Content-Type: multipart/mixed; boundary="-------------0010030445415" This is a multi-part message in MIME format. ---------------0010030445415 Content-Type: text/plain; name="00-389.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-389.keywords" random Schr dinger operators, density of states, Lifshitz tails ---------------0010030445415 Content-Type: application/postscript; name="err-il.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="err-il.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: err-il.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips err-il.dvi -o /home1/klopp/err-il.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.10.03:1140 %%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 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Fl(0)1362 535 y Fh(+)1437 427 y Fq(log)17 b Fn(j)e Fq(log)q(\()p Fo(N)10 b Fq(\()p Fo(E)5 b Fq(\))22 b Fn(\000)e Fo(N)10 b Fq(\(0\)\))p Fn(j)p 1437 467 923 4 v 1796 551 a Fq(log)16 b Fo(E)2395 488 y Fq(=)25 b Fn(\000)2652 427 y Fo(d)p 2572 467 210 4 v 2572 551 a(\027)g Fn(\000)20 b Fo(d)2791 488 y(:)-2698 b Fq(\(0.1\))243 732 y Fn(\017)42 b Fi(if)32 b Fo(V)53 b Fi(satis\014es)34 b(\(H.2bis.2\),)e(then,)h(we)g(have)398 840 y Ft({)42 b Fi(if)32 b(\(H.4\))h(hold)h(then,)f(we)g(have)626 1092 y Fq(lim)15 b(inf)646 1158 y Fk(E)t Fm(!)p Fl(0)808 1139 y Fh(+)903 1031 y Fq(log)h Fn(j)f Fq(log)r(\()p Fo(N)10 b Fq(\()p Fo(E)5 b Fq(\))22 b Fn(\000)e Fo(N)10 b Fq(\(0)1729 998 y Fm(\000)1788 1031 y Fq(\)\))p Fn(j)p 903 1071 982 4 v 1291 1155 a Fq(log)17 b Fo(E)1920 1092 y Fn(\025)24 b(\000)2096 1031 y Fo(d)p 2096 1071 48 4 v 2097 1155 a Fq(2)1894 1271 y Fi(and)553 1477 y Fq(lim)15 b(sup)586 1561 y Fk(E)t Fm(!)p Fl(0)748 1542 y Fh(+)857 1416 y Fq(log)q(\()p Fo(n)p Fq(\()p Fo(E)5 b Fq(\))21 b Fn(\000)f Fo(n)p Fq(\(0\)\))p 857 1456 668 4 v 1088 1539 a(log)c Fo(E)1559 1477 y(<)1665 1416 y(d)p 1665 1456 48 4 v 1666 1539 a Fq(2)1748 1477 y(=)-15 b Fn(\))14 b Fq(lim)h(inf)1929 1543 y Fk(E)t Fm(!)p Fl(0)2091 1524 y Fh(+)2187 1416 y Fq(log)h Fn(j)f Fq(log)r(\()p Fo(N)10 b Fq(\()p Fo(E)5 b Fq(\))22 b Fn(\000)d Fo(N)10 b Fq(\(0)3012 1383 y Fm(\000)3072 1416 y Fq(\)\))p Fn(j)p 2187 1456 982 4 v 2575 1539 a Fq(log)17 b Fo(E)3203 1477 y(>)25 b Fn(\000)3380 1416 y Fo(d)p 3380 1456 48 4 v 3381 1539 a Fq(2)3437 1477 y Fo(;)398 1733 y Ft({)42 b Fi(if)32 b(\(H.4s\))h(holds)i(then,)e(0)g (is)f(a)h(c)-5 b(ontinuity)34 b(p)-5 b(oint)34 b(for)f Fo(N)42 b Fi(and)34 b(we)f(have)656 1981 y Fq(lim)613 2046 y Fk(E)t Fm(!)p Fl(0)775 2028 y Fh(+)850 1919 y Fq(log)17 b Fn(j)e Fq(log)r(\()p Fo(N)10 b Fq(\()p Fo(E)5 b Fq(\))21 b Fn(\000)f Fo(N)10 b Fq(\(0\)\))p Fn(j)p 850 1960 923 4 v 1209 2043 a Fq(log)17 b Fo(E)1808 1981 y Fq(=)25 b Fn(\000)1985 1919 y Fo(d)p 1985 1960 48 4 v 1986 2043 a Fq(2)2067 1981 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Fo(:)118 3986 y Fq(The)30 b(computations)g(done)g(in)f(pages)i(376{378)j(giv)m(e)956 4233 y(lim)15 b(inf)1013 4291 y Fk(")p Fm(!)p Fl(0)1021 4348 y Fk(">)p Fl(0)1233 4172 y Fq(log)i Fn(j)e Fq(log)q(\()p Fo(N)10 b Fq(\()p Fo(")p Fq(\))22 b Fn(\000)e Fo(N)10 b Fq(\(0)2029 4139 y Fm(\000)2089 4172 y Fq(\)\))p Fn(j)p 1233 4212 952 4 v 1621 4295 a Fq(log)17 b Fo(")2220 4233 y Fn(\025)25 b(\000)2476 4172 y Fo(\027)p 2397 4212 210 4 v 2397 4295 a(\027)g Fn(\000)20 b Fo(d)2636 4233 y Fn(\000)g Fq(\()p Fo(d)h Fn(\000)f Fq(1\))p Fo(\013)1187 4653 y Fq(2.)51 b Fp(Pr)n(oof)34 b(of)g Fq(\(0.1\))i Fp(when)d(\(H.4s\))h(holds)331 4815 y Fq(First)j(note)i(that)g(to)g (pro)m(v)m(e)g(\(0.1\))h(w)m(e)f(only)e(need)h(to)h(pro)m(v)m(e)h(an)e (upp)s(er)e(b)s(ound)g(on)i(the)h(densit)m(y)e(of)118 4923 y(states)d Fo(N)43 b Fq(that)34 b(coalesces)h(with)c(the)j (already)e(pro)m(v)m(en)i(lo)m(w)m(er)f(b)s(ound.)47 b(Therefore)33 b(w)m(e)g(will)e(use)i(a)g(metho)s(d)118 5031 y(di\013eren)m(t)39 b(from)g(the)g(one)h(dev)m(elop)s(ed)e(in)g ([9)q(];)44 b(it)39 b(will)d(b)s(e)j(related)g(to)h(the)g(metho)s(d)e (used)h(in)f([5)q(,)h(6].)68 b(The)118 5139 y(pro)s(of)31 b(go)s(es)i(roughly)e(as)h(follo)m(ws.)44 b(First)32 b(w)m(e)g(will)e(pro)m(v)m(e)i(that)h(the)f(p)s(erio)s(dic)d(appro)m (ximations)i(in)m(tro)s(duced)118 5247 y(in)26 b(section)i(5.1)g(of)f ([9)q(])h(are)f(exp)s(onen)m(tially)f(close)i(to)g(the)f(true)g(densit) m(y)g(of)g(states.)41 b(Then,)27 b(w)m(e)h(will)d(study)h(the)118 5355 y(densit)m(y)k(of)g(states)i(of)e(these)h(appro)m(ximations.)1990 5454 y Fl(2)p eop %%Page: 3 3 3 2 bop 118 241 a Fq(2.1.)54 b Ft(P)m(erio)s(dic)46 b(appro)m (ximations.)f Fq(Let)40 b Fo(H)1759 255 y Fk(!)r(;n)1910 241 y Fq(b)s(e)e(the)i(p)s(erio)s(dic)c(appro)m(ximation)i(de\014ned)g (in)f([9)q(])i(and)118 349 y Fo(N)191 363 y Fk(!)r(;n)334 349 y Fq(b)s(e)30 b(their)f(in)m(tegrated)j(densit)m(y)d(of)i(state.)42 b(Then,)29 b(w)m(e)i(pro)m(v)m(e)118 500 y Ft(Theorem)j(2.1.)46 b Fi(Assume)29 b(\(H.1\),)h(\(H.3\))g(and)g(\(H.2bis.1\).)41 b(Pick)28 b Fo(I)33 b Fn(\032)24 b Fj(R)38 b Fi(a)30 b(c)-5 b(omp)g(act)31 b(interval.)42 b(Then,)30 b(for)118 608 y(any)36 b Fo(\021)340 622 y Fl(0)408 608 y Fo(>)29 b Fq(0)p Fi(,)36 b(ther)-5 b(e)36 b(exists)f Fo(\027)1142 622 y Fl(0)1211 608 y Fo(>)29 b Fq(0)35 b Fi(and)h Fo(")1612 622 y Fl(0)1681 608 y Fo(>)29 b Fq(0)35 b Fi(such)g(that,)i(for)e Fq(0)30 b Fo(<)f(")g(<)g(")2822 622 y Fl(0)2862 608 y Fi(,)35 b Fo(E)g Fn(2)29 b Fo(I)42 b Fi(and)36 b Fo(n)29 b Fn(\025)g Fo(")3604 575 y Fm(\000)p Fk(\027)3694 584 y Fh(0)3732 608 y Fi(,)35 b(we)118 716 y(have)567 855 y Fj(E)10 b Fq(\()q Fo(N)730 869 y Fk(!)r(;n)848 855 y Fq(\()p Fo(E)26 b Fq(+)20 b Fo("=)p Fq(2\))i Fn(\000)e Fo(N)1420 869 y Fk(!)r(;n)1533 855 y Fq(\()p Fo(E)26 b Fn(\000)20 b Fo("=)p Fq(2\)\))i Fn(\000)e Fo(e)2109 817 y Fm(\000)p Fk(")2197 794 y Fe(\000)p Ff(\021)2277 809 y Fh(0)1165 995 y Fn(\024)25 b Fo(N)10 b Fq(\()p Fo(E)26 b Fq(+)20 b Fo(")p Fq(\))h Fn(\000)f Fo(N)10 b Fq(\()p Fo(E)5 b Fq(\))1666 1159 y Fn(\024)25 b Fj(E)10 b Fq(\()q Fo(N)1925 1173 y Fk(!)r(;n)2044 1159 y Fq(\()p Fo(E)26 b Fq(+)19 b(2)p Fo(")p Fq(\))j Fn(\000)e Fo(N)2570 1173 y Fk(!)r(;n)2683 1159 y Fq(\()p Fo(E)26 b Fn(\000)19 b Fq(2)p Fo(")p Fq(\)\))j(+)e Fo(e)3213 1122 y Fm(\000)p Fk(")3301 1098 y Fe(\000)p Ff(\021)3381 1113 y Fh(0)3424 1159 y Fo(:)118 1002 y Fq(\(2.1\))118 1311 y(Theorem)35 b(2.1)h(states)h(that)e(the)h(densit)m(y)e(of)h(states)i(of)e(the)g (random)g(Hamiltonian)e Fo(H)3186 1325 y Fk(!)3271 1311 y Fq(is)h(exp)s(onen)m(tially)118 1419 y(w)m(ell)22 b(appro)m(ximated)h (b)m(y)g(the)g(exp)s(ectation)h(of)f(the)h(densities)d(of)i(states)i (of)e(the)g(p)s(erio)s(dic)e(Hamiltonians)g Fo(H)3785 1433 y Fk(!)r(;n)118 1526 y Fq(ev)m(en)39 b(for)e Fo(n)h Fq(not)g(v)m(ery)g(large)g(\(i.e.)63 b(for)38 b Fo(n)f Fq(of)h(size)g(roughly)f(in)m(v)m(erse)g(p)s(olynomial)f(of)i(the)g (appro)m(ximation)118 1634 y(parameter\).)118 1742 y Ft(Pro)s(of.)i Fq(Theorem)27 b(2.1)h(is)e(a)h(consequence)g(of)g(a)h (similar)c(result)i(for)g(random)g(op)s(erators)h(with)f(a)h(compactly) 118 1850 y(supp)s(orted)38 b(single)h(site)h(p)s(oten)m(tial.)69 b(More)41 b(precisely)-8 b(,)42 b(de\014ne)d Fo(V)2415 1864 y Fk(")2452 1850 y Fq(\()p Fo(x)p Fq(\))j(=)f Fo(V)20 b Fq(\()p Fo(x)p Fq(\))p Ft(1)2975 1869 y Fk(")p Fm(\001j)p Fk(x)p Fm(j\025)p Fl(1)3242 1850 y Fq(and)39 b(the)h(random)118 1958 y(op)s(erator)1197 2117 y Fo(H)1273 2131 y Fk(";!)1400 2117 y Fq(=)25 b Fo(H)i Fq(+)20 b Fo(V)1743 2131 y Fk(";!)1871 2117 y Fq(=)25 b Fo(H)i Fq(+)2179 2031 y Fg(X)2161 2236 y Fk(\015)t Fm(2)p Fd(Z)2299 2217 y Ff(d)2344 2117 y Fo(!)2401 2131 y Fk(\015)2445 2117 y Fo(V)2498 2131 y Fk(")2535 2117 y Fq(\()p Fn(\001)20 b(\000)g Fo(\015)5 b Fq(\))p Fo(:)118 2367 y Fq(Let)25 b Fo(N)348 2381 y Fk(")408 2367 y Fq(b)s(e)e(the)h(in)m(tegrated)h(densit)m(y)e(of)h (states)h(of)g Fo(H)1931 2381 y Fk(";!)2033 2367 y Fq(.)38 b(W)-8 b(e)25 b(de\014ne)e Fo(H)2578 2381 y Fk(";!)r(;n)2743 2367 y Fq(,)i(the)g(p)s(erio)s(dic)c(appro)m(ximations)118 2475 y(of)28 b Fo(H)295 2489 y Fk(";!)425 2475 y Fq(in)f(the)h(same)g (w)m(a)m(y)h(as)f(for)f Fo(H)1411 2489 y Fk(!)1489 2475 y Fq(\(just)h(replace)g Fo(V)48 b Fq(b)m(y)27 b Fo(V)2285 2489 y Fk(")2322 2475 y Fq(\).)40 b(Let)29 b Fo(N)2656 2489 y Fk(";!)r(;n)2849 2475 y Fq(b)s(e)e(their)g(in)m(tegrated)h (densit)m(y)118 2583 y(of)j(states.)118 2691 y(W)-8 b(e)23 b(\014rst)f(notice)g(that)g(using)f(the)h(deca)m(y)h(assumption)e (\(H.2bis.1\))h(and)g(the)g(fact)h(that)g(the)f(random)f(v)-5 b(ariables)118 2799 y(\()p Fo(!)210 2813 y Fk(\015)254 2799 y Fq(\))289 2813 y Fk(\015)365 2799 y Fq(are)30 b(b)s(ounded,)f(uniformly)e(in)i Fo(n)h Fq(and)g Fo(!)s Fq(,)g(w)m(e)h(ha)m(v)m(e)718 2955 y Fn(k)p Fq(\(1)22 b Fn(\000)e Fq(\001\))1067 2918 y Fm(\000)p Fl(1)1161 2955 y Fq(\()p Fo(H)1272 2969 y Fk(";!)1395 2955 y Fn(\000)g Fo(H)1562 2969 y Fk(!)1612 2955 y Fq(\))p Fn(k)h Fq(+)f Fn(k)p Fq(\(1)h Fn(\000)f Fq(\001\))2152 2918 y Fm(\000)p Fl(1)2246 2955 y Fq(\()p Fo(H)2357 2969 y Fk(";!)r(;n)2543 2955 y Fn(\000)g Fo(H)2710 2969 y Fk(!)r(;n)2822 2955 y Fq(\))p Fn(k)26 b(\024)f Fo(C)7 b(")3138 2918 y Fk(\027)t Fm(\000)p Fk(d)3272 2955 y Fo(:)118 3101 y Fq(This)29 b(implies)e(that,)k(uniformly)d(in)h Fo(n)h Fq(and)g Fo(!)s Fq(,)g(and)g(lo)s(cally)f(uniformly)e(in)i Fo(E)5 b Fq(,)31 b(one)g(has)1354 3248 y Fo(N)1427 3262 y Fk(")1464 3248 y Fq(\()p Fo(E)26 b Fn(\000)19 b Fo(C)7 b(")1796 3211 y Fk(\027)t Fm(\000)p Fk(d)1931 3248 y Fq(\))25 b Fn(\024)g Fo(N)10 b Fq(\()p Fo(E)5 b Fq(\))27 b Fn(\024)d Fo(N)2507 3262 y Fk(")2544 3248 y Fq(\()p Fo(E)i Fq(+)20 b Fo(C)7 b(")2877 3211 y Fk(\027)t Fm(\000)p Fk(d)3011 3248 y Fq(\))969 3404 y Fo(N)1042 3418 y Fk(";!)r(;n)1207 3404 y Fq(\()p Fo(E)26 b Fn(\000)20 b Fo(C)7 b(")1540 3366 y Fk(\027)t Fm(\000)p Fk(d)1674 3404 y Fq(\))26 b Fn(\024)f Fo(N)1904 3418 y Fk(!)r(;n)2017 3404 y Fq(\()p Fo(E)5 b Fq(\))26 b Fn(\024)f Fo(N)2354 3418 y Fk(";!)r(;n)2519 3404 y Fq(\()p Fo(E)h Fq(+)20 b Fo(C)7 b(")2852 3366 y Fk(\027)t Fm(\000)p Fk(d)2986 3404 y Fq(\))p Fo(:)118 3325 y Fq(\(2.2\))118 3569 y(So)29 b(to)g(pro)m(v)m(e)h(Theorem)e(2.1,) j(w)m(e)e(only)f(need)g(to)i(pro)m(v)m(e)f(it)f(for)h Fo(H)2345 3583 y Fk(!)r(;")2476 3569 y Fq(and)f(replace)g(`)p Fo(")i Fq(b)m(y)e Fo(")3220 3536 y Fl(1)p Fk(=)p Fl(\()p Fk(\027)t Fm(\000)p Fk(d)p Fl(\))3481 3569 y Fq(.)40 b(The)28 b(gain)118 3677 y(is)36 b(that,)i(for)f(the)f(new)g(op)s (erators,)i(the)f(single)e(site)h(p)s(oten)m(tial)g(is)g(compactly)h (supp)s(orted.)57 b(The)36 b(pro)s(of)f(of)118 3785 y(Theorem)30 b(2.1)i(in)d(this)g(case)i(relies)f(up)s(on)e(the)118 3936 y Ft(Lemma)33 b(2.1.)46 b Fi(Assume)32 b(that)i(the)e(single)g (site)h(p)-5 b(otential)34 b Fo(V)53 b Fi(is)32 b(c)-5 b(omp)g(actly)35 b(supp)-5 b(orte)g(d)35 b(in)d(a)h(b)-5 b(al)5 b(l)33 b(of)f(r)-5 b(adius)118 4044 y Fo(R)31 b Fi(c)-5 b(enter)g(e)g(d)32 b(at)g(0.)41 b(Pick)30 b Fo(I)7 b Fi(,)31 b(a)g(r)-5 b(elatively)32 b(c)-5 b(omp)g(act)33 b(op)-5 b(en)31 b(interval)h(in)e Fj(R)40 b Fi(.F)-7 b(or)31 b(any)g Fo(\014)g Fn(2)25 b Fq(\(0)p Fo(;)15 b Fq(1\))p Fi(,)33 b(ther)-5 b(e)31 b(exists)118 4152 y Fo(C)j(>)28 b Fq(1)34 b Fi(and)h Fo(\032)28 b(>)f Fq(0)34 b Fi(\(dep)-5 b(ending)36 b(only)f(on)f Fo(d)p Fi(\))g(such)h(that,)g (for)g(any)f Fo(')28 b Fn(2)g(C)2710 4119 y Fm(1)2705 4177 y Fl(0)2785 4152 y Fq(\()p Fo(I)7 b Fq(\))p Fi(,)34 b(for)h Fo(k)c Fn(2)c Fj(N)3337 4119 y Fm(\003)3416 4152 y Fi(and)35 b Fo(n)27 b Fn(2)h Fj(N)3824 4119 y Fm(\003)3870 4152 y Fi(,)118 4260 y Fo(n)d(>)g(R)q Fi(,)32 b(we)h(have)520 4462 y Fn(j)p Fj(E)11 b Fq(\(\()q Fo(';)16 b(dN)891 4476 y Fk(!)r(;n)1010 4462 y Fq(\)\))21 b Fn(\000)f Fq(\()p Fo(';)15 b(dN)10 b Fq(\))p Fn(j)27 b(\024)e Fo(C)7 b Fn(j)p Fo(n)19 b Fn(\000)h Fo(R)q Fn(j)1996 4425 y Fm(\000)p Fl(\(1)p Fm(\000)p Fk(\014)s Fl(\))p Fk(k)2382 4462 y Fq(sup)2383 4541 y Fk(x)p Fm(2)p Fd(R)2297 4600 y Fl(0)p Fm(\024)p Fk(j)t Fm(\024)p Fk(k)r Fl(+)p Fk(\032)2619 4331 y Fg(\014)2619 4385 y(\014)2619 4440 y(\014)2619 4494 y(\014)2650 4462 y Fq(\()p Fn(j)p Fo(x)p Fn(j)h Fq(+)e Fo(C)7 b Fq(\))3005 4425 y Fk(\032)p Fl(+)p Fk(k)3149 4401 y Fo(d)3196 4368 y Fk(j)3233 4401 y Fo(')p 3149 4441 144 4 v 3153 4525 a(d)3200 4498 y Fk(j)3237 4525 y Fo(x)3302 4462 y Fq(\()p Fo(x)p Fq(\))3424 4331 y Fg(\014)3424 4385 y(\014)3424 4440 y(\014)3424 4494 y(\014)3470 4462 y Fo(:)-3377 b Fq(\(2.3\))118 4742 y Ft(Pro)s(of.)77 b Fq(This)40 b(lemma)h(w)m(as)i(pro)m(v)m(ed)f(in)f([5)q(,)h(6,)h(10])g (in)d(a)j(di\013eren)m(t)e(setting)i(but)e(essen)m(tially)g(the)h(same) 118 4850 y(argumen)m(ts)29 b(apply)f(in)g(our)g(case)i(to)s(o.)41 b(The)29 b(pro)s(of)f(relies)g(up)s(on)f(the)j(follo)m(wing)d(estimate) j(on)f(the)g(di\013erence)118 4958 y(b)s(et)m(w)m(een)i(the)g(resolv)m (en)m(ts)g(of)f Fo(H)1223 4972 y Fk(!)r(;n)1366 4958 y Fq(and)g Fo(H)1619 4972 y Fk(!)1699 4958 y Fq(at)h(an)f(energy)h Fo(z)e Fn(2)c Fj(C)44 b Fn(n)21 b Fj(R)39 b Fq(:)118 5138 y(\(2.4\))403 5061 y Fg(\015)403 5115 y(\015)454 5138 y Ft(1)506 5156 y Fk(C)5 b Fl(\(0)p Fk(;)p Fl(1\))725 5064 y Fg(\000)767 5138 y Fq(\()p Fo(z)25 b Fn(\000)20 b