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b(alues)56 b(in)g(the)g(b)5 b(ounded)56 b(in)-5 b(terv)c(al)56 b([)p Fp(M)5031 7722 y Fj(\000)5143 7697 y Fp(;)28 b(M)5378 7722 y Fm(+)5488 7697 y Fr(],)56 b(whose)g(common)758 7896 y(probabilit)-5 b(y)79 b(distribution)h(has)g(a)f(b)5 b(ounded)79 b(densit)-5 b(y)80 b Fp(g)85 b Fr(with)78 b Fp(g)92 b(>)85 b Fr(0)79 b(near)g Fp(M)6005 7921 y Fj(\000)6117 7896 y Fr(;)91 b Fp(u)p Fr(\()p Fp(x)p Fr(\))78 b(a)758 8096 y(measurable)62 b(function)e(with)h(compact)f(supp)5 b(ort,)63 b(0)55 b Fn(\024)g Fp(u)g Fn(2)g Fr(L)4750 8035 y Fo(p)4827 8096 y Fr(\()p Fl(R)5002 8035 y Fo(d)5088 8096 y Fp(;)28 b Fr(d)p Fp(x)p Fr(\))61 b(with)f Fp(p)55 b Fn(2)g Fr([1)p Fp(;)28 b Fn(1)p Fr(])60 b(if)758 8300 y Fp(d)g Fr(=)f(1)64 b(or)f Fp(M)1611 8325 y Fj(\000)1783 8300 y Fn(\025)d Fr(0)j(,)j Fp(p)59 b Fn(2)h Fr(\()2629 8234 y Fo(d)p 2628 8261 70 7 v 2629 8357 a Fm(2)2717 8300 y Fp(;)28 b Fn(1)p Fr(])64 b(if)f Fp(d)d Fn(\025)g Fr(2)j(and)h Fp(M)4200 8325 y Fj(\000)4372 8300 y Fp(<)c Fr(0,)65 b(and)f(0)c Fp(<)g(U)5531 8325 y Fj(\000)5703 8300 y Fn(\024)f Fp(U)18 b Fr(\()p Fp(x)p Fr(\))63 b(where)758 8499 y Fp(U)18 b Fr(\()p Fp(x)p Fr(\))45 b(=)1335 8375 y Fk(P)1510 8549 y Fo(i)p Fj(2)1667 8504 y Fi(1)p 1666 8522 58 6 v 1666 8589 a Fh(q)1744 8549 y Fg(Z)1829 8516 y Fh(d)1927 8499 y Fp(u)p Fr(\()p Fp(x)36 b Fn(\000)h Fp(i)p Fr(\).)957 8746 y(This)44 b(mo)5 b(del)43 b(w)-5 b(as)44 b(studied)g(b)-5 b(y)43 b(Com)-5 b(b)5 b(es)44 b(and)f(Hislop)h([CH1)o(])g(and)f(Kirsc)-5 b(h)44 b([Ki)q(])f(with)g Fp(M)6448 8771 y Fj(\000)6606 8746 y Fr(=)758 8945 y(0)65 b(and)g Fp(V)1335 8970 y Fm(p)t(er)1594 8945 y Fr(=)d(0,)68 b(who)d(pro)-5 b(v)g(ed)66 b(that)f(for)g(an)-5 b(y)65 b(\014xed)g(energy)g Fp(E)4924 8970 y Fm(1)5060 8945 y Fp(>)e Fr(0)i(w)-5 b(e)65 b(ha)-5 b(v)g(e)66 b(Anderson)758 9145 y(lo)5 b(calization)65 b(in)h(the)f(in)-5 b(terv)c(al)65 b([0)p Fp(;)28 b(E)3085 9170 y Fm(1)3159 9145 y Fr(])66 b(for)f(su\016cien)-5 b(tly)67 b(large)e(disorder.)106 b(\(In)65 b(this)h(case)g(the)758 9344 y(almost-sure)57 b(sp)5 b(ectrum)55 b(of)g Fp(H)2698 9369 y Fo(\025;!)2968 9344 y Fr(is)h([0)p Fp(;)28 b Fn(1)p Fr(])55 b(for)g(all)h Fp(\025)46 b(>)g Fr(0.