(3.8\))1828 5637 y Fs(11)p eop %%Page: 12 12 12 11 bop 265 100 a Fz(T)-8 b(o)37 b(estimate)h(the)f(op)s(erator)g (norms)f(on)h(the)g(righ)m(t)f(hand)g(side)g(it)g(is)g(su\016cien)m(t)g (to)i(b)s(ound)265 247 y(them)d(b)m(y)h(the)g(Hilb)s(ert-Sc)m(hmidt)c (norms)j Fx(k)p Fy(:)p Fx(k)1864 261 y Fw(2)1904 247 y Fz(.)56 b(Using)35 b(b)s(ounds)e(\(A.5\))k(on)f(the)f(k)m(ernels)g (of)265 394 y Fy(@)318 361 y Fu(n)313 416 y(x)365 394 y Fy(R)434 408 y Fw(0)474 394 y Fz(\()p Fy(z)t Fz(\))c(for)f Fy(n)25 b Fz(=)g(0)p Fy(;)15 b Fz(1,)32 b(and)e(the)g(prop)s(erties)f (of)h(the)h(functions)e Fy(J)2586 408 y Fu(i)2615 394 y Fz(,)2692 371 y(~)2670 394 y Fy(J)2720 408 y Fu(i)2779 394 y Fz(w)m(e)i(obtain)265 622 y Fx(k)p Fz(\()p Fy(@)398 585 y Fw(2)p Fv(\000)p Fu(n)393 645 y(x)536 622 y Fy(J)586 636 y Fu(i)615 622 y Fz(\))p Fy(@)703 585 y Fu(n)698 645 y(x)751 622 y Fy(R)820 636 y Fw(0)859 622 y Fz(\()p Fy(z)t Fz(\))998 599 y(~)975 622 y Fy(J)1025 636 y Fu(i)1055 622 y Fx(k)1100 585 y Fw(2)1100 645 y(2)1222 622 y Fz(=)1376 498 y Fp(Z)1427 705 y Fw(supp)12 b Fu(@)1625 675 y Ft(2)p Fg(\000)p Fi(n)1621 715 y(x)1746 705 y Fu(J)1785 715 y Fi(i)1830 622 y Fz(d)p Fd(x)p Fx(j)p Fy(@)2019 585 y Fw(2)p Fv(\000)p Fu(n)2014 645 y(x)2156 622 y Fy(J)2206 636 y Fu(i)2235 622 y Fz(\()p Fy(x)p Fz(\))p Fx(j)2382 585 y Fw(2)2437 498 y Fp(Z)2488 705 y Fw(supp)2662 688 y(~)2645 705 y Fu(J)2684 715 y Fi(i)2729 622 y Fz(d)p Fd(x)2840 585 y Fv(0)2863 622 y Fx(j)p Fy(@)2941 585 y Fu(n)2936 645 y(x)2988 622 y Fy(R)3057 636 y Fw(0)3097 622 y Fz(\()p Fd(x)p Fy(;)j Fd(x)3293 585 y Fv(0)3316 622 y Fz(;)g Fy(z)t Fz(\))p Fx(j)3462 585 y Fw(2)1222 874 y Fx(\024)83 b Fz(4)p Fy(C)1493 837 y Fw(2)1486 897 y Fu(n)1534 874 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))1779 751 y Fp(Z)1830 957 y Fw(supp)12 b Fu(@)2028 927 y Ft(2)p Fg(\000)p Fi(n)2024 967 y(x)2149 957 y Fu(J)2188 967 y Fi(i)2234 874 y Fz(d)o Fd(x)2359 751 y Fp(Z)2410 957 y Fw(supp)2583 940 y(~)2567 957 y Fu(J)2606 967 y Fi(i)2651 874 y Fz(d)p Fd(x)2761 837 y Fv(0)2785 874 y Fy(e)2827 837 y Fv(\000)2892 810 y Fi(B)p 2892 822 49 3 v 2901 863 a Ft(4)2950 837 y Fw(\()p Fu(x)p Fv(\000)p Fu(x)3112 813 y Fg(0)3135 837 y Fw(\))3162 813 y Ft(2)1222 1127 y Fx(\024)83 b Fz(4)p Fy(C)1493 1089 y Fw(2)1486 1149 y Fu(n)1534 1127 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))p Fy(e)1806 1088 y Fv(\000)1871 1061 y Fi(B)p 1872 1073 V 1881 1114 a Ft(8)1930 1094 y Fz(\()1975 1061 y Fi(D)p 1975 1073 52 3 v 1986 1114 a Ft(4)2037 1088 y Fw(+1)2127 1094 y Fz(\))2162 1047 y Ft(2)2216 1003 y Fp(Z)2267 1209 y Fw(supp)12 b Fu(@)2465 1179 y Ft(2)p Fg(\000)p Fi(n)2461 1220 y(x)2586 1209 y Fu(J)2625 1219 y Fi(i)2670 1127 y Fz(d)p Fd(x)2796 1003 y Fp(Z)2846 1209 y Fh(R)2913 1127 y Fz(d)p Fd(x)3024 1089 y Fv(0)3047 1127 y Fy(e)3089 1089 y Fv(\000)3154 1062 y Fi(B)p 3154 1074 49 3 v 3163 1115 a Ft(8)3213 1089 y Fw(\()p Fu(x)p Fv(\000)p Fu(x)3375 1066 y Fg(0)3397 1089 y Fw(\))3424 1066 y Ft(2)1222 1366 y Fx(\024)83 b Fz(16)1466 1263 y Fp(q)p 1558 1263 77 4 v 1575 1330 a Fu(\031)p 1568 1345 57 4 v 1568 1397 a(B)1634 1366 y Fy(C)1706 1328 y Fw(2)1699 1388 y Fu(n)1746 1366 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))p Fy(L)2038 1328 y Fw(2)2079 1366 y Fy(e)2121 1328 y Fv(\000)2207 1301 y Fi(B)p 2186 1313 92 3 v 2186 1354 a Ft(128)2287 1328 y Fu(D)2347 1305 y Ft(2)2411 1366 y Fy(:)880 b Fz(\(3.9\))265 1594 y(F)-8 b(or)31 b(the)g(norms)e(in)m(v)m(olving)g(the)i(p)s(oten)m (tials)e Fy(W)1909 1608 y Fu(i)1968 1594 y Fz(w)m(e)i(obtain)e(in)g(a)i (similar)d(w)m(a)m(y)711 1822 y Fx(k)p Fy(@)809 1785 y Fw(2)p Fv(\000)p Fu(n)804 1845 y(x)947 1822 y Fy(J)997 1836 y Fu(i)1025 1822 y Fy(@)1078 1785 y Fu(n)1073 1845 y(x)1125 1822 y Fy(R)1194 1836 y Fw(0)1234 1822 y Fz(\()p Fy(z)t Fz(\))p Fy(W)1436 1836 y Fu(i)1465 1822 y Fx(k)1510 1785 y Fw(2)1510 1845 y(2)557 1994 y Fz(=)711 1870 y Fp(Z)761 2076 y Fw(supp)12 b Fu(@)959 2046 y Ft(2)p Fg(\000)p Fi(n)955 2087 y(x)1080 2076 y Fu(J)1119 2086 y Fi(i)1165 1994 y Fz(d)o Fd(x)p Fx(j)p Fy(@)1353 1956 y Fw(2)p Fv(\000)p Fu(n)1348 2016 y(x)1491 1994 y Fy(J)1541 2008 y Fu(i)1569 1994 y Fz(\()p Fy(x)p Fz(\))p Fx(j)1716 1956 y Fw(2)1772 1870 y Fp(Z)1822 2076 y Fw(supp)g Fu(W)2046 2086 y Fi(i)2092 1994 y Fz(d)o Fd(x)2202 1956 y Fv(0)2225 1994 y Fx(j)p Fy(@)2303 1956 y Fu(\013)2298 2016 y(x)2353 1994 y Fy(R)2422 2008 y Fw(0)2462 1994 y Fz(\()p Fd(x)p Fy(;)j Fd(x)2658 1956 y Fv(0)2681 1994 y Fz(;)g Fy(z)t Fz(\))p Fx(j)2827 1956 y Fw(2)2868 1994 y Fx(j)p Fy(W)2979 2008 y Fu(i)3007 1994 y Fz(\()p Fd(x)3102 1956 y Fv(0)3125 1994 y Fz(\))p Fx(j)3185 1956 y Fw(2)557 2246 y Fx(\024)83 b Fz(4)p Fy(C)828 2209 y Fw(2)821 2269 y Fu(n)868 2246 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))p Fy(e)1140 2209 y Fv(\000)1228 2181 y Fi(B)p 1206 2193 V 1206 2235 a Ft(128)1307 2209 y Fu(D)1367 2185 y Ft(2)1421 2122 y Fp(Z)1472 2329 y Fw(supp)12 b Fu(@)1670 2299 y Ft(2)p Fg(\000)p Fi(n)1666 2339 y(x)1791 2329 y Fu(J)1830 2339 y Fi(i)1875 2246 y Fz(d)p Fd(x)2001 2122 y Fp(Z)2051 2329 y Fw(supp)g Fu(W)2275 2339 y Fi(i)2321 2246 y Fz(d)o Fd(x)2431 2209 y Fv(0)2454 2246 y Fy(e)2496 2209 y Fv(\000)2561 2181 y Fi(B)p 2561 2193 49 3 v 2571 2235 a Ft(8)2620 2209 y Fw(\()p Fu(x)p Fv(\000)p Fu(x)2782 2185 y Fg(0)2804 2209 y Fw(\))2831 2185 y Ft(2)2870 2246 y Fx(j)p Fy(W)2981 2260 y Fu(i)3010 2246 y Fz(\()p Fd(x)3105 2209 y Fv(0)3128 2246 y Fz(\))p Fx(j)3188 2209 y Fw(2)3253 2246 y Fy(:)-23 b Fz(\(3.10\))265 2494 y(It)21 b(is)e(clear)i(that)g(since)e Fy(V)1095 2508 y Fu(!)1166 2494 y Fz(is)h(b)s(ounded,)g(and)g Fy(U)1865 2509 y Fu(`)1898 2494 y Fz(,)j Fy(U)2008 2508 y Fu(r)2066 2494 y Fz(do)d(not)h(gro)m(w)g(faster)g(than)f(p)s (olynomials,)265 2641 y(the)31 b(double)e(in)m(tegral)i(in)e(the)i (righ)m(t)g(hand)e(side)h(of)h(the)g(last)g(inequalit)m(y)e(is)h(b)s (ounded)e(ab)s(o)m(v)m(e)265 2788 y(b)m(y)f Fy(L)450 2755 y Fw(2)516 2788 y Fz(times)g(a)g(constan)m(t)i(dep)s(ending)24 b(only)i(on)h Fy(B)32 b Fz(and)26 b Fy(V)2271 2802 y Fw(0)2311 2788 y Fz(.)39 b(F)-8 b(rom)28 b(this)e(result,)h(\(3.8\))r (,)h(\(3.9\))265 2935 y(and)h(dist)n(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)874 2950 y Fu(`)909 2935 y Fz(\))j Fx([)f Fy(\033)s Fz(\()p Fy(H)1206 2949 y Fu(r)1244 2935 y Fz(\))h Fx([)f Fy(\033)s Fz(\()p Fy(H)1541 2950 y Fu(b)1576 2935 y Fz(\)\))26 b Fx(\025)f Fy(e)1810 2899 y Fv(\000)1896 2872 y Fi(B)p 1875 2884 92 3 v 1875 2925 a Ft(512)1976 2899 y Fw(\(log)13 b Fu(L)p Fw(\))2181 2875 y Ft(2)2250 2935 y Fz(w)m(e)29 b(obtain)g(\()2720 2912 y(~)2699 2935 y Fy(C)7 b Fz(\()p Fy(B)e(;)15 b(V)2973 2949 y Fw(0)3012 2935 y Fz(\))30 b(a)f(constan)m(t)265 3081 y(indep)s(enden)m(t)f(of)j Fy(L)p Fz(\))1134 3310 y Fx(k)p Fy(K)1256 3324 y Fu(i)1285 3310 y Fz(\()p Fy(z)t Fz(\))p Fx(k)26 b(\024)1589 3287 y Fz(~)1568 3310 y Fy(C)7 b Fz(\()p Fy(B)e(;)15 b(V)1842 3324 y Fw(0)1881 3310 y Fz(\))p Fy(")1958 3272 y Fv(\000)p Fw(1)2053 3310 y Fy(Le)2157 3272 y Fv(\000)2244 3245 y Fi(B)p 2223 3257 V 2223 3298 a Ft(512)2324 3272 y Fw(\(log)e Fu(L)p Fw(\))2529 3249 y Ft(2)2594 3310 y Fy(;)636 b Fz(\(3.11\))265 3538 y(where)24 b(w)m(e)g(used)g(the)g(expression)f (for)h Fy(C)1631 3552 y Fu(n)1678 3538 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))25 b(in)e(App)s(endix)f(A)i(and)g(the)g(fact)h(that)g Fx(R)p Fy(e)15 b(z)30 b Fx(2)265 3685 y Fz(\001)341 3699 y Fu(")378 3685 y Fz(.)p 3422 3831 4 62 v 3426 3773 55 4 v 3426 3831 V 3479 3831 4 62 v 265 4204 a Fr(4)161 b(Estimates)53 b(of)h(Eigenpro)9 b(jectors)51 b(of)j Fc(H)2896 4226 y Fb(!)265 4441 y Fz(In)26 b(this)h(section)g(w)m(e)g (use)g(the)h(decoupling)d(form)m(ula)h(\(3.5\))j(to)f(giv)m(e)g (deterministic)d(estimates)265 4588 y(for)i(the)g(di\013erence)f(b)s (et)m(w)m(een)i(pro)5 b(jectors)27 b(of)g Fy(H)1905 4602 y Fu(!)1982 4588 y Fz(and)g Fy(H)2232 4603 y Fu(b)2265 4588 y Fz(,)h Fy(H)2394 4603 y Fu(`)2454 4588 y Fz(and)e Fy(H)2703 4602 y Fu(r)2741 4588 y Fz(.)39 b(W)-8 b(e)28 b(then)f(com)m(bine)265 4735 y(this)33 b(information)g(with)g(a)i (probabilistic)c(estimate)k(\(W)-8 b(egner)36 b(estimate\))g(to)f (deduce)f(that)265 4881 y(the)j(sp)s(ectrum)g(of)g Fy(H)1019 4895 y Fu(!)1106 4881 y Fz(is)f(the)i(union)d(of)j(the)f(three)g(sets)h (\006)2391 4896 y Fu(`)2424 4881 y Fz(,)h(\006)2554 4895 y Fu(r)2629 4881 y Fz(and)d(\006)2878 4896 y Fu(b)2949 4881 y Fz(satisfying)g(the)265 5028 y(parts)30 b Fy(a)p Fz(\))h(and)f Fy(c)p Fz(\))h(of)f(Theorem)g(1.)265 5241 y FA(Prop)s(osition)j(1.)39 b Fj(Assume)30 b(that)h Fz(\()p Fy(H)7 b Fz(1\))31 b Fj(holds.)43 b(T)-7 b(ake)30 b Fy(p)25 b Fx(\025)g Fz(7)30 b Fj(and)h(any)f Fy(e)2812 5206 y Fv(\000)2899 5178 y Fi(B)p 2878 5190 92 3 v 2878 5232 a Ft(512)2979 5206 y Fw(\(log)13 b Fu(L)p Fw(\))3184 5182 y Ft(2)3249 5241 y Fy(<)25 b(\032)g(<)275 5343 y Fu(d)p Fw(\()p Fu(")p Fw(\))p 275 5367 124 4 v 319 5419 a(2)409 5388 y Fy(L)471 5355 y Fv(\000)p Fu(p)565 5388 y Fj(.)57 b(F)-7 b(or)39 b Fy(L)c(>)f(L)p Fz(\()p Fy(")p Fz(\))39 b Fj(let)f Fz(\012)1439 5332 y Fg(00)1439 5415 y Fw(\003)1529 5388 y Fj(b)-5 b(e)38 b(the)g(set)g(of)g(r)-5 b(e)g(alizations)41 b(of)d(the)g(r)-5 b(andom)40 b(p)-5 b(otential)1828 5637 y Fs(12)p eop %%Page: 13 13 13 12 bop 265 100 a Fj(such)41 b(that)g(for)g(e)-5 b(ach)42 b Fy(!)g Fx(2)d Fz(\012)1304 67 y Fv(00)1304 127 y Fw(\003)1397 100 y Fz(dist)14 b(\()q Fy(\033)s Fz(\()p Fy(H)1761 115 y Fu(b)1795 100 y Fz(\))21 b Fx(\\)e Fz(\001)2007 114 y Fu(")2044 100 y Fy(;)c(E)2156 67 y Fu(\013)2151 128 y Fw(0)p Fu(k)2229 100 y Fz(\))40 b Fx(\025)f Fy(d)p Fz(\()p Fy(")p Fz(\))p Fy(L)2635 67 y Fv(\000)p Fu(p)2772 100 y Fj(for)i(al)5 b(l)41 b Fy(E)3135 67 y Fu(\013)3130 128 y Fw(0)p Fu(k)3247 100 y Fx(2)e Fz(\001)3423 114 y Fu(")3460 100 y Fj(,)265 247 y Fy(\013)26 b Fz(=)f Fy(`;)15 b(r)s Fj(.)41 b(Then)366 451 y(i\))49 b(If)32 b Fy(P)13 b Fz(\()p Fy(E)753 418 y Fu(\013)748 478 y Fw(0)p Fu(k)827 451 y Fz(\))33 b Fj(is)f(the)i(eigenpr)-5 b(oje)g(ctor)34 b(of)f Fy(H)1912 465 y Fu(\013)1994 451 y Fj(asso)-5 b(ciate)g(d)35 b(to)f(the)f(eigenvalue)f Fy(E)3191 418 y Fu(\013)3186 478 y Fw(0)p Fu(k)3289 451 y Fx(2)25 b Fz(\001)3451 465 y Fu(")480 598 y Fj(and)31 b Fy(P)725 565 y Fu(\013)712 625 y(k)805 598 y Fj(the)g(eigenpr)-5 b(oje)g(ctor)32 b(of)f Fy(H)1718 612 y Fu(!)1798 598 y Fj(for)g(the)g(intervals)h Fy(I)2506 565 y Fu(\013)2499 625 y(k)2580 598 y Fz(=)25 b([)p Fy(E)2773 565 y Fu(\013)2768 625 y Fw(0)p Fu(k)2862 598 y Fx(\000)15 b Fy(\032;)g(E)3107 565 y Fu(\013)3102 625 y Fw(0)p Fu(k)3196 598 y Fz(+)h Fy(\032)p Fz(])30 b Fj(we)480 744 y(have)1081 891 y Fx(k)p Fy(P)1197 854 y Fu(\013)1184 914 y(k)1268 891 y Fx(\000)19 b Fy(P)13 b Fz(\()p Fy(E)1536 854 y Fu(\013)1531 914 y Fw(0)p Fu(k)1610 891 y Fz(\))p Fx(k)26 b(\024)f Fy(")1854 854 y Fv(\000)p Fw(1)1949 891 y Fy(C)2021 854 y Fv(0)2043 891 y Fz(\()p Fy(B)5 b(;)15 b(V)2245 905 y Fw(0)2285 891 y Fz(\))p Fy(Le)2424 854 y Fv(\000)2511 826 y Fi(B)p 2489 838 92 3 v 2489 880 a Ft(512)2591 854 y Fw(\(log)e Fu(L)p Fw(\))2796 830 y Ft(2)2861 891 y Fy(:)415 b Fz(\(4.1\))338 1100 y Fj(ii\))49 b(L)-5 b(et)656 1077 y Fz(\026)641 1100 y(\001)32 b Fx(\032)h Fz(\001)929 1114 y Fu(")1002 1100 y Fj(b)-5 b(e)37 b(an)g(interval)h(such)f(that)h Fz(dist)o(\()2188 1077 y(\026)2173 1100 y(\001)p Fy(;)15 b(\033)s Fz(\()p Fy(H)2455 1115 y Fu(`)2488 1100 y Fz(\))24 b Fx([)f Fy(\033)s Fz(\()p Fy(H)2797 1114 y Fu(r)2835 1100 y Fz(\)\))34 b(=)3052 1055 y Fu(d)p Fw(\()p Fu(")p Fw(\))p 3052 1079 124 4 v 3096 1131 a(2)3186 1100 y Fy(L)3248 1067 y Fv(\000)p Fu(p)3342 1100 y Fj(.)55 b(If)480 1247 y Fy(P)538 1262 y Fu(b)572 1247 y Fz(\()623 1224 y(\026)607 1247 y(\001\))31 b Fj(is)g(the)g(eigenpr)-5 b(oje)g(ctor)32 b(of)f Fy(H)1758 1262 y Fu(b)1823 1247 y Fj(for)g(the)g(interval)2461 1224 y Fz(\026)2446 1247 y(\001)f Fj(and)i Fy(P)13 b Fz(\()2849 1224 y(\026)2833 1247 y(\001\))31 b Fj(the)g(eigenpr)-5 b(o-)480 1393 y(je)g(ctor)33 b(of)g Fy(H)917 1407 y Fu(!)999 1393 y Fj(for)h(the)f(interval)1644 1370 y Fz(\026)1629 1393 y(\001)f Fj(we)h(have)1037 1612 y Fx(k)p Fy(P)13 b Fz(\()1204 1589 y(\026)1188 1612 y(\001\))21 b Fx(\000)f Fy(P)1469 1627 y Fu(b)1504 1612 y Fz(\()1554 1589 y(\026)1539 1612 y(\001\))p Fx(k)26 b(\024)f Fy(")1859 1575 y Fv(\000)p Fw(3)1953 1612 y Fy(C)2025 1575 y Fv(0)2048 1612 y Fz(\()p Fy(B)5 b(;)15 b(V)2250 1626 y Fw(0)2290 1612 y Fz(\))p Fy(L)2387 1575 y Fu(p)2427 1612 y Fy(e)2469 1575 y Fv(\000)2555 1547 y Fi(B)p 2534 1559 92 3 v 2534 1601 a Ft(512)2635 1575 y Fw(\(log)f Fu(L)p Fw(\))2841 1551 y Ft(2)2905 1612 y Fy(:)371 b Fz(\(4.2\))265 1849 y Fj(Pr)-5 b(o)g(of.)50 b Fz(W)-8 b(e)29 b(start)g(b)m(y)f(pro)m(ving)h(\(4.1\))g(for)f Fy(\013)e Fz(=)f Fy(r)s Fz(.)39 b(The)28 b(case)h Fy(\013)d Fz(=)f Fy(`)j Fz(is)f(iden)m(tical.)39 b(F)-8 b(rom)28 b(the)265 1995 y(decoupling)h(form)m(ula)g(w)m(e)i(ha)m(v)m(e)570 2252 y Fy(R)q Fz(\()p Fy(z)t Fz(\))21 b Fx(\000)f Fy(R)937 2266 y Fu(r)975 2252 y Fz(\()p Fy(z)t Fz(\))84 b(=)1328 2097 y Fp( )1400 2166 y(X)1409 2363 y Fu(i)p Fv(2I)1547 2252 y Fy(J)1597 2266 y Fu(i)1625 2252 y Fy(R)1694 2266 y Fu(i)1723 2252 y Fz(\()p Fy(z)t Fz(\))1862 2229 y(~)1839 2252 y Fy(J)1889 2266 y Fu(i)1918 2097 y Fp(!)15 b( )2109 2139 y Fv(1)2078 2166 y Fp(X)2077 2360 y Fu(n)p Fw(=1)2225 2252 y Fx(K)q Fz(\()p Fy(z)t Fz(\))2411 2215 y Fu(n)2460 2097 y Fp(!)2552 2252 y Fx(\000)20 b Fz(\(1)h Fx(\000)f Fy(J)2885 2266 y Fu(r)2923 2252 y Fz(\))p Fy(R)3027 2266 y Fu(r)3066 2252 y Fz(\()p Fy(z)t Fz(\))1175 2493 y Fx(\000)82 b Fy(J)1378 2507 y Fu(r)1417 2493 y Fy(R)1486 2507 y Fu(r)1524 2493 y Fz(\()p Fy(z)t Fz(\)\(1)22 b Fx(\000)1855 2470 y Fz(~)1833 2493 y Fy(J)1883 2507 y Fu(r)1921 2493 y Fz(\))f(+)f Fy(J)2118 2508 y Fu(`)2151 2493 y Fy(R)2220 2508 y Fu(`)2253 2493 y Fz(\()p Fy(z)t Fz(\))2392 2470 y(~)2369 2493 y Fy(J)2419 2508 y Fu(`)2473 2493 y Fz(+)g Fy(J)2614 2508 y Fu(b)2649 2493 y Fy(R)2718 2508 y Fu(b)2753 2493 