Fo(H)1036 5152 y Fk(!)r(;n)1148 5138 y Fq(\))1183 5100 y Fm(\000)p Fl(1)1298 5138 y Fn(\000)g Fq(\()p Fo(z)25 b Fn(\000)20 b Fo(H)1658 5152 y Fk(!)1708 5138 y Fq(\))1743 5100 y Fm(\000)p Fl(1)1837 5064 y Fg(\001)1894 5138 y Ft(1)1946 5156 y Fk(C)5 b Fl(\(0)p Fk(;)p Fl(1\))2151 5061 y Fg(\015)2151 5115 y(\015)2226 5138 y Fq(=)881 5318 y(=)977 5241 y Fg(\015)977 5295 y(\015)1028 5318 y Ft(1)1080 5337 y Fk(C)g Fl(\(0)p Fk(;)p Fl(1\))1299 5244 y Fg(\000)1341 5318 y Fq(\()p Fo(z)25 b Fn(\000)20 b Fo(H)1610 5332 y Fk(!)r(;n)1722 5318 y Fq(\))1757 5281 y Fm(\000)p Fl(1)1852 5318 y Fq(\()p Fo(V)1940 5332 y Fk(!)r(;n)2073 5318 y Fn(\000)g Fo(V)2217 5332 y Fk(!)2267 5318 y Fq(\)\()p Fo(z)26 b Fn(\000)20 b Fo(H)2572 5332 y Fk(!)2622 5318 y Fq(\))2657 5281 y Fm(\000)p Fl(1)2751 5244 y Fg(\001)2808 5318 y Ft(1)2860 5337 y Fk(C)5 b Fl(\(0)p Fk(;)p Fl(1\))3065 5241 y Fg(\015)3065 5295 y(\015)3140 5318 y Fn(\024)25 b Fo(C)7 b(e)3350 5281 y Fm(\000)3427 5247 y Fh(1+)p Fe(j)p Ff(z)r Fe(j)p 3415 5265 170 3 v 3415 5307 a Ff(C)s Fe(\001)p Fh(Im)p Ff(z)3595 5281 y Fm(j)p Fk(n)p Fm(\000)p Fk(R)p Fm(j)3789 5318 y Fo(:)1990 5454 y Fl(3)p eop %%Page: 4 4 4 3 bop 118 241 a Fq(Estimate)32 b(\(2.4\))i(is)c(a)i(consequence)h(of) f(the)g(computation)f(done)h(to)g(pro)m(v)m(e)h(Theorem)e(5.1)i(in)d ([9)q(])i(\(pp)f(358{)118 349 y(360\),)39 b(of)d(the)g(fact)h(that,)h (b)m(y)e(construction,)h(\()p Fo(V)1810 363 y Fk(!)r(;n)1947 349 y Fn(\000)23 b Fo(V)2094 363 y Fk(!)2145 349 y Fq(\))36 b(v)-5 b(anishes)34 b(in)h(the)h(cub)s(e)f(cen)m(tered)i(at)f(0)h(of)f (side)118 457 y(length)25 b Fo(n)10 b Fn(\000)g Fo(R)25 b Fq(\(as)h Fo(V)45 b Fq(is)24 b(supp)s(orted)g(in)g(a)h(ball)f(of)h (size)g Fo(R)q Fq(\),)i(and)e(of)g(the)g(exp)s(onen)m(tial)g(deca)m(y)h (of)f(the)h(resolv)m(en)m(t)118 565 y(k)m(ernels)33 b(\()p Fo(z)26 b Fn(\000)c Fo(H)697 579 y Fk(!)747 565 y Fq(\))782 532 y Fm(\000)p Fl(1)910 565 y Fq(and)32 b(\()p Fo(z)27 b Fn(\000)22 b Fo(H)1362 579 y Fk(!)r(;n)1474 565 y Fq(\))1509 532 y Fm(\000)p Fl(1)1637 565 y Fq(\(this)32 b(is)g(a)i(consequence)g (of)f(Com)m(b)s(es-Thomas)g(estimates)h(\(see)118 672 y(e.g.)j([1)q(,)f(4)q(]\)\).)58 b(T)-8 b(o)36 b(obtain)g(Lemma)g(2.1)h (from)f(\(2.4\))q(,)i(one)e(then)g(uses)f(the)i(represen)m(tation)f (form)m(ula)f(\(5.9\))118 780 y(in)23 b([9])h(for)g Fo(')p Fq(\()p Fo(H)7 b Fq(\))25 b(in)e(terms)g(of)h(almost)g(analytic)g (extensions)f(of)h Fo(')p Fq(,)i(and)d(the)h(standard)f(estimates)i (for)f(almost)118 888 y(analytic)30 b(extension)g(\(5.8\))i(in)d([9)q (].)p 3832 888 4 62 v 3836 830 55 4 v 3836 888 V 3890 888 4 62 v 331 996 a(Let)40 b(us)f(no)m(w)h(deriv)m(e)f(Theorem)h(2.1)h (from)e(Lemma)h(2.1.)70 b(Let)40 b Fo(')h Fq(b)s(e)e(a)h(Gevrey)g (class)g(function)e(of)118 1104 y(Gevrey)31 b(exp)s(onen)m(t)f Fo(\013)25 b(>)g Fq(1)30 b(\(see)h([2)q(]\);)g(assume)e(moreo)m(v)m(er) j(that)e Fo(')h Fq(has)e(compact)i(supp)s(ort)e(in)f(\()p Fn(\000)p Fq(2)p Fo(;)15 b Fq(2\),)32 b(that)118 1212 y(0)26 b Fn(\024)f Fo(')g Fn(\024)g Fq(1)31 b(and)f(that)h Fo(')26 b Fn(\021)f Fq(1)30 b(on)h([)p Fn(\000)p Fq(1)p Fo(;)15 b Fq(1].)42 b(Let)31 b Fo(E)f Fn(2)25 b Fj(R)39 b Fq(and)30 b(set)1575 1423 y Fo(')1634 1437 y Fk(E)t(;")1746 1423 y Fq(\()p Fn(\001)p Fq(\))c(=)f Fo(')2037 1295 y Fg(\022)2115 1362 y Fn(\001)20 b(\000)g Fo(E)p 2115 1402 209 4 v 2198 1486 a(")2334 1295 y Fg(\023)2416 1423 y Fo(:)118 1629 y Fq(Fix)29 b Fo(I)k Fn(\032)25 b Fj(R)38 b Fq(compact.)j(Then,)29 b(b)m(y)h(Lemma)g(2.1)g(and)f(the)h(Gevrey)g (estimates)h(on)e(the)h(deriv)-5 b(ativ)m(es)29 b(of)h Fo(')p Fq(,)g(w)m(e)118 1737 y(get)i(that,)f(there)f(exist)h Fo(C)g(>)25 b Fq(1)31 b(suc)m(h)f(that,)h(for)f Fo(E)h Fn(2)25 b Fo(I)7 b Fq(,)31 b Fo(n)24 b(>)h(R)q Fq(,)31 b Fo(k)d Fn(\025)d Fq(1)31 b(and)f(0)25 b Fo(<)g(")h(<)f Fq(1,)31 b(w)m(e)g(ha)m(v)m(e)697 1899 y Fn(j)p Fj(E)11 b Fq(\(\()p Fo(')907 1913 y Fk(E)t(;")1025 1899 y Fo(;)k(dN)1195 1861 y Fk(n)1185 1921 y(!)1242 1899 y Fq(\)\))21 b Fn(\000)f Fq(\()p Fo(')1518 1913 y Fk(E)t(;")1631 1899 y Fo(;)15 b(dN)10 b Fq(\))p Fn(j)26 b(\024)f Fo(")2025 1861 y Fm(\000)p Fk(k)r Fm(\000)p Fk(\032)2214 1899 y Fq(\()p Fo(k)f Fq(+)c Fo(\032)p Fq(\))2493 1861 y Fk(\013)p Fl(\()p Fk(k)r Fl(+)p Fk(\032)p Fl(\))2727 1899 y Fq(\()p Fo(n)g Fn(\000)g Fo(R)q Fq(\))3033 1861 y Fm(\000)p Fl(\(1)p Fm(\000)p Fk(\014)s Fl(\))p Fk(k)118 2075 y Fq(W)-8 b(e)35 b(pic)m(k)e Fo(k)h Fq(=)c(\()p Fo(n)22 b Fn(\000)g Fo(R)q Fq(\))965 2042 y Fl(\(1)p Fm(\000)p Fk(\014)s Fl(\))p Fk(=)p Fl(2)p Fk(\013)1307 2075 y Fq(and)33 b(get)h(that,)h(there)f(exist)g Fo(C)i(>)31 b Fq(1)j(suc)m(h)f(that,)i(for)e Fo(E)j Fn(2)30 b Fo(I)7 b Fq(,)35 b Fo(n)30 b(>)g(R)23 b Fq(+)f(2)118 2183 y(and)30 b(0)c Fo(<)f(")g(<)g Fq(1,)31 b(w)m(e)g(ha)m(v)m(e)738 2450 y Fn(j)p Fj(E)11 b Fq(\(\()p Fo(')948 2464 y Fk(E)t(;")1066 2450 y Fo(;)k(dN)1236 2413 y Fk(n)1226 2473 y(!)1283 2450 y Fq(\)\))21 b Fn(\000)f Fq(\()p Fo(')1559 2464 y Fk(E)t(;")1672 2450 y Fo(;)15 b(dN)10 b Fq(\))p Fn(j)26 b(\024)2024 2295 y Fg( )2106 2389 y Fq(\()p Fo(n)20 b Fn(\000)g Fo(R)q Fq(\))2412 2356 y Fl(\(1)p Fm(\000)p Fk(\014)s Fl(\))p Fk(=)p Fl(3)p 2106 2429 569 4 v 2350 2513 a Fo(")2392 2486 y Fl(2)2685 2295 y Fg(!)2757 2318 y Fl(\()p Fk(n)p Fm(\000)p Fk(R)p Fl(\))2962 2294 y Fh(\(1)p Fe(\000)p Ff(\014)s Fh(\))p Ff(=)p Fh(2)p Ff(\013)3253 2450 y Fo(:)118 2703 y Fq(No)m(w,)31 b(in)e(the)i(case)g(w)m(e)g(are)g (in)m(terested)f(in)f Fo(R)d Fq(=)f Fo(R)1874 2717 y Fk(")1936 2703 y Fn(\030)g Fo(")2074 2670 y Fl(1)p Fk(=)p Fl(\()p Fk(d)p Fm(\000)p Fk(\027)t Fl(\))2334 2703 y Fq(.)41 b(Hence,)31 b(w)m(e)g(c)m(ho)s(ose)g Fo(n)25 b Fn(\025)g Fo(")3335 2670 y Fm(\000)p Fl(3)p Fm(\000)p Fk(\021)3542 2703 y Fq(+)20 b Fo(R)3702 2717 y Fk(")3739 2703 y Fq(,)30 b(w)m(e)118 2811 y(get)i(that)f(there)f(exist)g Fo(")956 2825 y Fl(0)1021 2811 y Fo(>)25 b Fq(0)31 b(suc)m(h)f(that,)h (for)g(0)25 b Fo(<)g(")h(<)f(")2132 2825 y Fl(0)2172 2811 y Fq(,)30 b(w)m(e)h(ha)m(v)m(e)1083 2984 y Fn(j)p Fj(E)11 b Fq(\(\()p Fo(')1293 2998 y Fk(E)t(;")1411 2984 y Fo(;)k(dN)1581 2946 y Fk(n)1624 2954 y Ff(")1571 3006 y Fk(!)1662 2984 y Fq(\)\))21 b Fn(\000)f Fq(\()p Fo(')1938 2998 y Fk(E)t(;")2050 2984 y Fo(;)15 b(dN)10 b Fq(\))p Fn(j)27 b(\024)e Fo(e)2445 2946 y Fm(\000)p Fk(")2533 2923 y Fe(\000)p Ff(\021)r Fe(\001)p Fh(\(1)p Fe(\000)p Ff(\014)s Fh(\))p Ff(=)p Fh(3)p Ff(\013)2908 2984 y Fo(:)-2815 b Fq(\(2.5\))118 3140 y(By)31 b(the)f(de\014nition)e(of)j Fo(')p Fq(,)g(as)g Fo(dN)1272 3154 y Fk(";!)r(;n)1467 3140 y Fq(and)f Fo(dN)1764 