\))957 9549 y(But)65 b(the)g(case)h Fp(M)2116 9574 y Fj(\000)2291 9549 y Fn(6)p Fr(=)e(0)h(cannot)h(b)5 b(e)65 b(treated)g(in)h(the)f(same)h(w)-5 b(a)g(y)-14 b(.)105 b(If)65 b Fp(E)5659 9489 y Fm(inf)5649 9596 y Fo(\025)5894 9549 y Fr(denotes)h(the)758 9754 y(b)5 b(ottom)64 b(of)i(the)f(sp)5 b(ectrum)65 b(of)g Fp(H)2921 9779 y Fo(\025;!)3136 9754 y Fr(,)j(w)-5 b(e)65 b(no)-5 b(w)66 b(ha)-5 b(v)g(e)66 b Fp(E)4386 9693 y Fm(inf)4376 9801 y Fo(\025)4618 9754 y Fn(!)c(\0061)k Fr(as)g Fp(\025)d Fn(!)f(1)k Fr(if)f Fp(M)6364 9779 y Fj(\000)6496 9705 y Fh(>)6496 9796 y(<)6606 9754 y Fr(0.)758 9953 y(Th)-5 b(us)77 b(ev)-5 b(en)76 b(for)g(a)h(\014xed)f(in)-5 b(terv)c(al)76 b(at)g(the)g(b)5 b(ottom)75 b(of)h(the)g(sp)5 b(ectrum)77 b(\(i.e.,)k(of)76 b(the)g(form)758 10152 y([)p Fp(E)937 10092 y Fm(inf)927 10199 y Fo(\025)1106 10152 y Fp(;)28 b(E)1313 10092 y Fm(inf)1303 10199 y Fo(\025)1509 10152 y Fr(+)g Fp(\016)6 b Fr(])51 b(with)g(a)g(\014xed)g Fp(\016)h(>)46 b Fr(0\),)52 b(b)5 b(oth)50 b(the)h(constan)-5 b(t)51 b(in)g(W)-14 b(egner's)52 b(estimate)e(and)i(the)758 10351 y(constan)-5 b(t)50 b(in)h(the)e(Simon-Lieb)j(inequalit)-5 b(y)50 b(increase)g(as)h(the)f(disorder)h Fp(\025)f Fr(increases,)i(so) f(as)f(w)-5 b(e)p 758 10491 997 7 v 957 10674 a Fu(2000)45 b Ft(Mathematics)50 b(Subje)-7 b(ct)49 b(Classi\014c)-7 b(ation.)63 b Fu(Primary)46 b(82B44;)f(Secondary)j(47B80,)d(60H25.)957 10840 y(A.K.)i(w)l(as)f(supp)t(orted)h(in)g(part)g(b)l(y)h(NSF)f(Gran)l (t)g(DMS-0200710.)3713 11039 y Fm(1)p eop %%Page: 2 2 2 1 bop 758 861 a Fm(2)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fr(increase)46 b Fp(\025)f Fr(the)g(initial)h(length)g(scale)g(for)f(the)g(m)-5 b(ultiscale)47 b(analysis)g(also)f(increases.)71 b(Th)-5 b(us)47 b(w)-5 b(e)758 1458 y(cannot)49 b(\014x)h(a)g(length)g(scale)g (and)h(satisfy)f(the)f(initial)i(condition)f(for)g(the)f(m)-5 b(ultiscale)51 b(analysis)758 1658 y(at)j(the)h(disorder)h(dep)5 b(enden)-5 b(t)55 b(energy)f(in)-5 b(terv)c(al)55 b(b)-5 b(y)55 b(taking)g(the)f(disorder)i(large)f(enough;)h(the)758 1857 y(initial)f(length)h(scale)f(is)h(a)g(mo)-5 b(ving)56 b(target.)957 2056 y(T)-14 b(o)53 b(deal)h(with)f(suc)-5 b(h)55 b(di\016culties,)g(w)-5 b(e)53 b(dev)-5 b(elopp)5 b(ed)54 b(in)g([GK4])g(explicit)f(\014nite)g(v)-5 b(olume)54 b(cri-)758 2255 y(teria)48 b(for)h(pro)-5 b(ving)49 b(lo)5 b(calization)48 b(in)i(situations)f(where)g(the)f(crucial)h(quan)-5 b(tities)49 b(of)g(the)f(mo)5 b(del)758 2455 y(that)48 b(en)-5 b(ter)49 b(the)g(m)-5 b(ultiscale)50 b(analysis)h(\(e.g.,)f (the)f(constan)-5 b(t)50 b(in)f(W)-14 b(egner's)50 