Content-Type: multipart/mixed; boundary="-------------9907071136616" This is a multi-part message in MIME format. ---------------9907071136616 Content-Type: text/plain; name="99-259.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-259.keywords" Localization, Random Operators, Anderson Model ---------------9907071136616 Content-Type: application/postscript; name="nm2mparc.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nm2mparc.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.76 (MiKTeX 1.11) Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: nm2mparc.dvi %%CreationDate: Wed Jul 07 11:19:46 1999 %%Pages: 9 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: DVIPS.EXE nm2mparc %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 1999.07.07:1116 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 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Fn(\021)28 b Fo(=)d(0)185 b(\(2.4\))0 5317 y(has)32 b(a)i(non-trivial)c(solution)h (for)i(all)e Fn(\025)e Fm(2)g Fl(R)s Fo(.)54 b(Therefore)32 b Fn(E)38 b Fo(is)32 b(an)g(eigen)m(v)-5 b(alue)33 b(of)g(the)g (selfadjoin)m(t)e(Sturm-)0 5433 y(Liouville)h(op)s(erator)k Fn(T)802 5448 y Fk(\025)882 5433 y Fo(corresp)s(onding)d(to)j(\(2.4\))h (for)e(ev)m(ery)g Fn(\025)p Fo(.)55 b(The)35 b(op)s(erators)g Fn(T)2982 5448 y Fk(\025)3063 5433 y Fo(are)g(b)s(ounded)e(from)1867 5682 y(3)p eop %%Page: 4 4 4 3 bop 0 91 a Fo(b)s(elo)m(w,)37 b(ha)m(v)m(e)h(simple)c(discrete)i (sp)s(ectrum,)g(and)g(all)f(eigen)m(v)-5 b(alues)36 b(dep)s(end)e (analytically)h(on)h Fn(\025)p Fo(.)58 b(Since)36 b(no)0 208 y(eigen)m(v)-5 b(alue)25 b(crossings)g(are)h(allo)m(w)m(ed)f(in)f (dimension)f(one,)k(there)f(is)e(an)h(in)m(teger)h Fn(N)36 b Fo(suc)m(h)25 b(that)h Fn(T)3269 223 y Fk(\025)3340 208 y Fo(has)f(exactly)0 324 y Fn(N)40 b Fo(eigen)m(v)-5 b(alues)30 b(b)s(elo)m(w)g Fn(E)36 b Fo(for)30 b(all)f Fn(\025)p 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Fo(and)f(\(in)g(form)g (sense\))h Fm(h)p Fn(f)2646 803 y Fk(i)2674 789 y Fn(;)15 b(T)2767 804 y Fk(\025)2813 789 y Fn(f)2858 803 y Fk(i)2885 789 y Fm(i)27 b Fo(=)e Fn(C)3108 803 y Fk(i)3157 789 y Fo(+)20 b Fn(\025)3316 720 y Fi(R)3387 789 y Fn(w)r Fm(j)p Fn(f)3524 803 y Fk(i)3552 789 y Fm(j)3577 756 y Fj(2)3617 789 y Fo(.)42 b(So)0 905 y(there)26 b(is)f(\003)h Fm(2)e Fl(R)35 b Fo(su\016cien)m(tly)25 b(large)h(suc)m(h)g(that)g Fm(h)p Fn(f)1723 919 y Fk(i)1752 905 y Fn(;)15 b(T)1845 919 y Fj(\003)1898 905 y Fn(f)1943 919 y Fk(i)1971 905 y Fm(i)25 b Fn(<)g(E)5 b Fm(k)p Fn(f)2289 919 y Fk(i)2318 905 y Fm(k)2363 872 y Fj(2)2429 905 y Fo(for)25 b(all)g Fn(i)p Fo(,)j(and)d(th)m(us)h Fm(h)p Fn(v)s(;)15 b(T)e(v)s Fm(i)27 b Fn(<)e(E)5 b Fm(k)p Fn(v)s Fm(k)3739 872 y Fj(2)0 1021 y Fo(for)27 b(all)e Fn(v)k Fm(2)c Fo(lin)n Fm(f)p Fn(f)608 1035 y Fj(1)647 1021 y Fn(;)15 b(:)g(:)g(:)i(;)e(f)894 1035 y Fk(N)7 b Fj(+1)1051 1021 y Fm(g)13 b(n)g(f)p Fo(0)p Fm(g)p Fo(.)42 b(By)27 b(the)g(v)-5 b(ariational)25 b(principle)f Fn(T)2533 1035 y Fj(\003)2613 1021 y Fo(has)i(at)i(least)f Fn(N)c Fo(+)13 b(1)27 b(eigen)m(v)-5 b(alues)0 1137 y(b)s(elo)m(w)30 b Fn(E)5 b Fo(,)30 b(a)h(con)m(tradiction.)2703 b Fa(2)141 1324 y Fo(W)-8 b(e)32 b(note)f(the)f(form)m(ula)g(\(a)h(pro)s(of)f(is)f (included)f(in)h(Section)h(4\))1012 1565 y Fn(@)1060 1580 y Fk(\025)1105 1565 y Fn(\036)1159 1579 y Fk(a)1201 