y Fz(\()p Fy(z)t Fz(\))2891 2470 y(~)2869 2493 y Fy(J)2919 2508 y Fu(b)2979 2493 y Fy(:)297 b Fz(\(4.3\))265 2712 y(Let)36 b(\000)f(b)s(e)g(a)g(circle)g(of)g(radius)f Fy(\032)h Fz(in)f(the)i(complex)f(plane,)h(cen)m(tered)g(at)g Fy(E)2908 2679 y Fu(r)2903 2740 y Fw(0)p Fu(k)2981 2712 y Fz(.)56 b(Because)36 b(of)265 2859 y(\()p Fy(H)7 b Fz(1\))33 b(and)f(dist)14 b(\()p Fy(\033)s Fz(\()p Fy(H)1038 2874 y Fu(b)1073 2859 y Fz(\))20 b Fx(\\)g Fz(\001)1285 2873 y Fu(")1322 2859 y Fy(;)15 b(E)1434 2826 y Fu(r)1429 2886 y Fw(0)p Fu(k)1507 2859 y Fz(\))29 b Fx(\025)f Fy(d)p Fz(\()p Fy(")p Fz(\))p Fy(L)1891 2826 y Fv(\000)p Fu(p)1986 2859 y Fz(,)33 b Fy(R)2113 2874 y Fu(b)2148 2859 y Fz(\()p Fy(z)t Fz(\))g(and)e Fy(R)2544 2874 y Fu(`)2577 2859 y Fz(\()p Fy(z)t Fz(\))i(ha)m(v)m(e)h(no)e(p)s(oles)f(in)g(\000.)265 3006 y(Moreo)m(v)m(er)e(the)f(only)e(p)s(ole)g(of)i Fy(R)1372 3020 y Fu(r)1410 3006 y Fz(\()p Fy(z)t Fz(\))g(is)e(precisely)g Fy(E)2083 2973 y Fu(r)2078 3033 y Fw(0)p Fu(k)2156 3006 y Fz(.)39 b(Th)m(us)27 b(in)m(tegrating)h(\(4.3\))h(along)e(the)265 3152 y(circle)j(\000)777 3392 y Fy(P)848 3354 y Fu(r)835 3415 y(k)906 3392 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)1175 3354 y Fu(r)1170 3415 y Fw(0)p Fu(k)1249 3392 y Fz(\))83 b(=)1574 3330 y(1)p 1531 3371 132 4 v 1531 3454 a(2)p Fy(\031)s(i)1688 3268 y Fp(I)1738 3474 y Fw(\000)1802 3236 y Fp( )1873 3305 y(X)1882 3502 y Fu(i)p Fv(2I)2020 3392 y Fy(J)2070 3406 y Fu(i)2099 3392 y Fy(R)2168 3406 y Fu(i)2196 3392 y Fz(\()p Fy(z)t Fz(\))2335 3369 y(~)2312 3392 y Fy(J)2362 3406 y Fu(i)2391 3236 y Fp(!)2510 3278 y Fv(1)2479 3305 y Fp(X)2478 3500 y Fu(n)p Fw(=1)2627 3392 y Fx(K)q Fz(\()p Fy(z)t Fz(\))2813 3354 y Fu(n)2876 3392 y Fz(d)p Fy(z)1367 3633 y Fx(\000)g Fz(\(1)21 b Fx(\000)f Fy(J)1763 3647 y Fu(r)1801 3633 y Fz(\))p Fy(P)13 b Fz(\()p Fy(E)2014 3595 y Fu(r)2009 3656 y Fw(0)p Fu(k)2088 3633 y Fz(\))20 b Fx(\000)g Fy(J)2284 3647 y Fu(r)2323 3633 y Fy(P)13 b Fz(\()p Fy(E)2501 3595 y Fu(r)2496 3656 y Fw(0)p Fu(k)2574 3633 y Fz(\)\(1)21 b Fx(\000)2823 3610 y Fz(~)2801 3633 y Fy(J)2851 3647 y Fu(r)2890 3633 y Fz(\))k Fy(:)326 b Fz(\(4.4\))265 3851 y(W)-8 b(e)38 b(pro)s(ceed)e(to)h(estimate)g(the)g(norms)e(of)i(the)f(three)h(con)m (tributions)e(on)h(the)h(righ)m(t)f(hand)265 3998 y(side)29 b(of)38 b(\(4.4\))r(.)i(The)30 b(norm)g(of)h(the)f(\014rst)g(term)g(is) g(smaller)e(than)403 4260 y Fy(\032)465 4105 y Fp( )537 4174 y(X)546 4371 y Fu(i)p Fv(2I)684 4260 y Fz(sup)689 4339 y Fu(z)s Fv(2)p Fw(\000)836 4260 y Fx(k)p Fy(R)950 4274 y Fu(i)979 4260 y Fz(\()p Fy(z)t Fz(\))p Fx(k)1140 4105 y Fp(!)1316 4199 y Fz(sup)1453 4221 y Fu(z)s Fv(2)p Fw(\000)1599 4199 y Fx(kK)q Fz(\()p Fy(z)t Fz(\))p Fx(k)p 1238 4239 718 4 v 1238 4323 a Fz(1)21 b Fx(\000)f Fz(sup)1531 4344 y Fu(z)s Fv(2)p Fw(\000)1678 4323 y Fx(kK)q Fz(\()p Fy(z)t Fz(\))p Fx(k)1991 4260 y(\024)25 b Fz(6)p Fy(")2174 4223 y Fv(\000)p Fw(1)2269 4260 y Fy(C)7 b Fz(\()p Fy(B)e(;)15 b(V)2543 4274 y Fw(0)2583 4260 y Fz(\))p Fy(Le)2722 4223 y Fv(\000)2808 4196 y Fi(B)p 2787 4208 92 3 v 2787 4249 a Ft(512)2888 4223 y Fw(\(log)f Fu(L)p Fw(\))3094 4199 y Ft(2)3158 4260 y Fy(:)118 b Fz(\(4.5\))265 4540 y(Indeed,)33 b(for)f Fy(i)e Fz(=)f Fy(r)35 b Fz(w)m(e)e(ha)m(v)m(e)h(sup)1452 4562 y Fu(z)s Fv(2)p Fw(\000)1598 4540 y Fx(k)p Fy(R)1712 4554 y Fu(r)1750 4540 y Fz(\()p Fy(z)t Fz(\))p Fx(k)d Fz(=)e Fy(\032)2089 4507 y Fv(\000)p Fw(1)2216 4540 y Fz(b)m(y)k(construction.)47 b(F)-8 b(or)34 b Fy(i)29 b Fz(=)g Fy(`;)15 b(b)33 b Fz(w)m(e)265 4687 y(ha)m(v)m(e)i(sup)615 4709 y Fu(z)s Fv(2)p Fw(\000)761 4687 y Fx(k)p Fy(R)875 4701 y Fu(i)903 4687 y Fz(\()p Fy(z)t Fz(\))p Fx(k)e Fy(<)1252 4651 y Fw(2)p 1208 4666 124 4 v 1208 4720 a Fu(d)p Fw(\()p Fu(")p Fw(\))1342 4687 y Fy(L)1404 4654 y Fu(p)1443 4687 y Fz(.)51 b(Since)32 b Fy(\032)f(<)1948 4642 y Fu(d)p Fw(\()p Fu(")p Fw(\))p 1948 4666 V 1993 4718 a(2)2082 4687 y Fy(L)2144 4654 y Fv(\000)p Fu(p)2272 4687 y Fz(w)m(e)j(note)h(that)f(in)e(all)h(three)h(cases)265 4834 y(\()p Fy(i)d Fx(2)f(I)7 b Fz(\))33 b Fy(\032)15 b Fz(sup)777 4856 y Fu(z)s Fv(2)p Fw(\000)923 4834 y Fx(k)p Fy(R)1037 4848 y Fu(i)1066 4834 y Fz(\()p Fy(z)t Fz(\))p Fx(k)31 b(\024)f Fz(1.)51 b(F)-8 b(urthermore,)34 b(since)f Fy(\032)d(>)g(e)2482 4798 y Fv(\000)2569 4771 y Fi(B)p 2547 4783 92 3 v 2547 4824 a Ft(512)2649 4798 y Fw(\(log)13 b Fu(L)p Fw(\))2854 4775 y Ft(2)2893 4834 y Fz(,)34 b(using)e(Lemma)265 4981 y(1)h(w)m(e)g(get)h(\(4.5\))r(.)47 b(T)-8 b(o)34 b(estimate)f(the)g(second)g(term)f(in)g(\(4.4\))i(w)m(e)f (note)h(that)f(b)m(y)g(the)f(second)265 5127 y(resolv)m(en)m(t)f(form)m (ula)751 5283 y Fy(P)13 b Fz(\()p Fy(E)929 5250 y Fu(r)924 5311 y Fw(0)p Fu(k)1003 5283 y Fz(\))p 708 5325 374 4 v 708 5409 a(\()p Fy(z)25 b Fx(\000)20 b Fy(E)973 5377 y Fu(r)968 5438 y Fw(0)p Fu(k)1046 5409 y Fz(\))1117 5346 y(=)25 b(\()p Fy(z)f Fx(\000)c Fy(H)1481 5360 y Fw(0)1520 5346 y Fz(\))1555 5309 y Fv(\000)p Fw(1)1650 5346 y Fy(P)1708 5360 y Fu(r)1746 5346 y Fz(\()p Fy(E)1853 5309 y Fu(r)1848 5369 y Fw(0)p Fu(k)1927 5346 y Fz(\))g(+)g(\()p Fy(z)25 b Fx(\000)20 b Fy(H)2342 5360 y Fw(0)2381 5346 y Fz(\))2416 5309 y Fv(\000)p Fw(1)2511 5346 y Fy(U)2573 5360 y Fu(r)2664 5283 y Fy(P)13 b Fz(\()p Fy(E)2842 5250 y Fu(r)2837 5311 y Fw(0)p Fu(k)2915 5283 y Fz(\))p 2621 5325 V 2621 5409 a(\()p Fy(z)24 b Fx(\000)c Fy(E)2885 5377 y Fu(r)2880 5438 y Fw(0)p Fu(k)2958 5409 y Fz(\))3029 5346 y Fy(:)247 b Fz(\(4.6\))1828 5637 y Fs(13)p eop %%Page: 14 14 14 13 bop 265 100 a Fz(In)m(tegrating)32 b(\(4.6\))g(along)f(\000)f(w)m (e)h(obtain)f(the)g(iden)m(tit)m(y)1211 324 y Fy(P)13 b Fz(\()p Fy(E)1389 287 y Fu(r)1384 347 y Fw(0)p Fu(k)1462 324 y Fz(\))26 b(=)f(\()p Fy(E)1726 287 y Fu(r)1721 347 y Fw(0)p Fu(k)1819 324 y Fx(\000)20 b Fy(H)1986 338 y Fw(0)2025 324 y Fz(\))2060 287 y Fv(\000)p Fw(1)2155 324 y Fy(U)2217 338 y Fu(r)2255 324 y Fy(P)13 b Fz(\()p Fy(E)2433 287 y Fu(r)2428 347 y Fw(0)p Fu(k)2506 324 y Fz(\))760 b(\(4.7\))265 548 y(this)29 b(implies)366 771 y Fx(k)p Fz(\(1)22 b Fx(\000)e Fy(J)654 785 y Fu(r)692 771 y Fz(\))p Fy(P)13 b Fz(\()p Fy(E)905 734 y Fu(r)900 794 y Fw(0)p Fu(k)979 771 y Fz(\))p Fx(k)84 b(\024)e(k)p Fz(\(1)22 b Fx(\000)e Fy(J)1584 785 y Fu(r)1622 771 y Fz(\))p Fy(R)1726 785 y Fw(0)1766 771 y Fz(\()p Fy(E)1873 734 y Fu(r)1868 794 y Fw(0)p Fu(k)1946 771 y Fz(\))p Fy(U)2043 785 y Fu(r)2082 771 y Fx(k)26 b(\024)e(k)p Fz(\(1)e Fx(\000)e Fy(J)2536 785 y Fu(r)2574 771 y Fz(\))p Fy(R)2678 785 y Fw(0)2718 771 y Fz(\()p Fy(E)2825 734 y Fu(r)2820 794 y Fw(0)p Fu(k)2898 771 y Fz(\))p Fy(U)2995 785 y Fu(r)3034 771 y Fx(k)3079 785 y Fw(2)3301 771 y Fz(\(4.8\))1143 983 y(=)1296 855 y Fp(\032)1365 859 y(Z)1486 983 y Fz(d)o Fd(x)p Fx(j)p Fz(1)h Fx(\000)f Fy(J)1828 997 y Fu(r)1866 983 y Fz(\()p Fy(x)p Fz(\))p Fx(j)2013 945 y Fw(2)2069 859 y Fp(Z)2190 983 y Fz(d)p Fd(x)2301 945 y Fv(0)2324 983 y Fx(j)p Fy(R)2418 997 y Fw(0)2458 983 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)2653 945 y Fv(0)2676 983 y Fz(;)g Fy(E)2788 945 y Fu(r)2783 1006 y Fw(0)p Fu(k)2862 983 y Fz(\))p Fy(U)2959 997 y Fu(r)2997 983 y Fz(\()p Fy(x)3084 945 y Fv(0)3108 983 y Fz(\))p Fx(j)3168 945 y Fw(2)3208 855 y Fp(\033)3276 877 y Fw(1)p Fu(=)p Fw(2)265 1223 y Fz(since)30 b(the)g(distance)g(\(in)f(the)h Fy(x)h Fz(direction\))e(b)s(et)m(w)m(een)i(the)f(supp)s(orts)e(of)j (\(1)20 b Fx(\000)g Fy(J)3016 1237 y Fu(r)3054 1223 y Fz(\))31 b(and)e Fy(U)3358 1237 y Fu(r)3426 1223 y Fz(is)265 1370 y(greater)35 b(than)802 1334 y Fu(D)p 802 1349 60 4 v 814 1401 a Fw(2)894 1370 y Fz(+)22 b(1)34 b(w)m(e)g(can)g(pro)s (ceed)g(in)e(a)i(similar)d(w)m(a)m(y)j(as)g(in)e(the)i(estimate)g(of)41 b(\(3.10\))265 1516 y(to)31 b(obtain)1051 1663 y Fx(k)p Fz(\(1)21 b Fx(\000)f Fy(J)1338 1677 y Fu(r)1376 1663 y Fz(\))p Fy(P)13 b Fz(\()p Fy(E)1589 1626 y Fu(r)1584 1686 y Fw(0)p Fu(k)1663 1663 y Fz(\))p Fx(k)26 b(\024)f Fy(")1907 1626 y Fv(\000)p Fw(1)2022 1640 y Fz(\026)2002 1663 y Fy(C)6 b Fz(\()p Fy(B)f Fz(\))p Fy(Le)2321 1626 y Fv(\000)2393 1598 y Fi(B)p 2387 1610 61 3 v 2387 1652 a Ft(64)2457 1626 y Fw(\(log)14 b Fu(L)p Fw(\))2663 1602 y Ft(2)3301 1663 y Fz(\(4.9\))265 1857 y(where)554 1834 y(\026)533 1857 y Fy(C)6 b Fz(\()p Fy(B)f Fz(\))36 b(is)f(a)g(constan)m (t)i(dep)s(ending)c(only)i(on)g Fy(B)5 b Fz(.)56 b(F)-8 b(or)36 b(the)f(third)f(term)i(in)e(\(4.4\))j(w)m(e)265 2003 y(use)30 b(the)h(adjoin)m(t)f(of)37 b(\(4.7\))1211 2227 y Fy(P)13 b Fz(\()p Fy(E)1389 2189 y Fu(r)1384 2250 y Fw(0)p Fu(k)1462 2227 y Fz(\))26 b(=)f Fy(P)13 b Fz(\()p Fy(E)1797 2189 y Fu(r)1792 2250 y Fw(0)p Fu(k)1870 2227 y Fz(\))p Fy(U)1967 2241 y Fu(r)2006 2227 y Fz(\()p Fy(E)2113 2189 y Fu(r)2108 2250 y Fw(0)p Fu(k)2206 2227 y Fx(\000)20 b Fy(H)2373 2241 y Fw(0)2412 2227 y Fz(\))2447 2189 y Fv(\000)p Fw(1)3255 2227 y Fz(\(4.10\))265 2451 y(to)31 b(get)926 2674 y Fx(k)p Fy(J)1021 2688 y Fu(r)1060 2674 y Fy(P)13 b Fz(\()p Fy(E)1238 2637 y Fu(r)1233 2697 y Fw(0)p Fu(k)1312 2674 y Fz(\)\(1)21 b Fx(\000)1561 2651 y Fz(~)1539 2674 y Fy(J)1589 2688 y Fu(r)1627 2674 y Fz(\))p Fx(k)26 b(\024)f(k)p Fy(U)1936 2688 y Fu(r)1975 2674 y Fy(R)2044 2688 y Fw(0)2083 2674 y Fz(\()p Fy(E)2190 2637 y Fu(r)2185 2697 y Fw(0)p Fu(k)2264 2674 y Fz(\)\(1)c Fx(\000)2513 2651 y Fz(~)2491 2674 y Fy(J)2541 2688 y Fu(r)2579 2674 y Fz(\))p Fx(k)596 b Fz(\(4.11\))265 2898 y(from)38 b(whic)m(h)g(w)m(e)h(obtain)f(the)h(same)h(b)s(ound)c(as)j (in)f(\(4.9\))r(.)66 b(Com)m(bining)36 b(this)i(result)g(with)265 3045 y(\(4.4\))r(,)30 b(\(4.5\))r(,)h(\(4.9\))h(w)m(e)f(obtain)f (\(4.1\))i(in)d(the)i(prop)s(osition.)401 3192 y(Let)g(us)f(no)m(w)g (sk)m(etc)m(h)i(the)f(pro)s(of)e(of)38 b(\(4.2\))r(.)i(F)-8 b(rom)31 b(the)g(decoupling)d(form)m(ula)i(w)m(e)h(ha)m(v)m(e)575 3458 y Fy(R)q Fz(\()p Fy(z)t Fz(\))21 b Fx(\000)f Fy(R)942 3473 y Fu(b)977 3458 y Fz(\()p Fy(z)t Fz(\))84 b(=)1330 3303 y Fp( )1402 3372 y(X)1411 3569 y Fu(i)p Fv(2I)1549 3458 y Fy(J)1599 3472 y Fu(i)1627 3458 y Fy(R)1696 3472 y Fu(i)1724 3458 y Fz(\()p Fy(z)t Fz(\))1863 3436 y(~)1840 3458 y Fy(J)1890 3472 y Fu(i)1920 3303 y Fp(!)15 b( )2110 3345 y Fv(1)2080 3372 y Fp(X)2079 3566 y Fu(n)p Fw(=1)2227 3458 y Fx(K)q Fz(\()p Fy(z)t Fz(\))2413 3421 y Fu(n)2462 3303 y Fp(!)2554 3458 y Fx(\000)20 b Fz(\(1)h Fx(\000)f Fy(J)2887 3473 y Fu(b)2922 3458 y Fz(\))p Fy(R)3026 3473 y Fu(b)3060 3458 y Fz(\()p Fy(z)t Fz(\))1177 3699 y Fx(\000)82 b Fy(J)1380 3714 y Fu(b)1415 3699 y Fy(R)1484 3714 y Fu(b)1518 3699 y Fz(\()p Fy(z)t Fz(\)\(1)22 b Fx(\000)1849 3676 y Fz(~)1827 3699 y Fy(J)1877 3714 y Fu(b)1912 3699 y Fz(\))f(+)e Fy(J)2108 3714 y Fu(`)2142 3699 y Fy(R)2211 3714 y Fu(`)2244 3699 y Fz(\()p Fy(z)t Fz(\))2383 3676 y(~)2360 3699 y Fy(J)2410 3714 y Fu(`)2464 3699 y Fz(+)h Fy(J)2605 3713 y Fu(r)2644 3699 y Fy(R)2713 3713 y Fu(r)2751 3699 y Fz(\()p Fy(z)t Fz(\))2890 3676 y(~)2867 3699 y Fy(J)2917 3713 y Fu(r)2981 3699 y Fy(:)249 b Fz(\(4.12\))265 3923 y(Giv)m(en)29 b(an)g(in)m(terv)-5 b(al)994 3900 y(\026)979 3923 y(\001)25 b Fx(\032)g Fz(\001)1252 3937 y Fu(")1318 3923 y Fz(suc)m(h)k(that)g(dist)o(\()1915 3900 y(\026)1899 3923 y(\001)p Fy(;)15 b(\033)s Fz(\()p Fy(H)2181 3938 y Fu(`)2215 3923 y Fz(\))j Fx([)f Fy(\033)s Fz(\()p Fy(H)2512 3937 y Fu(r)2550 3923 y Fz(\)\))26 b(=)2752 3878 y Fu(d)p Fw(\()p Fu(")p Fw(\))p 2752 3902 124 4 v 2796 3954 a(2)2886 3923 y Fy(L)2948 3890 y Fv(\000)p Fu(p)3042 3923 y Fz(,)k(w)m(e)f(c)m(ho)s(ose)265 4070 y(a)j(circle)587 4047 y(\026)581 4070 y(\000)g(in)f(the)h(complex)f (plane)g(with)g(diameter)h(equal)f(to)i Fx(j)2521 4047 y Fz(\026)2506 4070 y(\001)p Fx(j)p Fz(.)45 b(Then)31 b(if)g(w)m(e)h(in)m(tegrate)265 4216 y(o)m(v)m(er)459 4193 y(\026)453 4216 y(\000)23 b(the)h(last)g(t)m(w)m(o)h(terms)e(on)h (the)g(righ)m(t)f(hand)f(side)h(do)h(not)g(con)m(tribute)f(while)f(the) h(second)265 4363 y(and)30 b(third)e(ones)j(giv)m(e)g(\(1)21 b Fx(\000)f Fy(J)1301 4378 y Fu(b)1335 4363 y Fz(\))p Fy(P)1428 4378 y Fu(b)1463 4363 y Fz(\()1514 4340 y(\026)1498 4363 y(\001)q(\))30 b(and)g Fy(J)1867 4378 y Fu(b)1902 4363 y Fy(P)1960 4378 y Fu(b)1994 4363 y Fz(\()2045 4340 y(\026)2029 4363 y(\001)q(\)\(1)21 b Fx(\000)f Fy(J)2383 4378 y Fu(b)2418 4363 y Fz(\).)41 b(Therefore)681 4630 y Fx(k)p Fy(P)33 b Fx(\000)20 b Fy(P)966 4645 y Fu(b)1001 4630 y Fz(\()1052 4607 y(\026)1036 4630 y(\001\))p Fx(k)84 b(\024)f(j)1470 4607 y Fz(\026)1455 4630 y(\001)p Fx(j)1571 4475 y Fp( )1643 4544 y(X)1652 4741 y Fu(i)p Fv(2I)1790 4630 y Fz(sup)1795 4718 y Fu(z)s Fv(2)1882 4701 y Fw(\026)1878 4718 y(\000)1942 4630 y Fx(k)p Fy(R)2056 4644 y Fu(i)2084 4630 y Fz(\()p Fy(z)t Fz(\))p Fx(k)2245 4475 y Fp(!)2422 4567 y Fz(sup)2559 4591 y Fu(z)s Fv(2)2646 4574 y Fw(\026)2642 4591 y(\000)2705 4567 y Fx(kK)q Fz(\()p Fy(z)t Fz(\))p Fx(k)p 2344 4609 718 4 v 2344 4693 a Fz(1)20 b Fx(\000)g Fz(sup)2637 4716 y Fu(z)s Fv(2)2724 4699 y Fw(\026)2720 4716 y(\000)2783 4693 y Fx(kK)q Fz(\()p Fy(z)t Fz(\))p Fx(k)1276 4871 y Fz(+)83 b Fx(k)p Fz(\(1)21 b Fx(\000)f Fy(J)1717 4886 y Fu(b)1752 4871 y Fz(\))p Fy(P)1845 4886 y Fu(b)1880 4871 y Fz(\()1930 4848 y(\026)1915 4871 y(\001\))p Fx(k)h Fz(+)f Fx(k)p Fy(J)2278 4886 y Fu(b)2313 4871 y Fy(P)2371 4886 y Fu(b)2406 4871 y Fz(\()2456 4848 y(\026)2441 4871 y(\001\)\(1)h Fx(\000)2766 4848 y Fz(~)2744 4871 y Fy(J)2794 4886 y Fu(b)2829 4871 y Fz(\))p Fx(k)26 b Fy(:)295 b Fz(\(4.13\))265 5095 y(F)-8 b(rom)42 b(Lemma)g(1,)j Fx(j)1006 5072 y Fz(\026)991 5095 y(\001)p Fx(j)f Fy(<)g(d)p Fz(\()p Fy(")p Fz(\))p Fy(L)1472 5062 y Fv(\000)p Fw(1)1609 5095 y Fz(and)d(sup)1934 5118 y Fu(z)s Fv(2)2021 5101 y Fw(\026)2017 5118 y(\000)2080 5095 y Fx(k)p Fy(R)2194 5109 y Fu(i)2223 5095 y Fz(\()p Fy(z)t Fz(\))p Fx(k)k Fy(<)2598 5059 y Fw(2)p 2554 5074 124 4 v 2554 5127 a Fu(d)p Fw(\()p Fu(")p Fw(\))2688 5095 y Fy(L)2750 5062 y Fu(p)2831 5095 y Fz(the)d(\014rst)f(term)g(is)265 5241 y(b)s(ounded)28 b(ab)s(o)m(v)m(e)k(b)m(y)1286 5388 y(12)p