3154 y Fk(")1831 3140 y Fq(are)h(p)s(ositiv) m(e)f(measures,)g(w)m(e)h(ha)m(v)m(e)160 3296 y Fj(E)10 b Fq(\()q Fo(N)323 3310 y Fk(";!)r(;n)494 3296 y Fq(\()p Fo(E)26 b Fq(+)20 b Fo(")p Fq(\))h Fn(\000)e Fo(N)974 3310 y Fk(";!)r(;n)1140 3296 y Fq(\()p Fo(E)25 b Fn(\000)20 b Fo(")p Fq(\)\))27 b Fn(\024)e Fj(E)10 b Fq(\()q(\()p Fo(')1777 3310 y Fk(E)t(;")1895 3296 y Fo(;)15 b(dN)2055 3310 y Fk(";!)r(;n)2221 3296 y Fq(\)\))26 b Fn(\024)f Fj(E)11 b Fq(\()p Fo(N)2576 3310 y Fk(";!)r(;n)2747 3296 y Fq(\()p Fo(E)26 b Fq(+)20 b(2)p Fo(")p Fq(\))h Fn(\000)f Fo(N)3273 3310 y Fk(";!)r(;n)3438 3296 y Fq(\()p Fo(E)26 b Fn(\000)20 b Fq(2)p Fo(")p Fq(\)\))p Fo(;)698 3436 y(N)771 3450 y Fk(")808 3436 y Fq(\()p Fo(E)26 b Fq(+)20 b Fo(")p Fq(\))h Fn(\000)f Fo(N)1289 3450 y Fk(")1326 3436 y Fq(\()p Fo(E)26 b Fn(\000)20 b Fo(")p Fq(\))26 b Fn(\024)e Fq(\()p Fo(')1837 3450 y Fk(E)t(;")1950 3436 y Fo(;)15 b(dN)10 b Fq(\))26 b Fn(\024)f Fo(N)2350 3450 y Fk(")2387 3436 y Fq(\()p Fo(E)h Fq(+)20 b(2)p Fo(")p Fq(\))h Fn(\000)f Fo(N)2913 3450 y Fk(")2950 3436 y Fq(\()p Fo(E)26 b Fn(\000)20 b Fq(2)p Fo(")p Fq(\))p Fo(:)118 3597 y Fq(This,)29 b(equation)h(\(2.2\))i(and)e(\(2.5\))i(complete)f (the)g(pro)s(of)e(of)i(Theorem)f(2.1)h(for)g Fo(H)2961 3611 y Fk(";!)3093 3597 y Fq(hence)g(for)f Fo(H)3561 3611 y Fk(!)3611 3597 y Fq(.)p 3832 3597 4 62 v 3836 3539 55 4 v 3836 3597 V 3890 3597 4 62 v 118 3780 a(2.2.)54 b Ft(Pro)s(of)41 b(of)g(the)f(upp)s(er)h(b)s(ound.)46 b Fq(W)-8 b(e)37 b(pic)m(k)e Fo(\021)1993 3794 y Fl(0)2066 3780 y Fo(>)e(\027)6 b(=)p Fq(\()p Fo(\027)30 b Fn(\000)23 b Fo(d)p Fq(\))36 b(in)e(Theorem)h(2.1)i(and)e Fo(n)e Fn(\030)g Fo(")3656 3747 y Fm(\000)p Fk(\021)3789 3780 y Fq(for)118 3888 y Fo(\021)j(>)31 b(\027)346 3902 y Fl(0)420 3888 y Fq(\()p Fo(\027)500 3902 y Fl(0)574 3888 y Fq(is)j(giv)m(en)g(in)f(Theorem)i(2.1)g(and)f Fo(\021)k Fq(will)32 b(b)s(e)h(c)m(hosen)i(later)g(on\).)53 b(So)34 b(to)i(pro)m(v)m(e)f(\(0.1\))r(,)h(w)m(e)e(only)118 3996 y(need)c(to)h(pro)m(v)m(e)h(that)1013 4200 y(lim)15 b(sup)1058 4284 y Fk(")p Fm(!)p Fl(0)1197 4265 y Fh(+)1317 4138 y Fq(log)h Fn(j)f Fq(log)r(\()p Fj(E)c Fq(\()p Fo(N)1805 4152 y Fk(!)s(;n)1924 4138 y Fq(\()p Fo(")p Fq(\))21 b Fn(\000)f Fo(N)2221 4152 y Fk(!)r(;n)2334 4138 y Fq(\(0\)\)\))p Fn(j)p 1317 4180 1230 4 v 1844 4263 a Fq(log)d Fo(")2582 4200 y Fn(\024)25 b(\000)2839 4139 y Fo(d)p 2759 4180 210 4 v 2759 4263 a(\027)g Fn(\000)20 b Fo(d)2977 4200 y(:)-2884 b Fq(\(2.6\))118 4418 y(Indeed,)27 b(b)m(y)g(assumption)e (\(H.4s\))j(and)e(the)h(computations)f(done)h(in)e(section)i(5.2)h(in)e ([9],)i(w)m(e)f(kno)m(w)g(that)h(,for)118 4526 y Fo(")j Fq(small)e(enough,)h(for)g(all)f Fo(!)34 b Fq(and)29 b Fo(n)p Fq(,)i(one)f(has)g Fo(N)1811 4540 y Fk(!)r(;n)1924 4526 y Fq(\(0\))d(=)e Fo(N)2235 4540 y Fk(!)r(;n)2347 4526 y Fq(\()p Fn(\000)p Fo(")p Fq(\).)331 4634 y(W)-8 b(e)31 b(pro)m(v)m(e)118 4802 y Ft(Lemma)i(2.2.)46 b Fi(De\014ne)29 b(the)h(p)-5 b(otential)32 b Fo(G)e Fi(by)f Fo(G)p Fq(\()p Fn(\001)p Fq(\))e(=)2003 4733 y Fg(P)2099 4828 y Fk(\015)t Fm(2)p Fl(\000)2249 4802 y Fo(g)2292 4816 y Fm(\000)2352 4802 y Fq(\()p Fn(\001)14 b(\000)g Fo(\015)5 b Fq(\))p Fi(.)40 b(Then,)31 b(if)e Fo(n)c Fn(\030)g Fo(")3232 4769 y Fm(\000)p Fk(\032)3357 4802 y Fi(with)30 b Fo(\032)c(>)3777 4766 y Fl(1)p 3730 4781 131 4 v 3730 4833 a Fk(\027)t Fm(\000)p Fk(d)3870 4802 y Fi(,)118 4918 y(the)33 b(event)f Fn(f)p Fo(!)s Fq(;)49 b Fo(V)740 4932 y Fk(!)r(;n)878 4918 y Fn(\025)25 b Fo("G)p Fn(g)34 b Fi(has)f(pr)-5 b(ob)g(ability)35 b(at)e(le)-5 b(ast)34 b Fq(1)21 b Fn(\000)f Fo(P)2297 4932 y Fk(")2366 4918 y Fi(wher)-5 b(e)34 b Fo(P)2681 4932 y Fk(")2751 4918 y Fi(satis\014es)1387 5131 y Fq(lim)15 b(sup)1432 5214 y Fk(")p Fm(!)p Fl(0)1571 5195 y Fh(+)1691 5069 y Fq(log)h Fn(j)f Fq(log)r(\()p Fo(P)2074 5083 y Fk(")2112 5069 y Fq(\))p Fn(j)p 1691 5110 482 4 v 1844 5193 a Fq(log)i Fo(")2207 5131 y Fn(\024)25 b(\000)2465 5069 y Fo(d)p 2384 5110 210 4 v 2384 5193 a(\027)h Fn(\000)20 b Fo(d)2603 5131 y(:)-2510 b Fq(\(2.7\))118 5355 y(and)1990 5454 y Fl(4)p eop %%Page: 5 5 5 4 bop 118 241 a Ft(Lemma)33 b(2.3.)46 b Fi(Ther)-5 b(e)33 b(exists)g Fo(C)f(>)25 b Fq(0)33 b Fi(and)g Fo(")1696 255 y Fl(0)1761 241 y Fo(>)25 b Fq(0)33 b Fi(\(indep)-5 b(endent)35 b(of)e Fo(n)f Fi(and)h Fo(!)s Fi(\))g(such)g(that,)g(if)g Fq(0)25 b Fo(<)g(")h(<)f(")3858 255 y Fl(0)118 349 y Fi(and)34 b Fo(V)348 363 y Fk(!)r(;n)486 349 y Fn(\025)25 b Fo(C)7 b("G)33 b Fi(then,)g(for)g Fo(n)25 b Fn(2)f Fj(N)7 b Fi(,)39 b(one)32 b(has)i Fo(N)1874 363 y Fk(!)r(;n)1987 349 y Fq(\(0\))26 b(=)f Fo(N)2297 363 y Fk(!)r(;n)2410 349 y Fq(\()p Fo(")p Fq(\))p Fi(.)118 513 y Fq(No)m(w,)32 b(estimate)f(\(2.6\))i(is)c(an)i(immediate)f(consequence)h(of)g(Lemmas) f(2.2)i(and)e(2.3.)43 b(Indeed,)30 b(pic)m(king)f Fo(C)37 b Fq(as)118 621 y(in)29 b(Lemma)i(2.3,)g(one)g(computes)684 758 y Fj(E)10 b Fq(\()q Fo(N)847 772 y Fk(!)r(;n)965 758 y Fq(\()p Fo(")p Fq(\))22 b Fn(\000)e Fo(N)1263 772 y Fk(!)r(;n)1376 758 y Fq(\(0\)\))26 b(=)f Fj(E)11 b Fq(\([)q Fo(N)1837 772 y Fk(!)r(;n)1955 758 y Fq(\()p Fo(")p Fq(\))22 b Fn(\000)e Fo(N)2253 772 y Fk(!)r(;n)2365 758 y Fq(\(0\)])p Ft(1)2557 777 y Fm(f)p Fk(!)r Fl(;)33 b Fk(V)2732 785 y Ff(!)r(;n)2834 777 y Fm(\025)p Fk(C)5 b("G)p Fm(g)3071 758 y Fq(\))1784 898 y(+)19 b Fj(E)11 b Fq(\([)q Fo(N)2063 912 y Fk(!)r(;n)2182 898 y Fq(\()p Fo(")p Fq(\))21 b Fn(\000)f Fo(N)2479 912 y Fk(!)r(;n)2592 898 y Fq(\(0\)])p Ft(1)2784 917 y Fm(f)p Fk(!)r Fl(;)32 b Fk(V)2958 925 y Ff(!)r(;n)3060 917 y Fm(6\025)p Fk(C)5 b("G)p Fm(g)3297 898 y Fq(\))1552 1038 y(=)25 b Fj(E)11 b Fq(\([)q Fo(N)1837 1052 y Fk(!)r(;n)1955 1038 y Fq(\()p Fo(")p Fq(\))22 b Fn(\000)e Fo(N)2253 1052 y Fk(!)r(;n)2365 1038 y Fq(\(0\)])p Ft(1)2557 1056 y Fm(f)p Fk(!)r Fl(;)33 b Fk(V)2732 1064 y Ff(!)r(;n)2834 1056 y Fm(6\025)p Fk(C)5 b("G)p Fm(g)3071 1038 y Fq(\))1552 1178 y Fn(\024)25 b Fo(C)1713 1192 y Fl(0)1753 1178 y Fj(P)p Fq(\()p Fn(f)p Fo(!)s Fq(;)47 b Fo(V)2073 1192 y Fk(!)r(;n)2211 1178 y Fn(6\025)25 b Fo(C)7 b("G)p Fn(g)p Fq(\))1552 1318 y(=)25 b Fo(C)1713 1332 y Fl(0)1753 1318 y Fq(\(1)c Fn(\000)f Fj(P)p Fq(\()p Fn(f)p Fo(!)s Fq(;)47 b Fo(V)2265 1332 y Fk(!)r(;n)2403 1318 y Fn(\025)25 b Fo(C)7 b("G)p Fn(g)p Fq(\)\))26 b(=)f Fo(C)2986 1332 y Fl(0)3026 1318 y Fo(P)3084 1332 y Fk(C)5 b(")118 1474 y Fq(and)38 b(w)m(e)g(conclude)g(using)g (\(2.7\))i(and)d(taking)h(\()p Fo(\027)32 b Fn(\000)25 b Fo(d)p Fq(\))p Fo(\021)42 b(>)c Fq(1.)65 b(Here,)41 b(w)m(e)d(ha)m(v)m(e)i(used)d(the)i(fact)g(that)f(the)118 