1565 y Fo(\()p Fn(b;)15 b(\025)p Fo(\))26 b(=)f Fm(\000)1746 1503 y Fo(1)p 1606 1544 326 4 v 1606 1627 a Fn(r)1647 1641 y Fk(a)1689 1627 y Fo(\()p Fn(b;)15 b(\025)p Fo(\))1891 1601 y Fj(2)1956 1450 y Fi(Z)2039 1476 y Fk(b)2002 1639 y(a)2089 1565 y Fn(w)r Fo(\()p Fn(x)p Fo(\))p Fn(u)2330 1579 y Fk(a)2373 1565 y Fo(\()p Fn(x;)g(\025)p Fo(\))2588 1527 y Fj(2)2643 1565 y Fn(dx:)826 b Fo(\(2.5\))0 1818 y(Th)m(us,)30 b(Lemma)g(3)h(sa)m(ys)g(that)g(if)1129 1749 y Fi(R)1184 1776 y Fj(1)1168 1846 y(0)1239 1818 y Fn(w)r Fo(\()p Fn(x)p Fo(\))p Fn(u)1480 1832 y Fk(a)1523 1818 y Fo(\()p Fn(x;)15 b(\025)p 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y Fj(2)654 2283 y Fo(.)0 2482 y Fe(Lemma)33 b(4)45 b Fd(L)-5 b(et)40 b Fn(E)699 2496 y Fj(0)775 2482 y Fm(2)d Fl(R)48 b Fd(b)-5 b(e)39 b(\014xe)-5 b(d.)62 b(Then)39 b(ther)-5 b(e)40 b(exists)f(a)h(discr)-5 b(ete)40 b(set)f Fn(M)47 b Fm(\032)37 b Fl(R)47 b Fd(such)40 b(that)g(for)g(every)0 2598 y Fn(\025)53 2612 y Fj(0)118 2598 y Fm(2)25 b Fl(R)k Fm(n)20 b Fn(M)43 b Fd(ther)-5 b(e)34 b(is)e(a)h(discr)-5 b(ete)33 b(set)g Fn(N)10 b Fo(\()p Fn(\025)1535 2612 y Fj(0)1575 2598 y Fo(\))26 b Fm(\032)f Fl(R)41 b Fd(with)33 b(the)h(fol)5 b(lowing)33 b(pr)-5 b(op)g(erty:)141 2715 y(If)33 b Fn(\026)292 2729 y Fj(0)356 2715 y Fm(2)25 b Fl(R)k Fm(n)21 b Fn(N)10 b Fo(\()p Fn(\025)765 2729 y Fj(0)805 2715 y Fo(\))p Fd(,)32 b(then)h(ther)-5 b(e)34 b(exist)f(close)-5 b(d)34 b(sets)e Fn(T)2037 2729 y Fj(1)2102 2715 y Fm(\032)25 b Fl(R)42 b Fd(and)33 b Fn(T)2526 2729 y Fj(2)2591 2715 y Fm(\032)25 b Fl(R)41 b Fd(such)33 b(that)141 2831 y(\(i\))136 b Fn(T)432 2845 y Fj(1)492 2831 y Fm([)20 b Fn(T)626 2845 y Fj(2)691 2831 y Fo(=)25 b Fl(R)s Fd(,)141 2947 y(\(ii\))136 b Fn(@)455 2962 y Fk(\025)501 2947 y Fn(\036)555 2961 y Fj(0)594 2947 y Fo(\(1)p Fn(;)15 b(\022)s(;)g(E)867 2961 y Fj(0)908 2947 y Fn(;)g(V)1001 2962 y Fk(\025)1046 2947 y Fo(\))26 b Fm(6)p Fo(=)f(0)33 b Fd(at)g Fn(\025)25 b Fo(=)g Fn(\025)1617 2961 y Fj(0)1689 2947 y Fd(for)34 b(al)5 b(l)33 b Fn(\022)27 b Fm(2)e Fn(T)2174 2961 y Fj(1)2214 2947 y Fd(,)141 3063 y(\(iii\))136 b Fn(@)483 3077 y Fk(\026)530 3063 y Fn(\036)584 3077 y Fj(0)624 3063 y Fo(\(2)p Fn(;)15 b(\022)s(;)g(E)897 3077 y Fj(0)937 3063 y Fn(;)g(V)1030 3078 y Fk(\025)1071 3087 y Fc(0)1106 3078 y Fk(;\026)1172 3063 y Fo(\))26 b Fm(6)p Fo(=)f(0)33 b Fd(at)g Fn(\026)25 b Fo(=)g Fn(\026)1747 3077 y Fj(0)1819 3063 y Fd(for)33 b(al)5 b(l)33 b Fn(\022)27 b Fm(2)e Fn(T)2303 3077 y Fj(2)2343 3063 y Fd(.)0 3262 y Fe(Pro)s(of:)41 b Fo(Let)31 b Fn(E)553 3276 y Fj(0)618 3262 y Fm(2)25 b Fl(R)39 b Fo(b)s(e)30 b(\014xed,)g(pic)m(k)g(an)g(arbitrary)f Fn(\022)1917 3276 y Fj(0)1982 3262 y Fm(2)24 b Fo([0)p Fn(;)15 b(\031)s Fo(\))32 b(and)e(c)m(ho)s(ose)1110 3453 y Fn(M)35 b Fo(:=)25 b Fm(f)p Fn(\025)h Fm(2)f Fl(R)34 b Fo(:)41 b Fn(@)1769 3468 y Fk(\025)1814 3453 y Fn(\036)1868 3467 y Fj(0)1908 3453 y Fo(\(1)p Fn(;)15 b(\022)2071 3467 y Fj(0)2111 3453 y Fn(;)g(E)2218 3467 y Fj(0)2258 3453 y Fn(;)g(V)2351 3468 y Fk(\025)2397 3453 y Fo(\))26 b(=)f(0)p Fm(g)p Fn(:)0 3644 y(M)40 b Fo(is)30 b(discrete)g(b)m(y)g (Lemma)h(3.)41 b(Let)31 b Fn(\025)1332 3658 y Fj(0)1396 3644 y Fm(2)25 b Fl(R)k Fm(n)21 b Fn(M)40 b Fo(and)30 b(de\014ne)868 3835 y Fn(T)13 b Fo(\()p Fn(\025)1022 3849 y Fj(0)1061 3835 y Fo(\))26 b(:=)f Fm(f)p Fn(\022)j