Fy(")1418 5351 y Fv(\000)p Fw(1)1513 5388 y Fy(C)7 b Fz(\()p Fy(B)e(;)15 b(V)1787 5402 y Fw(0)1827 5388 y Fz(\))p Fy(L)1924 5351 y Fu(p)1963 5388 y Fy(e)2005 5351 y Fv(\000)2092 5323 y Fi(B)p 2071 5335 92 3 v 2071 5377 a Ft(512)2172 5351 y Fw(\(log)e Fu(L)p Fw(\))2377 5327 y Ft(2)2442 5388 y Fy(:)788 b Fz(\(4.14\))1828 5637 y Fs(14)p eop %%Page: 15 15 15 14 bop 265 100 a Fz(In)34 b(order)g(to)h(estimate)g(the)g(second)f (norm)g(in)f(\(4.13\))k(w)m(e)e(notice)f(that)i(\(in)d(the)i(same)g(w)m (a)m(y)265 247 y(as)c(in)e(\(4.6\))r(,)h(\(4.7\))r(\))1252 394 y Fy(P)1310 409 y Fu(b)1344 394 y Fz(\()1395 371 y(\026)1379 394 y(\001)q(\))25 b(=)1646 308 y Fp(X)1612 517 y Fu(E)1668 494 y Fi(b)1664 540 y(\014)1705 517 y Fv(2)1764 500 y Fw(\026)1752 517 y(\001)1826 394 y Fy(R)1895 408 y Fw(0)1935 394 y Fz(\()p Fy(E)2042 356 y Fu(b)2037 417 y(\014)2084 394 y Fz(\))p Fy(V)2172 408 y Fu(!)2223 394 y Fy(P)2281 409 y Fu(b)2316 394 y Fz(\()p Fy(E)2423 356 y Fu(b)2418 417 y(\014)2466 394 y Fz(\))754 b(\(4.15\))265 684 y(th)m(us)943 831 y Fx(k)p Fz(\(1)21 b Fx(\000)f Fy(J)1230 846 y Fu(b)1265 831 y Fz(\))p Fy(P)1358 846 y Fu(b)1393 831 y Fz(\()1443 808 y(\026)1428 831 y(\001\))p Fx(k)26 b(\024)1740 745 y Fp(X)1706 954 y Fu(E)1762 931 y Fi(b)1758 978 y(\014)1799 954 y Fv(2)1858 937 y Fw(\026)1846 954 y(\001)1920 831 y Fx(k)p Fz(\(1)c Fx(\000)e Fy(J)2208 846 y Fu(b)2242 831 y Fz(\))p Fy(R)2346 845 y Fw(0)2386 831 y Fz(\()p Fy(E)2493 794 y Fu(b)2488 854 y(\014)2536 831 y Fz(\))p Fy(V)2624 845 y Fu(!)2675 831 y Fx(k)2720 845 y Fw(2)2785 831 y Fy(:)445 b Fz(\(4.16\))265 1127 y(Eac)m(h)36 b(term)f(of)g(the)g(sum)f(can)h(b)s(e)g(b)s(ounded)e(in)g (a)j(w)m(a)m(y)g(similar)c(to)k(\(3.10\))r(,)h(and)d(since)g(the)265 1274 y(n)m(um)m(b)s(er)29 b(of)i(terms)f(in)f(the)i(sum)e(is)g(equal)h (to)h(T)-8 b(r)15 b Fy(P)2017 1289 y Fu(b)2052 1274 y Fz(\()2102 1251 y(\026)2087 1274 y(\001\))31 b(w)m(e)g(get)716 1512 y Fx(k)p Fz(\(1)22 b Fx(\000)e Fy(J)1004 1527 y Fu(b)1039 1512 y Fz(\))p Fy(P)1132 1527 y Fu(b)1167 1512 y Fz(\()1217 1489 y(\026)1202 1512 y(\001\))p Fx(k)84 b(\024)e Fy(")1637 1474 y Fv(\000)p Fw(1)1732 1512 y Fy(C)7 b Fz(\()p Fy(B)e(;)15 b(V)2006 1526 y Fw(0)2045 1512 y Fz(\))p Fy(Le)2184 1474 y Fv(\000)2256 1447 y Fi(B)p 2250 1459 61 3 v 2250 1500 a Ft(64)2321 1474 y Fw(\(log)e Fu(L)p Fw(\))2526 1451 y Ft(2)2580 1512 y Fz(T)-8 b(r)15 b Fy(P)2747 1527 y Fu(b)2782 1512 y Fz(\()2832 1489 y(\026)2817 1512 y(\001\))1442 1683 y Fx(\024)82 b Fz(2)p Fy(")1682 1646 y Fv(\000)p Fw(3)1778 1683 y Fy(c)p Fz(\()p Fy(B)5 b Fz(\))1961 1646 y Fw(2)2001 1683 y Fy(C)i Fz(\()p Fy(B)e(;)15 b(V)2275 1697 y Fw(0)2314 1683 y Fz(\))p Fy(V)2422 1646 y Fw(2)2402 1706 y(0)2462 1683 y Fy(L)2524 1646 y Fw(5)2563 1683 y Fy(e)2605 1646 y Fv(\000)2676 1619 y Fi(B)p 2670 1631 V 2670 1672 a Ft(64)2741 1646 y Fw(\(log)e Fu(L)p Fw(\))2946 1622 y Ft(2)3011 1683 y Fy(:)219 b Fz(\(4.17\))265 1921 y(The)45 b(second)g(inequalit)m(y)e(follo)m(ws)h(from)h(Lemma)g(4)g(in)f(App)s (endix)e(B)j(\(where)g(w)m(e)h(need)265 2068 y Fy(B)30 b(>)25 b Fz(4)p Fy(V)558 2082 y Fw(0)598 2068 y Fz(\).)40 b(F)-8 b(or)30 b Fx(k)p Fy(J)955 2083 y Fu(b)990 2068 y Fy(P)1048 2083 y Fu(b)1083 2068 y Fz(\()1133 2045 y(\026)1118 2068 y(\001\)\(1)18 b Fx(\000)1437 2045 y Fz(~)1415 2068 y Fy(J)1465 2083 y Fu(b)1499 2068 y Fz(\))p Fx(k)30 b Fz(one)f(uses)f(the)h(adjoin)m(t)g(of)g(iden)m(tit)m(y)g(\(4.15\))j(to) d(obtain)265 2215 y(the)39 b(same)g(result.)64 b(The)38 b(result)g(\(4.2\))i(of)f(the)g(prop)s(osition)d(then)j(follo)m(ws)e(b) m(y)i(com)m(bining)265 2362 y(\(4.13\))r(,)31 b(\(4.14\))i(and)c (\(4.17\))s(.)p 3422 2362 4 62 v 3426 2304 55 4 v 3426 2362 V 3479 2362 4 62 v 401 2583 a(In)35 b(app)s(endix)d(B)k(w)m(e)f (adapt)g(the)g(metho)s(d)g(of)g([CH])g(to)h(our)e(geometry)j(to)e(get)i (the)e(fol-)265 2730 y(lo)m(wing)29 b(W)-8 b(egner)32 b(estimate.)265 2952 y FA(Prop)s(osition)k(2.)42 b Fj(L)-5 b(et)33 b Fy(B)d Fx(\025)25 b Fz(4)p Fy(V)1403 2966 y Fw(0)1475 2952 y Fj(and)34 b Fy(E)c Fx(2)25 b Fz(\001)1911 2966 y Fu(")917 3190 y Fq(P)972 3204 y Fw(\003)1041 3190 y Fz(\(dist)o(\()p Fy(\033)s Fz(\()p Fy(H)1424 3205 y Fu(b)1459 3190 y Fz(\))p Fy(;)15 b(E)5 b Fz(\))27 b Fy(<)d(\016)s Fz(\))j Fx(\024)e Fz(4)p Fy(c)p Fz(\()p Fy(B)5 b Fz(\))p Fx(k)p Fy(h)p Fx(k)2334 3204 y Fv(1)2411 3190 y Fy(\016)s(")2496 3152 y Fv(\000)p Fw(2)2591 3190 y Fy(V)2644 3204 y Fw(0)2684 3190 y Fy(L)2746 3152 y Fw(4)2810 3190 y Fy(:)420 b Fz(\(4.18\))265 3428 y Fj(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(1,)f(p)-5 b(art)35 b(a\))e(and)g(c\).)49 b Fz(Let)44 b Fy(!)51 b Fx(2)c Fz(\012)2196 3371 y Fg(00)2196 3455 y Fw(\003)2292 3428 y Fz(where)d(\012)2635 3371 y Fg(00)2635 3455 y Fw(\003)2731 3428 y Fz(is)f(the)h(set)g(giv)m(en)g(in)265 3575 y(Prop)s(osition)37 b(1.)65 b(Since)38 b(for)g Fy(L)h Fz(large)f(enough)h(the)g(righ)m(t)f(hand)f(side)h(of)46 b(\(4.1\))40 b(is)e(strictly)265 3721 y(smaller)29 b(than)h(one)g(the)h (t)m(w)m(o)g(pro)5 b(jectors)31 b(necessarily)e(ha)m(v)m(e)j(the)e (same)h(dimension.)38 b(There-)265 3868 y(fore)h Fy(\033)s Fz(\()p Fy(H)619 3882 y Fu(!)669 3868 y Fz(\))26 b Fx(\\)g Fy(I)864 3835 y Fu(\013)857 3896 y(k)952 3868 y Fz(con)m(tains)38 b(a)h(unique)e(energy)i(lev)m(el)f Fy(E)2290 3835 y Fu(\013)2285 3896 y(k)2378 3868 y Fz(for)h(eac)m(h)h Fy(I)2786 3835 y Fu(\013)2779 3896 y(k)2874 3868 y Fz(of)e(radius)f Fy(\032)p Fz(.)65 b(In)265 4015 y(particular)31 b(b)m(y)h(taking)g(the) g(smallest)f(v)-5 b(alue)32 b Fy(\032)c Fz(=)g Fy(e)2059 3979 y Fv(\000)2146 3952 y Fi(B)p 2124 3964 92 3 v 2124 4005 a Ft(512)2226 3979 y Fw(\(log)13 b Fu(L)p Fw(\))2431 3956 y Ft(2)2502 4015 y Fz(w)m(e)33 b(get)g(\(2.15\))r(.)46 b(The)32 b(n)m(um-)265 4162 y(b)s(er)j(of)h(suc)m(h)g(lev)m(els)f(is)h Fx(O)s Fz(\()p Fy(L)p Fz(\))g(since)f(they)i(are)f(in)f(one)h(to)h(one) f(corresp)s(ondence)g(with)e(the)265 4308 y(energy)d(lev)m(els)f(of)g Fy(H)978 4322 y Fu(\013)1027 4308 y Fz(.)41 b(The)30 b(sets)g(\006)1523 4322 y Fu(\013)1603 4308 y Fz(of)g(Theorem)h(1)f (are)h(precisely)948 4547 y(\006)1014 4561 y Fu(\013)1088 4547 y Fz(=)1184 4460 y Fp([)1215 4658 y Fu(k)1300 4547 y Fz(\()p Fy(\033)s Fz(\()p Fy(H)1501 4561 y Fu(!)1552 4547 y Fz(\))21 b Fx(\\)e Fy(I)1735 4509 y Fu(\013)1728 4570 y(k)1805 4547 y Fx(\\)h Fz(\001)1962 4561 y Fu(")1998 4547 y Fz(\))c Fy(;)379 b(\013)26 b Fz(=)f Fy(`;)15 b(r)28 b(:)450 b Fz(\(4.19\))265 4835 y(The)43 b(set)h(of)g(all)f(other)h (eigen)m(v)-5 b(alues)43 b(in)g Fy(\033)s Fz(\()p Fy(H)1897 4849 y Fu(!)1947 4835 y Fz(\))30 b Fx(\\)e Fz(\001)2177 4849 y Fu(")2214 4835 y Fz(,)47 b(de\014nes)c(\006)2664 4850 y Fu(b)2698 4835 y Fz(,)k(and)c(is)g(necessarily)265 4982 y(con)m(tained)d(in)f(in)m(terv)-5 b(als)1192 4959 y(\026)1176 4982 y(\001)40 b(suc)m(h)g(that)h(dist)o(\()1911 4959 y(\026)1896 4982 y(\001)p Fy(;)15 b(\033)s Fz(\()p Fy(H)2178 4997 y Fu(`)2212 4982 y Fz(\))27 b Fx([)f Fy(\033)s Fz(\()p Fy(H)2527 4996 y Fu(r)2565 4982 y Fz(\)\))42 b(=)2800 4937 y Fu(d)p Fw(\()p Fu(")p Fw(\))p 2800 4961 124 4 v 2844 5013 a(2)2933 4982 y Fy(L)2995 4949 y Fv(\000)p Fu(p)3090 4982 y Fz(.)70 b(In)39 b(view)265 5128 y(of)46 b(\(2.15\))41 b(this)c(implies)f(\(2.17\))s(.)65 b(Since)38 b(the)h(t)m(w)m(o)h(pro)5 b(jectors)39 b(in)e(\(4.2\))k(necessarily)c (ha)m(v)m(e)265 5275 y(the)j(same)h(dimension,)f(the)g(n)m(um)m(b)s(er) e(of)j(eigenstates)g(in)e(\006)2408 5290 y Fu(b)2481 5275 y Fz(is)g(the)i(same)f(than)g(that)g(of)1828 5637 y Fs(15)p eop %%Page: 16 16 16 15 bop 265 100 a Fy(\033)s Fz(\()p Fy(H)431 115 y Fu(b)465 100 y Fz(\))13 b Fx(\\)g Fz(\001)663 114 y Fu(")701 100 y Fz(.)39 b(It)27 b(remains)f(to)i(estimate)f(the)g(probabilit)m(y) e(of)i(the)g(set)h(\012)2737 44 y Fg(00)2737 127 y Fw(\003)2789 100 y Fz(.)40 b(The)26 b(realizations)265 247 y(of)31 b(the)f(complemen)m(tary)h(set)g(are)f(suc)m(h)g(that)h(for)g(at)g (least)f(one)h Fy(E)2559 214 y Fu(\013)2554 275 y Fw(0)p Fu(k)2657 247 y Fx(2)25 b Fz(\001)2819 261 y Fu(")1338 481 y Fz(dist)o(\()p Fy(\033)s Fz(\()p Fy(H)1686 496 y Fu(b)1721 481 y Fz(\))p Fy(;)15 b(E)1868 443 y Fu(\013)1863 504 y Fw(0)p Fu(k)1941 481 y Fz(\))26 b Fy(<)f(d)p Fz(\()p Fy(")p Fz(\))p Fy(L)2319 443 y Fv(\000)p Fu(p)3255 481 y Fz(\(4.20\))265 714 y(but)30 b(from)g(Prop)s(osition)e(2)j(this)e (has)h(a)h(probabilit)m(y)d(smaller)h(than)1193 948 y(4)p Fy(c)p Fz(\()p Fy(B)5 b Fz(\))p Fx(k)p Fy(h)p Fx(k)1563 962 y Fv(1)1640 948 y Fy(d)p Fz(\()p Fy(")p Fz(\))p Fy(L)1861 910 y Fv(\000)p Fu(p)1956 948 y Fy(")1998 910 y Fv(\000)p Fw(2)2093 948 y Fy(V)2146 962 y Fw(0)2186 948 y Fy(L)2248 910 y Fw(4)2307 948 y Fx(\001)21 b(O)s Fz(\()p Fy(L)p Fz(\))695 b(\(4.21\))265 1181 y(where)28 b Fx(O)s Fz(\()p Fy(L)p Fz(\))h(comes)g(from)f(the)h(n)m(um)m(b)s(er)e(of)h(lev)m(els)g (in)g([)p Fy(\033)s Fz(\()p Fy(H)2363 1196 y Fu(`)2396 1181 y Fz(\))17 b Fx([)f Fy(\033)s Fz(\()p Fy(H)2691 1195 y Fu(r)2729 1181 y Fz(\)])h Fx(\\)f Fz(\001)2959 1195 y Fu(")2995 1181 y Fz(.)40 b(Th)m(us)27 b(for)i Fy(L)265 1328 y Fz(large)h(enough)1467 1475 y Fq(P)1522 1489 y Fw(\003)1576 1475 y Fz(\(\012)1677 1414 y Fg(00)1677 1497 y Fw(\003)1730 1475 y Fz(\))c Fx(\025)f Fz(1)20 b Fx(\000)g Fy(L)2105 1437 y Fw(6)p Fv(\000)p Fu(p)2260 1475 y Fy(:)970 b Fz(\(4.22\))265 1673 y(W)-8 b(e)32 b(recall)d(that)i Fy(p)25 b Fx(\025)g Fz(7.)p 3422 1673 4 62 v 3426 1615 55 4 v 3426 1673 V 3479 1673 4 62 v 265 2047 a Fr(5)161 b(Estimates)53 b(of)h(Curren)l(ts)265 2283 y Fz(In)22 b(this)g(section)h(w)m(e)h(c)m(haracterize)g(the)f (eigen)m(v)-5 b(alues)23 b(of)g Fy(H)2243 2297 y Fu(!)2316 2283 y Fz(in)f(terms)g(of)h(the)h(curren)m(t)e(carried)265 2430 y(b)m(y)30 b(the)h(corresp)s(onding)d(eigenstates.)42 b(This)29 b(will)e(yield)i(parts)h Fy(b)p Fz(\))h(and)f Fy(d)p Fz(\))h(of)f(Theorem)g(1.)265 2650 y Fj(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(1,)f(p)-5 b(art)35 b(b\).)48 b Fz(Let)21 b Fy(E)1670 2617 y Fu(\013)1665 2678 y(k)1746 2650 y Fx(2)j Fz(\006)1897 2664 y Fu(\013)1947 2650 y Fz(.)37 b(The)20 b(asso)s(ciated)h(curren)m(t)g(is)e(b)m(y)i (de\014nition)1604 2884 y Fy(J)1663 2846 y Fu(\013)1654 2907 y(k)1738 2884 y Fz(=)k(T)-8 b(r)15 b Fy(v)1987 2898 y Fu(y)2028 2884 y Fy(P)2099 2846 y Fu(\013)2086 2907 y(k)3301 2884 y Fz(\(5.1\))265 3117 y(and)30 b(will)e(b)s(e)h(compared) i(to)g(that)g(of)f Fy( )1625 3084 y Fu(\013)1622 3145 y Fw(0)p Fu(k)1486 3351 y Fy(J)1545 3313 y Fu(\013)1536 3374 y Fw(0)p Fu(k)1639 3351 y Fz(=)25 b(T)-8 b(r)15 b Fy(v)1888 3365 y Fu(y)1930 3351 y Fy(P)e Fz(\()p Fy(E)2108 3313 y Fu(\013)2103 3374 y Fw(0)p Fu(k)2181 3351 y Fz(\))26 b Fy(:)1034 b Fz(\(5.2\))265 3584 y(The)34 b(di\013erence)h(b)s(et)m(w) m(een)g(these)g(t)m(w)m(o)i(curren)m(ts)d(will)e(b)s(e)i(estimated)h(b) m(y)g Fx(k)p Fy(P)2963 3551 y Fu(\013)2950 3612 y(k)3036 3584 y Fx(\000)23 b Fy(P)13 b Fz(\()p Fy(E)3308 3551 y Fu(\013)3303 3612 y Fw(0)p Fu(k)3382 3584 y Fz(\))p Fx(k)p Fz(.)265 3731 y(First)37 b(w)m(e)i(observ)m(e)f(that)h Fy(v)1216 3745 y Fu(y)1257 3731 y Fy(P)1328 3698 y Fu(\013)1315 3758 y(k)1415 3731 y Fz(is)e(trace)i(class.)63 b(Indeed,)40 b Fy(v)2395 3745 y Fu(y)2436 3731 y Fy(P)2507 3698 y Fu(\013)2494 3758 y(k)2594 3731 y Fz(=)e Fy(v)2747 3745 y Fu(y)2788 3731 y Fy(P)2859 3698 y Fu(\013)2846 3758 y(k)2909 3731 y Fy(P)2980 3698 y Fu(\013)2967 3758 y(k)3067 3731 y Fz(with)f Fy(v)3326 3745 y Fu(y)3367 3731 y Fy(P)3438 3698 y Fu(\013)3425 3758 y(k)265 3878 y Fz(b)s(ounded)28 b(and)i Fx(k)p Fy(P)929 3845 y Fu(\013)916 3905 y(k)979 3878 y Fx(k)1024 3892 y Fw(1)1089 3878 y Fz(=)25 b(T)-8 b(r)15 b Fy(P)1365 3845 y Fu(\013)1352 3905 y(k)1440 3878 y Fz(=)25 b(1)415 4111 y Fx(k)p Fy(v)504 4125 y Fu(y)546 4111 y Fy(P)617 4073 y Fu(\013)604 4134 y(k)667 4111 y Fx(k)712 4073 y Fw(2)712 4133 y(1)777 4111 y Fx(\024)g(k)p Fy(v)962 4125 y Fu(y)1004 4111 y Fy(P)1075 4073 y Fu(\013)1062 4134 y(k)1124 4111 y Fx(k)1169 4073 y Fw(2)1234 4111 y Fx(\024)g(k)p Fy(P)1446 4073 y Fu(\013)1433 4134 y(k)1496 4111 y Fy(v)1543 4073 y Fw(2)1540 4133 y Fu(y)1583 4111 y Fy(P)1654 4073 y Fu(\013)1641 4134 y(k)1703 4111 y Fx(k)h(\024)f Fz(2)p Fx(k)p Fy(P)2031 4073 y Fu(\013)2018 4134 y(k)2081 4111 y Fz(\()p Fy(H)2192 4125 y Fu(!)2263 4111 y Fx(\000)20 b Fy(V)2407 4125 y Fu(!)2457 4111 y Fz(\))p Fy(P)2563 4073 y Fu(\013)2550 4134 y(k)2613 4111 y Fx(k)26 b(\024)f Fz(2)p Fy(E)2897 4073 y Fu(\013)2892 4134 y(k)2967 4111 y Fz(+)20 b Fy(V)3111 4125 y Fw(0)3301 4111 y Fz(\(5.3\))265 4344 y(to)30 b(get)h(the)f(second)g(inequalit)m (y)e(one)i(has)f(simply)e(added)i(p)s(ositiv)m(e)g(terms)h(to)g Fy(v)3026 4311 y Fw(2)3023 4367 y Fu(y)3066 4344 y Fz(.)40 b(Similarly)819 4578 y Fx(k)p Fy(v)908 4592 y Fu(y)950 4578 y Fy(P)13 b Fz(\()p Fy(E)1128 4540 y Fu(\013)1123 4601 y Fw(0)p Fu(k)1201 4578 y Fz(\))p Fx(k)1281 4540 y Fw(2)1281 4600 y(1)1405 4578 y Fx(\024)82 b(k)p Fy(v)1647 4592 y Fu(y)1689 4578 y Fy(P)13 b Fz(\()p Fy(E)1867 4540 y Fu(\013)1862 4601 y Fw(0)p Fu(k)1941 4578 y Fz(\))p Fx(k)2021 4540 y Fw(2)2086 4578 y Fx(\024)25 b(k)p Fy(P)13 b Fz(\()p Fy(E)2405 4540 y Fu(\013)2400 4601 y Fw(0)p Fu(k)2479 4578 y Fz(\))p Fy(v)2561 4540 y Fw(2)2558 4600 y Fu(y)2601 4578 y Fy(P)g Fz(\()p Fy(E)2779 4540 y Fu(\013)2774 4601 y Fw(0)p Fu(k)2853 4578 y Fz(\))p Fx(k)1405 4750 y(\024)82 b Fz(2)p Fx(k)p Fy(P)13 b Fz(\()p Fy(E)1826 4712 y Fu(\013)1821 4773 y Fw(0)p Fu(k)1901 4750 y Fz(\))p Fy(H)2012 4764 y Fu(\013)2061 4750 y Fy(P)g Fz(\()p Fy(E)2239 4712 y Fu(\013)2234 4773 y Fw(0)p Fu(k)2313 4750 y Fz(\))p Fx(k)26 b(\024)f Fz(2)p Fy(E)2632 4712 y Fu(\013)2627 4773 y Fw(0)p Fu(k)2730 4750 y Fy(:)546 b Fz(\(5.4\))265 4983 y(The)30 b(iden)m(tit)m(y)711 5216 y Fy(P)782 5179 y Fu(\013)769 5240 y(k)852 5216 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)1121 5179 y Fu(\013)1116 5240 y Fw(0)p Fu(k)1194 5216 y Fz(\))84 b(=)e([)p Fy(P)1562 5179 y Fu(\013)1549 5240 y(k)1632 5216 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)1901 