1582 y(in)m(tegrated)32 b(densit)m(y)f(of)g(states)h(of)g Fo(H)1410 1596 y Fk(!)r(;n)1553 1582 y Fq(is)f(b)s(ounded)e(lo)s(cally) g(uniformly)g(in)h(energy)-8 b(,)32 b(uniformly)c(in)i Fo(!)k Fq(and)118 1689 y Fo(n)p Fq(.)118 1861 y(2.3.)54 b Ft(The)28 b(pro)s(of)h(of)g(Lemma)e(2.3.)46 b Fq(W)-8 b(e)26 b(will)c(use)j(the)g(notations)g(and)g(results)e(of)i(section)h (5.2)g(in)d([9)q(].)39 b(The)118 1969 y(op)s(erator)29 b Fo(H)24 b Fq(+)16 b Fo("G)29 b Fq(is)f(\000-p)s(erio)s(dic.)38 b(Hence,)30 b(it)e(is)g(also)g(as)h(\(2)p Fo(n)17 b Fq(+)f(1\)\000-p)s (erio)s(dic.)39 b(When)28 b(doing)g(the)h(Flo)s(quet)118 2077 y(reduction)34 b(using)g(this)g(p)s(erio)s(d,)g(w)m(e)i(will)d (call)h(it)h(\()p Fo(H)30 b Fq(+)24 b Fo("G)p Fq(\))2233 2091 y Fk(n)2281 2077 y Fq(,)36 b(and)f(\()p Fo(H)c Fq(+)23 b Fo("G)p Fq(\))2908 2092 y Fk(n;\022)3046 2077 y Fq(denotes)35 b(this)f(op)s(erator)118 2185 y(considered)g(with)g(quasi-p)s(erio)s (dic)e(b)s(oundary)h(conditions)g(giv)m(en)j(b)m(y)f Fo(\022)i Fq(\(see)f(the)f(in)m(tro)s(duction)e(of)j([9])g(for)118 2293 y(more)31 b(details\).)331 2401 y(Recall)37 b(that)h Fo(#)p Fq(\(\()p Fo(H)33 b Fq(+)24 b Fo("G)p Fq(\))1292 2415 y Fk(n)1341 2401 y Fo(;)15 b(E)5 b Fq(\))38 b(denotes)g(the)g(n)m (um)m(b)s(er)e(of)i(Flo)s(quet)g(eigen)m(v)-5 b(alues)37 b(of)h(\()p Fo(H)32 b Fq(+)25 b Fo("G)p Fq(\))3795 2416 y Fk(n;\022)118 2509 y Fq(b)s(elo)m(w)d(energy)g Fo(E)28 b Fq(for)22 b(Flo)s(quet)h(parameter)g Fo(\022)s Fq(.)37 b(Standard)21 b(p)s(erturbation)f(theory)j(and)f(the)h(results)e(of)h (section)118 2616 y(5.2)32 b(in)d([9])i(\(see)g(\(5.24\)\))i(tell)d(us) f(that,)j(for)e Fo(")25 b(>)g Fq(0)31 b(su\016cien)m(tly)e(small,)1040 2768 y Fo(#)p Fq(\(\()p Fo(H)f Fq(+)20 b Fo("G)p Fq(\))1507 2783 y Fk(n;\022)1609 2768 y Fo(;)15 b Fq(0\))27 b(=)e Fo(#)p Fq(\(\()p Fo(H)7 b Fq(\))2094 2783 y Fk(n;\022)2196 2768 y Fo(;)15 b Fq(0\))26 b(=)f Fo(#)p Fq(\()p Fo(H)2603 2783 y Fk(!)r(;n;\022)2770 2768 y Fo(;)15 b Fq(0)2855 2730 y Fm(\000)2915 2768 y Fq(\))p Fo(:)-2857 b Fq(\(2.8\))118 2918 y(Hence,)32 b(if)d Fo(V)550 2932 y Fk(!)r(;n)688 2918 y Fn(\025)c Fo("G)p Fq(,)31 b(this)f(immediately)e(implies)g (that,)j(for)f Fo(E)h Fn(\025)25 b Fq(0,)31 b(one)g(has)1198 3069 y(0)26 b Fn(\024)f Fo(N)1438 3083 y Fk(!)r(;n)1550 3069 y Fq(\()p Fo(E)5 b Fq(\))22 b Fn(\000)e Fo(N)1878 3083 y Fk(!)r(;n)1991 3069 y Fq(\(0\))26 b Fn(\024)f Fo(N)2301 3083 y Fk(")2338 3069 y Fq(\()p Fo(E)5 b Fq(\))21 b Fn(\000)f Fo(N)2665 3083 y Fk(")2702 3069 y Fq(\(0\))118 3219 y(where)27 b Fo(N)451 3233 y Fk(")516 3219 y Fq(is)f(the)i(densit) m(y)f(of)h(states)h(of)f(the)g(p)s(erio)s(dic)d(op)s(erator)j Fo(H)22 b Fq(+)15 b Fo("G)p Fq(.)40 b(So)28 b(Lemma)f(2.2)i(will)c(b)s (e)i(pro)m(v)m(ed)118 3327 y(if)f(w)m(e)i(pro)m(v)m(e)g(that)g(there)g (exists)f Fo(C)32 b(>)25 b Fq(0)j(\(indep)s(enden)m(t)d(of)j Fo(")p Fq(\))g(suc)m(h)f(that,)i(for)e Fo(")g Fq(su\016cien)m(tly)f (small,)h(one)h(has)118 3435 y Fo(N)191 3449 y Fk(")228 3435 y Fq(\()p Fo("=C)7 b Fq(\))21 b Fn(\000)f Fo(N)642 3449 y Fk(")679 3435 y Fq(\(0\))26 b(=)f(0.)41 b(This)29 b(is)g(a)i(consequence)g(of)118 3600 y Ft(Lemma)i(2.4.)46 b Fi(L)-5 b(et)34 b Fo(H)41 b Fi(b)-5 b(e)34 b(a)g Fq(\000)p Fi(-p)-5 b(erio)g(dic)35 b(Schr\177)-46 b(odinger)36 b(op)-5 b(er)g(ator,)37 b(and)e Fo(G)f Fi(b)-5 b(e)34 b(a)h(non)f(ne)-5 b(gative)34 b Fq(\000)p Fi(-p)-5 b(erio)g(dic)118 3708 y(p)g(otential)42 b(that)f(is)f(lo)-5 b(c)g(al)5 b(ly)41 b(uniformly)g(in)f Fo(L)1681 3675 y Fk(p)1720 3708 y Fq(\()p Fj(R)1815 3675 y Fk(d)1862 3708 y Fq(\))g Fi(\(wher)-5 b(e)41 b Fo(p)e Fi(is)h(chosen)h(as)f(in)g(assumption)h (\(H.2\)\))g(and)118 3816 y(such)c(that)h Fo(G)33 b(>)g Fq(0)k Fi(on)g(some)h(op)-5 b(en)37 b(set.)55 b(Assume)36 b Fq(\()p Fn(\000)p Fo(\016)n(;)15 b Fq(0\))39 b Fi(do)-5 b(es)38 b(not)f(interse)-5 b(ct)38 b(the)f(sp)-5 b(e)g(ctrum)38 b(of)f Fo(H)44 b Fi(\(for)118 3924 y(some)33 b Fo(\016)d(>)24 b Fq(0)p Fi(\).)331 4032 y(Then,)43 b(ther)-5 b(e)42 b(exists)f Fo(C)46 b(>)40 b Fq(0)h Fi(and)h Fo(")1633 4046 y Fl(0)1713 4032 y Fo(>)e Fq(0)h Fi(such)g(that,)j(for)e Fq(0)e Fn(\024)g Fo(")h Fn(\024)f Fo(")2935 4046 y Fl(0)2974 4032 y Fi(,)j Fq(\()p Fn(\000)p Fo(\016)s(=)p Fq(2)p Fo(;)15 b("=C)7 b Fq(\))44 b Fi(do)-5 b(es)42 b(not)118 4139 y(interse)-5 b(ct)33 b(the)g(sp)-5 b(e)g(ctrum)35 b(of)d Fo(H)c Fq(+)19 b Fo("G)p Fi(.)118 4304 y Ft(Pro)s(of.)53 b Fq(Let)35 b(us)e(\014rst)h(assume)g(that)g(0)h(is)e(not)i(in)e(the)h (sp)s(ectrum)f(of)h Fo(H)7 b Fq(;)37 b(then,)e(there)f(exists)g Fo(\016)3485 4271 y Fm(0)3541 4304 y Fo(>)d Fq(0)k(suc)m(h)118 4412 y(that)26 b(\()p Fn(\000)p Fo(\016)n(;)15 b(\016)537 4379 y Fm(0)563 4412 y Fq(\))26 b(do)s(es)f(not)h(in)m(tersect)g(the)g (sp)s(ectrum)e(of)i Fo(H)7 b Fq(,)27 b(and)e(regular)g(p)s(erturbation) e(theory)j(tells)f(us)g(that,)118 4520 y(for)k Fo(")h Fq(su\016cien)m(tly)e(small,)g(\()p Fn(\000)p Fo(\016)s(=)p Fq(2)p Fo(;)15 b(\016)1366 4487 y Fm(0)1393 4520 y Fo(=)p Fq(2\))30 b(do)s(es)f(not)h(in)m(tersect)f(the)h(sp)s(ectrum)e(of)i Fo(H)25 b Fq(+)18 b Fo("G)p Fq(.)41 b(This)27 b(completes)118 4628 y(the)k(pro)s(of)e(of)i(Lemma)f(2.4)i(in)d(this)g(case.)331 4736 y(W)-8 b(e)37 b(no)m(w)e(assume)h(that)g(0)g(b)s(elongs)f(to)h (the)g(sp)s(ectrum)e(of)i Fo(H)7 b Fq(.)57 b(Let)36 b(\()p Fo(E)2846 4750 y Fk(n)2893 4736 y Fq(\()p Fo(";)15 b(\022)s Fq(\)\))3126 4750 y Fl(1)p Fm(\024)p Fk(n)3300 4736 y Fq(b)s(e)35 b(the)h(Flo)s(quet)118 4844 y(eigen)m(v)-5 b(alues)38 b(of)g(\()p Fo(H)33 b Fq(+)25 b Fo("G)p Fq(\))1095 4859 y Fk(\022)1173 4844 y Fq(ordered)37 b(increasingly;)j(they)e(are)h (con)m(tin)m(uous)e(in)g Fo(")h Fq(and)g Fo(\022)s Fq(.)62 b(Hence,)41 b(there)118 4952 y(exists)36 b Fo(")417 4966 y Fl(0)491 4952 y Fo(>)e Fq(0)i(and)f Fo(n)914 4966 y Fl(0)987 4952 y Fn(\025)g Fq(1)h(suc)m(h)f(that,)j(for)e(0)f Fn(\024)f Fo(")g Fn(\024)g Fq(0,)k(one)e(has)g(that)g Fo(E)2841 4966 y Fk(n)2888 4952 y Fq(\()p Fo(";)15 b(\022)s Fq(\))35 b Fo(<)f(E)3293 4966 y Fl(0)3369 4952 y Fq(if)g(and)i(only)f (if)118 5060 y Fo(n)26 b(<)g(n)351 5074 y Fl(0)390 5060 y Fq(.)43 b(Then,)31 b(to)h(pro)m(v)m(e)g(Lemma)f(2.4,)i(w)m(e)e(need)g (to)h(pro)m(v)m(e)g(that,)g(there)g(exists)e Fo(C)j(>)27 b Fq(0)k(and)g Fo(")3483 5074 y Fl(0)3549 5060 y Fo(>)26 b Fq(0)32 b(suc)m(h)118 5168 y(that,)f(for)f(0)c Fn(\024)f Fo(")h Fn(\024)f Fo(")852 5182 y Fl(0)892 5168 y Fq(,)30 b(one)h(has)1684 5318 y Fo(E)1751 5332 y Fk(n)1794 5341 y Fh(0)1832 5318 y Fq(\()p Fo(";)15 b(\022)s