Fm(2)d Fl(R)34 b Fo(:)41 b Fn(@)1650 3850 y Fk(\025)1695 3835 y Fn(\036)1749 3849 y Fj(0)1789 3835 y Fo(\(1)p Fn(;)15 b(\022)s(;)g(E)2062 3849 y Fj(0)2102 3835 y Fn(;)g(V)2195 3850 y Fk(\025)2241 3835 y Fo(\))26 b(=)f(0)20 b(at)31 b Fn(\025)26 b Fo(=)f Fn(\025)2802 3849 y Fj(0)2841 3835 y Fm(g)p Fn(:)0 4025 y(@)48 4040 y Fk(\025)94 4025 y Fn(\036)148 4039 y Fj(0)187 4025 y Fo(\(1)p Fn(;)15 b(\022)s(;)g(E)460 4039 y Fj(0)501 4025 y Fn(;)g(V)594 4040 y Fk(\025)639 4025 y Fo(\))32 b(is)e(analytic)h(and)g Fn(\031)s Fo(-p)s(erio)s(dic)e (in)h Fn(\022)k Fo(\(this)d(can)g(b)s(e)g(seen)h(from)f(\(2.5\),)i (noting)e(that)h Fn(u)3640 4039 y Fk(a)3709 4025 y Fo(=)0 4154 y Fn(u)52 4121 y Fk(N)52 4177 y(a)134 4154 y Fo(sin)14 b Fn(\022)28 b Fo(+)e Fn(u)481 4121 y Fk(D)481 4177 y(a)560 4154 y Fo(cos)16 b Fn(\022)s Fo(,)41 b(where)d Fn(u)1132 4106 y Fk(N)q(=D)1132 4165 y(a)1328 4154 y Fo(are)h(solutions)e(with)h (Neumann)g(resp)s(ectiv)m(ely)h(Diric)m(hlet)e(b)s(oundary)0 4270 y(conditions)29 b(at)i Fn(a)p Fo(\).)41 b(Since)29 b Fn(\022)975 4284 y Fj(0)1040 4270 y Fm(62)c Fn(T)13 b Fo(\()p Fn(\025)1280 4284 y Fj(0)1319 4270 y Fo(\),)31 b(this)e(implies)f(that)j Fn(T)13 b Fo(\()p Fn(\025)2247 4284 y Fj(0)2287 4270 y Fo(\))30 b(is)g(discrete)g(and)f Fn(\031)s Fo(-p)s(erio)s(dic.)141 4387 y(F)-8 b(or)31 b(arbitrary)e Fn(\022)s Fo(,)h(the)h(set)1076 4577 y Fn(M)1164 4592 y Fk(\022)1229 4577 y Fo(:=)25 b Fm(f)p Fn(\026)h Fm(2)e Fl(R)35 b Fo(:)40 b Fn(@)1766 4591 y Fk(\026)1813 4577 y Fn(\036)1867 4591 y Fj(0)1907 4577 y Fo(\(2)p Fn(;)15 b(\022)s(;)g(E)2180 4591 y Fj(0)2220 4577 y Fn(;)g(V)2313 4592 y Fk(\025)2354 4601 y Fc(0)2389 4592 y Fk(;\026)2456 4577 y Fo(\))25 b(=)g(0)p Fm(g)0 4768 y Fo(is)k(discrete)i(b)m(y)f(Lemma)g(3.)41 b(Th)m(us)30 b(the)g(\014nite)g(union)1395 4959 y Fn(N)10 b Fo(\()p Fn(\025)1566 4973 y Fj(0)1605 4959 y Fo(\))26 b(:=)1969 4878 y Fi([)1787 5066 y Fk(\022)r Fs(2)p Fk(T)10 b Fj(\()p Fk(\025)1988 5075 y Fc(0)2023 5066 y Fj(\))p Fs(\\)p Fj([0)p Fk(;\031)r Fj(\))2258 4959 y Fn(M)2346 4974 y Fk(\022)0 5242 y Fo(is)29 b(discrete.)41 b(If)30 b Fn(\026)607 5256 y Fj(0)671 5242 y Fm(2)25 b Fl(R)k Fm(n)21 b Fn(N)10 b Fo(\()p Fn(\025)1080 5256 y Fj(0)1120 5242 y Fo(\))30 b(then)1184 5433 y Fn(@)1232 5447 y Fk(\026)1279 5433 y Fn(\036)1333 5447 y Fj(0)1372 5433 y Fo(\(2)p Fn(;)15 b(\022)s(;)g(E)1645 5447 y Fj(0)1686 5433 y Fn(;)g(V)1779 5448 y Fk(\025)1820 5457 y Fc(0)1855 5448 y Fk(;\026)1921 5433 y Fo(\))26 b Fm(6)p Fo(=)f(0)91 b(at)31 b Fn(\026)25 b Fo(=)g Fn(\026)2556 5447 y Fj(0)3593 5433 y Fo(\(2.6\))1867 5682 y(4)p eop %%Page: 5 5 5 4 bop 0 91 a Fo(for)35 b(all)e Fn(\022)i Fm(2)e Fn(T)13 b Fo(\()p Fn(\025)600 105 y Fj(0)639 91 y Fo(\).)55 b(By)35 b(con)m(tin)m(uit)m(y)g(of)g Fn(@)1491 105 y Fk(\026)1538 91 y Fn(\036)1592 105 y Fj(0)1666 91 y Fo(in)e Fn(\022)k Fo(there)e(is)f(an)h Fn(\021)h(>)d Fo(0)i(suc)m(h)f(that)i(\(2.6\))h (holds)c(for)i(all)e Fn(\022)0 208 y Fo(with)25 b(dist\()p Fn(\022)s(;)15 b(T)e Fo(\()p Fn(\025)626 222 y Fj(0)665 208 y Fo(\)\))26 b Fm(\024)f Fn(\021)s Fo(.)40 b(If)26 b(dist\()p Fn(\022)s(;)15 b(T)e Fo(\()p Fn(\025)1480 222 y Fj(0)1518 208 y Fo(\)\))26 b Fm(\025)f Fn(\021)s Fo(,)j(then)f Fn(@)2063 223 y Fk(\025)2108 208 y Fn(\036)2162 222 y Fj(0)2202 208 y Fo(\(1)p Fn(;)15 b(\022)s(;)g(E)2475 222 y Fj(0)2515 208 y Fn(;)g(V)2608 