5179 y Fu(\013)1896 5240 y Fw(0)p Fu(k)1975 5216 y Fz(\)])2035 5179 y Fw(2)2095 5216 y Fz(+)20 b([)p Fy(P)2282 5179 y Fu(\013)2269 5240 y(k)2352 5216 y Fx(\000)g Fy(P)13 b Fz(\()p Fy(E)2621 5179 y Fu(\013)2616 5240 y Fw(0)p Fu(k)2694 5216 y Fz(\)])p Fy(P)g Fz(\()p Fy(E)2932 5179 y Fu(\013)2927 5240 y Fw(0)p Fu(k)3006 5216 y Fz(\))1313 5388 y(+)82 b Fy(P)13 b Fz(\()p Fy(E)1644 5351 y Fu(\013)1639 5411 y Fw(0)p Fu(k)1718 5388 y Fz(\)[)p Fy(P)1849 5351 y Fu(\013)1836 5411 y(k)1919 5388 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)2188 5351 y Fu(\013)2183 5411 y Fw(0)p Fu(k)2261 5388 y Fz(\)])980 b(\(5.5\))1828 5637 y Fs(16)p eop %%Page: 17 17 17 16 bop 265 100 a Fz(implies)650 328 y Fx(j)p Fy(J)734 291 y Fu(\013)725 351 y(k)804 328 y Fx(\000)20 b Fy(J)954 291 y Fu(\013)945 351 y Fw(0)p Fu(k)1024 328 y Fx(j)83 b Fz(=)1286 224 y Fp(\014)1286 278 y(\014)1286 333 y(\014)1316 328 y Fz(T)-8 b(r)15 b Fy(v)1469 342 y Fu(y)1510 328 y Fz([)p Fy(P)1606 291 y Fu(k)1593 351 y(\013)1669 328 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)1938 291 y Fu(\013)1933 351 y Fw(0)p Fu(k)2012 328 y Fz(\)])2072 224 y Fp(\014)2072 278 y(\014)2072 333 y(\014)2128 328 y Fx(\024)2224 251 y Fp(\014)2224 306 y(\014)2254 328 y Fz(T)-8 b(r)15 b Fy(v)2407 342 y Fu(y)2449 328 y Fz([)p Fy(P)2545 291 y Fu(\013)2532 351 y(k)2614 328 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)2883 291 y Fu(\013)2878 351 y Fw(0)p Fu(k)2957 328 y Fz(\)])3017 291 y Fw(2)3057 251 y Fp(\014)3057 306 y(\014)1132 500 y Fz(+)83 b Fx(j)p Fz(T)-8 b(r)15 b Fy(v)1464 514 y Fu(y)1505 500 y Fz([)p Fy(P)1601 462 y Fu(\013)1588 523 y(k)1671 500 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)1940 462 y Fu(\013)1935 523 y Fw(0)p Fu(k)2014 500 y Fz(\)])p Fy(P)g Fz(\()p Fy(E)2252 462 y Fu(\013)2247 523 y Fw(0)p Fu(k)2326 500 y Fz(\))p Fx(j)1132 672 y Fz(+)83 b Fx(j)p Fz(T)-8 b(r)15 b Fy(v)1464 686 y Fu(y)1505 672 y Fy(P)e Fz(\()p Fy(E)1683 634 y Fu(\013)1678 695 y Fw(0)p Fu(k)1757 672 y Fz(\)[)p Fy(P)1888 634 y Fu(\013)1875 695 y(k)1958 672 y Fx(\000)20 b Fy(P)13 b Fz(\()p Fy(E)2227 634 y Fu(\013)2222 695 y Fw(0)p Fu(k)2300 672 y Fz(\)])p Fx(j)42 b Fy(:)849 b Fz(\(5.6\))265 899 y(F)-8 b(rom)31 b(\(5.6\))r(,)f(\(5.3\))j(and)c(\(5.4\))j(w)m(e)f(get)578 1127 y Fx(j)p Fy(J)662 1090 y Fu(\013)653 1150 y(k)732 1127 y Fx(\000)20 b Fy(J)882 1090 y Fu(\013)873 1150 y Fw(0)p Fu(k)951 1127 y Fx(j)83 b(\024)g Fz(2)15 b(\()q Fx(k)p Fy(v)1398 1141 y Fu(y)1440 1127 y Fy(P)1511 1090 y Fu(\013)1498 1150 y(k)1560 1127 y Fx(k)1605 1141 y Fw(1)1666 1127 y Fz(+)k Fx(k)p Fy(v)1845 1141 y Fu(y)1887 1127 y Fy(P)13 b Fz(\()p Fy(E)2065 1090 y Fu(\013)2060 1150 y Fw(0)p Fu(k)2139 1127 y Fz(\))p Fx(k)2219 1141 y Fw(1)2259 1127 y Fz(\))j Fx(k)p Fy(P)2426 1090 y Fu(\013)2413 1150 y(k)2496 1127 y Fx(\000)k Fy(P)13 b Fz(\()p Fy(E)2765 1090 y Fu(\013)2760 1150 y Fw(0)p Fu(k)2838 1127 y Fz(\))p Fx(k)1059 1299 y(\024)83 b Fz(2)1273 1198 y Fp(\020)1328 1299 y Fz(\()p Fy(B)25 b Fz(+)20 b(3)p Fy(V)1646 1313 y Fw(0)1686 1299 y Fz(\))1721 1261 y Fw(1)p Fu(=)p Fw(2)1851 1299 y Fz(+)g(\()p Fy(B)25 b Fz(+)20 b(2)p Fy(V)2260 1313 y Fw(0)2300 1299 y Fz(\))2335 1261 y Fw(1)p Fu(=)p Fw(2)2446 1198 y Fp(\021)2515 1299 y Fx(k)p Fy(P)2631 1261 y Fu(\013)2618 1322 y(k)2701 1299 y Fx(\000)g Fy(P)13 b Fz(\()p Fy(E)2970 1261 y Fu(\013)2965 1322 y Fw(0)p Fu(k)3044 1299 y Fz(\))p Fx(k)26 b Fy(:)126 b Fz(\(5.7\))265 1527 y(Com)m(bining)41 b(this)g(last)i(inequalit)m(y)e(with)h(\(4.1\))j (w)m(e)e(get)h(the)f(result)f(\(2.16\))j(of)e(Theorem)265 1674 y(1.)p 3422 1674 4 62 v 3426 1616 55 4 v 3426 1674 V 3479 1674 4 62 v 401 1892 a(In)30 b(order)g(to)h(pro)m(v)m(e)g(part)f Fy(d)p Fz(\))h(of)g(Theorem)f(1)h(w)m(e)g(need)f(the)g(follo)m(wing)f (lemma.)265 2105 y FA(Lemma)34 b(2.)43 b Fj(Fix)34 b Fy(!)c Fx(2)c Fz(\012)1164 2048 y Fg(0)1164 2132 y Fw(\003)1251 2105 y Fj(the)34 b(set)g(of)g(r)-5 b(e)g(alizations)36 b(in)e Fz(\()p Fy(H)7 b Fz(2\))p Fj(.)46 b(L)-5 b(et)34 b Fy( )2737 2072 y Fu(b)2734 2129 y Fw(1)2773 2105 y Fj(,)g Fy( )2897 2072 y Fu(b)2894 2129 y Fw(2)2967 2105 y Fj(b)-5 b(e)34 b(two)h(eigen-)265 2251 y(states)f(of)e Fy(H)703 2266 y Fu(b)770 2251 y Fj(with)h(eigenvalues)g Fy(E)1516 2218 y Fu(b)1511 2276 y Fw(1)1583 2251 y Fj(and)g Fy(E)1831 2218 y Fu(b)1826 2276 y Fw(2)1866 2251 y Fj(.)42 b(Then)1122 2501 y Fx(j)p Fz(\()p Fy( )1244 2463 y Fu(b)1241 2523 y Fw(1)1281 2501 y Fy(;)15 b(v)1365 2515 y Fu(y)1407 2501 y Fy( )1469 2463 y Fu(b)1466 2523 y Fw(2)1506 2501 y Fz(\))p Fx(j)26 b(\024)f Fz(2)p Fx(j)p Fy(E)1830 2463 y Fu(b)1825 2523 y Fw(1)1885 2501 y Fx(\000)20 b Fy(E)2048 2463 y Fu(b)2043 2523 y Fw(2)2083 2501 y Fx(j)p Fy(L)g Fz(+)g Fy(e)2323 2460 y Fv(\000)2388 2415 y Fu(\026)p Fw(\()p Fu(")p Fw(\))p 2388 2439 130 4 v 2435 2491 a(4)2528 2460 y Fu(L)2605 2501 y Fy(:)671 b Fz(\(5.8\))401 2729 y(F)-8 b(or)41 b Fy( )636 2696 y Fu(b)633 2753 y Fw(1)714 2729 y Fz(=)g Fy( )888 2696 y Fu(b)885 2753 y Fw(2)924 2729 y Fz(,)i Fy(E)1064 2696 y Fu(b)1059 2753 y Fw(1)1139 2729 y Fz(=)e Fy(E)1323 2696 y Fu(b)1318 2753 y Fw(2)1398 2729 y Fz(this)d(sho)m(ws)i(that)h(eigenstates)g(of)f Fy(H)2721 2744 y Fu(b)2794 2729 y Fz(do)g(not)g(carry)g(an)m(y)265 2875 y(appreciable)26 b(curren)m(t.)39 b(The)27 b(main)f(idea)h(of)g (the)h(pro)s(of)e(sk)m(etc)m(hed)j(b)s(elo)m(w)d(is)g(that)i Fy(v)3121 2889 y Fu(y)3190 2875 y Fz(is)e(equal)265 3022 y(to)31 b(the)g(comm)m(utator)h([)p Fx(\000)p Fy(iy)s(;)15 b(H)1337 3037 y Fu(b)1371 3022 y Fz(])31 b(up)e(to)i(a)g(small)e(b)s (oundary)f(term.)265 3240 y Fj(Pr)-5 b(o)g(of.)50 b Fz(The)23 b(w)m(a)m(v)m(efunctions)h Fy( )1372 3207 y Fu(b)1369 3265 y Fw(1)1432 3240 y Fz(and)e Fy( )1663 3207 y Fu(b)1660 3265 y Fw(2)1723 3240 y Fz(are)i(de\014ned)e(on)h Fq(R)9 b Fx(\002)d Fz([)p Fx(\000)2542 3204 y Fu(L)p 2542 3219 48 4 v 2548 3272 a Fw(2)2606 3240 y Fy(;)2656 3204 y Fu(L)p 2656 3219 V 2662 3272 a Fw(2)2714 3240 y Fz(],)25 b(are)f(p)s(erio)s(dic)c(along)265 3387 y Fy(y)34 b Fz(and)c(are)h(t)m (wice)h(di\013eren)m(tiable)d(in)g Fy(y)s Fz(.)42 b(Here)32 b(w)m(e)f(will)d(w)m(ork)j(with)f(p)s(erio)s(dized)e(v)m(ersions)i(of) 265 3534 y(these)h(functions)e(where)h(the)g Fy(y)k Fz(direction)29 b(is)g(in\014nite)f(\(but)i(w)m(e)h(k)m(eep)g(the)g(same)g(notation\).) 265 3681 y(This)e(allo)m(ws)g(us)h(to)h(shift)e(in)m(tegrals)h(o)m(v)m (er)i Fy(y)h Fz(from)d([)p Fx(\000)2137 3645 y Fu(L)p 2137 3660 V 2143 3712 a Fw(2)2195 3681 y Fy(;)2245 3645 y Fu(L)p 2245 3660 V 2251 3712 a Fw(2)2303 3681 y Fz(])h(to)g([)6 b(\026)-51 b Fy(y)2540 3695 y Fw(2)2579 3681 y Fy(;)22 b Fz(\026)-52 b Fy(y)2664 3695 y Fw(2)2724 3681 y Fz(+)19 b Fy(L)p Fz(].)41 b(W)-8 b(e)32 b(ha)m(v)m(e)762 3944 y(\()p Fy( )859 3907 y Fu(b)856 3967 y Fw(1)896 3944 y Fy(;)15 b(v)980 3958 y Fu(y)1021 3944 y Fy( )1083 3907 y Fu(b)1080 3967 y Fw(2)1120 3944 y Fz(\))26 b(=)1277 3820 y Fp(Z)1327 4027 y Fh(R)1394 3944 y Fz(d)p Fy(x)1512 3820 y Fp(Z)1608 3847 y Fw(\026)-40 b Fu(y)1638 3856 y Ft(2)1672 3847 y Fw(+)p Fu(L)1568 4027 y Fw(\026)g Fu(y)1598 4036 y Ft(2)1795 3944 y Fz(d)o Fy(y)s Fz([)p Fy( )1980 3907 y Fu(b)1977 3967 y Fw(1)2017 3944 y Fz(\()p Fd(x)p Fz(\)])2172 3907 y Fv(\003)2212 3944 y Fz(\()p Fx(\000)p Fy(i@)2397 3958 y Fu(y)2459 3944 y Fx(\000)20 b Fy(B)5 b(x)p Fz(\))p Fy( )2773 3907 y Fu(b)2770 3967 y Fw(2)2810 3944 y Fz(\()p Fd(x)p Fz(\))26 b Fy(:)310 b Fz(\(5.9\))265 4192 y(An)30 b(in)m(tegration)g(b)m(y)h(parts)f (yields)548 4442 y Fy(i)p Fz(\()p Fy( )676 4404 y Fu(b)673 4464 y Fw(1)713 4442 y Fy(;)15 b(v)797 4456 y Fu(y)839 4442 y Fy( )901 4404 y Fu(b)898 4464 y Fw(2)938 4442 y Fz(\))83 b(=)1220 4406 y Fw(1)p 1220 4421 36 4 v 1220 4473 a(2)1280 4318 y Fp(Z)1331 4524 y Fh(R)1398 4442 y Fz(d)o Fy(x)1515 4318 y Fp(Z)1611 4344 y Fw(\026)-40 b Fu(y)1641 4353 y Ft(2)1676 4344 y Fw(+)p Fu(L)1571 4524 y Fw(\026)g Fu(y)1601 4533 y Ft(2)1798 4442 y Fz(d)p Fy(y)s Fz([)p Fy( )1984 4404 y Fu(b)1981 4464 y Fw(1)2020 4442 y Fz(\()p Fd(x)q Fz(\)])2176 4404 y Fv(\003)2216 4442 y Fy(y)s Fz(\()p Fx(\000)p Fy(i@)2449 4456 y Fu(y)2511 4442 y Fx(\000)20 b Fy(B)5 b(x)p Fz(\))2763 4404 y Fw(2)2802 4442 y Fy( )2864 4404 y Fu(b)2861 4464 y Fw(2)2901 4442 y Fz(\()p Fd(x)p Fz(\))1056 4713 y Fx(\000)1220 4677 y Fw(1)p 1220 4692 V 1220 4745 a(2)1280 4589 y Fp(Z)1331 4796 y Fh(R)1398 4713 y Fz(d)o Fy(x)1515 4589 y Fp(Z)1611 4616 y Fw(\026)-40 b Fu(y)1641 4625 y Ft(2)1676 4616 y Fw(+)p Fu(L)1571 4796 y Fw(\026)g Fu(y)1601 4805 y Ft(2)1798 4713 y Fz(d)p Fy(y)s Fz([\()p Fx(\000)p Fy(i@)2107 4727 y Fu(y)2169 4713 y Fx(\000)20 b Fy(B)5 b(x)p Fz(\))2421 4676 y Fw(2)2460 4713 y Fy( )2522 4676 y Fu(b)2519 4736 y Fw(1)2559 4713 y Fz(\()p Fd(x)p Fz(\)])2714 4676 y Fv(\003)2754 4713 y Fy(y)s( )2864 4676 y Fu(b)2861 4736 y Fw(2)2901 4713 y Fz(\()p Fd(x)p Fz(\))20 b(+)g Fx(B)53 b Fz(\(5.10\))265 4961 y(where)30 b Fx(B)j Fz(is)c(a)i(b)s(oundary)d (term)j(giv)m(en)f(b)m(y)919 5189 y Fx(B)86 b Fz(=)c Fy(i)1259 5153 y Fu(L)p 1259 5168 48 4 v 1265 5220 a Fw(2)1333 5065 y Fp(Z)1383 5271 y Fh(R)1451 5189 y Fz(d)o Fy(x)p Fz([\()p Fx(\000)p Fy(i@)1763 5203 y Fu(y)1826 5189 y Fx(\000)19 b Fy(B)5 b(x)p Fz(\))p Fy( )2139 5151 y Fu(b)2136 5211 y Fw(1)2176 5189 y Fz(\()p Fy(x;)22 b Fz(\026)-52 b Fy(y)2348 5203 y Fw(2)2388 5189 y Fz(\)])2448 5151 y Fv(\003)2488 5189 y Fy( )2550 5151 y Fu(b)2547 5211 y Fw(2)2586 5189 y Fz(\()p Fy(x;)22 b Fz(\026)-52 b Fy(y)2758 5203 y Fw(2)2798 5189 y Fz(\))1065 5388 y(+)82 b([)p Fy( )1305 5351 y Fu(b)1302 5411 y Fw(1)1342 5388 y Fz(\()p Fy(x;)22 b Fz(\026)-52 b Fy(y)1514 5402 y Fw(2)1554 5388 y Fz(\)])1614 5351 y Fv(\003)1654 5388 y Fz(\()p Fx(\000)p Fy(i@)1839 5402 y Fu(y)1901 5388 y Fx(\000)20 b Fy(B)5 b(x)p Fz(\))p Fy( )2215 5351 y Fu(b)2212 5411 y Fw(2)2252 5388 y Fz(\()p Fy(x;)22 b Fz(\026)-52 b Fy(y)2424 5402 y Fw(2)2463 5388 y Fz(\))26 b Fy(:)706 b Fz(\(5.11\))1828 5637 y Fs(17)p eop %%Page: 18 18 18 17 bop 265 100 a Fz(W)-8 b(e)32 b(can)g(add)f(a)g(p)s(erio)s(dized)e (v)m(ersion)h(of)i Fy(V)1752 114 y Fu(!)1833 100 y Fz(and)2021 65 y Fw(1)p 2021 80 36 4 v 2021 132 a(2)2066 100 y Fy(p)2112 67 y Fw(2)2112 123 y Fu(x)2187 100 y Fz(to)g(the)f(kinetic)g(energy)g (op)s(erator)h(in)265 247 y(b)s(oth)27 b(terms)g(on)g(the)g(righ)m(t)g (hand)f(side)h(of)34 b(\(5.10\))c(and)c(use)h(that)h Fy( )2574 214 y Fu(b)2571 272 y Fw(1)2638 247 y Fz(and)f Fy( )2874 214 y Fu(b)2871 272 y Fw(2)2937 247 y Fz(are)h(eigenfunc-)265 394 y(tions)i(of)g Fy(H)667 409 y Fu(b)731 394 y Fz(to)i(obtain)597 651 y Fy(i)p Fz(\()p Fy( )725 613 y Fu(b)722 673 y Fw(1)762 651 y Fy(;)15 b(v)846 665 y Fu(y)888 651 y Fy( )950 613 y Fu(b)947 673 y Fw(2)986 651 y Fz(\))26 b(=)f(\()p Fy(E)1250 613 y Fu(b)1245 673 y Fw(2)1305 651 y Fx(\000)20 b Fy(E)1468 613 y Fu(b)1463 673 y Fw(1)1503 651 y Fz(\))1553 527 y Fp(Z)1604 733 y Fh(R)1671 651 y Fz(d)p Fy(x)1789 527 y Fp(Z)1885 553 y Fw(\026)-40 b Fu(y)1915 562 y Ft(2)1949 553 y Fw(+)p Fu(L)1844 733 y Fw(\026)g Fu(y)1874 742 y Ft(2)2071 651 y Fz(d)p Fy(y)s(y)s Fz([)p Fy( )2305 613 y Fu(b)2302 673 y Fw(1)2341 651 y Fz(\()p Fd(x)q Fz(\)])2497 613 y Fv(\003)2537 651 y Fy( )2599 613 y Fu(b)2596 673 y Fw(2)2635 651 y Fz(\()p Fd(x)q Fz(\))20 b(+)g Fx(B)28 b Fy(:)265 b Fz(\(5.12\))265 913 y(F)-8 b(rom)31 b Fx(j)p Fy(y)s Fx(j)26 b(\024)e(j)6 b Fz(\026)-51 b Fy(y)793 927 y Fw(2)833 913 y Fx(j)20 b Fz(+)g Fy(L)25 b Fx(\024)g Fz(2)p Fy(L)31 b Fz(and)f(the)g(Sc)m(h)m(w)m(arz)h (inequalit)m(y)e(w)m(e)i(obtain)1215 1151 y Fx(j)p Fz(\()p Fy( )1337 1113 y Fu(b)1334 1173 y Fw(1)1374 1151 y Fy(;)15 b(v)1458 1165 y Fu(y)1500 1151 y Fy( )1562 1113 y Fu(b)1559 1173 y Fw(2)1599 1151 y Fz(\))p Fx(j)26 b(\024)f Fz(2)p Fy(L)p Fx(j)p Fy(E)1985 1113 y Fu(b)1980 1173 y Fw(2)2040 1151 y Fx(\000)20 b Fy(E)2203 1113 y Fu(b)2198 1173 y Fw(1)2238 1151 y Fx(j)g Fz(+)g Fx(jB)s(j)25 b Fy(:)718 b Fz(\(5.13\))265 1388 y(With)30 b(the)g(help)f(of)38 b(\(C.6\))q(,)31 b(\(C.7\))g(in)e(App)s(endix)f(C)i(w)m(e)h(get)1610 1646 y Fx(jB)s(j)25 b(\024)g Fy(e)1886 1605 y Fv(\000)1951 1560 y Fu(\026)p Fw(\()p Fu(")p Fw(\))p 1951 1584 130 4 v 1998 1636 a(4)2091 1605 y Fu(L)3255 1646 y Fz(\(5.14\))265 1883 y(this)k(concludes)h(the)g(pro)s(of)g(of)38 b(\(5.8\))r(.)p 3422 1883 4 62 v 3426 1825 55 4 v 3426 1883 V 3479 1883 4 62 v 265 2105 a Fj(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(1,)f(p)-5 b(art)35 b(d\).)49 b Fz(Let)1633 2082 y(\026)1618 2105 y(\001)35 b(an)h(in)m(terv)-5 b(al)34 b(lik)m(e)h(in)g(part)g Fy(ii)p Fz(\))i(of)e(Prop)s(osition)f(1.)265 2252 y(W)-8 b(e)36 b(consider)e(the)h(maximal)e(set)j(of)f(in)m(terv)-5 b(als)33 b Fx(F)2009 2267 y Fu(k)2085 2252 y Fx(\032)2204 2229 y Fz(\026)2188 2252 y(\001)i(suc)m(h)f(that)i Fx(jF)2800 2267 y Fu(k)2843 2252 y Fx(j)d Fz(=)f Fy(e)3046 2216 y Fv(\000)3148 2189 y Fi(B)p 3111 2201 122 3 v 3111 2242 a Ft(1024)3243 2216 y Fw(\(log)13 b Fu(L)p Fw(\))3448 2192 y Ft(2)265 2398 y Fz(and)35 b(dist)o(\()p Fx(F)694 2413 y Fu(k)737 2398 y Fy(;)15 b Fx(F)842 2413 y Fu(\025)888 2398 y Fz(\))34 b Fx(\025)g Fz(4)p Fy(e)1149 2362 y Fv(\000)1236 2335 y Fi(B)p 1214 2347 92 3 v 1214 2389 a Ft(512)1316 2362 y Fw(\(log)13 b Fu(L)p Fw(\))1521 2339 y Ft(2)1560 2398 y Fz(,)37 b Fy(k)g Fx(6)p Fz(=)c Fy(\025)p Fz(.)56 b(Since)34 b(the)i(n)m(um)m(b)s(er)e(of)i(gaps)f(b)s(et)m(w)m(een)h (the)265 2545 y Fx(F)330 2560 y Fu(k)407 2545 y Fz(in)531 2522 y(\026)516 2545 y(\001)d(is)g(less)g(than)g Fy(e)1195 2482 y Fi(B)p 1158 2494 122 3 v 1158 2535 a Ft(1024)1290 2509 y Fw(\(log)13 b Fu(L)p Fw(\))1495 2486 y Ft(2)1534 2545 y Fx(j)1575 2522 y Fz(\026)1559 2545 y(\001)p Fx(j)34 b Fz(and)f Fx(j)1915 2522 y Fz(\026)1899 2545 y(\001)p Fx(j)e Fy(<)2142 2500 y Fu(d)p Fw(\()p Fu(")p Fw(\))p 2142 2524 124 4 v 2180 2576 a Fu(L)2276 2545 y Fz(,)k(it)e(follo)m(ws)f (from)i(Prop)s(osition)d(2)265 2692 y(that)576 2950 y Fq(P)631 2964 y Fw(\003)684 2950 y Fz(\(\012)785 2964 y Fw(\003)838 2912 y Fv(000)900 2950 