Fq(\))26 b Fn(\025)f Fo("=C)q(:)-2212 b Fq(\(2.9\))1990 5454 y Fl(5)p eop %%Page: 6 6 6 5 bop 331 241 a Fq(De\014ne)32 b Fn(S)i Fq(=)27 b Fn(f)p Fo(\022)j Fn(2)d Fj(T)1066 208 y Fm(\003)1107 241 y Fq(;)47 b Fo(E)1246 255 y Fk(n)1289 264 y Fh(0)1328 241 y Fq(\(0)p Fo(;)15 b(\022)s Fq(\))28 b(=)f(0)p Fn(g)p Fq(;)33 b Fn(S)39 b Fq(is)31 b(compact.)46 b(Outside)30 b(of)i(a)g(neigh)m(b)s (orho)s(o)s(d)d(of)j Fn(S)7 b Fq(,)32 b(\(2.9\))118 349 y(is)e(certainly)h(satis\014ed)f(as,)i(there,)g(one)f(has)g Fo(E)1732 363 y Fk(n)1775 372 y Fh(0)1814 349 y Fq(\()p Fo(";)15 b(\022)s Fq(\))28 b Fn(\025)e Fo(E)2204 363 y Fk(n)2247 372 y Fh(0)2286 349 y Fq(\(0)p Fo(;)15 b(\022)s Fq(\))27 b Fn(\025)f Fo(\016)2654 316 y Fm(0)2710 349 y Fq(\(for)31 b(some)h Fo(\016)3157 316 y Fm(0)3208 349 y Fo(>)26 b Fq(0\).)44 b(So)31 b(w)m(e)h(need)118 457 y(only)e(to)h(pro)m(v)m(e)g(\(2.9\))h(in)d(a)i(neigh)m(b)s(orho)s(o)s (d)d(of)i Fn(S)7 b Fq(.)118 565 y(Therefore,)25 b(pic)m(k)e Fo(\022)778 579 y Fl(0)842 565 y Fn(2)i(S)7 b Fq(.)38 b(Let)24 b Fo(p)p Fq(\()p Fo(\022)1333 579 y Fl(0)1373 565 y Fq(\))f(b)s(e)g(the)h(m)m(ultiplicit)m(y)d(of)i(0)h(as)g(an)f (eigen)m(v)-5 b(alue)24 b(of)f Fo(H)3168 580 y Fk(\022)3232 565 y Fq(=)i(\()p Fo(H)14 b Fq(+)6 b Fo("G)p Fq(\))3678 583 y Fk(\022)15 b Fm(j)p Fk(")p Fl(=0)3872 565 y Fq(;)118 672 y(it)30 b(is)f(\014nite.)40 b(F)-8 b(or)31 b Fo(\022)i Fq(close)d(to)i Fo(\022)1179 686 y Fl(0)1248 672 y Fq(and)e Fo(")g Fq(small,)g(de\014ne)f(the)i(Riesz)f(pro)5 b(jector)1387 915 y(\005)1455 930 y Fk(";\022)1572 915 y Fq(=)1668 791 y Fg(I)1719 997 y Fm(C)1756 1006 y Fh(0)1795 915 y Fq(\()p Fo(z)24 b Fn(\000)c Fq(\()p Fo(H)28 b Fq(+)20 b Fo("G)p Fq(\))2365 930 y Fk(\022)2405 915 y Fq(\))2440 877 y Fm(\000)p Fl(1)2535 915 y Fo(dz)118 1177 y Fq(where)j Fn(C)422 1191 y Fl(0)487 1177 y Fq(=)i Fn(fj)p Fo(z)t Fn(j)h Fq(=)f Fo(\016)889 1144 y Fm(0)913 1177 y Fn(g)g Fq(is)d(a)j(small)d(lo)s(op)h(in)f(the)i(complex)g(plane.)37 b(\005)2503 1192 y Fk(";\022)2618 1177 y Fq(is)23 b(the)h(pro)5 b(jector)24 b(on)m(to)h(the)f(sp)s(ectral)118 1285 y(space)37 b(asso)s(ciated)g(to)h(the)e(in)m(terv)-5 b(al)36 b(\()p Fn(\000)p Fo(\016)1570 1252 y Fm(0)1594 1285 y Fo(;)15 b(\016)1677 1252 y Fm(0)1702 1285 y Fq(\))37 b(and)f(the)g(op)s(erator) h(\()p Fo(H)32 b Fq(+)24 b Fo("G)p Fq(\))2878 1300 y Fk(\022)2918 1285 y Fq(.)59 b(It)37 b(is)f(real)g(analytic)g(in)f Fo(\022)118 1393 y Fq(and)d Fo(")p Fq(;)h(hence)f(of)h(constan)m(t)g (rank.)45 b(Moreo)m(v)m(er,)35 b Fo(E)1869 1407 y Fk(n)1912 1416 y Fh(0)1951 1393 y Fq(\()p Fo(";)15 b(\022)s Fq(\))33 b(is)e(the)h(smallest)f(eigen)m(v)-5 b(alue)32 b(of)h(the)f(\014nite)f (rank)118 1503 y(op)s(erator)g Fo(O)554 1518 y Fk(";\022)671 1503 y Fq(=)25 b(\005)835 1518 y Fk(";\022)926 1503 y Fq(\()p Fo(H)i Fq(+)20 b Fo("G)p Fq(\))1303 1518 y Fk(\022)1344 1503 y Fq(\005)1412 1518 y Fk(";\022)1533 1503 y Fq(\(acting)31 b(on)g(\005)2036 1518 y Fk(";\022)2127 1503 y Fo(L)2189 1470 y Fl(2)2189 1531 y(lo)r(c)2281 1503 y Fq(\()p Fo(R)2386 1470 y Fk(d)2427 1503 y Fq(\)\).)118 1611 y(The)f(op)s(erator)h Fo(O)741 1626 y Fk(";\022)863 1611 y Fq(is)e(real)h(analytic)g(in)f Fo(")i Fq(and)e Fo(\022)s Fq(;)h(di\013eren)m(tiating)f(it)h(in)f Fo(")p Fq(,)i(w)m(e)g(obtain)921 1808 y Fo(dO)1037 1823 y Fk(";\022)p 921 1850 208 4 v 980 1934 a Fo(d")1164 1871 y Fq(=)25 b(\005)1328 1886 y Fk(";\022)1419 1871 y Fo(G)p Fq(\005)1558 1886 y Fk(";\022)1772 1871 y Fq(and)2054 1743 y Fg(\022)2131 1808 y Fo(dO)2247 1823 y Fk(";\022)p 2131 1850 V 2190 1934 a Fo(d")2349 1743 y Fg(\023)2416 1962 y Fm(j)p Fk(")p Fl(=0)2588 1871 y Fq(=)g(\005)2752 1886 y Fl(0)p Fk(;\022)2846 1871 y Fo(G)p Fq(\005)2985 1886 y Fl(0)p Fk(;\022)3079 1871 y Fo(:)-2986 b Fq(\(2.10\))118 2141 y(One)30 b(pro)m(v)m(es)118 2334 y Ft(Lemma)j(2.5.)46 b Fi(F)-7 b(or)34 b(some)f Fo(C)f(>)25 b Fq(0)33 b Fi(and)g Fo(\022)i Fi(close)e(to)g Fo(\022)1999 2348 y Fl(0)2038 2334 y Fi(,)g(one)g(has)1575 2535 y Fq(\005)1643 2550 y Fl(0)p Fk(;\022)1737 2535 y Fo(G)p Fq(\005)1876 2550 y Fl(0)p Fk(;\022)1995 2535 y Fn(\025)25 b Fq(1)p Fo(=C)7 b Fq(\005)2321 2550 y Fl(0)p Fk(;\022)2416 2535 y Fo(:)-2323 b Fq(\(2.11\))118 2733 y(This)29 b(and)g(\(2.10\))k(implies)28 b(that,)j(for)f Fo(")h Fq(p)s(ositiv)m(e,)e(su\016cien)m(tly)g(small,)h (one)g(has)1742 2957 y Fo(O)1811 2972 y Fk(";\022)1927 2957 y Fn(\025)2048 2896 y Fo(")p 2033 2936 72 4 v 2033 3019 a(C)2115 2957 y Fq(\005)2183 2972 y Fk(";\022)118 2957 y Fq(\(2.12\))118 3190 y(as,)h(for)f(an)m(y)h Fo(")p Fq(,)g Fo(")706 3157 y Fm(0)760 3190 y Fq(su\016cien)m(tly)e(small)g (and)g Fo(\022)s Fq(,)h Fo(\022)1776 3157 y Fm(0)1829 3190 y Fq(su\016cien)m(tly)f(close)i(to)g Fo(\022)2656 3204 y Fl(0)2695 3190 y Fq(,)g(one)f(has)1572 3429 y(\005)1640 3444 y Fk(";\022)1731 3429 y Fq(\005)1799 3444 y Fk(";\022)1887 3425 y Fe(0)1913 3429 y Fq(\005)1981 3444 y Fk(";\022)2098 3429 y Fn(\025)2204 3367 y Fq(1)p 2204 3408 46 4 v 2204 3491 a(2)2259 3429 y(\005)2327 3444 y Fk(";\022)2419 3429 y Fo(:)-2326 b Fq(\(2.13\))118 3660 y(Equation)27 b(\(2.12\))j(then)e(immediately)e(implies)f(\(2.9\))k(for)f Fo(")g Fq(su\016cien)m(tly)e(small)g(and)h Fo(\022)j Fq(su\016cien)m(tly)d(close)h(to)118 3768 y Fo(\022)161 3782 y Fl(0)200 3768 y Fq(.)47 b(W)-8 b(e)34 b(the)f(obtain)f(\(2.9\))i (for)f(a)g(neigh)m(b)s(orho)s(o)s(d)d(of)i Fn(S)40 b Fq(as)33 b Fn(S)39 b Fq(is)32 b(compact.)48 b(This)31 b(completes)i(the)g(pro)s(of)e(of)118 3876 y(Lemma)g(2.4.)p 3832 3876 4 62 v 3836 3818 55 4 v 3836 3876 V 3890 3876 4 62 v 118 3984 a Ft(Pro)s(of)36 b(of)f(Lemma)e(2.5.)41 b Fq(First,)30 b(w)m(e)g(note)h(that)754 4184 y(\005)822 4199 y Fl(0)p Fk(;\022)916 4184 y Fo(G)p Fq(\005)1055 4199 y Fl(0)p Fk(;\022)1175 4184 y Fq(=)25 b(\005)1339 4199 y Fl(0)p Fk(;\022)1427 4208 y Fh(0)1466 4184 y Fo(G)p Fq(\005)1605 4199 y Fl(0)p Fk(;\022)1693 4208 y Fh(0)1752 4184 y Fn(\000)20 b Fq(\005)1911 4199 y Fl(0)p Fk(;\022)2020 4184 y Fq(\()q(\005)2124 4199 y Fl(0)p Fk(;\022)2212 4208 y Fh(0)2251 4184 y Fo(G)p Fq(\005)2390 4199 y Fl(0)p Fk(;\022)2478 4208 y Fh(0)2537 4184 y Fn(\000)g Fq(\005)2696 4199 y Fl(0)p Fk(;\022)2790 4184 y Fo(G)p Fq(\005)2929 4199 y Fl(0)p Fk(;\022)3024 4184 y Fq(\))15 b(\005)3142 4199 y Fl(0)p Fk(;\022)3236 4184 y Fo(:)118 4390 y Fq(Hence,)34 b(as)e Fo(\022)e Fn(7!)e Fq(\005)789 4405 y Fl(0)p Fk(;\022)884 4390 y Fo(G)p Fq(\005)1023 4405 y Fl(0)p Fk(;\022)1149 4390 y Fq(is)j(con)m(tin)m(uous,)i(b)m(y)g(\(2.13\))r(,)g(w)m(e)g(just) e(need)h(to)h(pro)m(v)m(e)g(\(2.11\))h(for)e