223 y Fk(\025)2654 208 y Fo(\))25 b Fm(6)p Fo(=)g(0)j(at)f Fn(\025)e Fo(=)g Fn(\025)3217 222 y Fj(0)3283 208 y Fo(b)m(y)i(de\014nition)0 324 y(of)k Fn(T)13 b Fo(\()p Fn(\025)258 338 y Fj(0)297 324 y Fo(\).)41 b(Th)m(us,)30 b(with)f(the)h(c)m(hoice)1139 520 y Fn(T)1192 534 y Fj(1)1315 520 y Fo(:=)83 b Fm(f)p Fn(\022)28 b Fm(2)d Fl(R)34 b Fo(:)41 b(dist)n(\()p Fn(\022)s(;)15 b(T)e Fo(\()p Fn(\025)2274 534 y Fj(0)2314 520 y Fo(\)\))26 b Fm(\025)f Fn(\021)s Fm(g)p Fn(;)1139 661 y(T)1192 675 y Fj(2)1315 661 y Fo(:=)83 b Fm(f)p Fn(\022)28 b Fm(2)d Fl(R)34 b Fo(:)41 b(dist)n(\()p Fn(\022)s(;)15 b(T)e Fo(\()p Fn(\025)2274 675 y Fj(0)2314 661 y Fo(\)\))26 b Fm(\024)f Fn(\021)s Fm(g)0 858 y Fo(w)m(e)31 b(ha)m(v)m(e)g(pro)m(v)m (en)g(Lemma)g(4.)2678 b Fa(2)0 1133 y Fe(Lemma)33 b(5)45 b Fd(L)-5 b(et)42 b Fn(E)701 1147 y Fj(0)740 1133 y Fd(,)h Fn(\025)864 1147 y Fj(0)944 1133 y Fd(and)f Fn(\026)1184 1147 y Fj(0)1264 1133 y Fd(b)-5 b(e)41 b(given)f(as)h(in)g(L)-5 b(emma)42 b(4.)67 b(Then)42 b(ther)-5 b(e)41 b(exist)g Fn(")g(>)f Fo(0)p Fd(,)j Fn(\016)h(>)c Fo(0)h Fd(and)0 1250 y Fn(C)32 b Fo(=)25 b Fn(C)7 b Fo(\()p Fn(E)367 1264 y Fj(0)406 1250 y Fn(;)15 b(\025)499 1264 y Fj(0)539 1250 y Fn(;)g(\026)634 1264 y Fj(0)674 1250 y Fn(;)g(";)g(\016)s Fo(\))34 b Fd(such)f(that)1048 1381 y Fi(Z)1131 1407 y Fk(\025)1172 1416 y Fc(0)1207 1407 y Fj(+)p Fk(\016)1095 1569 y(\025)1136 1578 y Fc(0)1170 1569 y Fs(\000)p Fk(\016)1315 1381 y Fi(Z)1398 1407 y Fk(\026)1440 1416 y Fc(0)1475 1407 y Fj(+)p Fk(\016)1361 1569 y(\026)1403 1578 y Fc(0)1438 1569 y Fs(\000)p Fk(\016)1583 1496 y Fn(r)1624 1510 y Fj(0)1664 1496 y Fo(\()p Fn(N)5 b(;)15 b(\022)s(;)g(E)5 b(;)15 b(V)2068 1511 y Fk(\025;\026)2176 1496 y Fo(\))2211 1458 y Fs(\000)p Fj(2)2321 1496 y Fn(d\026)g(d\025)26 b Fm(\024)f Fn(C)868 b Fo(\(2.7\))0 1739 y Fd(uniformly)34 b(in)e Fn(N)j(>)25 b Fo(0)p Fd(,)33 b Fn(\022)28 b Fm(2)d Fo([0)p Fn(;)15 b(\031)s Fo(\))p Fd(,)33 b(and)h Fn(E)d Fm(2)25 b Fo([)p Fn(E)1707 1753 y Fj(0)1767 1739 y Fm(\000)19 b Fn(";)c(E)2006 1753 y Fj(0)2067 1739 y Fo(+)20 b Fn(")p Fo(])p Fd(.)0 1944 y Fe(Pro)s(of:)49 b Fo(W)-8 b(e)36 b(will)31 b(frequen)m(tly)j(use)g(that)g Fn(r)1503 1958 y Fj(0)1543 1944 y Fo(\()p Fn(N)5 b(;)15 b(\022)s(;)g(E)5 b(;)15 b(V)1947 1959 y Fk(\025;\026)2055 1944 y Fo(\))35 b(is)e(join)m(tly)g(con)m(tin)m(uous)h(in)f(all)g(its)h(parameters)0 2060 y(\()p Fn(N)5 b(;)15 b(\022)s(;)g(E)5 b(;)15 b(\025;)g(\026)p Fo(\),)32 b(a)d(consequence)i(of)f(standard)e(facts)j(from)e(ODE)g (theory)-8 b(.)42 b(Th)m(us)28 b(w)m(e)i(ma)m(y)g(assume)g Fn(N)35 b(>)25 b Fo(2.)141 2176 y(With)34 b Fn(T)433 2190 y Fj(1)506 2176 y Fo(and)f Fn(T)739 2190 y Fj(2)813 2176 y Fo(c)m(hosen)h(as)g(in)f(Lemma)h(4,)h(assume)f Fn(\022)f Fm(2)e Fn(T)2304 2190 y Fj(1)2377 2176 y Fo(\014rst.)51 b(F)-8 b(rom)34 b(the)g(de\014nition)e(of)i(Pr)s(\177)-48 b(ufer)0 2292 y(v)-5 b(ariables)29 b(w)m(e)i(get)g(\(omitting)f(the)h (parameter)g Fn(E)5 b Fo(\))709 2489 y Fn(r)750 2503 y Fj(0)790 2489 y Fo(\()p 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Fk(\025;\026)2544 2826 y Fo(=)25 b Fn(V)40 b Fo(+)20 b Fn(\026W)2965 2840 y Fj(2)3035 2826 y Fo(in)29 b([1)p Fn(;)15 b Fm(1)p Fo(\)\).)141 2943 y(Let)32 b Fn(\013)c Fo(:=)g Fn(\036)569 2957 y Fj(0)609 2943 y Fo(\(1)p Fn(;)15 b(\022)s(;)g(V)868 2958 y Fk(\025)914 2943 y Fo(\).)45 