y Fz(\))83 b Fx(\021)g Fq(P)1227 2964 y Fw(\003)1296 2795 y Fp( )1368 2950 y Fy(!)28 b Fx(2)d Fz(\012)1605 2964 y Fw(\003)1698 2950 y Fz(:)41 b Fy(\033)s Fz(\()p Fy(H)1930 2965 y Fu(b)1964 2950 y Fz(\))21 b Fx(\\)2116 2927 y Fz(\026)2100 2950 y(\001)k Fx(\032)2297 2864 y Fp([)2329 3061 y Fu(k)2414 2950 y Fx(F)2479 2965 y Fu(k)2522 2795 y Fp(!)1018 3202 y Fx(\025)83 b Fz(1)21 b Fx(\000)f Fz(16)p Fy(c)p Fz(\()p Fy(B)5 b Fz(\))p Fx(k)p Fy(h)p Fx(k)1744 3216 y Fv(1)1821 3202 y Fy(")1863 3165 y Fv(\000)p Fw(2)1958 3202 y Fy(V)2011 3216 y Fw(0)2050 3202 y Fy(L)2112 3165 y Fw(4)2152 3202 y Fy(e)2194 3165 y Fv(\000)2280 3138 y Fi(B)p 2259 3150 92 3 v 2259 3191 a Ft(512)2360 3165 y Fw(\(log)13 b Fu(L)p Fw(\))2565 3141 y Ft(2)2605 3202 y Fy(e)2694 3138 y Fi(B)p 2657 3150 122 3 v 2657 3191 a Ft(1024)2789 3165 y Fw(\(log)g Fu(L)p Fw(\))2994 3141 y Ft(2)3043 3158 y Fu(d)p Fw(\()p Fu(")p Fw(\))p 3043 3181 124 4 v 3081 3234 a Fu(L)1018 3374 y Fz(=)83 b(1)21 b Fx(\000)f Fz(16)p Fy(c)p Fz(\()p Fy(B)5 b Fz(\))p Fx(k)p Fy(h)p Fx(k)1744 3388 y Fv(1)1821 3374 y Fy(")1863 3336 y Fv(\000)p Fw(2)1958 3374 y Fy(V)2011 3388 y Fw(0)2050 3374 y Fy(d)p Fz(\()p Fy(")p Fz(\))p Fy(L)2271 3336 y Fw(3)2312 3374 y Fy(e)2354 3336 y Fv(\000)2456 3309 y Fi(B)p 2419 3321 122 3 v 2419 3363 a Ft(1024)2551 3336 y Fw(\(log)13 b Fu(L)p Fw(\))2756 3313 y Ft(2)2821 3374 y Fy(:)409 b Fz(\(5.15\))265 3612 y(No)m(w)40 b(supp)s(ose)e(that) i Fy( )1097 3627 y Fu(\014)1184 3612 y Fz(is)f(an)g(eigenstate)i(of)e Fy(H)2042 3626 y Fu(!)2132 3612 y Fz(corresp)s(onding)e(to)j Fy(E)2912 3627 y Fu(\014)3000 3612 y Fx(2)3116 3589 y Fz(\026)3101 3612 y(\001.)68 b(F)-8 b(or)40 b(a)265 3758 y(giv)m(en)31 b Fy(!)f Fx(2)c Fz(\012)743 3702 y Fg(000)743 3785 y Fw(\003)836 3758 y Fz(one)32 b(can)f(sho)m(w)g(that)h Fy(E)1662 3773 y Fu(\014)1740 3758 y Fz(is)f(necessarly)f(included)f (in)h(one)h(of)g(the)h(fattened)265 3917 y(in)m(terv)-5 b(als)659 3894 y(~)634 3917 y Fx(F)699 3932 y Fu(k)776 3917 y Fx(\021)34 b(F)946 3932 y Fu(k)1012 3917 y Fz(+)23 b Fy(e)1148 3876 y Fv(\000)1238 3840 y Fu(B)p 1213 3855 106 4 v 1213 3907 a Fw(512)1329 3876 y(\(log)14 b Fu(L)p Fw(\))1535 3852 y Ft(2)1574 3917 y Fz(.)56 b(In)34 b(order)h(to)i(c)m (hec)m(k)g(this)d(it)h(is)f(su\016cien)m(t)h(to)h(adapt)265 4063 y(the)i(estimates)h(\(4.13\))h(to)e(\(4.17\))j(to)d(the)g (di\013erence)g(of)g(pro)5 b(jectors)38 b Fx(k)p Fy(P)13 b Fz(\()2916 4040 y(~)2890 4063 y Fx(F)2955 4078 y Fu(k)2999 4063 y Fz(\))21 b Fx(\000)e Fy(P)3203 4078 y Fu(b)3238 4063 y Fz(\()3298 4040 y(~)3273 4063 y Fx(F)3338 4078 y Fu(k)3382 4063 y Fz(\))p Fx(k)p Fz(.)265 4210 y(The)40 b(main)g(p)s(oin)m(t)g(is)g(to)i(c)m(hec)m(k)g(that)f(with)f(our)g(c)m (hoice)i(of)f(in)m(terv)-5 b(als)40 b(one)h(is)f(allo)m(w)m(ed)h(to)265 4357 y(replace)31 b(the)h(circle)976 4334 y(\026)970 4357 y(\000)g(b)m(y)f(circles)1466 4334 y(\026)1460 4357 y(\000)1517 4372 y Fu(k)1591 4357 y Fz(cen)m(tered)h(at)g(the)g(midp)s (oin)m(t)d(of)i Fx(F)2782 4372 y Fu(k)2857 4357 y Fz(and)f(of)i (diameter)265 4504 y Fy(e)307 4468 y Fv(\000)409 4441 y Fi(B)p 372 4453 122 3 v 372 4494 a Ft(1024)504 4468 y Fw(\(log)13 b Fu(L)p Fw(\))709 4445 y Ft(2)769 4504 y Fz(+)20 b(2)p Fy(e)947 4468 y Fv(\000)1034 4441 y Fi(B)p 1012 4453 92 3 v 1012 4494 a Ft(512)1113 4468 y Fw(\(log)14 b Fu(L)p Fw(\))1319 4445 y Ft(2)1358 4504 y Fz(.)41 b(W)-8 b(e)31 b(do)g(not)f(giv)m(e)h(the)g(details)e(here.)41 b(One)29 b(\014nds)892 4741 y Fx(k)p Fy(P)13 b Fz(\()1069 4718 y(~)1043 4741 y Fx(F)1108 4756 y Fu(k)1152 4741 y Fz(\))21 b Fx(\000)f Fy(P)1357 4756 y Fu(b)1391 4741 y Fz(\()1451 4718 y(~)1426 4741 y Fx(F)1491 4756 y Fu(k)1535 4741 y Fz(\))p Fx(k)26 b(\024)f Fy(")1779 4704 y Fv(\000)p Fw(3)1873 4741 y Fy(C)1945 4704 y Fv(00)1987 4741 y Fz(\()p Fy(B)5 b(;)15 b(V)2189 4755 y Fw(0)2229 4741 y Fz(\))p Fy(Le)2368 4704 y Fv(\000)2470 4677 y Fi(B)p 2433 4689 122 3 v 2433 4730 a Ft(1024)2565 4704 y Fw(\(log)f Fu(L)p Fw(\))2771 4680 y Ft(2)2835 4741 y Fy(:)395 b Fz(\(5.16\))265 4979 y(Therefore)30 b Fy(P)13 b Fz(\()808 4956 y(~)783 4979 y Fx(F)848 4994 y Fu(k)891 4979 y Fz(\))p Fy( )985 4994 y Fu(\014)1058 4979 y Fz(=)25 b Fy( )1213 4994 y Fu(\014)1291 4979 y Fz(for)30 b(some)h Fy(k)j Fz(and)29 b(w)m(e)i(ha)m(v)m(e)401 5216 y Fy(J)451 5231 y Fu(\014)581 5216 y Fz(=)83 b(\()p Fy( )829 5231 y Fu(\014)877 5216 y Fy(;)15 b(v)961 5230 y Fu(y)1002 5216 y Fy( )1061 5231 y Fu(\014)1109 5216 y Fz(\))25 b(=)g(\()p Fy( )1359 5231 y Fu(\014)1407 5216 y Fy(;)15 b(v)1491 5230 y Fu(y)1533 5216 y Fy(P)e Fz(\()1664 5193 y(~)1639 5216 y Fx(F)1704 5231 y Fu(k)1747 5216 y Fz(\))p Fy( )1841 5231 y Fu(\014)1889 5216 y Fz(\))26 b(=)f(\()p Fy(P)2139 5231 y Fu(b)2174 5216 y Fz(\()2234 5193 y(~)2209 5216 y Fx(F)2274 5231 y Fu(k)2317 5216 y Fz(\))p Fy( )2411 5231 y Fu(\014)2459 5216 y Fy(;)15 b(v)2543 5230 y Fu(y)2585 5216 y Fy(P)2643 5231 y Fu(b)2677 5216 y Fz(\()2737 5193 y(~)2712 5216 y Fx(F)2777 5231 y Fu(k)2821 5216 y Fz(\))p Fy( )2915 5231 y Fu(\014)2963 5216 y Fz(\))257 b(\(5.17\))581 5388 y(+)83 b(\([)p Fy(P)13 b Fz(\()926 5365 y(~)901 5388 y Fx(F)966 5403 y Fu(k)1010 5388 y Fz(\))20 b Fx(\000)g Fy(P)1214 5403 y Fu(b)1249 5388 y Fz(\()1309 5365 y(~)1284 5388 y Fx(F)1349 5403 y Fu(k)1392 5388 y Fz(\)])p Fy( )1511 5403 y Fu(\014)1559 5388 y Fy(;)15 b(v)1643 5402 y Fu(y)1685 5388 y Fy(P)1743 5403 y Fu(b)1778 5388 y Fz(\()1838 5365 y(~)1813 5388 y Fx(F)1878 5403 y Fu(k)1921 5388 y Fz(\))p Fy( )2015 5403 y Fu(\014)2063 5388 y Fz(\))21 b(+)e(\()p Fy( )2303 5403 y Fu(\014)2351 5388 y Fy(;)c(v)2435 5402 y Fu(y)2477 5388 y Fz([)p Fy(P)e Fz(\()2633 5365 y(~)2608 5388 y Fx(F)2673 5403 y Fu(k)2717 5388 y Fz(\))20 b Fx(\000)g Fy(P)2921 5403 y Fu(b)2956 5388 y Fz(\()3016 5365 y(~)2991 5388 y Fx(F)3056 5403 y Fu(k)3099 5388 y Fz(\)])p Fy( )3218 5403 y Fu(\014)3266 5388 y Fz(\))26 b Fy(:)1828 5637 y Fs(18)p eop %%Page: 19 19 19 18 bop 265 100 a Fz(T)-8 b(o)39 b(estimate)g(the)f(\014rst)g(term)g (on)g(the)h(righ)m(t)f(hand)f(side)g(of)46 b(\(5.17\))40 b(w)m(e)f(use)f(the)g(sp)s(ectral)265 247 y(decomp)s(osition)29 b(in)g(terms)h(of)h(eigenstates)g(of)g Fy(H)1967 262 y Fu(b)2001 247 y Fz(,)1277 485 y Fy(P)1335 500 y Fu(b)1370 485 y Fz(\()1430 462 y(~)1405 485 y Fx(F)1470 500 y Fu(k)1513 485 y Fz(\))p Fy( )1607 500 y Fu(\014)1680 485 y Fz(=)1824 399 y Fp(X)1776 612 y Fu(E)1832 593 y Fi(b)1828 628 y(\034)1867 612 y Fv(2)1933 595 y Fw(~)1914 612 y Fv(F)1964 624 y Fi(k)2003 485 y Fz(\()p Fy( )2100 448 y Fu(b)2097 508 y(\034)2140 485 y Fy(;)15 b( )2239 500 y Fu(\014)2287 485 y Fz(\))p Fy( )2384 448 y Fu(b)2381 508 y(\034)2450 485 y Fy(:)780 b Fz(\(5.18\))265 800 y(W)-8 b(e)32 b(ha)m(v)m(e)517 1038 y(\()p Fy(P)610 1053 y Fu(b)645 1038 y Fz(\()705 1015 y(~)680 1038 y Fx(F)745 1053 y Fu(k)788 1038 y Fz(\))p Fy( )882 1053 y Fu(\014)930 1038 y Fy(;)15 b(v)1014 1052 y Fu(y)1056 1038 y Fy(P)1114 1053 y Fu(b)1149 1038 y Fz(\()1209 1015 y(~)1184 1038 y Fx(F)1249 1053 y Fu(k)1292 1038 y Fz(\))p Fy( )1386 1053 y Fu(\014)1434 1038 y Fz(\))25 b(=)1694 952 y Fp(X)1590 1165 y Fu(E)1646 1146 y Fi(b)1642 1181 y(\034)1681 1165 y Fu(;E)1757 1146 y Fi(b)1753 1181 y(\033)1794 1165 y Fv(2)1860 1148 y Fw(~)1841 1165 y Fv(F)1891 1177 y Fi(k)1930 1038 y Fz(\()p Fy( )2024 1053 y Fu(\014)2071 1038 y Fy(;)15 b( )2173 1001 y Fu(b)2170 1061 y(\034)2214 1038 y Fz(\)\()p Fy( )2346 1001 y Fu(b)2343 1061 y(\033)2391 1038 y Fy(;)g( )2490 1053 y Fu(\014)2538 1038 y Fz(\)\()p Fy( )2670 1001 y Fu(b)2667 1061 y(\034)2711 1038 y Fy(;)g(v)2795 1052 y Fu(y)2837 1038 y Fy( )2899 1001 y Fu(b)2896 1061 y(\033)2943 1038 y Fz(\))p Fy(:)252 b Fz(\(5.19\))265 1353 y(F)-8 b(rom)31 b(Lemma)f(2)h(and)f(Lemma)g(4)h (in)e(App)s(endix)f(B)j(w)m(e)f(get)657 1591 y Fx(j)p Fz(\()p Fy(P)775 1606 y Fu(b)811 1591 y Fz(\()871 1568 y(~)846 1591 y Fx(F)911 1606 y Fu(k)954 1591 y Fz(\))p Fy( )1048 1606 y Fu(\014)1096 1591 y Fy(;)15 b(v)1180 1605 y Fu(y)1222 1591 y Fy(P)1280 1606 y Fu(b)1314 1591 y Fz(\()1374 1568 y(~)1349 1591 y Fx(F)1414 1606 y Fu(k)1458 1591 y Fz(\))p Fy( )1552 1606 y Fu(\014)1599 1591 y Fz(\))p Fx(j)84 b(\024)f Fz(\(T)-8 b(r)15 b Fy(P)2099 1606 y Fu(b)2134 1591 y Fz(\()p Fx(F)2234 1606 y Fu(k)2277 1591 y Fz(\)\))2347 1554 y Fw(2)2387 1591 y Fz(4)p Fy(Le)2536 1554 y Fv(\000)2638 1527 y Fi(B)p 2602 1539 122 3 v 2602 1580 a Ft(1024)2733 1554 y Fw(\(log)f Fu(L)p Fw(\))2939 1530 y Ft(2)1743 1763 y Fx(\024)83 b Fz(16)p Fy(c)p Fz(\()p Fy(B)5 b Fz(\))2170 1726 y Fw(4)2211 1763 y Fy(")2253 1726 y Fv(\000)p Fw(4)2347 1763 y Fy(V)2421 1726 y Fw(4)2400 1786 y(0)2460 1763 y Fy(L)2522 1726 y Fw(9)2561 1763 y Fy(e)2603 1726 y Fv(\000)2705 1698 y Fi(B)p 2669 1710 V 2669 1752 a Ft(1024)2800 1726 y Fw(\(log)14 b Fu(L)p Fw(\))3006 1702 y Ft(2)3070 1763 y Fy(:)160 b Fz(\(5.20\))265 2001 y(The)39 b(second)h(term)f(on)h(the)g(righ)m(t)f(hand)f(side)h(of) 47 b(\(5.17\))42 b(is)c(estimated)i(b)m(y)g(the)g(Sc)m(h)m(w)m(arz)265 2148 y(inequalit)m(y)554 2386 y(\([)p Fy(P)13 b Fz(\()746 2363 y(~)720 2386 y Fx(F)785 2401 y Fu(k)829 2386 y Fz(\))21 b Fx(\000)f Fy(P)1034 2401 y Fu(b)1069 2386 y Fz(\()1129 2363 y(~)1104 2386 y Fx(F)1169 2401 y Fu(k)1212 2386 y Fz(\)])p Fy( )1331 2401 y Fu(\014)1379 2386 y Fy(;)15 b(v)1463 2400 y Fu(y)1505 2386 y Fy(P)1563 2401 y Fu(b)1598 2386 y Fz(\()1657 2363 y(~)1633 2386 y Fx(F)1698 2401 y Fu(k)1741 2386 y Fz(\))p Fy( )1835 2401 y Fu(\014)1883 2386 y Fz(\))1918 2349 y Fw(2)1983 2386 y Fx(\024)25 b(k)p Fy(v)2168 2400 y Fu(y)2210 2386 y Fy(P)2268 2401 y Fu(b)2302 2386 y Fz(\()2362 2363 y(~)2337 2386 y Fx(F)2402 2401 y Fu(k)2446 2386 y Fz(\))p Fy( )2540 2401 y Fu(\014)2587 2386 y Fx(k)2632 2349 y Fw(2)2672 2386 y Fx(k)p Fy(P)13 b Fz(\()2849 2363 y(~)2823 2386 y Fx(F)2888 2401 y Fu(k)2932 2386 y Fz(\))21 b Fx(\000)f Fy(P)3137 2401 y Fu(b)3171 2386 y Fz(\()3231 2363 y(~)3206 2386 y Fx(F)3271 2401 y Fu(k)3315 2386 y Fz(\))p Fx(k)3395 2349 y Fw(2)401 2558 y Fx(\024)82 b Fz(2\()p Fy(P)692 2573 y Fu(b)728 2558 y Fz(\()788 2535 y(~)763 2558 y Fx(F)828 2573 y Fu(k)871 2558 y Fz(\))p Fy( )965 2573 y Fu(\014)1013 2558 y Fy(;)15 b Fz(\()p Fy(H)1164 2573 y Fu(b)1219 2558 y Fx(\000)20 b Fy(V)1363 2572 y Fu(!)1413 2558 y Fz(\))p Fy(P)1506 2573 y Fu(b)1541 2558 y Fz(\()1601 2535 y(~)1576 2558 y Fx(F)1641 2573 y Fu(k)1685 2558 y Fz(\))p Fy( )1779 2573 y Fu(\014)1826 2558 y Fz(\))p Fx(k)p Fy(P)13 b Fz(\()2038 2535 y(~)2012 2558 y Fx(F)2077 2573 y Fu(k)2122 2558 y Fz(\))20 b Fx(\000)g Fy(P)2326 2573 y Fu(b)2361 2558 y Fz(\()2421 2535 y(~)2396 2558 y Fx(F)2461 2573 y Fu(k)2504 2558 y Fz(\))p Fx(k)2584 2520 y Fw(2)401 2729 y Fx(\024)82 b Fz(\()p Fy(B)26 b Fz(+)19 b(3)p Fy(V)872 2743 y Fw(0)912 2729 y Fz(\))p Fx(k)p Fy(P)13 b Fz(\()1124 2706 y(~)1098 2729 y Fx(F)1163 2744 y Fu(k)1208 2729 y Fz(\))20 b Fx(\000)g Fy(P)1412 2744 y Fu(b)1447 2729 y Fz(\()1507 2706 y(~)1482 2729 y Fx(F)1547 2744 y Fu(k)1590 2729 y Fz(\))p Fx(k)1670 2692 y Fw(2)1736 2729 y Fy(:)1494 b Fz(\(5.21\))265 2968 y(The)30 b(third)e(term)j(is)e(treated)j(in)d(a)h(similar)e(w)m(a)m(y) 421 3206 y(\()p Fy( )515 3221 y Fu(\014)562 3206 y Fy(;)15 b(v)646 3220 y Fu(y)688 3206 y Fz([)p Fy(P)e Fz(\()844 3183 y(~)819 3206 y Fx(F)884 3221 y Fu(k)928 3206 y Fz(\))20 b Fx(\000)g Fy(P)1132 3221 y Fu(b)1167 3206 y Fz(\()1227 3183 y(~)1202 3206 y Fx(F)1267 3221 y Fu(k)1310 3206 y Fz(\)])p Fy( )1429 3221 y Fu(\014)1477 3206 y Fz(\))1512 3168 y Fw(2)1635 3206 y Fx(\024)83 b(k)p Fy(v)1878 3220 y Fu(y)1920 3206 y Fy( )1979 3221 y Fu(\014)2026 3206 y Fx(k)2071 3168 y Fw(2)2111 3206 y Fx(k)p Fy(P)13 b Fz(\()2287 3183 y(~)2262 3206 y Fx(F)2327 3221 y Fu(k)2371 3206 y Fz(\))20 b Fx(\000)g Fy(P)2575 3221 y Fu(b)2610 3206 y Fz(\()2670 3183 y(~)2645 3206 y Fx(F)2710 3221 y Fu(k)2753 3206 y Fz(\))p Fx(k)2833 3168 y Fw(2)1635 3377 y Fx(\024)83 b Fz(2\()p Fy( )1928 3392 y Fu(\014)1976 3377 y Fy(;)15 b Fz(\()p Fy(H)2127 3391 y Fu(!)2198 3377 y Fx(\000)20 b Fy(V)2342 3391 y Fu(!)2392 3377 y Fz(\))p Fy( )2486 3392 y Fu(\014)2534 3377 y Fz(\))p Fx(k)p Fy(P)13 b Fz(\()2746 3354 y(~)2720 3377 y Fx(F)2785 3392 y Fu(k)2829 3377 y Fz(\))21 b Fx(\000)f Fy(P)3034 3392 y Fu(b)3068 3377 y Fz(\()3128 3354 y(~)3103 3377 y Fx(F)3168 3392 y Fu(k)3212 3377 y Fz(\))p Fx(k)3292 3340 y Fw(2)1635 3549 y Fx(\024)83 b Fz(\()p Fy(B)25 b Fz(+)20 b(3)p Fy(V)2107 3563 y Fw(0)2147 3549 y Fz(\))p Fx(k)p Fy(P)13 b Fz(\()2358 3526 y(~)2333 3549 y Fx(F)2398 3564 y Fu(k)2442 3549 y Fz(\))20 b Fx(\000)g Fy(P)2646 3564 y Fu(b)2681 3549 y Fz(\()2741 3526 y(~)2716 3549 y Fx(F)2781 3564 y Fu(k)2824 3549 y Fz(\))p Fx(k)2904 3512 y Fw(2)2970 3549 y Fy(:)260 b Fz(\(5.22\))265 3787 y(The)27 b(last)h(estimate)h(\(2.18\))h(of)e (Theorem)f(1)h(then)g(follo)m(ws)f(from)g(\(5.16\))s(,)h(\(5.20\))r(,)h (\(5.21\))h(and)265 3934 y(\(5.22\))r(.)p 3422 3934 4 62 v 3426 3876 55 4 v 3426 3934 V 3479 3934 4 62 v 401 4155 a Fj(R)-5 b(emark)p Fz(.)64 b(The)37 b(set)1146 4132 y(^)1135 4155 y(\012)1201 4169 y Fw(\003)1292 4155 y Fz(in)f(Theorem)h(1)h(ma)m(y)g(b)s(e)e(tak)m(en)j(equal)e(to)h(\012) 2899 4099 y Fg(0)2899 4182 y Fw(\003)2977 4155 y Fx(\\)24 b Fz(\012)3128 4169 y Fw(\003)3181 4122 y Fv(00)3248 4155 y Fx(\\)h Fz(\012)3400 4099 y Fg(000)3400 4182 y Fw(\003)3462 4155 y Fz(.)265 4302 y(This)k(set)i(has)f(a)g(probabilit)m (y)e(larger)i(than)g(1)21 b Fx(\000)f Fz(3)p Fy(L)2051 4269 y Fv(\000)p Fu(s)2173 4302 y Fz(with)29 b Fy(s)c Fz(=)g(min)o(\()p Fy(\022)s(;)15 b(p)20 b Fx(\000)g Fz(6\).)265 4677 y Fr(A)161 b(Resolv)l(en)l(t)53 b(of)g(the)g(Landau)h(Hamiltonian) 265 4913 y Fz(The)22 b(k)m(ernel)g Fy(R)773 4927 y Fw(0)812 4913 y Fz(\()p Fd(x)q Fy(;)15 b Fd(x)1008 4880 y Fv(0)1031 4913 y Fz(;)g Fy(z)t Fz(\))24 b(of)e(the)h(resolv)m(en)m(t)g Fy(R)1863 4927 y Fw(0)1902 4913 y Fz(\()p Fy(z)t Fz(\))k(=)e(\()p Fy(z)8 b Fx(\000)t Fy(H)2377 4927 y Fw(0)2417 4913 y Fz(\))2452 4880 y Fv(\000)p Fw(1)2569 4913 y Fz(with)21 b(p)s(erio)s(dic)f(b)s(oundary)265 5060 