b(estimate,)g(the)f (con-)758 2654 y(stan)-5 b(t)62 b(in)h(the)f(Simon-Lieb)h(inequalit)-5 b(y\))62 b(dep)5 b(end)62 b(on)h(the)f(parameters)g(of)g(the)g(mo)5 b(del)62 b(\(e.g.,)758 2853 y(the)52 b(disorder)h(parameter,)g(the)f (energy)g(where)g(lo)5 b(calization)52 b(is)h(to)f(b)5 b(e)52 b(pro)-5 b(v)g(en,)54 b(the)e(strength)758 3052 y(of)f(the)g(magnetic)g(\014eld\).)72 b(The)52 b(emphasis)h(w)-5 b(as)52 b(on)f(pro)-5 b(viding)53 b(explicit)e(conditions,)i(dep)5 b(end-)758 3252 y(ing)63 b(on)h(the)f(v)-9 b(arious)64 b(parameters)g(of)f(the)g(mo)5 b(del,)66 b(for)d(starting)g(the)g(b)5 b(o)g(otstrap)63 b(m)-5 b(ultiscale)758 3451 y(analysis)61 b([GK1)q(].)89 b(These)60 b(criteria)g(th)-5 b(us)61 b(yield)g(Anderson)g(lo)5 b(calization,)61 b(strong)g(dynamical)758 3650 y(lo)5 b(calization,)74 b(SULE,)c(etc.)119 b(\(See)71 b(also)g([GK3)q(])f(for)h(a)g(discussion)i(of)d(the)g(consequences)h (of)758 3849 y(the)d(b)5 b(o)g(otstrap)68 b(m)-5 b(ultiscale)70 b(analysis;)76 b(see)69 b([DRJLS])g(for)f(a)h(discussion)i(of)d (SULE.\))g(These)758 4049 y(criteria)55 b(are)g(review)-5 b(ed)55 b(in)h(Section)f(2.)957 4248 y(In)f(Section)f(3)h(w)-5 b(e)54 b(apply)g(our)g(explicit)f(criteria)g(to)h(Anderson)g (Hamiltonians)h Fp(H)6127 4273 y Fo(\025;!)6395 4248 y Fr(as)f(in)758 4447 y(\(1.1\))o(,)d(pro)-5 b(ving)50 b(lo)5 b(calization)49 b(at)h(high)g(disorder)g(in)g(an)g(in)-5 b(terv)c(al)49 b(of)h(size)g(prop)5 b(ortional)49 b(to)h(the)758 4646 y(disorder)j(at)g(the)f(b)5 b(ottom)52 b(of)h(the)f(sp)5 b(ectrum.)73 b(T)-14 b(o)53 b(do)g(so)g(w)-5 b(e)53 b(deriv)-5 b(e)53 b(explicit)f(b)5 b(ounds)54 b(on)f(the)758 4846 y(lo)5 b(cation)55 b(of)h(the)f(b)5 b(ottom)55 b(of)h(the)f(sp)5 b(ectrum,)56 b Fp(E)3833 4785 y Fm(inf)3823 4893 y Fo(\025)4002 4846 y Fr(,)g(as)g(a)g(function)g(of)g(the)f(disorder)i Fp(\025)p Fr(.)75 b(\(See)758 5045 y(Theorem)44 b(3.1.\))70 b(Using)44 b(these)g(b)5 b(ounds)46 b(and)e(our)h(explicit)e(criteria,) j(w)-5 b(e)45 b(pro)-5 b(v)g(e)44 b(lo)5 b(calization)44 b(in)758 5244 y(the)50 b(in)-5 b(terv)c(al)51 b([)p Fp(E)1814 5184 y Fm(inf)1804 5291 y Fo(\025)1983 5244 y Fp(;)28 b(E)2190 5184 y Fm(inf)2180 5291 y Fo(\025)2386 5244 y Fr(+)h Fp(c\025)p Fr(])50 b(for)h(large)g(disorder)h Fp(\025)p Fr(,)g(where)e Fp(c)h Fr(is)g(a)g(constan)-5 b(t)52 b(indep)5 b(enden)-5 b(t)758 5443 y(of)61 b Fp(\025)p Fr(.)91 b(In)61 b(addition,)i(w)-5 b(e)61 b(sho)-5 