b(Then)31 b Fn(@)1306 2958 y Fk(\025)1351 2943 y Fn(\013)d Fm(6)p Fo(=)g(0)k(at)g Fn(\025)c Fo(=)f Fn(\025)1957 2957 y Fj(0)1997 2943 y Fo(,)32 b Fn(E)h Fo(=)27 b Fn(E)2319 2957 y Fj(0)2390 2943 y Fo(b)m(y)32 b(Lemma)g(4.)45 b(Con)m(tin)m(uit)m(y)31 b(of)h Fn(@)3568 2958 y Fk(\025)3614 2943 y Fn(\013)g Fo(in)0 3059 y(\()p Fn(\025;)15 b(\022)s(;)g(E)5 b Fo(\))28 b(\(use)f(\(2.5\)\))j(implies) 24 b(the)k(existence)f(of)h Fn(")d(>)g Fo(0)j(and)e Fn(\016)j(>)c Fo(0)j(suc)m(h)f(that)g Fm(j)p Fn(@)2846 3074 y Fk(\025)2892 3059 y Fn(\013)p Fm(j)f(\025)f Fn(C)3162 3073 y Fj(2)3227 3059 y Fn(>)g Fo(0)i(uniformly)0 3175 y(in)j Fn(\022)e Fm(2)e Fn(T)318 3189 y Fj(1)357 3175 y Fo(,)31 b Fn(E)h Fm(2)26 b Fo([)p Fn(E)691 3189 y Fj(0)751 3175 y Fm(\000)20 b Fn(";)15 b(E)991 3189 y Fj(0)1052 3175 y Fo(+)21 b Fn(")p Fo(])31 b(and)f Fn(\025)c Fm(2)g Fo([)p Fn(\025)1663 3189 y Fj(0)1724 3175 y Fm(\000)20 b Fn(\016)n(;)15 b(\025)1946 3189 y Fj(0)2007 3175 y Fo(+)20 b Fn(\016)s Fo(].)44 b(Th)m(us)29 b(w)m(e)j(can)f(estimate)h(the)f Fn(\025)p Fo(-in)m(tegral)g(in)0 3291 y(\(2.7\))h(b)m(y)e(using)f(\(2.8\))j(and)e (substituting)e Fn(\013)j Fo(for)f Fn(\025)h Fo(to)g(get)686 3422 y Fi(Z)769 3449 y Fk(\025)810 3458 y Fc(0)844 3449 y Fj(+)p Fk(\016)732 3611 y(\025)773 3620 y Fc(0)808 3611 y Fs(\000)p Fk(\016)952 3537 y Fn(r)993 3551 y Fj(0)1033 3537 y Fo(\()p Fn(N)5 b(;)15 b(\022)s(;)g(V)1325 3552 y Fk(\025;\026)1432 3537 y Fo(\))1467 3500 y Fs(\000)p Fj(2)1577 3537 y Fn(d\025)26 b Fm(\024)1809 3476 y Fn(C)1874 3490 y Fj(1)p 1809 3516 105 4 v 1809 3600 a Fn(C)1874 3614 y Fj(2)1938 3422 y Fi(Z)2021 3449 y Fk(\013)2066 3458 y Fc(2)1984 3611 y Fk(\013)2029 3620 y Fc(1)2121 3537 y Fn(r)2162 3551 y Fj(1)2201 3537 y Fo(\()p Fn(N)5 b(;)15 b(\013;)g(V)42 b Fo(+)20 b Fn(\026W)2779 3551 y Fj(2)2818 3537 y Fo(\))2853 3500 y Fs(\000)p Fj(2)2963 3537 y Fn(d\013:)500 b Fo(\(2.9\))0 3777 y(Here)38 b(w)m(e)g(ha)m(v)m (e)g(c)m(hosen)g(an)g(in)m(terv)-5 b(al)36 b([)p Fn(\013)1429 3791 y Fj(1)1469 3777 y Fn(;)15 b(\013)1567 3791 y Fj(2)1607 3777 y Fo(])38 b(suc)m(h)f(that)h Fn(\013)p Fo(\([)p Fn(\025)2257 3791 y Fj(0)2322 3777 y Fm(\000)25 b Fn(\016)n(;)15 b(\025)2549 3791 y Fj(0)2614 3777 y Fo(+)25 b Fn(\016)s Fo(]\))38 b Fm(\032)f Fo([)p Fn(\013)3042 3791 y Fj(1)3082 3777 y Fn(;)15 b(\013)3180 3791 y Fj(2)3220 3777 y Fo(])37 b(uniformly)e(in)0 3893 y Fn(\022)30 b Fm(2)d Fo([0)p Fn(;)15 b(\031)s Fo(\))34 b(and)d Fn(E)i Fm(2)28 b Fo([)p Fn(E)854 3907 y Fj(0)915 3893 y Fm(\000)21 b Fn(";)15 b(E)1156 3907 y Fj(0)1217 3893 y Fo(+)21 b Fn(")p Fo(].)46 b(Sub)s(dividing)27 b([)p Fn(\013)2037 3907 y Fj(1)2077 3893 y Fn(;)15 b(\013)2175 3907 y Fj(2)2215 3893 y Fo(])32 b(in)m(to)g(in)m(terv)-5 b(als)31 b(of)h(length)f Fm(\024)d Fn(\031)35 b Fo(and)c(using)0 4009 y(the)g(w)m(ell)e(kno)m(wn)h(form)m (ula)f(\(e.g.)j(Corollary)d(12)j(of)e([12)q(]\))941 4140 y Fi(Z)1024 4167 y Fk(\022)r Fj(+)p Fk(\031)987 4329 y(\022)1176 4255 y Fn(r)1217 4269 y Fj(0)1256 4255 y Fo(\()p Fn(x;)15 b(\013;)g(E)5 b(;)15 b(Q)p Fo(\))1700 4218 y Fs(\000)p Fj(2)1812 4255 y Fn(d\013)26 b Fo(=)f Fn(\031)94 b Fo(for)31 b(all)e Fn(\022)s(;)15 b(x;)g(E)5 b(;)15 b(Q;)0 4486 y Fo(w)m(e)34 b(get)g(the)g(estimate)g(\()p Fm(j)p Fn(\013)936 4500 y Fj(2)998 4486 y Fm(\000)22 b Fn(\013)1149 4500 y Fj(1)1189 4486 y Fm(j)g Fo(+)g Fn(\031)s Fo(\))p