Fo(\022)e Fq(=)e Fo(\022)3655 4404 y Fl(0)3694 4390 y Fq(.)46 b(Let)118 4497 y(\()p Fo( )212 4511 y Fk(j)249 4497 y Fq(\))284 4516 y Fl(1)p Fm(\024)p Fk(j)t Fm(\024)p Fk(p)p Fl(\()p Fk(\022)558 4525 y Fh(0)592 4516 y Fl(\))660 4497 y Fq(b)s(e)35 b(a)h(basis)e(of)i(eigen)m(v)m(ector)i(in)c(the)i(k)m(ernel)f(of)h Fo(H)2416 4512 y Fk(\022)2449 4521 y Fh(0)2487 4497 y Fq(.)57 b(Then)34 b(\005)2879 4512 y Fl(0)p Fk(;\022)3007 4497 y Fq(=)3112 4429 y Fg(P)3208 4524 y Fl(1)p Fm(\024)p Fk(j)t Fm(\024)p Fk(p)p Fl(\()p Fk(\022)3482 4533 y Fh(0)3516 4524 y Fl(\))3563 4497 y Fo( )3622 4511 y Fk(j)3682 4497 y Fn(\012)24 b Fo( )3836 4511 y Fk(j)3872 4497 y Fq(.)118 4617 y(Hence,)33 b(\(2.11\))g(will)c(b)s(e)i(satis\014ed)f(if)g(and)h (only)g(if)f(the)h(matrix)g(\(\()p Fn(h)p Fo(G )2562 4631 y Fk(i)2592 4617 y Fo(;)15 b( )2691 4631 y Fk(j)2728 4617 y Fn(i)p Fq(\)\))2833 4631 y Fk(i;j)2946 4617 y Fq(is)30 b(p)s(ositiv)m(e)h(de\014nite.)42 b(It)32 b(is)118 4725 y(non)g(negativ)m(e.)48 b(If)32 b(0)h(w)m(ere)g(an)f(eigen)m(v)-5 b(alue)33 b(of)f(this)g(matrix,)g(there)h(w)m(ould)e(exist)h Fo( )g Fn(2)p Fq(Span\(\()p Fo( )3451 4739 y Fk(j)3487 4725 y Fq(\))3522 4744 y Fl(1)p Fm(\024)p Fk(j)t Fm(\024)p Fk(p)p Fl(\()p Fk(\022)3796 4753 y Fh(0)3831 4744 y Fl(\))3862 4725 y Fq(\))118 4892 y(suc)m(h)i(that)528 4769 y Fg(Z)579 4975 y Fk(C)629 4984 y Fh(0)683 4892 y Fo(G)p Fq(\()p Fo(x)p Fq(\))p Fn(j)p Fo( )s Fq(\()p Fo(x)p Fq(\))p Fn(j)1110 4855 y Fl(2)1152 4892 y Fo(dx)e Fq(=)g(0.)53 b(Hence,)37 b(as)d Fo(G)f(>)f Fq(0)i(on)h(some)g(op)s(en)e(set,)k Fo( )g Fq(w)m(ould)d(v)-5 b(anish)32 b(on)j(this)118 5061 y(op)s(en)g(set:)51 b(this)35 b(is)f(not)i(p)s(ossible)d(b)m(y)j (the)f(unique)f(con)m(tin)m(uation)h(principle)d(as)k Fo( )j Fq(satis\014es)c Fo(H)7 b( )37 b Fq(=)d(0)i(\(see)118 5169 y(e.g.)c([3,)f(11)q(]\).)p 3832 5169 V 3836 5111 55 4 v 3836 5169 V 3890 5169 4 62 v 1990 5454 a Fl(6)p eop %%Page: 7 7 7 6 bop 118 241 a Fq(2.4.)54 b Ft(The)41 b(pro)s(of)i(of)f(Lemma)e (2.2.)46 b Fq(Let)37 b(us)e(write)h(out)h Fo(V)2278 255 y Fk(!)r(;n)2427 241 y Fq(and)f(use)g(assumption)f(\(H.2bis.1\).)60 b(One)118 349 y(computes)602 513 y Fo(V)655 527 y Fk(!)r(;n)768 513 y Fq(\()p Fo(x)p Fq(\))26 b(=)1012 427 y Fg(X)1012 624 y Fk(\015)t Fm(2)p Fl(\000)1158 513 y Fo(!)1215 532 y Fl([)p Fk(\015)t Fl(])1298 513 y Fo(V)21 b Fq(\()p Fo(x)f Fn(\000)g Fo(\015)5 b Fq(\))31 b(where)f([)p Fo(\015)5 b Fq(])26 b(=)f Fo(\015)35 b Fq(mo)s(d)30 b(\(2)p Fo(n)20 b Fq(+)g(1\)\000)916 777 y Fn(\025)1012 690 y Fg(X)1012 887 y Fk(\015)t Fm(2)p Fl(\000)1158 777 y Fo(!)1215 795 y Fl([)p Fk(\015)t Fl(])1316 690 y Fg(X)1314 887 y Fk(\013)p Fm(2)p Fl(\000)1465 777 y Fo(g)1508 791 y Fm(\000)1568 777 y Fq(\()p Fo(x)g Fn(\000)g Fo(\015)25 b Fn(\000)20 b Fo(\013)p Fq(\))h Fn(\001)g Fq(\(1+)26 b Fn(j)f Fo(\013)h Fn(j)p Fq(\))2460 739 y Fm(\000)p Fk(\027)916 1040 y Fq(=)1014 953 y Fg(X)1012 1150 y Fk(\013)p Fm(2)p Fl(\000)1163 1040 y Fo(A)1231 1054 y Fk(\013)1281 1040 y Fq(\()p Fo(!)s Fq(\))p Fo(g)1454 1054 y Fm(\000)1514 1040 y Fq(\()p Fo(x)20 b Fn(\000)g Fo(\013)p Fq(\))32 b(where)d Fo(A)2167 1054 y Fk(\013)2217 1040 y Fq(\()p Fo(!)s Fq(\))d(=)2469 953 y Fg(X)2469 1150 y Fk(\015)t Fm(2)p Fl(\000)2615 1040 y Fo(!)2672 1058 y Fl([)p Fk(\015)t Fl(])2756 1040 y Fq(\(1+)f Fn(j)h Fo(\013)21 b Fn(\000)e Fo(\015)31 b Fn(j)p Fq(\))3290 1002 y Fm(\000)p Fk(\027)3388 1040 y Fo(:)118 1292 y Fq(One)37 b(c)m(hec)m(ks)i(that)f Fo(A)877 1306 y Fk(\013)926 1292 y Fq(\()p Fo(!)s Fq(\))g(=)e Fo(A)1269 1307 y Fk(\013)p Fl(+)p Fk(\014)1417 1292 y Fq(\()p Fo(!)s Fq(\))h(for)h Fo(\014)k Fn(2)36 b Fq(\(2)p Fo(n)25 b Fq(+)g(1\)\000.)62 b(De\014ne)37 b(\000)2746 1306 y Fl(2)p Fk(n)p Fl(+1)2955 1292 y Fq(=)g(\000)p Fo(=)p Fq(\(2)p Fo(n)25 b Fq(+)g(1\)\000)37 b(=)g Fn(f)p Fo(\015)42 b Fn(2)118 1400 y Fq(\000;)15 b Fn(j)p Fo(\015)5 b Fn(j)26 b(\024)f Fo(n)p Fn(g)p Fq(.)41 b(Then,)29 b(w)m(e)i(ha)m(v)m (e)574 1569 y Fj(P)p Fq(\()p Fn(f)p Fo(V)762 1583 y Fk(!)r(;n)901 1569 y Fn(\025)25 b Fo("G)p Fn(g)p Fq(\))i Fn(\025)e Fj(P)p Fq(\()p Fn(f8)p Fo(\015)31 b Fn(2)25 b Fq(\000)1720 1583 y Fl(2)p Fk(n)p Fl(+1)1893 1569 y Fq(;)45 b Fo(A)2031 1583 y Fk(\015)2076 1569 y Fq(\()p Fo(!)s Fq(\))26 b Fn(\025)f Fo(")p Fn(g)p Fq(\))h Fn(\025)f Fq(1)c Fn(\000)2804 1483 y Fg(X)2729 1680 y Fk(\015)t Fm(2)p Fl(\000)2860 1689 y Fh(2)p Ff(n)p Fh(+1)3025 1569 y Fj(P)p Fq(\()p Fn(f)p Fo(A)3228 1583 y Fk(\015)3274 1569 y Fq(\()p Fo(!)s Fq(\))26 b Fn(\024)f Fo(")p Fn(g)p Fq(\))p Fo(:)118 1825 y Fq(Note)36 b(that,)h(as)f(the)f(\()p Fo(!)940 1839 y Fk(\015)984 1825 y Fq(\))1019 1839 y Fk(\015)1099 1825 y Fq(are)g(stationary)-8 b(,)38 b(the)d(distribution)c(of)k(the)g (random)g(v)-5 b(ariable)34 b Fo(A)3395 1839 y Fk(\015)3439 1825 y Fq(\()p Fo(!)s Fq(\))i(is)e(inde-)118 1933 y(p)s(enden)m(t)c(of) g Fo(\015)5 b Fq(.)41 b(Hence)1054 2085 y Fj(P)p Fq(\()p Fn(f)p Fo(V)1242 2099 y Fk(!)r(;n)1381 2085 y Fn(\025)25 b Fo("G)p Fn(g)p Fq(\))i Fn(\025)e Fq(1)c Fn(\000)f Fq(\(2)p Fo(n)g Fq(+)g(1\))2276 2048 y Fk(d)2318 2085 y Fj(P)p Fq(\()p Fn(f)p Fo(A)2521 2099 y Fl(0)2562 2085 y Fq(\()p Fo(!)s Fq(\))25 b Fn(\024)g Fo(")p Fn(g)p Fq(\))p Fo(:)-2842 b Fq(\(2.14\))118 2242 y(So)30 b(w)m(e)h(only)f(need)g(to)h(estimate)g Fj(P)p Fq(\()p Fn(f)p Fo(A)1468 2256 y Fl(0)1509 2242 y Fq(\()p Fo(!)s Fq(\))26 b Fn(\024)f Fo(")p Fn(g)p Fq(\).)42 b(Therefore,)30 b(w)m(e)h(compute)832 2411 y Fo(A)900 2425 y Fl(0)939 2411 y Fq(\()p Fo(!)s Fq(\))26 b(=)1304 2325 y Fg(X)1191 2526 y Fk(\014)s Fm(2)p Fl(\(2)p Fk(n)p Fl(+1\)\000)1639 2325 y Fg(X)1564 2522 y Fk(\015)t Fm(2)p Fl(\000)1695 2531 y Fh(2)p Ff(n)p Fh(+1)1860 2411 y Fo(!)1917 2425 y Fk(\015)1961 2411 y Fq(\(1+)g Fn(j)f Fo(\014)h Fq(+)20 b Fo(\015)30 b Fn(j)p Fq(\))2493 2374 y Fm(\000)p Fk(\027)1095 2683 y Fq(=)1266 2596 y Fg(X)1191 2793 y Fk(\015)t Fm(2)p Fl(\000)1322 2802 y Fh(2)p Ff(n)p Fh(+1)1488 2683 y Fo(!)1545 2697 y Fk(\015)1589 2683 y Fo(u)1641 2697 y Fk(\015)1715 2683 y Fq(where)g Fo(u)2030 2697 y Fk(\015)2100 2683 y Fq(=)2309 2596 y Fg(X)2196 2798 y Fk(\014)s Fm(2)p Fl(\(2)p Fk(n)p Fl(+1\)\000)2553 2683 y Fq(\(1+)c Fn(j)g Fo(\014)f Fq(+)20 b Fo(\015)30 b Fn(j)p Fq(\))3085 2645 y Fm(\000)p Fk(\027)118 2938 y Fq(Using)g(Mark)m(o)m (v's)i(inequalit)m(y)-8 b(,)29 b(for)h(an)m(y)h Fo(\025)25 b(>)g Fq(0,)31 b(one)g(obtains)1201 3111 y Fj(P)p Fq(\()p Fo(A)1359 3125 y Fl(0)1400 3111 y Fq(\()p Fo(!)s Fq(\))26 b Fn(\024)f Fo(")p Fq(\))h Fn(\024)f Fj(E)1920 3010 y Fg(\020)1981 3111 y Fo(e)2023 3059 y Fk(\025)p Fl(\()p