b(w)63 b(that)d(for)h(large)g(disorder)h(eigenfunctions)g(with)f(eigen)-5 b(v)c(alues)758 5647 y(in)67 b(the)f(in)-5 b(terv)c(al)66 b([)p Fp(E)2050 5587 y Fm(inf)2040 5694 y Fo(\025)2219 5647 y Fp(;)28 b(E)2426 5587 y Fm(inf)2416 5694 y Fo(\025)2639 5647 y Fr(+)45 b Fp(c\025)p Fr(])66 b(deca)-5 b(y)67 b(exp)5 b(onen)-5 b(tially)66 b(with)h(a)g(rate)f Fn(\025)f Fp(c)5761 5587 y Fj(0)5807 5506 y Fn(p)p 5946 5506 97 7 v 5946 5647 a Fp(\025)h Fr(for)h(some)758 5846 y(constan)-5 b(t)72 b Fp(c)1516 5786 y Fj(0)1562 5846 y Fr(.)123 b(\(See)71 b(Theorem)h(3.2.\))122 b(W)-14 b(e)72 b(had)g(previously)g(obtained)g (suc)-5 b(h)73 b(results)f(when)758 6046 y Fp(M)919 6071 y Fj(\000)1077 6046 y Fr(=)46 b(0)56 b(and)f Fp(u)h Fr(b)5 b(ounded)56 b([GK4].)957 6245 y(Lo)5 b(calization)47 b(for)h(con)-5 b(tin)g(uous)49 b(random)g(op)5 b(erators)48 b(has)g(b)5 b(een)48 b(usually)g(pro)-5 b(v)g(en)49 b(b)-5 b(y)48 b(a)g(m)-5 b(ul-)758 6444 y(tiscale)41 b(analysis.)70 b(\(But)40 b(note)h(that)f(the)h(fractional)g(momen)-5 b(t)42 b(metho)5 b(d)40 b([AM,)h(ASFH])g(has)h(just)758 6643 y(b)5 b(een)51 b(extended)f(to)h(the)g(con)-5 b(tin)g(uum)53 b([AENSS)o(].\))72 b(The)52 b(m)-5 b(ultiscale)52 b(analysis)g(is)g(a)f (tec)-5 b(hnique,)758 6843 y(initially)58 b(dev)-5 b(elop)5 b(ed)58 b(b)-5 b(y)58 b(F)-14 b(r\177)-83 b(ohlic)-5 b(h)59 b(and)g(Sp)5 b(encer)58 b([FS])h(and)f(F)-14 b(r\177)-83 b(ohlic)-5 b(h,)60 b(Martinelli,)f(Sp)5 b(encer)758 7042 y(and)39 b(Scopp)5 b(olla)39 b([FMSS)r(],)j(and)d(simpli\014ed)i(b)-5 b(y)39 b(v)-5 b(on)39 b(Dreifus)g([vD])g(and)g(v)-5 b(on)39 b(Dreifus)g(and)h(Klein)758 7241 y([vDK)o(],)56 b(for)f(the)g(purp)5 b(ose)57 b(of)e(pro)-5 b(ving)56 b(Anderson)g(lo)5 b(calization.)74 b(i.e.,)56 b(pure)f(p)5 b(oin)-5 b(t)56 b(sp)5 b(ectrum)758 7440 y(and)49 b(exp)5 b(onen)-5 b(tial)49 b(deca)-5 b(y)49 b(of)g(eigenfunctions.)72 b(\(See)49 b(also)h([HM)o(,)g(CKM,)f(Sp)q(,)g (KLS,)g(vDK2,)g(K,)758 7640 y(CH1)o(,)65 b(Klo1,)g(FK1)q(,)g(FK2,)g (KSS1)q(,)g(KSS2)q(,)g(CHT)o(,)g(FLM,)g(W)-14 b(a,)65 b(St,)g(GK1,)g(U,)g(KK1,)g(KK2].\))758 7839 y(It)55 b(w)-5 b(as)57 b(later)f(sho)-5 b(wn)57 b(to)f(also)g(yield)g(dynamical)h(lo)5 b(calization)56 b(\(non)g(spreading)h(of)f(the)g(w)-5 b(a)g(v)g(e)758 8038 y(pac)g(k)g(ets\))79 b([GDB,)g(Ge,)h(DS,)g(GK1].) 