Fn(C)1484 4500 y Fj(1)1524 4486 y Fn(=C)1634 4500 y Fj(2)1707 4486 y Fo(for)33 b(\(2.9\).)51 b(Carrying)32 b(out)h(the)h(no)m(w)f(trivial)f Fn(\026)p Fo(-in)m(tegral)h(in)0 4602 y(\(2.7\))f(w)m(e)f(get)999 4653 y Fi(Z)1082 4679 y Fk(\026)1124 4688 y Fc(0)1159 4679 y Fj(+)p Fk(\016)1045 4841 y(\026)1087 4850 y Fc(0)1122 4841 y Fs(\000)p Fk(\016)1267 4653 y Fi(Z)1350 4679 y Fk(\025)1391 4688 y Fc(0)1426 4679 y Fj(+)p Fk(\016)1313 4841 y(\025)1354 4850 y Fc(0)1389 4841 y Fs(\000)p Fk(\016)1533 4768 y Fn(r)1574 4782 y Fj(0)1614 4768 y Fo(\()p Fn(N)5 b(;)15 b(\022)s(;)g(V)1906 4783 y Fk(\025;\026)2014 4768 y Fo(\))2049 4730 y Fs(\000)p Fj(2)2158 4768 y Fn(d\025)g(d\026)26 b Fm(\024)f Fn(C)32 b(<)25 b Fm(1)766 b Fo(\(2.10\))0 4979 y(uniformly)27 b(in)i Fn(N)36 b(>)25 b Fo(2,)31 b Fn(\022)c Fm(2)e Fn(T)1035 4993 y Fj(1)1075 4979 y Fo(,)30 b(and)g Fn(E)g Fm(2)25 b Fo([)p Fn(E)1582 4993 y Fj(0)1642 4979 y Fm(\000)20 b Fn(";)15 b(E)1882 4993 y Fj(0)1943 4979 y Fo(+)20 b Fn(")p Fo(].)141 5096 y(F)-8 b(or)31 b Fn(\022)d Fm(2)d Fn(T)514 5110 y Fj(2)583 5096 y Fo(w)m(e)31 b(will)d(carry)i(out)h(the) g Fn(\026)p Fo(-in)m(tegral)f(\014rst.)40 b(Use)691 5292 y Fn(r)732 5306 y Fj(0)771 5292 y Fo(\()p Fn(N)5 b(;)15 b(\022)s(;)g(V)1063 5307 y Fk(\025;\026)1171 5292 y Fo(\))1206 5254 y Fs(\000)p Fj(2)1384 5292 y Fo(=)83 b Fn(r)1579 5306 y Fj(0)1618 5292 y Fo(\(2)p Fn(;)15 b(\022)s(;)g(V)1877 5307 y Fk(\025;\026)1985 5292 y Fo(\))2020 5254 y Fs(\000)p Fj(2)2115 5292 y Fn(r)2156 5306 y Fj(2)2195 5292 y Fo(\()p Fn(N)5 b(;)15 b(\036)2402 5306 y Fj(0)2443 5292 y Fo(\(1)p Fn(;)g(\022)s(;)g(V)2702 5307 y Fk(\025;\026)2810 5292 y Fo(\))p Fn(;)g(V)21 b Fo(\))2994 5254 y Fs(\000)p Fj(2)1384 5433 y Fm(\024)83 b Fn(C)1603 5447 y Fj(3)1642 5433 y Fn(r)1683 5447 y Fj(2)1722 5433 y Fo(\()p Fn(N)5 b(;)15 b(\036)1929 5447 y Fj(0)1970 5433 y Fo(\(2)p Fn(;)g(\022)s(;)g(V)2229 5448 y Fk(\025;\026)2337 5433 y Fo(\))p Fn(;)g(V)21 b Fo(\))2521 5396 y Fs(\000)p Fj(2)1867 5682 y Fo(5)p eop %%Page: 6 6 6 5 bop 0 91 a Fo(lo)s(cally)30 b(uniformly)f(in)h(\()p Fn(\025;)15 b(\026;)g(\022)s(;)g(E)5 b Fo(\).)46 b(With)31 b Fn(\014)5 b Fo(\()p Fn(\026)p Fo(\))28 b(:=)g Fn(\036)1922 105 y Fj(0)1961 91 y Fo(\(2)p Fn(;)15 b(\022)s(;)g(V)2220 106 y Fk(\025;\026)2328 91 y Fo(\))32 b(w)m(e)g(ha)m(v)m(e)h Fn(@)2789 105 y Fk(\026)2836 91 y Fn(\014)g Fm(6)p Fo(=)27 b(0)32 b(at)g(\()p Fn(\025)3295 105 y Fj(0)3335 91 y Fn(;)15 b(\026)3430 105 y Fj(0)3470 91 y Fn(;)g(E)3577 105 y Fj(0)3617 91 y Fo(\))32 b(b)m(y)0 208 y(Lemma)e(4.)41 b(Con)m(tin)m(uit)m(y)28 b(of)i Fn(@)1039 222 y Fk(\026)1086 208 y Fn(\014)35 b Fo(in)28 b(\()p Fn(\025;)15 b(\026;)g(\022)s(;)g(E)5 b Fo(\))31 b(allo)m(ws)e(to)h(argue)g(as)g(ab)s(o)m(v)m(e)g(and)f (conclude)g(the)h(existence)0 324 y(of)h Fn(")25 b(>)g Fo(0)31 b(and)f Fn(\016)f(>)c Fo(0)30 b(suc)m(h)h(that)999 441 y Fi(Z)1082 467 y Fk(\025)1123 476 y Fc(0)1158 467 y Fj(+)p Fk(\016)1045 629 y(\025)1086 638 y Fc(0)1121 629 y Fs(\000)p Fk(\016)1265 441 y Fi(Z)1348 467 y Fk(\026)1390 476 y Fc(0)1426 467 y Fj(+)p Fk(\016)1312 629 y(\026)1354 638 y Fc(0)1389 629 y Fs(\000)p Fk(\016)1533 556 y Fn(r)1574 570 y Fj(0)1614 556 y Fo(\()p Fn(N)5 b(;)15 b(\022)s(;)g(V)1906 571 y Fk(\025;\026)2014 556 y Fo(\))2049 518 y Fs(\000)p Fj(2)2158 556 y Fn(d\026)g(d\025)26 b Fm(\024)f Fn(C)32 b(<)25 b Fm(1)766 