Fk(")p Fm(\000)2179 3009 y Fc(P)2254 3080 y Ff(\015)s Fe(2)p Fh(\000)2369 3095 y(2)p Ff(n)p Fh(+1)2534 3059 y Fk(!)2578 3067 y Ff(\015)2618 3059 y Fk(u)2659 3067 y Ff(\015)2699 3059 y Fl(\))2730 3010 y Fg(\021)1755 3310 y Fn(\024)g Fo(e)1893 3273 y Fk(\025")2069 3224 y Fg(Y)1986 3421 y Fk(\015)t Fm(2)p Fl(\000)2117 3430 y Fh(2)p Ff(n)p Fh(+1)2283 3310 y Fj(E)2352 3209 y Fg(\020)2413 3310 y Fo(e)2455 3273 y Fm(\000)p Fk(\025!)2595 3281 y Ff(\015)2635 3273 y Fk(u)2676 3281 y Ff(\015)2720 3209 y Fg(\021)2789 3310 y Fo(:)118 3249 y Fq(\(2.15\))118 3561 y(Moreo)m(v)m(er,)33 b(there)d(exists)g Fo(\021)f(>)c Fq(0)31 b(suc)m(h)f(that,)h(if)e Fo(\025u)1887 3575 y Fk(\015)1957 3561 y Fo(<)c(\021)s Fq(,)31 b(then)627 3743 y Fj(E)697 3642 y Fg(\020)757 3743 y Fo(e)799 3706 y Fm(\000)p Fk(\025!)939 3714 y Ff(\015)979 3706 y Fk(u)1020 3714 y Ff(\015)1064 3642 y Fg(\021)1144 3743 y Fn(\024)25 b Fo(e)1282 3706 y Fm(\000)p 1337 3668 79 3 v Fk(!)1381 3715 y Fh(0)1415 3706 y Fk(\025u)1497 3714 y Ff(\015)1537 3706 y Fk(=)p Fl(2)1642 3743 y Fq(where)p 1905 3693 97 4 v 30 w Fo(!)1962 3757 y Fl(0)2026 3743 y Fq(=)g Fj(E)11 b Fq(\()p Fo(!)2269 3757 y Fl(0)2314 3743 y Fq(\))26 b Fo(>)f Fq(0)31 b(b)m(y)f(assumption)f(\(H.3\))q Fo(:)-3270 b Fq(\(2.16\))118 3927 y(Indeed,)29 b(using)e(the)i(T)-8 b(a)m(ylor)29 b(expansion)f(of)h Fo(x)c Fn(7!)h Fo(e)1872 3894 y Fm(\000)p Fk(x)1971 3927 y Fq(,)j(as)g(the)g(random)g(v)-5 b(ariables)27 b(\()p Fo(!)3090 3941 y Fk(\015)3135 3927 y Fq(\))3170 3941 y Fk(\015)3243 3927 y Fq(are)i(b)s(ounded,)f(for)118 4035 y Fo(\025u)223 4049 y Fk(\015)298 4035 y Fq(su\016cien)m(tly)h (small,)g(one)i(obtains)565 4217 y Fj(E)634 4116 y Fg(\020)695 4217 y Fo(e)737 4180 y Fm(\000)p Fk(\025!)877 4188 y Ff(\015)917 4180 y Fk(u)958 4188 y Ff(\015)1002 4116 y Fg(\021)1081 4217 y Fn(\024)25 b Fq(1)c Fn(\000)f Fj(E)10 b Fq(\()q Fo(!)1481 4231 y Fk(\015)1531 4217 y Fq(\))p Fo(\025u)1671 4231 y Fk(\015)1736 4217 y Fq(+)20 b Fo(C)7 b Fq(\()p Fo(\025u)2039 4231 y Fk(\015)2083 4217 y Fq(\))2118 4180 y Fl(2)2183 4217 y Fn(\024)25 b Fq(1)c Fn(\000)f Fj(E)10 b Fq(\()q Fo(!)2583 4231 y Fk(\015)2633 4217 y Fq(\))p Fo(\025u)2773 4231 y Fk(\015)2817 4217 y Fq(\(1)21 b Fn(\000)f Fo(C)7 b(\021)s Fq(\))26 b Fn(\024)f Fo(e)3328 4180 y Fm(\000)p 3383 4142 79 3 v Fk(!)3427 4189 y Fh(0)3461 4180 y Fk(\025u)3543 4188 y Ff(\015)3583 4180 y Fk(=)p Fl(2)3658 4217 y Fo(:)118 4396 y Fq(Plugging)30 b(\(2.16\))j(in)m(to)d (\(2.15\))s(,)g(w)m(e)h(compute)1159 4595 y(log)r(\()p Fj(P)p Fq(\()p Fo(A)1470 4609 y Fl(0)1511 4595 y Fq(\()p Fo(!)s Fq(\))26 b Fn(\024)f Fo(")p Fq(\)\))h Fn(\024)f Fo(\025")c Fn(\000)2213 4533 y Fo(\025)p 2266 4483 97 4 v(!)2323 4547 y Fl(0)p 2213 4574 150 4 v 2265 4657 a Fq(2)2393 4595 y Fn(\001)2513 4508 y Fg(X)2438 4705 y Fk(\015)t Fm(2)p Fl(\000)2569 4714 y Fh(2)p Ff(n)p Fh(+1)2471 4776 y Fk(\025u)2553 4784 y Ff(\015)2593 4776 y Fk(<\021)2735 4595 y Fo(u)2787 4609 y Fk(\015)2831 4595 y Fo(:)-2738 b Fq(\(2.17\))118 4938 y(Assume)30 b(that)h Fo(n)25 b(>>)f(\025)951 4905 y Fl(1)p Fk(=\027)1065 4938 y Fq(;)31 b(then,)f(the)h(de\014nition)d(of)i Fo(u)2069 4952 y Fk(\015)2144 4938 y Fq(implies)e(that,)j(for)f(some)h Fo(C)g(>)25 b Fq(0,)31 b(one)g(has)1084 5049 y Fg(X)1009 5246 y Fk(\015)t Fm(2)p Fl(\000)1140 5255 y Fh(2)p Ff(n)p Fh(+1)1043 5317 y Fk(\025u)1125 5325 y Ff(\015)1165 5317 y Fk(<\021)1306 5136 y Fo(u)1358 5150 y Fk(\015)1427 5136 y Fn(\025)1771 5049 y Fg(X)1771 5246 y Fk(\015)t Fm(2)p Fl(\000)1523 5329 y Fk(C)5 b(\025)1619 5305 y Fh(1)p Ff(=\027)1720 5329 y Fm(\024j)p Fk(\015)t Fm(j\024)p Fk(n)p Fm(\000)p Fk(\025)2049 5305 y Fh(1)p Ff(=\027)2149 5136 y Fq(\(1)21 b(+)f Fn(j)p Fo(\015)5 b Fn(j)p Fq(\))2478 5098 y Fm(\000)p Fk(\027)2602 5136 y Fn(\025)2721 5074 y Fq(1)p 2708 5115 72 4 v 2708 5198 a Fo(C)2790 5136 y(\025)2853 5070 y Ff(d)p Fe(\000)p Ff(\027)p 2853 5083 115 3 v 2893 5124 a(\027)2981 5136 y Fo(:)1990 5454 y Fl(7)p eop %%Page: 8 8 8 7 bop 118 249 a Fq(Plugging)29 b(this)g(bac)m(k)j(in)m(to)e(\(2.17\)) j(and)c(pic)m(king)g Fo(\025)d Fq(=)f Fo(\032")2138 179 y Ff(\027)p 2098 191 115 3 v 2098 232 a(d)p Fe(\000)p Ff(\027)2226 249 y Fq(,)31 b(w)m(e)g(obtain)887 438 y(log)r(\()p Fj(P)p Fq(\()p Fo(A)1198 452 y Fl(0)1239 438 y Fq(\()p Fo(!)s Fq(\))25 b Fn(\024)g Fo(")p Fq(\)\))i Fn(\024)e(\000)p Fo(\032)p Fq(\()p 1878 388 97 4 v Fo(!)1935 452 y Fl(0)1974 438 y Fo(\032)2021 400 y Fk(d=\027)2136 438 y Fo(=C)i Fn(\000)20 b Fq(1\))p Fo(")2486 395 y Fm(\000)2593 368 y Ff(d)p 2552 380 115 3 v 2552 421 a(\027)s Fe(\000)p Ff(d)2705 438 y Fn(\024)2824 376 y Fq(1)p 2811 417 72 4 v 2811 500 a Fo(C)2893 438 y(")2935 395 y Fm(\000)3041 368 y Ff(d)p 3000 380 115 3 v 3000 421 a(\027)s Fe(\000)p Ff(d)118 438 y Fq(\(2.18\))118 610 y(for)30 b Fo(\032)h Fq(su\016cien)m(tly)e(large.)118 717 y(So)e(that,)i(if)d(w)m(e)i(pic)m (k)f(e.g.)41 b Fo(n)25 b Fn(\030)g Fo(")1255 684 y Fm(\000)p Fk(\032)1378 717 y Fq(with)h Fo(\032)f(>)1807 682 y Fl(1)p 1760 697 131 4 v 1760 749 a Fk(\027)t Fm(\000)p Fk(d)1900 717 y Fq(,)j(and)f(plug)f(estimate)i(\(2.18\))i(in)m(to)d(\(2.14\))r(,) h(w)m(e)g(complete)118 827 y(the)j(pro)s(of)e(of)i(Lemma)f(2.2.)p 3832 827 4 62 v 3836 769 55 4 v 3836 827 V 3890 827 4 62 v 1743 1011 a Fp(References)157 1156 y Fs([1])39 b(J.)18 b(M.)h(Com)n(b)r(es)e(and)g(L.)h(Thomas.)h(Asymptotic)d(b)r(eha)n(vior) h(of)i(eigenfunctions)g(for)f(m)n(ulti-particle)f(Sc)n(hr\177)-38 b(odinger)18 b(op)r(erators.)276 1247 y Fb(Communic)l(ations)28 b(in)f(Mathematic)l(al)i(Physics)p Fs(,)e(34:251{270,)i(1973.)157 1339 y([2])39 b(L.)26 b(H\177)-38 b(ormander.)25 b Fb(The)k(analysis)f (of)g(line)l(ar)f(p)l(artial)i(di\013er)l(ential)f(e)l(quations.)h(I)p Fs(,)d(v)n(olume)f(256)i(of)g Fb(Grund)t(lehr)l(en)j(der)f(Math-)276 1430 y(ematischen)g(Wissenschaften)p Fs(.)f(Springer)d(V)-6 b(erlag,)27 b(1990.)157 1521 y([3])39 b(D.)24 b(Jerison)h(and)f(C.)h (Kenig.)f(Unique)f(con)n(tin)n(uation)h(and)g(absence)g(of)h(p)r (ositiv)n(e)f(eigen)n(v)l(alues)h(for)g(Sc)n(hr\177)-38 b(odinger)24 b(op)r(erators.)276 1613 y Fb(A)n(nnals)k(of)f (Mathematics)p Fs(,)h(121:463{494,)i(1984.)157 1704 y([4])39 b(F.)22 b(Klopp.)f(An)f(asymptotic)h(expansion)g(for)i(the)d(densit)n (y)h(of)h(states)g(of)g(a)g(random)e(Sc)n(hr\177)-38 b(odinger)21 b(op)r(erator)i(with)e(Bernoulli)276 1795 y(disorder.)26 b Fb(R)l(andom)i(Op)l(er)l(ators)i(and)e(Sto)l(chastic)i (Equations)p Fs(,)d(3\(4\):315{332,)j(1995.)157 1887 y([5])39 b(F.)27 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