y(conditions)42 b(along)i Fy(y)i Fz(can)e(b)s(e)f(expressed)g(in)f(term)i(of)g(the)g(k) m(ernel)f Fy(R)2786 5027 y Fv(1)2785 5085 y Fw(0)2860 5060 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)3056 5027 y Fv(0)3079 5060 y Fz(;)g Fy(z)t Fz(\))45 b(of)e(the)265 5207 y(resolv)m(en)m(t)30 b(of)g(the)g(t)m(w)m(o)g(dimensional)d(Landau)i(Hamiltonian)e (de\014ned)i(on)g(the)h(whole)e(plane)265 5354 y Fq(R)325 5321 y Fw(2)370 5354 y Fz(.)85 b(Since)44 b(the)i(sp)s(ectrum)e(and)g (the)h(eigenfunctions)f(of)h Fy(H)2481 5368 y Fw(0)2566 5354 y Fz(are)g(exactly)h(kno)m(wn,)j(b)m(y)1828 5637 y Fs(19)p eop %%Page: 20 20 20 19 bop 265 100 a Fz(writing)27 b(do)m(wn)i(the)g(sp)s(ectral)f (decomp)s(osition)g(of)h Fy(R)2080 114 y Fw(0)2119 100 y Fz(\()p Fd(x)q Fy(;)15 b Fd(x)2315 67 y Fv(0)2338 100 y Fz(;)g Fy(z)t Fz(\))30 b(and)f(applying)e(the)i(P)m(oisson)265 247 y(summation)g(form)m(ula)h(w)m(e)h(get)g(for)f Fy(z)g Fx(2)25 b Fy(\032)p Fz(\()p Fy(H)1809 261 y Fw(0)1848 247 y Fz(\))1046 476 y Fy(R)1115 490 y Fw(0)1154 476 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)1350 438 y Fv(0)1373 476 y Fz(;)g Fy(z)t Fz(\))26 b(=)1627 389 y Fp(X)1616 587 y Fu(m)p Fv(2)p Fh(Z)1785 476 y Fy(R)1855 438 y Fv(1)1854 498 y Fw(0)1929 476 y Fz(\()p Fy(x)15 b(y)24 b Fx(\000)c Fy(mL;)15 b(x)2425 438 y Fv(0)2463 476 y Fy(y)2511 438 y Fv(0)2535 476 y Fz(;)g Fy(z)t Fz(\))26 b Fy(:)571 b Fz(\(A.1\))265 760 y(The)30 b(form)m(ula)f(for)i Fy(R)995 727 y Fv(1)994 784 y Fw(0)1069 760 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)1265 727 y Fv(0)1288 760 y Fz(;)g Fy(z)t Fz(\))31 b(is)f(\(see)h(for)f(example)g([DMP4)r(]\))508 988 y Fy(R)578 951 y Fv(1)577 1011 y Fw(0)653 988 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)848 951 y Fv(0)872 988 y Fz(;)g Fy(z)t Fz(\))26 b(=)1136 952 y Fu(B)p 1125 967 79 4 v 1125 1019 a Fw(2)p Fu(\031)1213 988 y Fz(\000\()p Fy(\013)1363 1002 y Fu(z)1403 988 y Fz(\))p Fy(U)1525 914 y Fp(\000)1567 988 y Fy(\013)1625 1002 y Fu(z)1665 988 y Fy(;)15 b Fz(1;)1801 952 y Fu(B)p 1801 967 57 4 v 1812 1019 a Fw(2)1868 988 y Fx(j)p Fd(x)20 b Fx(\000)g Fd(x)2124 951 y Fv(0)2147 988 y Fx(j)2172 951 y Fw(2)2212 914 y Fp(\001)2269 988 y Fy(e)2311 951 y Fv(\000)2376 923 y Fi(B)p 2376 935 49 3 v 2385 977 a Ft(4)2434 951 y Fv(j)p Fa(x)p Fv(\000)p Fa(x)2602 927 y Fg(0)2624 951 y Fv(j)2644 927 y Ft(2)2682 988 y Fy(M)10 b Fz(\()p Fd(x)q Fy(;)15 b Fd(x)2976 951 y Fv(0)2999 988 y Fz(\))244 b(\(A.2\))265 1217 y(where)30 b Fy(\013)586 1231 y Fu(z)651 1217 y Fz(=)25 b(\()792 1181 y Fw(1)p 792 1196 36 4 v 792 1248 a(2)858 1217 y Fx(\000)969 1181 y Fu(z)p 959 1196 57 4 v 959 1248 a(B)1025 1217 y Fz(\))31 b(and)1152 1445 y Fy(M)10 b Fz(\()p Fd(x)p Fy(;)15 b Fd(x)1445 1407 y Fv(0)1469 1445 y Fz(\))25 b(=)g(exp)1779 1371 y Fp(\000)1836 1409 y Fu(i)p 1831 1424 36 4 v 1831 1476 a Fw(2)1876 1445 y Fy(B)5 b Fz(\()p Fy(x)20 b Fz(+)g Fy(x)2200 1407 y Fv(0)2223 1445 y Fz(\)\()p Fy(y)k Fx(\000)c Fy(y)2501 1407 y Fv(0)2524 1445 y Fz(\))2559 1371 y Fp(\001)3278 1445 y Fz(\(A.3\))265 1673 y(is)36 b(the)i(phase)f(factor)h(in)e(the)i(Landau)e(gauge.)63 b(In)37 b(\(A.2\))i(the)e(Landau)g(lev)m(els)g(app)s(ear)f(as)265 1820 y(simple)27 b(p)s(oles)g(of)i(the)f(Euler)g(\000)g(function)f(and) h Fy(U)10 b Fz(\()p Fx(\000)p Fy(\025;)15 b(b)p Fz(;)g Fy(\032)p Fz(\))30 b(is)d(the)i(logarithmic)e(solution)g(of)265 1967 y(the)k(Kummer)e(equation)h(\(see)h(eqns.)41 b(\(13.1.1\))33 b(and)d(\(13.1.6\))j(of)e([AS)o(]\))1290 2217 y Fy(\032)1347 2156 y(d)1394 2123 y Fw(2)1434 2156 y Fy(U)p 1347 2196 159 4 v 1359 2279 a(d\032)1453 2253 y Fw(2)1536 2217 y Fz(+)20 b(\()p Fy(b)h Fx(\000)e Fy(\032)p Fz(\))1904 2156 y Fy(dU)p 1904 2196 120 4 v 1916 2279 a(d\032)2054 2217 y Fz(+)h Fy(\025\032)25 b Fz(=)g(0)h Fy(:)816 b Fz(\(A.4\))265 2468 y FA(Lemma)33 b(3.)42 b Fj(If)33 b Fx(jI)7 b Fy(m)15 b(z)t Fx(j)26 b(\024)g Fz(1)p Fj(,)34 b Fx(R)p Fy(e)15 b(z)31 b Fx(2)1641 2395 y Fp(\003)1689 2432 y Fw(1)p 1689 2447 36 4 v 1689 2500 a(2)1735 2468 y Fy(B)5 b(;)1858 2432 y Fw(3)p 1858 2447 V 1858 2500 a(2)1904 2468 y Fy(B)1978 2395 y Fp(\002)2048 2468 y Fj(and)2235 2432 y Fu(B)p 2235 2447 57 4 v 2246 2500 a Fw(2)2302 2468 y Fx(j)p Fy(x)20 b Fx(\000)g Fy(x)2542 2435 y Fv(0)2565 2468 y Fx(j)2590 2435 y Fw(2)2655 2468 y Fy(>)25 b Fz(1)33 b Fj(then,)h(for)g Fy(L)f Fj(lar)-5 b(ge)265 2615 y(enough,)33 b(ther)-5 b(e)33 b(exists)g Fy(C)1142 2629 y Fu(n)1189 2615 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))p Fj(,)34 b Fy(n)24 b Fz(=)h(0)p Fy(;)15 b Fz(1)34 b Fj(indep)-5 b(endent)34 b(of)f Fy(L)g Fj(such)f(that)1078 2843 y Fx(j)p Fy(@)1156 2806 y Fu(n)1151 2866 y(x)1204 2843 y Fy(R)1273 2857 y Fw(0)1312 2843 y Fz(\()p Fd(x)q Fy(;)15 b Fd(x)1508 2806 y Fv(0)1531 2843 y Fz(;)g Fy(z)t Fz(\))p Fx(j)85 b(\024)d Fy(C)1980 2857 y Fu(n)2027 2843 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))p Fy(e)2299 2806 y Fv(\000)2364 2779 y Fi(B)p 2365 2791 49 3 v 2374 2832 a Ft(8)2424 2806 y Fw(\()p Fu(x)p Fv(\000)p Fu(x)2586 2782 y Fg(0)2608 2806 y Fw(\))2635 2782 y Ft(2)3278 2843 y Fz(\(A.5\))265 3072 y Fj(wher)-5 b(e)26 b Fy(C)579 3086 y Fu(n)626 3072 y Fz(\()p Fy(z)t(;)15 b(B)5 b Fz(\))26 b(=)f Fy(C)1043 3086 y Fu(n)1090 3072 y Fy(B)1164 3036 y Fw(1+)1264 3009 y Fi(n)p 1264 3021 39 3 v 1268 3062 a Ft(2)1332 3072 y Fz(dist)o(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)1766 3086 y Fw(0)1806 3072 y Fz(\)\))1876 3039 y Fv(\000)p Fw(1)1996 3072 y Fj(with)27 b Fy(C)2252 3086 y Fu(n)2324 3072 y Fj(a)e(numeric)-5 b(al)27 b(p)-5 b(ositive)26 b(c)-5 b(onstant.)401 3285 y Fz(F)d(or)40 b(our)f(purp)s(oses)e(w)m(e)j(need)f(only)f(deca)m(y)j(in)d(the)h Fy(x)g Fz(direction)f(as)h(pro)m(vided)f(b)m(y)h(the)265 3432 y(lemma)g(but)f(in)g(fact)j(there)e(is)g(also)g(a)g(Gaussian)g (deca)m(y)h(in)e(the)i Fy(y)i Fz(direction)c(as)h(long)g(as)265 3579 y Fx(j)p Fy(y)26 b Fx(\000)c Fy(y)502 3546 y Fv(0)525 3579 y Fx(j)32 b Fy(<)694 3543 y Fu(L)p 694 3558 48 4 v 700 3610 a Fw(2)752 3579 y Fz(.)51 b(One)34 b(can)g(also)g(pro)m(v)m (e)h(similar)c(estimates)k(when)d Fx(R)p Fy(e)15 b(z)39 b Fz(is)33 b(b)s(et)m(w)m(een)i(higher)265 3726 y(Landau)27 b(lev)m(els)h(but)f(the)i(constan)m(t)g(is)e(not)i(uniform)c(with)i (resp)s(ect)h(to)h Fy(\027)6 b Fz(.)40 b(Finally)26 b(w)m(e)j(p)s(oin)m (t)265 3872 y(out)f(that)f(this)g(estimate)h(do)s(es)f(not)g(hold)f (for)1858 3837 y Fu(B)p 1858 3852 57 4 v 1869 3904 a Fw(2)1925 3872 y Fx(j)p Fd(x)14 b Fx(\000)g Fd(x)2169 3839 y Fv(0)2192 3872 y Fx(j)2217 3839 y Fw(2)2282 3872 y Fy(<)25 b Fz(1)j(b)s(ecause)f(of)g(the)h(logarithmic)265 4019 y(singularit)m(y)g(in)h(the)i(Kummer)e(function)g(for)h Fy(\032)25 b Fx(!)h Fz(0)k(\(see)i(also)e(App)s(endix)e(C\).)265 4238 y Fj(Pr)-5 b(o)g(of.)50 b Fz(The)30 b(pro)s(of)g(relies)f(on)h (the)h(estimate)g(\(6.10\))h(of)f([DMP4)q(])g(whic)m(h)e(w)m(e)i(state) h(here)e(for)265 4384 y(con)m(v)m(enience.)42 b(F)-8 b(or)31 b Fy(\025)25 b Fz(=)g Fy(x)20 b Fz(+)g Fy(iy)s Fz(,)31 b Fy(N)f Fx(\000)20 b Fz(1)26 b Fy(<)f(x)g(<)g(N)40 b Fz(\()p Fy(N)c Fx(\025)25 b Fz(1\),)31 b Fy(b)25 b Fx(2)g Fq(N)43 b Fz(and)30 b Fy(\032)25 b(>)g Fz(1)909 4613 y Fx(j)p Fy(U)10 b Fz(\()p Fx(\000)p Fy(\025;)15 b(b)p Fz(;)g Fy(\032)p Fz(\))p Fx(j)85 b(\024)e Fz(2)1675 4575 y Fu(b)p Fw(+)p Fu(N)7 b Fv(\000)p Fw(1)1918 4613 y Fy(\032)1965 4575 y Fu(x)2009 4613 y Fz(\()p Fy(b)20 b Fz(+)g Fy(N)30 b Fz(+)20 b Fx(j)p Fy(y)s Fx(j)p Fz(\))2521 4575 y Fu(N)2600 4568 y Fv(j)p Fw(\000\()p Fv(\000)p Fu(x)p Fw(\))p Fv(j)p 2599 4592 235 4 v 2599 4645 a(j)p Fw(\000\()p Fv(\000)p Fu(\025)p Fw(\))p Fv(j)1476 4784 y Fz(+)83 b Fy(e)1672 4747 y Fv(\000)p Fw(\()p Fu(\032)p Fv(\000)p Fw(2\))1912 4784 y Fz(\()p Fy(\032)21 b Fz(+)f(1)h(+)e Fx(j)p Fy(y)s Fx(j)p Fz(\))2395 4747 y Fu(N)2511 4740 y Fw(\()p Fu(b)p Fw(+)p Fu(N)7 b Fw(\)!)p 2473 4764 298 4 v 2473 4817 a Fv(j)p Fw(\000\()p Fu(N)g Fv(\000)p Fu(\025)p Fw(\))p Fv(j)2806 4784 y Fy(:)447 b Fz(\(A.6\))265 5013 y(Using)20 b(this)f(estimate)j(for)e Fy(N)35 b Fz(=)25 b(1,)e Fx(j)p Fy(y)s Fx(j)j Fy(<)f Fz(1)c(and)f Fy(b)25 b Fz(=)g Fy(n)20 b Fz(together)i(with)d(\000\(1)q Fx(\000)q Fy(\025)p Fz(\))26 b(=)f Fx(\000)p Fy(\025)p Fz(\000\()p Fx(\000)p Fy(\025)p Fz(\))265 5160 y(w)m(e)31 b(ha)m(v)m(e)g(\()p Fy(C)715 5127 y Fv(0)708 5182 y Fu(n)786 5160 y Fz(a)g(n)m(umerical)e (constan)m(t\))884 5388 y Fx(j)p Fz(\000\()p Fx(\000)p Fy(\025)p Fz(\))p Fx(jj)p Fy(U)10 b Fz(\()p Fx(\000)p Fy(\025;)15 b(n)22 b Fz(+)e(1;)15 b Fy(\032)p Fz(\))p Fx(j)26 b(\024)f Fy(C)2035 5351 y Fv(0)2028 5411 y Fu(n)2075 5388 y Fy(\032)2137 5314 y Fp(\010)2190 5388 y Fz(\000\()p Fx(\000)p Fy(x)p Fz(\))c(+)f Fx(j)p Fy(\025)p Fx(j)2655 5351 y Fv(\000)p Fw(1)2750 5314 y Fp(\011)2843 5388 y Fy(:)410 b Fz(\(A.7\))1828 5637 y Fs(20)p eop %%Page: 21 21 21 20 bop 265 100 a Fz(F)-8 b(rom)41 b(\(A.7\))h(if)d Fx(jI)7 b Fy(m)15 b(z)t Fx(j)42 b(\024)f Fz(1,)i Fx(R)p Fy(e)15 b(z)47 b Fx(2)p Fz(])1691 65 y Fw(1)p 1691 80 36 4 v 1691 132 a(2)1736 100 y Fy(B)5 b(;)1860 65 y Fw(3)p 1860 80 V 1860 132 a(2)1905 100 y Fy(B)g Fz([)40 b(and)2241 65 y Fu(B)p 2241 80 57 4 v 2252 132 a Fw(2)2307 100 y Fx(j)p Fd(x)27 b Fx(\000)g Fd(x)2577 67 y Fv(0)2600 100 y Fx(j)2625 67 y Fw(2)2706 100 y Fy(>)42 b Fz(1)f(w)m(e)f(deduce)g(the) 265 247 y(estimate)31 b(\()p Fy(C)736 214 y Fv(00)729 270 y Fu(n)809 247 y Fz(a)f(n)m(umerical)f(constan)m(t\))456 469 y Fx(j)p Fz(\000\()p Fy(\013)631 483 y Fu(z)672 469 y Fz(\))p Fy(U)794 395 y Fp(\000)836 469 y Fy(\013)894 483 y Fu(z)934 469 y Fy(;)15 b(n)20 b Fz(+)g(1;)1236 433 y Fu(B)p 1236 448 V 1247 500 a Fw(2)1302 469 y Fx(j)p Fd(x)h Fx(\000)f Fd(x)1558 431 y Fv(0)1582 469 y Fx(j)1607 431 y Fw(2)1646 395 y Fp(\001)1703 469 y Fx(j)26 b(\024)f Fy(B)5 b(C)1996 431 y Fv(00)1989 491 y Fu(n)2052 469 y Fz(dist)o(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)2486 483 y Fw(0)2527 469 y Fz(\)\))2597 431 y Fv(\000)p Fw(1)2692 469 y Fx(j)p Fd(x)20 b Fx(\000)g Fd(x)2948 431 y Fv(0)2971 469 y Fx(j)2996 431 y Fw(2)3061 469 y Fy(:)192 b Fz(\(A.8\))265 691 y(F)-8 b(rom)31 b(\(A.8\))h(for)e Fy(n)25 b Fz(=)g(0)30 b(and)g(\(A.1\))i(w)m(e)f(get)416 913 y Fx(j)p Fy(R)510 927 y Fw(0)550 913 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)746 875 y Fv(0)769 913 y Fz(;)g Fy(z)t Fz(\))p Fx(j)26 b(\024)f Fz(2)p Fy(B)5 b(C)1228 875 y Fv(00)1221 935 y Fw(0)1285 913 y Fz(dist)o(\()p Fy(z)t(;)15 b(\033)s Fz(\()p Fy(H)1719 927 y Fw(0)1760 913 y Fz(\)\))1830 875 y Fv(\000)p Fw(1)1925 913 y Fy(e)1967 875 y Fv(\000)2032 848 y Fi(B)p 2032 860 49 3 v 2041 901 a Ft(8)2091 875 y Fw(\()p Fu(x)p Fv(\000)p Fu(x)2253 852 y Fg(0)2275 875 y Fw(\))2302 852 y Ft(2)2367 826 y Fp(X)2356 1023 y Fu(m)p Fv(2)p Fh(Z)2525 913 y Fy(e)2567 875 y Fv(\000)2632 848 y Fi(B)p 2632 860 V 2641 901 a Ft(8)2691 875 y Fw(\()p Fu(y)r Fv(\000)p Fu(y)2847 852 y Fg(0)2870 875 y Fv(\000)p Fu(mL)p Fw(\))3062 852 y Ft(2)3278 913 y Fz(\(A.9\))265 1190 y(since)30 b Fx(j)p Fy(y)23 b Fx(\000)d Fy(y)720 1157 y Fv(0)744 1190 y Fx(j)25 b Fy(<)h(L)k Fz(the)h(last)f(sum)g(can)h(b)s (e)f(b)s(ounded)e(b)m(y)j(a)g(constan)m(t,)h(whic)m(h)d(yields)g (\(A.5\))265 1337 y(for)h Fy(n)25 b Fz(=)g(0.)401 1483 y(T)-8 b(o)31 b(estimate)g(the)g(\014rst)f(deriv)-5 b(ativ)m(e)29 b(it)h(is)g(con)m(v)m(enien)m(t)h(to)g(use)f(the)h(relation)f([AS])1075 1658 y(d)p Fy(U)10 b Fz(\()p Fx(\000)p Fy(\025;)15 b Fz(1;)g Fy(\032)p Fz(\))p 1075 1698 491 4 v 1272 1782 a(d)o Fy(\032)1601 1719 y Fz(=)25 b Fy(U)10 b Fz(\()p Fx(\000)p Fy(\025;)15 b Fz(1;)g Fy(\032)p Fz(\))22 b Fx(\000)e Fy(U)10 b Fz(\()p Fx(\000)p Fy(\025;)15 b Fz(2;)g Fy(\032)p Fz(\))547 b(\(A.10\))265 1941 y(whic)m(h)29 b(yields)293 2163 y Fy(@)341 2177 y Fu(x)385 2163 y Fy(R)455 2125 y Fv(1)454 2185 y Fw(0)529 2163 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)725 2125 y Fv(0)748 2163 y Fz(;)g Fy(z)t Fz(\))84 b(=)1117 2127 y Fu(B)p 1117 2142 57 4 v 1128 2194 a Fw(2)1198 2089 y Fp(\002)1236 2163 y Fz(\()p Fy(x)21 b Fx(\000)f Fy(x)1487 2125 y Fv(0)1510 2163 y Fz(\))h(+)e Fy(i)p Fz(\()p Fy(y)24 b Fx(\000)c Fy(y)1930 2125 y Fv(0)1953 2163 y Fz(\))1988 2089 y Fp(\003)2042 2163 y Fy(R)2112 2125 y Fv(1)2111 2185 y Fw(0)2186 2163 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)2382 2125 y Fv(0)2405 2163 y Fz(;)g Fy(z)t Fz(\))707 b(\(A.11\))953 2334 y Fx(\000)83 b Fy(B)5 b Fz(\()p Fy(x)20 b Fx(\000)g Fy(x)1431 2297 y Fv(0)1454 2334 y Fz(\))1510 2299 y Fu(B)p 1499 2314 79 4 v 1499 2366 a Fw(2)p Fu(\031)1588 2334 y Fz(\000\()p Fy(\013)1738 2348 y Fu(z)1778 2334 y Fz(\))p Fy(U)1900 2261 y Fp(\000)1942 2334 y Fy(\013)2000 2348 y Fu(z)2040 2334 y Fy(;)15 b Fz(2;)2176 2299 y Fu(B)p 2176 2314 57 4 v 2187 2366 a Fw(2)2242 2334 y Fx(j)p Fd(x)21 b Fx(\000)f Fd(x)2499 2297 y Fv(0)2522 2334 y Fx(j)2547 2297 y Fw(2)2587 2261 y Fp(\001)2643 2334 y Fy(e)2685 2297 y Fv(\000)2750 2270 y Fi(B)p 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2716 y Fg(0)2803 2739 y Fw(\))2830 2716 y Ft(2)2865 2739 y Fw(+\()p Fu(y)r Fv(\000)p Fu(y)3076 2716 y Fg(0)3100 2739 y Fv(\000)p Fu(mL)p Fw(\))3292 2716 y Ft(2)3327 2745 y Fz(])3233 2925 y(\(A.12\))265 3072 y(the)31 b(result)e(\(A.5\))j (for)e Fy(n)25 b Fz(=)g(1)30 b(then)h(follo)m(ws)e(from)h(\(A.12\))i (and)e(\(A.1\))r(.)p 3422 3072 4 62 v 3426 3014 55 4 v 3426 3072 V 3479 3072 4 62 v 265 3443 a Fr(B)161 b(Bounds)58 b(on)i(the)f(Num)l(b)t(er)g(of)h(Eigen)l(v)-9 b(alues)60 b(in)265 3681 y(Small)55 b(In)l(terv)-9 b(als)265 3918 y Fz(W)h(e)36 b(\014rst)e(pro)m(v)m(e)i(a)f(deterministic)e(Lemma)i(on) g(the)g(maximal)f(n)m(um)m(b)s(er)f(of)j(eigen)m(v)-5 b(alues)34 b(of)265 4064 y Fy(H)341 4079 y Fu(b)406 4064 y Fz(b)s(elonging)c(to)i(energy)g(in)m(terv)-5 b(als)30 b Fy(I)39 b Fz(con)m(tained)31 b(in)g(\001)2263 4078 y Fu(")2299 4064 y Fz(.)44 b(Then)30 b(w)m(e)i(sk)m(etc)m(h)h(the)f (pro)s(of)e(of)265 4211 y(Prop)s(osition)24 b(2.)40 b(The)26 b(ideas)f(in)g(this)g(app)s(endix)f(come)j(from)f(the)g(metho)s(d)g (used)f(b)m(y)h(Com)m(b)s(es)265 4358 y(and)38 b(Hislop)f(to)j(obtain)e (the)g(W)-8 b(egner)40 b(estimate)g(whic)m(h)d(giv)m(es)i(the)g(exp)s (ected)g(n)m(um)m(b)s(er)e(of)265 4505 y(eigen)m(v)-5 b(alues)33 b(in)f Fy(I)7 b Fz(.)49 b(Since)32 b(Lemma)h(4)h(do)s(es)e (not)i(app)s(ear)e(in)g([CH])h(and)g(w)m(e)h(need)e(to)i(adapt)265 4652 y(the)d(tec)m(hnique)f(to)h(our)f(geometry)h(w)m(e)g(giv)m(e)g (some)g(details)e(b)s(elo)m(w.)401 4798 y(W)-8 b(e)39 b(b)s(egin)d(with)h(some)h(preliminary)c(observ)-5 b(ations)37 b(on)h(the)g(k)m(ernel)f Fy(P)2919 4812 y Fw(0)2959 4798 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)3154 4765 y Fv(0)3178 4798 y Fz(\))37 b(of)h(the)265 4945 y(pro)5 b(jector)34 b(on)m(to)g(the)f(\014rst)g(Landau)f(lev)m(el)h(with)e(p)s(erio)s(dic)g (b)s(oundary)g(conditions)g(along)i Fy(y)s Fz(.)265 5092 y(Using)d(the)g(sp)s(ectral)g(decomp)s(osition)f(and)h(the)g(P)m (oisson)g(summation)f(form)m(ula)h(one)h(gets)1096 5314 y Fy(P)1154 5328 y Fw(0)1193 5314 y Fz(\()p Fy(x)15 b(y)s(;)g(x)1435 5276 y Fv(0)1474 5314 y Fy(y)1522 5276 y Fv(0)1546 5314 y Fz(\))25 b(=)1713 5227 y Fp(X)1702 5425 y Fu(m)p Fv(2)p Fh(Z)1871 5314 y Fy(P)1942 5276 y Fv(1)1929 5336 y Fw(0)2017 5314 y Fz(\()p Fy(x)15 b(y)23 b Fx(\000)d Fy(mL;)15 b(x)2512 5276 y Fv(0)2551 5314 y Fy(y)2599 5276 y Fv(0)2622 5314 y Fz(\))625 b(\(B.1\))1828 5637 y Fs(21)p eop %%Page: 22 22 22 21 bop 265 100 a Fz(where)1081 247 y Fy(P)1152 210 y Fv(1)1139 270 y Fw(0)1227 247 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)1422 210 y Fv(0)1446 247 y Fz(\))25 b(=)1623 211 y Fu(B)p 1612 226 79 4 v 1612 279 a Fw(2)p Fu(\031)1700 247 y Fy(e)1742 210 y Fv(\000)1807 183 y Fi(B)p 1808 195 49 3 v 1817 236 a Ft(4)1866 210 y Fv(j)p Fa(x)p Fv(\000)p Fa(x)2033 186 y Fg(0)2056 210 y Fv(j)2076 186 y Ft(2)2114 247 y Fy(e)2156 210 y Fu(i)2190 183 y Fi(B)p 2190 195 V 2200 236 a Ft(2)2249 210 y Fw(\()p Fu(x)p Fw(+)p Fu(x)2411 186 y Fg(0)2433 210 y Fw(\)\()p Fu(y)r Fv(\000)p Fu(y)2616 186 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Fz(\)\()p Fy(H)2012 4854 y Fu(b)2067 4839 y Fx(\000)2168 4804 y Fu(B)p 2168 4819 V 2179 4871 a Fw(2)2234 4839 y Fz(\))p Fy(P)2327 4854 y Fu(b)2362 4839 y Fz(\()p Fy(I)g Fz(\)\()p Fy(H)2590 4854 y Fu(b)2646 4839 y Fx(\000)2747 4804 y Fu(B)p 2747 4819 V 2758 4871 a Fw(2)2813 4839 y Fz(\))p Fy(P)2906 4854 y Fu(b)2941 4839 y Fz(\()p Fy(I)g Fz(\))p Fy(P)3116 4853 y Fw(0)3157 4839 y Fz(\))1159 5011 y(=)82 b Fy(")1354 4973 y Fv(\000)p Fw(2)1464 5011 y Fz(T)-8 b(r\()p Fy(P)1651 5025 y Fw(0)1691 5011 y Fy(V)1744 5025 y Fu(!)1794 5011 y Fy(P)1852 5026 y Fu(b)1887 5011 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)2057 5025 y Fu(!)2108 5011 y Fy(P)2166 5025 y Fw(0)2206 5011 y Fz(\))26 b Fx(\024)f Fy(")2405 4973 y Fv(\000)p Fw(2)2499 5011 y Fx(k)p Fy(P)2602 5025 y Fw(0)2643 5011 y Fy(V)2696 5025 y Fu(!)2746 5011 y Fx(k)2791 5025 y Fw(2)2831 5011 y Fx(k)p Fy(V)2929 5025 y Fu(!)2980 5011 y Fy(P)3038 5025 y Fw(0)3078 5011 y Fx(k)3123 5025 y Fw(2)3282 5011 y Fz(\(B.9\))265 5241 y(eac)m(h)38 b(Hilb)s(ert-Sc)m (hmidt)c(norm)i(in)f(\(B.9\))k(is)c(b)s(ounded)g(b)m(y)i Fy(c)p Fz(\()p Fy(B)5 b Fz(\))p Fy(V)2595 5255 y Fw(0)2635 5241 y Fy(L)2697 5208 y Fw(2)2736 5241 y Fz(.)60 b(This)35 b(observ)-5 b(ation)265 5388 y(together)32 b(with)d(\(B.7\))j(giv)m(es) f(the)f(result)g(of)g(the)h(lemma.)p 3422 5388 4 62 v 3426 5330 55 4 v 3426 5388 V 3479 5388 4 62 v 1828 5637 a Fs(22)p eop %%Page: 23 23 23 22 bop 401 100 a Fz(Let)31 b(us)f(no)m(w)g(sk)m(etc)m(h)i(the)f(pro) s(of)e(of)i(Prop)s(osition)d(2.)265 321 y Fj(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)35 b(2.)49 b Fz(Let)29 b Fy(E)h Fx(2)25 b Fz(\001)1642 335 y Fu(")1706 321 y Fz(and)i Fy(I)33 b Fz(=)25 b([)p Fy(E)20 b Fx(\000)15 b Fy(\016)n(;)g(E)21 b Fz(+)15 b Fy(\016)s Fz(])28 b(for)g Fy(\016)j Fz(small)26 b(enough)i(\(w)m(e)265 468 y(require)h(that)i Fy(I)37 b Fz(is)30 b(con)m(tained)g(in)g(\001)1531 482 y Fu(")1567 468 y Fz(\).)41 b(By)31 b(the)f(Cheb)m(yshev)g(inequalit)m(y)f(w)m(e)h (ha)m(v)m(e)617 703 y Fq(P)672 717 y Fw(\003)741 703 y Fz(\(dist)o(\()p Fy(\033)s Fz(\()p Fy(H)1124 718 y Fu(b)1159 703 y Fz(\))p Fy(;)15 b(E)5 b Fz(\))27 b Fy(<)e(\016)s Fz(\))h(=)f Fq(P)1719 717 y Fw(\003)1788 703 y Fz(\(T)-8 b(r)15 b Fy(P)1990 718 y Fu(b)2025 703 y Fz(\()p Fy(I)7 b Fz(\))26 b Fx(\025)f Fz(1\))h Fx(\024)f Fq(E)2520 717 y Fw(\003)2579 703 y Fz(\(T)-8 b(r)16 b Fy(P)2782 718 y Fu(b)2816 703 y Fz(\()p Fy(I)7 b Fz(\)\))268 b(\(B.10\))265 939 y(where)21 b Fq(E)573 953 y Fw(\003)654 939 y Fz(is)f(the)h(exp)s (ectation)h(with)e(resp)s(ect)h(to)h(the)g(random)e(v)-5 b(ariables)20 b(in)g(\003.)38 b(T)-8 b(o)22 b(estimate)265 1086 y(it)30 b(w)m(e)h(use)f(an)g(in)m(termediate)g(inequalit)m(y)f(of) h(the)h(previous)e(pro)s(of)994 1321 y Fq(E)1049 1335 y Fw(\003)1108 1321 y Fz(\(T)-8 b(r)15 b Fy(P)1310 1336 y Fu(b)1345 1321 y Fz(\()p Fy(I)7 b Fz(\)\))26 b Fx(\024)f Fz(2)p Fy(")1706 1284 y Fv(\000)p Fw(2)1801 1321 y Fq(E)1856 1335 y Fw(\003)1915 1321 y Fz(\(T)-8 b(r)15 b Fy(P)2117 1335 y Fw(0)2157 1321 y Fy(V)2210 1335 y Fu(!)2261 1321 y Fy(P)2319 1336 y Fu(b)2353 1321 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)2523 1335 y Fu(!)2575 1321 y Fy(P)2633 1335 y Fw(0)2672 1321 y Fz(\))26 b Fy(:)478 b Fz(\(B.11\))265 1557 y(W)-8 b(riting)30 b Fy(V)652 1571 y Fu(!)r(;)p Fw(\003)796 1557 y Fz(=)892 1489 y Fp(P)988 1584 y Fa(i)o Fv(2)p Fw(\003)1132 1557 y Fy(X)1207 1572 y Fa(i)1240 1557 y Fz(\()p Fy(!)s Fz(\))p Fy(V)1423 1572 y Fa(i)431 1792 y Fz(T)-8 b(r)15 b Fy(P)598 1806 y Fw(0)638 1792 y Fy(V)691 1806 y Fu(!)741 1792 y Fy(P)799 1807 y Fu(b)834 1792 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)1004 1806 y Fu(!)1055 1792 y Fy(P)1113 1806 y Fw(0)1236 1792 y Fz(=)1432 1706 y Fp(X)1390 1909 y Fa(i)o Fu(;)p Fa(j)t Fv(2)p Fw(\003)1571 1890 y Ft(2)1620 1792 y Fy(X)1695 1807 y Fa(i)1728 1792 y Fz(\()p Fy(!)s Fz(\))p Fy(X)1933 1807 y Fa(j)1975 1792 y Fz(\()p Fy(!)s Fz(\))15 b(T)-8 b(r)16 b Fy(P)2288 1806 y Fw(0)2328 1792 y Fy(V)2381 1807 y Fa(i)2413 1792 y Fy(P)2471 1807 y Fu(b)2506 1792 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)2676 1807 y Fa(j)2718 1792 y Fy(P)2776 1806 y Fw(0)3236 1792 y Fz(\(B.12\))1236 2062 y(=)1432 1975 y Fp(X)1390 2178 y Fa(i)o Fu(;)p Fa(j)t Fv(2)p Fw(\003)1571 2159 y Ft(2)1620 2062 y Fy(X)1695 2077 y Fa(i)1728 2062 y Fz(\()p Fy(!)s Fz(\))p Fy(X)1933 2077 y Fa(j)1975 2062 y Fz(\()p Fy(!)s Fz(\))15 b(T)-8 b(r)16 b Fy(V)2303 2014 y Fw(1)p Fu(=)p Fw(2)2283 2091 y Fa(j)2413 2062 y Fy(P)2471 2076 y Fw(0)2511 2062 y Fy(V)2584 2014 y Fw(1)p Fu(=)p Fw(2)2564 2091 y Fa(i)2694 2062 y Fy(V)2767 2014 y Fw(1)p Fu(=)p Fw(2)2747 2091 y Fa(i)2877 2062 y Fy(P)2935 2077 y Fu(b)2970 2062 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)3161 2014 y Fw(1)p Fu(=)p Fw(2)3140 2091 y Fa(j)3296 2062 y Fy(:)265 2395 y Fz(Since)25 b Fy(V)571 2347 y Fw(1)p Fu(=)p Fw(2)551 2424 y Fa(j)681 2395 y Fy(P)739 2409 y Fw(0)779 2395 y Fy(V)852 2347 y Fw(1)p Fu(=)p Fw(2)832 2424 y Fa(i)988 2395 y Fz(is)g(trace)i(class)e(w)m(e)h(can)g(in)m(tro)s (duce)f(the)h(singular)e(v)-5 b(alue)25 b(decomp)s(osition)1287 2685 y Fy(V)1360 2637 y Fw(1)p Fu(=)p Fw(2)1340 2715 y Fa(j)1470 2685 y Fy(P)1528 2699 y Fw(0)1568 2685 y Fy(V)1641 2637 y Fw(1)p Fu(=)p Fw(2)1621 2715 y Fa(i)1776 2685 y Fz(=)1903 2572 y Fv(1)1873 2599 y Fp(X)1872 2793 y Fu(n)p Fw(=0)2020 2685 y Fy(\026)2075 2699 y Fu(n)2122 2685 y Fz(\()p Fy( )2216 2699 y Fu(n)2264 2685 y Fy(;)15 b(:)p Fz(\))p Fy(\036)2418 2699 y Fu(n)3236 2685 y Fz(\(B.13\))265 2997 y(where)528 2929 y Fp(P)624 2955 y Fv(1)624 3024 y Fu(n)p Fw(=0)776 2997 y Fy(\026)831 3011 y Fu(n)903 2997 y Fz(=)25 b Fx(k)p Fy(V)1118 2949 y Fw(1)p Fu(=)p Fw(2)1097 3026 y Fa(j)1228 2997 y Fy(P)1286 3011 y Fw(0)1326 2997 y Fy(V)1399 2949 y Fw(1)p Fu(=)p Fw(2)1379 3026 y Fa(i)1509 2997 y Fx(k)1554 3011 y Fw(1)1594 2997 y Fz(.)40 b(Then)783 3288 y(T)-8 b(r)14 b Fy(V)965 3240 y Fw(1)p Fu(=)p Fw(2)944 3317 y Fa(j)1075 3288 y Fy(P)1133 3302 y Fw(0)1172 3288 y Fy(V)1246 3240 y Fw(1)p Fu(=)p Fw(2)1225 3317 y Fa(i)1356 3288 y Fy(V)1429 3240 y Fw(1)p Fu(=)p Fw(2)1409 3317 y Fa(i)1539 3288 y Fy(P)1597 3303 y Fu(b)1632 3288 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)1823 3240 y Fw(1)p Fu(=)p Fw(2)1802 3317 y Fa(j)1958 3288 y Fz(=)2085 3174 y Fv(1)2055 3201 y Fp(X)2054 3396 y Fu(n)p Fw(=0)2202 3288 y Fy(\026)2257 3302 y Fu(n)2304 3288 y Fz(\()p Fy(\036)2393 3302 y Fu(n)2440 3288 y Fy(;)15 b(V)2554 3240 y Fw(1)p Fu(=)p Fw(2)2533 3317 y Fa(i)2664 3288 y Fy(P)2722 3303 y Fu(b)2757 3288 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)2948 3240 y Fw(1)p Fu(=)p Fw(2)2927 3317 y Fa(j)3058 3288 y Fy( )3117 3302 y Fu(n)3164 3288 y Fz(\))629 3589 y Fx(\024)814 3476 y Fv(1)783 3503 y Fp(X)783 3697 y Fu(n)p Fw(=0)931 3589 y Fy(\026)986 3603 y Fu(n)1032 3589 y Fz(\()p Fy(\036)1121 3603 y Fu(n)1169 3589 y Fy(;)15 b(V)1283 3541 y Fw(1)p Fu(=)p Fw(2)1262 3619 y Fa(i)1393 3589 y Fy(P)1451 3604 y Fu(b)1485 3589 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)1676 3541 y Fw(1)p Fu(=)p Fw(2)1655 3619 y Fa(i)1786 3589 y Fy(\036)1840 3603 y Fu(n)1887 3589 y Fz(\))1922 3552 y Fw(1)p Fu(=)p Fw(2)2033 3589 y Fz(\()p Fy( )2127 3603 y Fu(n)2174 3589 y Fy(;)15 b(V)2288 3541 y Fw(1)p Fu(=)p Fw(2)2267 3619 y Fa(j)2398 3589 y Fy(P)2456 3604 y Fu(b)2491 3589 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)2682 3541 y Fw(1)p Fu(=)p Fw(2)2661 3619 y Fa(j)2792 3589 y Fy( )2851 3603 y Fu(n)2898 3589 y Fz(\))2933 3552 y Fw(1)p Fu(=)p Fw(2)629 3891 y Fx(\024)793 3855 y Fw(1)p 793 3870 36 4 v 793 3922 a(2)884 3777 y Fv(1)854 3805 y Fp(X)853 3999 y Fu(n)p Fw(=0)1001 3891 y Fy(\026)1056 3905 y Fu(n)1118 3790 y Fp(n)1179 3891 y Fz(\()p Fy(\036)1268 3905 y Fu(n)1315 3891 y Fy(;)15 b(V)1429 3843 y Fw(1)p Fu(=)p Fw(2)1408 3920 y Fa(i)1539 3891 y Fy(P)1597 3906 y Fu(b)1631 3891 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)1822 3843 y Fw(1)p Fu(=)p Fw(2)1801 3920 y Fa(i)1932 3891 y Fy(\036)1986 3905 y Fu(n)2034 3891 y Fz(\))20 b(+)g(\()p Fy( )2274 3905 y Fu(n)2322 3891 y Fy(;)15 b(V)2435 3843 y Fw(1)p Fu(=)p Fw(2)2415 3920 y Fa(j)2545 3891 y Fy(P)2603 3906 y Fu(b)2638 3891 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)2829 3843 y Fw(1)p Fu(=)p Fw(2)2808 3920 y Fa(j)2939 3891 y Fy( )2998 3905 y Fu(n)3045 3891 y Fz(\))3080 3790 y Fp(o)3182 3891 y Fy(:)29 b Fz(\(B.14\))265 4173 y(An)h(application)f (of)h(the)h(sp)s(ectral)e(a)m(v)m(eraging)j(theorem)f(of)g([CH])f(sho)m (ws)g(that)1120 4409 y Fq(E)1174 4423 y Fw(\003)1233 4409 y Fz(\(\()p Fy( )1362 4423 y Fu(n)1410 4409 y Fy(;)15 b(V)1524 4361 y Fw(1)p Fu(=)p Fw(2)1503 4438 y Fa(j)1634 4409 y Fy(P)1692 4424 y Fu(b)1727 4409 y Fz(\()p Fy(I)7 b Fz(\))p Fy(V)1918 4361 y Fw(1)p Fu(=)p Fw(2)1897 4438 y Fa(j)2028 4409 y Fy( )2087 4423 y Fu(n)2134 4409 y Fz(\)\))26 b Fx(\024)f(k)p Fy(h)p Fx(k)2468 4423 y Fv(1)2544 4409 y Fz(2)p Fy(\016)607 b Fz(\(B.15\))265 4644 y(as)27 b(w)m(ell)f(as)h(for)g(the)g(term)g(with)f Fd(i)g Fz(replacing)g Fd(j)32 b Fz(and)27 b Fy(\036)2121 4658 y Fu(n)2195 4644 y Fz(replacing)e Fy( )2639 4658 y Fu(n)2687 4644 y Fz(.)39 b(Com)m(bining)26 b(\(B.11\))r(,)265 4791 y(\(B.14\))r(,)31 b(\(B.15\))h(and)e(\(B.12\))j(w)m(e)e(get)425 5027 y Fq(E)480 5041 y Fw(\003)539 5027 y Fz(\(T)-8 b(r)15 b Fy(P)741 5042 y Fu(b)776 5027 y Fz(\()p Fy(I)7 b Fz(\)\))26 b Fx(\024)f Fz(4)p Fx(k)p Fy(h)p Fx(k)1237 5041 y Fv(1)1313 5027 y Fy(\016)s(")1398 4989 y Fv(\000)p Fw(2)1551 4940 y Fp(X)1509 5143 y Fa(i)p Fu(;)p Fa(j)s Fv(2)p Fw(\003)1690 5124 y Ft(2)1740 5027 y Fx(k)p Fy(V)1859 4979 y Fw(1)p Fu(=)p Fw(2)1838 5056 y Fa(j)1969 5027 y Fy(P)2027 5041 y Fw(0)2066 5027 y Fy(V)2140 4979 y Fw(1)p Fu(=)p Fw(2)2119 5056 y Fa(i)2250 5027 y Fx(k)2295 5041 y Fw(1)2360 5027 y Fx(\024)g Fz(4)p Fx(k)p Fy(h)p Fx(k)2643 5041 y Fv(1)2719 5027 y Fy(\016)s(")2804 4989 y Fv(\000)p Fw(2)2900 5027 y Fy(c)p Fz(\()p Fy(B)5 b Fz(\))p Fy(V)3136 5041 y Fw(0)3176 5027 y Fy(L)3238 4989 y Fw(4)3302 5027 y Fy(:)3236 5241 y Fz(\(B.16\))p 3422 5388 4 62 v 3426 5330 55 4 v 3426 5388 V 3479 5388 4 62 v 1828 5637 a Fs(23)p eop %%Page: 24 24 24 23 bop 265 100 a Fr(C)161 b(Estimate)53 b(on)g(the)g(Eigenfunction)g (of)h Fc(H)3054 122 y Fb(b)265 337 y Fz(In)61 b(this)g(section)h(w)m(e) g(pro)m(v)m(e)h(Gaussian)e(deca)m(y)i(of)f(the)g(eigenfunction)f Fy( )3077 304 y Fu(b)3074 365 y(\014)3183 337 y Fz(and)g(its)265 484 y Fy(y)s Fx(\000)p Fz(deriv)-5 b(ativ)m(e)38 b(outside)f(the)i (supp)s(ort)e(of)i(the)f(random)g(p)s(oten)m(tial)g Fy(V)2703 498 y Fu(!)2753 484 y Fz(.)66 b(F)-8 b(rom)39 b(the)f(eigen-)265 631 y(v)-5 b(alue)30 b(equation)g(\()p Fy(H)982 645 y Fw(0)1042 631 y Fz(+)20 b Fy(V)1186 645 y Fu(!)1236 631 y Fz(\))p Fy( )1333 598 y Fu(b)1330 658 y(\014)1403 631 y Fz(=)25 b Fy(E)1571 598 y Fu(b)1566 658 y(\014)1613 631 y Fy( )1675 598 y Fu(b)1672 658 y(\014)1750 631 y Fz(w)m(e)31 b(get)1486 859 y Fy( )1548 822 y Fu(b)1545 882 y(\014)1617 859 y Fz(=)25 b Fy(R)1782 873 y Fw(0)1822 859 y Fz(\()p Fy(E)1929 822 y Fu(b)1924 882 y(\014)1971 859 y Fz(\))p Fy(V)2059 873 y Fu(!)2110 859 y Fy( )2172 822 y Fu(b)2169 882 y(\014)2242 859 y Fy(:)1014 b Fz(\(C.1\))265 1088 y(Th)m(us)870 1316 y Fx(j)p Fy( )957 1279 y Fu(b)954 1340 y(\014)1001 1316 y Fz(\()p Fd(x)q Fz(\))p Fx(j)83 b(\024)1394 1193 y Fp(Z)1445 1399 y Fh(R)p Fv(\002)p Fu(I)1579 1407 y Fi(p)1633 1316 y Fx(j)p Fy(R)1727 1330 y Fw(0)1767 1316 y Fz(\()p Fd(x)p Fy(;)15 b Fd(x)1963 1279 y Fv(0)1986 1316 y Fz(;)g Fy(E)2098 1279 y Fu(b)2093 1340 y(\014)2141 1316 y Fz(\))p Fy(V)2229 1330 y Fu(!)2280 1316 y Fz(\()p Fd(x)2375 1279 y Fv(0)2398 1316 y Fz(\))p Fy( )2495 1279 y Fu(b)2492 1340 y(\014)2540 1316 y Fz(\()p Fd(x)2635 1279 y Fv(0)2658 1316 y Fz(\))p Fx(j)g Fz(d)q Fd(x)2845 1279 y Fv(0)1240 1605 y Fx(\024)83 b Fy(V)1447 1619 y Fw(0)1502 1477 y Fp(\032)1570 1482 y(Z)1620 1688 y