b Fo(\(2.11\))0 804 y(uniformly)27 b(in)i Fn(N)35 b(>)25 b Fo(0,)31 b Fn(\022)d 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b(con)m(v)m(enience,)i(w)m(e)d(include)e(a)j (pro)s(of)f(of)g(form)m(ula)f(\(2.5\).)41 b(The)25 b(Pr)s(\177)-48 b(ufer)24 b(v)-5 b(ariables)24 b(satisfy)h(the)g(di\013eren)m(tial)0 5317 y(equations)1195 5433 y Fn(@)1243 5447 y Fk(x)1287 5433 y Fn(r)j Fo(=)d(\(1)c(+)f Fn(q)j Fo(+)d Fn(\025w)j Fm(\000)d Fn(E)5 b Fo(\))p Fn(r)18 b Fo(sin)c Fn(\036)h Fo(cos)h Fn(\036)1867 5682 y Fo(7)p eop %%Page: 8 8 8 7 bop 0 91 a Fo(and)1190 208 y Fn(@)1238 222 y Fk(x)1282 208 y Fn(\036)26 b Fo(=)f(cos)1579 170 y Fj(2)1634 208 y Fn(\036)20 b Fm(\000)g Fo(\()p Fn(q)j Fo(+)d Fn(\025w)j Fm(\000)d Fn(E)5 b Fo(\))15 b(sin)2456 170 y Fj(2)2510 208 y Fn(\036;)0 378 y Fo(i.e.)30 b(also)733 585 y Fn(@)781 599 y Fk(x)825 585 y Fn(@)873 600 y Fk(\025)919 585 y Fn(\036)83 b Fo(=)f Fm(\000)p Fo(2\()p Fn(@)1408 600 y Fk(\025)1455 585 y Fn(\036)p Fo(\)\(1)21 b(+)f Fn(q)j Fo(+)d Fn(\025w)j Fm(\000)d Fn(E)5 b Fo(\))15 b(sin)f Fn(\036)h Fo(cos)h Fn(\036)21 b Fm(\000)f Fn(w)d Fo(sin)2938 548 y Fj(2)2993 585 y Fn(\036)546 b Fo(\(4.1\))1056 766 y(=)82 b Fm(\000)p Fo(2\()p Fn(@)1408 781 y Fk(\025)1455 766 y Fn(\036)p Fo(\))1554 704 y Fn(@)1602 718 y Fk(x)1646 704 y Fn(r)p 1554 745 136 4 v 1600 828 a(r)1720 766 y Fm(\000)20 b Fn(w)d Fo(sin)2005 728 y Fj(2)2060 766 y Fn(\036:)0 1004 y Fo(This)29 b(yields)f Fn(@)511 1018 y Fk(x)555 1004 y Fo(\()p Fn(r)634 971 y Fj(2)674 1004 y Fn(@)722 1019 y Fk(\025)767 1004 y Fn(\036)p Fo(\))e(=)f Fm(\000)p Fn(w)r(r)1160 971 y Fj(2)1214 1004 y Fo(sin)14 b Fn(\036)25 b Fo(=)g Fm(\000)p Fn(w)r(u)1706 971 y Fj(2)1776 1004 y Fo(and)30 b(therefore)h(\(2.5\))h(since)e(\()p Fn(r)2854 971 y Fj(2)2893 1004 y Fn(@)2941 1019 y Fk(\025)2987 1004 y Fn(\036)p Fo(\)\()p Fn(a)p Fo(\))c(=)f(0.)141 1120 y(W)-8 b(e)26 b(note)g(that)g(in)e(\(4.1\))j(w)m(e)e(ha)m(v)m(e)i (used)d(that)i Fn(@)1769 1134 y Fk(x)1813 1120 y Fn(@)1861 1135 y Fk(\025)1907 1120 y Fn(\036)f Fo(=)g Fn(@)2130 1135 y Fk(\025)2176 1120 y Fn(@)2224 1134 y Fk(x)2268 1120 y Fn(\036)p Fo(.)39 b(Under)24 b Fn(L)2718 1087 y Fj(1)2718 1146 y Fk(l)q(oc)2809 1120 y Fo(-assumptions)f(on)j Fn(q)h Fo(and)e Fn(w)0 1236 y Fo(this)k(holds)g(in)h(distributional)c (sense)31 b(and)f(th)m(us)g(for)g(almost)h(ev)m(ery)g(\()p Fn(x;)15 b(\025)p Fo(\),)32 b(whic)m(h)d(yields)g(\(2.5\))j(for)e (almost)0 1352 y(ev)m(ery)h Fn(\025)p Fo(.)41 b(By)31 b(con)m(tin)m(uit)m(y)f(this)f(extends)h(to)i(ev)m(ery)f Fn(\025)p Fo(.)0 1639 y Fp(References)45 1828 y Fo([1])47 b(D.)26 b(Busc)m(hmann)f(and)g(G.)h(Stolz:)38 b(Tw)m(o-P)m(arameter)27 b(Sp)s(ectral)e(Av)m(eraging)h(and)f(Lo)s(calization)g(for)g(Non-)187 1928 y(Monotoneous)31 b(Random)f(Sc)m(hr\177)-45 b(odinger)29 b(op)s(erators,)i(Preprin)m(t,)e(a)m(v)-5 b(ailable)30 b(from)g(mp-arc)g(98-617)45 2102 y([2])47 b(R.)28 b(Carmona)f(and)h(J.) f(Lacroix:)40 b(Sp)s(ectral)26 b(theory)i(of)g(random)f(Sc)m(hr\177)-45 b(odinger)27 b(op)s(erators.)h(Birkh\177)-45 b(auser,)187 2202 y(Basel{Berlin)29 b(1990)45 2376 y([3])47 b(E.A.)26 b(Co)s(ddington)d(and)i(N.)h(Levinson:)37 b(Theory)25 b(of)h(ordinary)d(di\013eren)m(tial)h(equations.)i(McGra)m(w-Hill,)187 2476 y(New)k(Y)-8 b(ork)31 b(1955)45 2650 y([4])47 b(D.J.)38 b(Gilb)s(ert:)54 b(On)37 b(Sub)s(ordinacy)e(and)i(Analysis)f(of)i(Sc)m (hr\177)-45 b(odinger)36 b(Op)s(erators)h(with)g(t)m(w)m(o)i(Singular) 187 2750 y(Endp)s(oin)m(ts.)28 b Fd(Pr)-5 b(o)g(c.)34 b(R)-5 b(oyal)34 b(So)-5 b(c.)33 b(Edinbur)-5 b(gh)31 b Fe(112A)p Fo(,)h(213)f({)g(229)h(\(1989\))45 2924 y([5])47 b(W.)33 b(Kirsc)m(h,)f(S.)h(Kotani)f(and)g(B.)h(Simon:)44 b(Absence)33 b(of)f(absolutely)g(con)m(tin)m(uous)g(sp)s(ectrum)g(for)g (some)187 3024 y(one)d(dimensional)d(random)h(but)h(deterministic)f(p)s (oten)m(tials.)h Fd(A)n(nn.)i(Inst.)h(Henri)g(Poinc)-5 b(ar)n(\023)-44 b(e)29 b Fe(42)p Fo(,)h(383)g({)187 3124 y(406)i(\(1985\))45 3298 y([6])47 b(F.)22 b(Klopp:)35 b(Lo)s(calization)22 b(for)g(some)g(con)m(tin)m(uous)g(random)f(Sc)m (hr\177)-45 b(odinger)21 b(op)s(erators.)h Fd(Commun.)k(Math.)187 3398 y(Phys.)k Fe(167)p Fo(,)i(553)g({)e(569)i(\(1995\))45 3572 y([7])47 b(S.)42 b(Kotani:)63 b(Ly)m(apuno)m(v)43 b(indices)d(determine)h(absolutely)g(con)m(tin)m(uous)h(sp)s(ectra)g (of)g(stationary)g(one)187 3672 y(dimensional)33 b(Sc)m(hr\177)-45 b(odinger)35 b(op)s(erators.)i(In:)52 b(Sto)s(c)m(hastic)36 b(Analysis,)h(ed.)f(K.)g(Ito,)j(225)e({)g(247,)i(North)187 3771 y(Holland,)29 b(Amsterdam)h(1984)45 3945 y([8])47 b(S.)31 b(Kotani:)43 b(Ly)m(apuno)m(v)31 b(exp)s(onen)m(ts)g(and)g(sp)s (ectra)h(for)f(one-)h(dimensional)d(random)h(Sc)m(hr\177)-45 b(odinger)30 b(op-)187 4045 y(erators.)d(In:)38 b(Random)25 b(Matrices)i(and)e(their)g(Applications,)g(eds.)h(J.E.)g(Cohen,)g(H.)h (Kesten)f(and)f(C.M.)187 4145 y(Newman,)k(American)f(Mathematical)h(So) s(ciet)m(y)-8 b(,)30 b Fd(Contemp)-5 b(or)g(ary)34 b(Math.)29 b Fe(50)p Fo(,)h(277)g({)e(286,)j(Pro)m(vidence)187 4244 y(1986)45 4419 y([9])47 b(S.)24 b(Kotani)h(and)f(B.)i(Simon:)36 b(Lo)s(calization)25 b(in)e(General)i(One-Dimensional)e(Random)h (Systems.)h Fd(Com-)187 4518 y(mun.)33 b(Math.)g(Phys.)d Fe(112)p Fo(,)i(103)f({)g(119)h(\(1987\))0 4693 y([10])47 b(L.)36 b(P)m(astur)g(and)g(A.)h(Figotin:)52 b(Sp)s(ectra)36 b(of)g(Random)g(and)f(Almost-P)m(erio)s(dic)g(Op)s(erators.)h (Springer-)187 4792 y(V)-8 b(erlag,)31 b(1991)0 4967 y([11])47 b(M.)36 b(Reed)g(and)f(B.)i(Simon:)50 b(Metho)s(ds)35 b(of)h(Mo)s(dern)f(Mathematical)i(Ph)m(ysics,)f(I)s(I.)g(F)-8 b(ourier)35 b(Analysis,)187 5066 y(Self-Adjoin)m(tness,)29 b(and)h(IV.)g(Analysis)f(of)h(Op)s(erators,)g(Academic)h(Press,)f(New)h (Y)-8 b(ork)31 b(1975,)h(1978)0 5241 y([12])47 b(G.)27 b(Stolz:)39 b(Lo)s(calization)26 b(for)h(random)f(Sc)m(hr\177)-45 b(odinger)25 b(op)s(erators)i(with)f(P)m(oisson)g(p)s(oten)m(tial.)h Fd(A)n(nn.)h(Inst.)187 5340 y(Henri)k(Poinc)-5 b(ar)n(\023)-44 b(e)31 b Fe(63)p Fo(,)g(297)h({)f(314)h(\(1995\))1867 5682 y(8)p eop %%Page: 9 9 9 8 bop 0 91 a Fo([13])47 b(W.)32 b(M.)g(W)-8 b(ang:)43 b(Microlo)s(calization,)31 b(P)m(ercolation,)h(and)f(Anderson)f(Lo)s (calization)h(for)g(the)g(Magnetic)187 191 y(Sc)m(hr\177)-45 b(odinger)29 b(Op)s(erator)h(with)f(a)i(Random)e(P)m(otan)m(tial,)j Fd(J.)g(F)-7 b(unct.)33 b(A)n(nal.)d Fe(146)p Fo(,)h(1)g({)g(26)g (\(1997\))0 365 y([14])47 b(J.)35 b(W)-8 b(eidmann:)50 b(Sp)s(ectral)34 b(Theory)h(of)g(Ordinary)e(Di\013eren)m(tial)i(Op)s (erators,)h Fd(L)-5 b(e)g(ct.)37 b(Notes)h(in)f(Math.)p Fo(,)187 465 y(V)-8 b(ol.)31 b Fe(1258)p Fo(,)g(Springer)e(V)-8 b(erlag,)31 b(1987)1867 5682 y(9)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9907071136616--