146 b(Explicit)79 b(\014nite)h(v)-5 b(olume)80 b(criteria)e(for)i (applying)g(the)758 8237 y(m)-5 b(ultiscale)56 b(analysis)h(w)-5 b(ere)55 b(pro)-5 b(vided)56 b(in)g([GK4].)1566 8646 y(2.)92 b Fq(Explicit)63 b(finite)g(v)l(olume)f(criteria)h(f)l(or)f (localiza)-11 b(tion)957 8945 y Fr(In)58 b(this)g(article)f(a)h Ff(r)-8 b(andom)61 b(Schr\177)-85 b(odinger)63 b(op)-8 b(er)g(ator)75 b Fr(will)58 b(b)5 b(e)57 b(a)h(random)h(op)5 b(erator)57 b(of)h(the)758 9145 y(form)2541 9438 y Fp(H)2679 9463 y Fo(!)2821 9438 y Fr(=)47 b Fn(\000)p Fr(\001)37 b(+)g Fp(V)3564 9463 y Fo(!)3799 9438 y Fr(on)138 b(L)4216 9370 y Fm(2)4290 9438 y Fr(\()p Fl(R)4466 9370 y Fo(d)4552 9438 y Fp(;)28 b Fr(d)p Fp(x)p Fr(\))p Fp(;)1470 b Fr(\(2.1\))758 9732 y(where)71 b(\001)g(is)h(the)e Fp(d)p Fr(-dimensional)k(Laplacian) d(op)5 b(erator)71 b(and)h Fp(V)4915 9757 y Fo(!)5082 9732 y Fr(is)f(a)g(random)h(p)5 b(oten)-5 b(tial,)758 9931 y(i.e.,)51 b Fn(f)p Fp(V)1247 9956 y Fo(!)1343 9931 y Fr(\()p Fp(x)p Fr(\);)77 b Fp(x)46 b Fn(2)g Fl(R)2100 9871 y Fo(d)2187 9931 y Fn(g)k Fr(is)h(a)f(real)g(v)-9 b(alued)50 b(measurable)i(pro)5 b(cess)51 b(on)f(a)g(complete)g (probabilit)-5 b(y)758 10130 y(space)55 b(\(\012)p Fp(;)28 b Fn(F)16 b Fp(;)28 b Fl(P)6 b Fr(\),)64 b(suc)-5 b(h)56 b(that:)970 10409 y Fv(\(i\):)82 b Fp(V)1404 10434 y Fo(!)1591 10409 y Fr(=)91 b Fp(V)1945 10323 y Fm(\(1\))1908 10428 y Fo(!)2178 10409 y Fr(+)54 b Fp(V)2495 10323 y Fm(\(2\))2458 10428 y Fo(!)2673 10409 y Fr(,)89 b(where)82 b Fn(f)p Fp(V)3532 10323 y Fm(\()p Fo(i)p Fm(\))3495 10428 y Fo(!)3691 10409 y Fr(\()p Fp(x)p Fr(\);)109 b Fp(x)91 b Fn(2)f Fl(R)4569 10349 y Fo(d)4655 10409 y Fn(g)p Fr(,)f Fp(i)i Fr(=)g(1)p Fp(;)28 b Fr(2,)88 b(are)82 b(real)g(v)-9 b(alued)1169 10608 y(measurable)57 b(pro)5 b(cesses)56 b(on)f(\(\012)p Fp(;)28 b Fn(F)16 b Fp(;)28 b Fl(P)6 b Fr(\))64 b(suc)-5 b(h)56 b(that)f(for)g Fl(P)6 b Fr(-a.e.)83 b Fp(!)61 b Fr(w)-5 b(e)56 b(ha)-5 b(v)g(e:)1282 10840 y Fv(\(i)1410 10865 y Fm(1)1483 10840 y Fv(\):)84 b Fr(0)46 b Fn(\024)g Fp(V)2132 10754 y Fm(\(1\))2095 10859 y Fo(!)2356 10840 y Fn(2)g Fr(L)2617 10780 y Fm(1)2617 10887 y(lo)t(c)2792 10840 y Fr(\()p Fl(R)2967 10780 y Fo(d)3054 10840 y Fp(;)28 b Fr(d)p Fp(x)p Fr(\).)p eop %%Page: 3 3 3 2 bop 2654 861 a Fm(HIGH)58 b(DISORDER)h(LOCALIZA)-11 b(TION)1830 b(3)1282 1266 y Fv(\(i)1410 1291 y Fm(2)1483 1266 y Fv(\):)84 b Fp(V)1827 1180 y Fm(\(2\))1791 1286 y Fo(!)2043 1266 y Fr(is)37 b(relativ)-5 b(ely)36 b(form-b)5 b(ounded)38 b(with)f(resp)5 b(ect)36 b(to)h Fn(\000)p