Fw(supp)13 b Fu(V)1819 1696 y Fi(!)1883 1605 y Fx(j)p Fy(R)1977 1619 y Fw(0)2016 1605 y Fz(\()p Fd(x)q Fy(;)i Fd(x)2212 1568 y Fv(0)2235 1605 y Fz(;)g Fy(E)2347 1568 y Fu(b)2342 1629 y(\014)2390 1605 y Fz(\))p Fx(j)2450 1568 y Fw(2)2505 1605 y Fz(d)p Fd(x)2616 1568 y Fv(0)2639 1477 y Fp(\033)2707 1500 y Fw(1)p Fu(=)p Fw(2)2858 1605 y Fy(;)398 b Fz(\(C.2\))265 1854 y(and)761 2001 y Fx(j)p Fy(@)834 2015 y Fu(y)876 2001 y Fy( )938 1963 y Fu(b)935 2024 y(\014)982 2001 y Fz(\()p Fd(x)q Fz(\))p Fx(j)25 b(\024)g Fy(V)1312 2015 y Fw(0)1367 2001 y Fz(sup)1412 2073 y Fa(x)1519 2001 y Fx(j)p Fy( )1606 1963 y Fu(b)1603 2024 y(\014)1651 2001 y Fz(\()p Fd(x)p Fz(\))p Fx(j)1821 1877 y Fp(Z)1872 2083 y Fw(supp)12 b Fu(V)2070 2091 y Fi(!)2135 2001 y Fx(j)p Fy(@)2208 2015 y Fu(y)2250 2001 y Fy(R)2319 2015 y Fw(0)2358 2001 y Fz(\()p Fd(x)p Fy(;)j Fd(x)2554 1963 y Fv(0)2577 2001 y Fz(;)g Fy(E)2689 1963 y Fu(b)2684 2024 y(\014)2732 2001 y Fz(\))p Fx(j)g Fz(d)p Fd(x)2918 1963 y Fv(0)2966 2001 y Fy(:)290 b Fz(\(C.3\))401 2217 y(W)-8 b(e)46 b(need)e(b)s(ounds)f(on)h(the)h(in)m(tegral)f(k)m (ernel)g Fy(R)2136 2231 y Fw(0)2220 2217 y Fz(and)f(its)h Fy(y)s Fx(\000)p Fz(deriv)-5 b(ativ)m(e)44 b(to)h(get)h(an)265 2364 y(estimate)d(of)g(the)g(eigenfunctions)e(and)h(their)g Fy(y)s Fx(\000)p Fz(deriv)-5 b(ativ)m(e.)76 b(F)-8 b(rom)43 b([DMP4)q(])g(w)m(e)g(ha)m(v)m(e)265 2510 y(\()p Fy(E)31 b Fx(2)25 b Fz(\001)560 2524 y Fu(")596 2510 y Fz(\))633 2739 y Fx(j)p Fy(R)728 2701 y Fv(1)727 2761 y Fw(0)803 2739 y Fz(\()p Fd(x)p Fy(;)15 b 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Fx(j)p Fd(x)20 b Fx(\000)g Fd(x)2992 3831 y Fv(0)3016 3864 y Fx(j)3041 3831 y Fw(2)3105 3864 y Fy(>)25 b Fz(1)763 3966 y Fp(\000)804 4040 y Fz(1)c(+)961 3963 y Fp(\014)961 4017 y(\014)991 4040 y Fz(ln)o(\()1112 4004 y Fu(B)p 1112 4019 V 1123 4071 a Fw(2)1179 4040 y Fx(j)p Fd(x)f Fx(\000)g Fd(x)1435 4007 y Fv(0)1458 4040 y Fx(j)1483 4007 y Fw(2)1523 4040 y Fz(\))1558 3963 y Fp(\014)1558 4017 y(\014)1589 3966 y(\001)1645 4040 y Fz(\(1)i(+)d Fx(j)p Fy(x)p Fx(j)i Fz(+)f Fx(j)p Fd(x)g Fx(\000)g Fd(x)2307 4007 y Fv(0)2330 4040 y Fx(j)2355 4007 y Fv(\000)p Fw(1)2450 4040 y Fz(\))91 b(if)2670 4004 y Fu(B)p 2670 4019 V 2681 4071 a Fw(2)2736 4040 y Fx(j)p Fd(x)20 b Fx(\000)g Fd(x)2992 4007 y Fv(0)3016 4040 y Fx(j)3041 4007 y Fw(2)3105 4040 y Fx(\024)25 b Fz(1)h Fy(:)3281 3939 y Fz(\(C.5\))401 4251 y(With)k(the)h(help)e(of)37 b(\(C.4\))32 b(and)e(\(C.5\))h(w)m(e)g(can)f(see)h(that)g(for)f Fy(L)h Fz(large)f(enough)541 4563 y Fx(j)p Fy( )628 4525 y Fu(b)625 4586 y(\014)673 4563 y Fz(\()p Fd(x)p 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Fp(\002)3014 5259 y Fx(\000)3095 5223 y Fu(L)p 3095 5238 V 3101 5290 a Fw(2)3153 5259 y Fy(;)3203 5223 y Fu(L)p 3203 5238 V 3209 5290 a Fw(2)3261 5185 y Fp(\003)3324 5259 y Fy(:)3281 5388 y Fz(\(C.7\))1828 5637 y Fs(24)p eop %%Page: 25 25 25 24 bop 265 100 a Fz(Indeed,)30 b(for)g Fx(j)p Fy(m)p Fx(j)25 b Fy(>)g Fz(1)1062 65 y Fu(B)p 1062 80 57 4 v 1073 132 a Fw(2)1129 100 y Fz([\()p Fy(x)20 b Fx(\000)g Fy(x)1404 67 y Fv(0)1428 100 y Fz(\))1463 67 y Fw(2)1523 100 y Fz(+)f(\()p Fy(y)24 b Fx(\000)c Fy(y)1856 67 y Fv(0)1899 100 y Fx(\000)g Fy(mL)p Fz(\))2167 67 y Fw(2)2206 100 y Fz(])26 b Fy(>)f Fz(1)31 b(th)m(us)f(w)m(e)g(ha)m(v)m(e)361 354 y Fx(j)p Fy(R)455 368 y Fw(0)494 354 y Fz(\()p Fd(x)q Fy(;)15 b Fd(x)690 316 y Fv(0)713 354 y Fz(;)g Fy(E)825 316 y Fu(b)820 377 y(\014)868 354 y Fz(\))p Fx(j)26 b(\024)1070 331 y Fz(~)1050 354 y Fy(C)6 b Fz(\()p Fy(B)f Fz(\))p Fy(")1307 316 y Fv(\000)p Fw(1)1402 354 y Fy(e)1444 313 y Fv(\000)1509 277 y Fu(B)p 1509 292 V 1520 344 a Fw(8)1576 313 y(\()p 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Fx(j)26 b(\024)1816 946 y Fz(~)1796 969 y Fy(C)1868 931 y Fv(0)1891 969 y Fz(\()p Fy(B)5 b Fz(\))p Fy(")2077 931 y Fv(\000)p Fw(1)2171 969 y Fy(e)2213 928 y Fv(\000)2278 892 y Fu(B)p 2278 907 57 4 v 2289 959 a Fw(8)2345 928 y(\()p Fu(x)p Fv(\000)p Fu(x)2507 904 y Fg(0)2529 928 y Fw(\))2556 904 y Ft(2)2621 969 y Fy(:)635 b Fz(\(C.9\))265 1169 y(Replacing)34 b(this)f(b)s(ound)f(in)h(\(C.2\))j (w)m(e)f(get)g(the)g(Gaussian)f(deca)m(y)h(in)f(\(C.6\))h(On)f(the)g (other)265 1316 y(hand)26 b(if)h Fy(x)e Fx(2)733 1243 y Fp(\002)771 1316 y Fx(\000)852 1280 y Fu(L)p 852 1295 48 4 v 858 1348 a Fw(2)909 1316 y Fy(;)960 1280 y Fu(L)p 960 1295 V 966 1348 a Fw(2)1018 1243 y Fp(\003)1083 1316 y Fz(w)m(e)j(can)g(use)f(the)h(logarithmic)e(b)s(ounds)f(for)j(the)f (terms)h Fx(j)p Fy(m)p Fx(j)d(\024)g Fz(1)j(and)265 1463 y(w)m(e)i(remark)g(they)g(are)g(in)m(tegrable)g(and)f(b)s(ounded)f(b)m (y)h Fy(L)2220 1430 y Fw(2)2275 1463 y Fz(ln)o(\()p Fy(B)5 b(L)2522 1430 y Fw(2)2561 1463 y Fz(\).)41 b(The)29 b(same)h(argumen)m (ts)265 1610 y(hold)f(for)h(the)h Fy(y)s Fx(\000)p Fz(deriv)-5 b(ativ)m(e.)265 1984 y Fr(Ac)l(kno)l(wledgemen)l(ts)265 2221 y Fz(N.M.)39 b(wishes)e(to)i(thank)f(J.)g(F)-8 b(r\177)-45 b(ohlic)m(h)37 b(and)h(E.)g(Mourre)g(for)g(helpful)e(discussions)f(and) j(F.)265 2368 y(Ben)m(tosela)45 b(for)f(suggesting)g(the)g(use)g(of)g (the)g(decoupling)f(form)m(ula.)80 b(W)-8 b(e)45 b(also)f(thank)g(a)265 2515 y(referee)31 b(for)g(correcting)g(a)g(sligh)m(t)f(mistak)m(e)i(in) d(one)i(of)g(the)g(pro)s(ofs.)42 b(The)30 b(w)m(ork)h(of)g(C.F.)g(w)m (as)265 2661 y(supp)s(orted)26 b(b)m(y)i(a)g(gran)m(t)h(from)e(the)h(F) -8 b(onds)28 b(National)f(Suisse)g(de)g(la)h(Rec)m(herc)m(he)h(Scien)m (ti\014que)265 2808 y(No.)41 b(20)32 b(-)e(55654.98.)265 3183 y Fr(References)265 3419 y Fz([AM])80 b(M.)26 b(Aizenman,)f(S.)g (Molc)m(hano)m(v:)39 b(Lo)s(calization)24 b(at)i(large)f(disorder)e (and)h(at)i(extreme)546 3566 y(energies:)38 b(an)24 b(elemen)m(tary)h (deriv)-5 b(ation.)23 b(Comm)m(un.)h(Math.)h(Ph)m(ys.)g FA(157)p Fz(,)h(245)g(\(1993\))265 3755 y([A)m(G])95 b(M.)24 b(Aizenman,)g(G.M.)h(Graf:)37 b(Lo)s(calization)23 b(b)s(ounds)e(for)i(an)g(electron)g(gas.)h(J.)g(Ph)m(ys.)546 3901 y(A:)31 b(Math.)g(Gen.)g FA(31)p Fz(,)g(6783)h(\(1998\))265 4090 y([AS])112 b(M.)22 b(Abramo)m(witz)f(and)f(I.A.)h(Stegun:)36 b Fj(Handb)-5 b(o)g(ok)26 b(of)e(mathematic)-5 b(al)27 b(functions)p Fz(.)21 b(New)546 4236 y(Y)-8 b(ork:)41 b(Do)m(v)m(er,)33 b(1965)265 4425 y([ASS])61 b(J.)27 b(Avron,)h(R.)f(Seiler,)g(B.)g(Simon:)38 b(Charge)27 b(de\014ciency)-8 b(,)28 b(c)m(harge)g(transp)s(ort)f(and)f(the)546 4571 y(comparison)k(of)g(dimensions.)e(Comm)m(un.)i(Math.)h(Ph)m(ys.)f FA(159)p Fz(,)i(399)f(\(1994\))265 4760 y([BCD])50 b(P)-8 b(.)42 b(Briet,)j(J.M.)d(Com)m(b)s(es,)j(P)-8 b(.)42 b(Duclos:)63 b(Sp)s(ectral)41 b(stabilit)m(y)f(under)h(th)m(unneling.) 546 4907 y(Comm)m(un.)30 b(Math.)h(Ph)m(ys.)f FA(126)p Fz(,)i(133)f(\(1989\))265 5095 y([BES])54 b(J.)31 b(Bellissard,)e(A.)i (v)-5 b(an)31 b(Elst,)f(H.)h(Sc)m(h)m(ulz-Baldes:)42 b(The)30 b(noncomm)m(utativ)m(e)i(geome-)546 5242 y(try)e(of)h(the)f (quan)m(tum)g(Hall)g(e\013ect.)i(J.)e(Math.)h(Ph)m(ys.)g FA(35)p Fz(,)g(5373)h(\(1994\))1828 5637 y Fs(25)p eop %%Page: 26 26 26 25 bop 265 100 a Fz([BG])96 b(F.)32 b(Ben)m(tosela,)i(V.)e(Grecc)m (hi:)44 b(Stark)32 b(W)-8 b(annier)31 b(Ladders.)g(Comm)m(un.)g(Math.)i (Ph)m(ys.)546 247 y FA(142)p Fz(,)e(169)h(\(1991\))265 435 y([dBP])54 b(S.)23 b(de)h(Bi)m(\022)-43 b(evre,)26 b(J.V.)e(Pul)m(\023)-43 b(e:)37 b(Propagating)24 b(edge)h(states)f(for) g(magnetic)g(Hamiltonian.)546 582 y(Math.)31 b(Ph)m(ys.)g(Electr.)f(J.) g FA(5)p Fz(,)h(no.)g(3)f(\(1999\))265 771 y([CH])97 b(J.M.)31 b(Com)m(b)s(es,)e(P)-8 b(.D.)32 b(Hislop:)39 b(Landau)29 b(Hamiltonians)f(with)h(random)g(p)s(oten)m(tials:)546 917 y(lo)s(calization)43 b(and)h(the)g(densit)m(y)f(of)h(states.)i (Comm)m(un.)d(Math.)i(Ph)m(ys.)f FA(177)p Fz(,)k(603)546 1064 y(\(1996\))265 1252 y([BCH])i(J.M.)28 b(Barbaroux,)h(J.M.)f(Com)m (b)s(es,)g(P)-8 b(.D.)30 b(Hislop:)38 b(Lo)s(calization)27 b(near)h(band)f(edges)546 1399 y(for)j(random)g(Sc)m(hr\177)-45 b(odinger)29 b(op)s(erators.)i(Helv.)f(Ph)m(ys.)g(Acta)i FA(70)p Fz(,)f(16)g(\(1997\))265 1588 y([DMP1])50 b(T.C.)28 b(Dorlas,)h(N.)f(Macris,)h(J.V.)f(Pul)m(\023)-43 b(e:)40 b(Lo)s(calisation)27 b(in)g(a)h(single-band)e(appro)m(x-)546 1734 y(imation)31 b(to)h(random)f(Sc)m(hr\177)-45 b(odinger)31 b(op)s(erators)g(in)g(a)h(magnetic)g(\014eld.)f(Helv.)h(Ph)m(ys.)546 1881 y(Acta)g FA(68)p Fz(,)f(330)h(\(1995\))265 2069 y([DMP2])50 b(T.C.)30 b(Dorlas,)g(N.)g(Macris,)g(J.V.)g(Pul)m(\023)-43 b(e:)40 b(Lo)s(calization)29 b(in)f(single)h(Landau)f(bands.)546 2216 y(J.)i(Math.)i(Ph)m(ys.)e FA(37)p Fz(,)h(1574)h(\(1996\))265 2404 y([DMP3])50 b(T.C.)41 b(Dorlas,)i(N.)e(Macris,)i(J.V.)e(Pul)m (\023)-43 b(e:)61 b(The)40 b(nature)g(of)h(the)f(sp)s(ectrum)g(for)g(a) 546 2551 y(Landau)30 b(Hamiltonian)e(whit)h(delta)i(impurities.)c(J.)j (Stat.)i(Ph)m(ys.)e FA(87)p Fz(,)h(847)h(\(1997\))265 2740 y([DMP4])50 b(T.C.)33 b(Dorlas,)h(N.)g(Macris,)f(J.V.)h(Pul)m (\023)-43 b(e:)46 b(Characterization)33 b(of)g(the)g(sp)s(ectrum)f(of) 546 2886 y(the)22 b(Landau)g(Hamiltonian)e(with)h(delta)h(impurities.)c (Comm)m(un.)k(Math.)h(Ph)m(ys.)f FA(204)p Fz(,)546 3033 y(367)32 b(\(1999\))265 3221 y([F])172 b(C.)69 b(F)-8 b(errari:)118 b(Dynamique)68 b(d'une)h(particule)e(quan)m(tique)i(dans) f(un)g(c)m(hamp)546 3368 y(magn)m(\023)-43 b(etique)31 b(inhomog)m(\022)-43 b(ene.)32 b(Diploma)d(w)m(ork,)i(EPFL)f(\(1999\).) 265 3556 y([FK])101 b(J.)31 b(F)-8 b(r\177)-45 b(ohlic)m(h,)32 b(T.)f(Kerler:)42 b(Univ)m(ersalit)m(y)30 b(in)g(quan)m(tum)h(Hall)g (systems.)h(Nucl.)f(Ph)m(ys.)546 3703 y(B)g FA(354)p Fz(,)g(369)h(\(1991\))265 3892 y([F)m(GW])51 b(J.)25 b(F)-8 b(r\177)-45 b(ohlic)m(h,)25 b(G.M.)h(Graf,)h(J.)e(W)-8 b(alc)m(her:)39 b(On)24 b(the)h(extended)g(nature)f(of)i(edge)f(states) 546 4038 y(of)31 b(quan)m(tum)e(Hall)h(Hamiltonians.)f(Ann.)h(Henri)f (P)m(oincar)m(\023)-43 b(e)32 b FA(1)p Fz(,)e(405)i(\(2000\))265 4227 y([FM1])50 b(C.)34 b(F)-8 b(errari,)34 b(N.)g(Macris:)46 b(Extended)33 b(energy)h(lev)m(els)f(in)f(the)i(gap)g(for)f (macroscopic)546 4373 y(Hall)d(systems.)g(Preprin)m(t)265 4562 y([FM2])50 b(C.)43 b(F)-8 b(errari,)45 b(N.)e(Macris:)65 b(Lo)s(calized)42 b(energy)h(lev)m(els)g(inside)d(Landau)i(bands.)f(In) 546 4708 y(preparation)265 4897 y([FM3])50 b(C.)31 b(F)-8 b(errari,)30 b(N.)h(Macris:)42 b(Sp)s(ectral)29 b(prop)s(erties)g(of)i (\014nite)e(quan)m(tum)i(Hall)e(systems.)546 5044 y(T)-8 b(o)31 b(app)s(ear)f(in)f(J.)h(Op)s(er.)f(Theor.)i(\(math-ph/0203016\)) 265 5232 y([GM])77 b(S.)25 b(Gyger,)i(P)-8 b(.A.)26 b(Martin:)37 b(Lifetime)24 b(of)i(impurit)m(y)c(states)27 b(in)c(crossed)j(magnetic) f(and)546 5379 y(electric)31 b(\014elds.)e(J.)h(Math.)h(Ph)m(ys.)f FA(40)p Fz(,)h(3275)i(\(1999\))1828 5637 y Fs(26)p eop %%Page: 27 27 27 26 bop 265 100 a Fz([H])163 b(B.I.)36 b(Halp)s(erin:)49 b(Quan)m(tized)36 b(Hall)e(conductance,)39 b(curren)m(t-carrying)c (edge)h(states,)546 247 y(and)f(the)g(existence)h(of)g(extended)f (states)i(in)d(a)h(t)m(w)m(o-dimensional)f(disordered)g(p)s(o-)546 394 y(ten)m(tial.)d(Ph)m(ys.)f(Rev.)h(B)g FA(25)p Fz(,)g(2185)h (\(1982\))265 582 y([JL])127 b(V.)45 b(Jak)-5 b(\024)-41 b(si)m(\023)e(c,)49 b(Y.)c(Last:)69 b(Sp)s(ectral)43 b(structure)h(of)g(Anderson)g(t)m(yp)s(e)g(Hamiltonians.)546 729 y(mp-arc/99-204)265 917 y([K])160 b(H.)41 b(Kunz:)60 b(The)40 b(quan)m(tum)g(Hall)g(e\013ect)i(for)e(electrons)h(in)e(a)i (random)f(p)s(oten)m(tial.)546 1064 y(Comm)m(un.)30 b(Math.)h(Ph)m(ys.) f FA(112)p Fz(,)i(121)f(\(1987\))265 1252 y([MMP])50 b(N.)35 b(Macris,)g(P)-8 b(.A.)36 b(Martin)d(and)h(J.V.)h(Pul)m(\023) -43 b(e:)49 b(On)33 b(Edge)i(States)g(In)f(Semi-In\014nite)546 1399 y(Quan)m(tum)c(Hall)f(Systems.)h(J.)h(Ph)m(ys.)f(A:)h(Math.)g (Gen.)g FA(32)p Fz(,)g(1985)h(\(1999\))265 1588 y([Mi])122 b(N.)30 b(Minami:)39 b(Lo)s(cal)30 b(\015uctuation)f(of)g(the)h(sp)s (ectrum)f(of)h(m)m(ultidimensional)25 b(Ander-)546 1734 y(son)30 b(tigh)m(t)h(binding)c(mo)s(del.)j(Comm)m(un.)f(Math.)j(Ph)m (ys.)e FA(177)p Fz(,)h(709)h(\(1996\))265 1923 y([M])148 b(N.)29 b(Macris:)39 b(Sp)s(ectral)28 b(\015o)m(w)g(and)g(lev)m(el)g (spacing)f(of)i(edge)g(states)h(for)e(quan)m(tum)f(Hall)546 2069 y(Hamiltonians.)i(Preprin)m(t)265 2258 y([PG])98 b(R.E.)25 b(Prange)h(and)e(S.M.)i(Girvin:)36 b Fj(The)28 b(Quantum)g(Hal)5 b(l)28 b(E\013e)-5 b(ct)p Fz(.)24 b(New)i(Y)-8 b(ork:)38 b(Grad-)546 2404 y(uate)31 b(T)-8 b(exts)31 b(in)e(Con)m(temp)s(orary)h(Ph)m(ysics,)g(Springer,)e(1987)265 2593 y([SKR1])49 b(H.)34 b(Sc)m(h)m(ulz-Baldes,)h(J.)f(Kellendonk,)f (T.)h(Ric)m(h)m(ter:)48 b(Sim)m(ultaneous)33 b(quan)m(tization)546 2740 y(of)28 b(edge)g(and)e(bulk)g(Hall)g(conductivit)m(y)-8 b(.)27 b(J.)h(Ph)m(ys.)f(A:)h(Math.)g(Gen.)f FA(33)p Fz(,)i(L27)f(\(2000\))265 2928 y([SKR2])49 b(H.)33 b(Sc)m(h)m (ulz-Baldes,)h(J.)e(Kellendonk,)g(T.)h(Ric)m(h)m(ter:)46 b(Edge)33 b(curren)m(t)g(c)m(hannels)f(and)546 3075 y(Chern)d(n)m(um)m (b)s(ers)g(in)g(the)i(in)m(teger)f(quan)m(tum)g(Hall)g(e\013ect.)i (mp-arc/00-266)265 3263 y([TKNN])49 b(D.J.)33 b(Thouless,)e(M.)h (Kohmoto,)h(M.P)-8 b(.)33 b(Nigh)m(tingale,)f(M.)h(den)e(Nijs:)43 b(Quan)m(tized)546 3410 y(Hall)25 b(conductance)i(in)e(a)i(t)m(w)m (o-dimensional)e(p)s(erio)s(dic)e(p)s(oten)m(tial.)j(Ph)m(ys.)g(Rev.)g (Lett.)546 3556 y FA(49)p Fz(,)31 b(405)h(\(1982\))265 3745 y([W)-8 b(a])101 b(W.M.)32 b(W)-8 b(ang:)44 b(Microlo)s (calization,)30 b(p)s(ercolation)g(and)h(Anderson)f(lo)s(calization)g (for)546 3892 y(the)35 b(magnetic)h(Sc)m(hr\177)-45 b(odinger)33 b(op)s(erator)j(with)d(a)j(random)e(p)s(oten)m(tial.)h(J.)g(of)g(F)-8 b(unct.)546 4038 y(Anal.)30 b FA(146)p Fz(,)h(1)g(\(1997\))265 4227 y([W)-8 b(e])106 b(X.G.)26 b(W)-8 b(en:)39 b(Gapless)25 b(b)s(oundary)e(excitations)i(in)f(the)h(quan)m(tum)g(Hall)f(states)i (in)e(the)546 4373 y(c)m(hiral)29 b(spin)g(states.)j(Ph)m(ys.)e(Rev.)h (B)f FA(43)p Fz(,)i(11025)g(\(1991\))1828 5637 y Fs(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0203130243742--