Fr(\001)g(with)g(relativ)-5 b(e)36 b(b)5 b(ound)1481 1466 y Fp(<)46 b Fr(1.)970 1665 y Fv(\(ii\):)81 b Fr(There)68 b(is)h(an)g(ergo)5 b(dic)67 b(family)i Fn(f)p Fp(\034)3543 1690 y Fo(y)3622 1665 y Fr(;)96 b Fp(y)74 b Fn(2)67 b Fl(Z)4199 1605 y Fo(d)4285 1665 y Fn(g)h Fr(of)g(measure)h(preserving)g (transfor-)1169 1902 y(mations)75 b(on)g(\(\012)p Fp(;)28 b Fn(F)16 b Fp(;)28 b Fl(P)5 b Fr(\))83 b(suc)-5 b(h)76 b(that)e Fp(V)3680 1816 y Fm(\()p Fo(i)p Fm(\))3644 1922 y Fo(\034)3705 1939 y Fh(y)3777 1922 y Fo(!)3873 1902 y Fr(\()p Fp(x)p Fr(\))j(=)h Fp(V)4515 1816 y Fm(\()p Fo(i)p Fm(\))4479 1922 y Fo(!)4675 1902 y Fr(\()p Fp(x)49 b Fn(\000)g Fp(y)6 b Fr(\))75 b(for)f Fp(i)k Fr(=)g(1)p Fp(;)28 b Fr(2)74 b(and)h(all)1169 2112 y Fp(y)53 b Fn(2)45 b Fl(Z)1561 2052 y Fo(d)1648 2112 y Fr(.)970 2311 y Fv(\(iii\):)81 b Fr(There)60 b(exists)i Fp(\045)55 b(>)h Fr(0)61 b(suc)-5 b(h)62 b(that)f(for)g(an)-5 b(y)61 b(b)5 b(ounded)62 b(subsets)g Fp(B)5549 2336 y Fm(1)5624 2311 y Fp(;)28 b(B)5824 2336 y Fm(2)5959 2311 y Fr(of)61 b Fl(R)6265 2251 y Fo(d)6412 2311 y Fr(with)1169 2511 y(dist)q(\()p Fp(B)1629 2536 y Fm(1)1703 2511 y Fp(;)28 b(B)1903 2536 y Fm(2)1977 2511 y Fr(\))69 b Fp(>)g(\045)g Fr(the)g(pro)5 b(cesses)70 b Fn(f)p Fp(V)3676 2536 y Fo(!)3772 2511 y Fr(\()p Fp(x)p Fr(\);)96 b Fp(x)70 b Fn(2)f Fp(B)4610 2536 y Fm(1)4684 2511 y Fn(g)g Fr(and)h Fn(f)p Fp(V)5353 2536 y Fo(!)5449 2511 y Fr(\()p Fp(x)p Fr(\);)96 b Fp(x)70 b Fn(2)f Fp(B)6287 2536 y Fm(2)6361 2511 y Fn(g)g Fr(are)1169 2710 y(indep)5 b(enden)-5 b(t.)957 2954 y(It)49 b(follo)-5 b(ws,)51 b(using)f(the)f(ergo)5 b(dicit)-5 b(y)-14 b(,)50 b(that)e(there)h(are)g(nonnegativ)-5 b(e)49 b(constan)-5 b(ts)50 b(\002)6040 2979 y Fm(1)6160 2954 y Fp(<)d Fr(1)i(and)758 3153 y(\002)887 3178 y Fm(2)1017 3153 y Fr(suc)-5 b(h)56 b(that)f(for)g(all)h Fp( )c Fn(2)46 b(D)5 b Fr(\()p Fn(r)p Fr(\))54 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)1901 3300 y Fk(\014)1901 3400 y(\014)1901 3499 y(\014)1957 3307 y(D)2058 3491 y Fp( )6 b(;)28 b(V)2380 3422 y Fm(\(2\))2343 3532 y Fo(!)2558 3491 y Fp( )2672 3307 y Fk(E)2774 3300 y(\014)2774 3400 y(\014)2774 3499 y(\014)2875 3491 y Fn(\024)47 b Fr(\002)3180 3516 y Fm(1)3254 3491 y Fn(kr)p Fp( )6 b Fn(k)3672 3422 y Fm(2)3784 3491 y Fr(+)37 b(\002)4079 3516 y Fm(2)4154 3491 y Fn(k)p Fp( )6 b Fn(k)4434 3422 y Fm(2)4647 3491 y Fr(for)55 b Fl(P)5 b Fr(-)q(a.e.)83 b Fp(!)34 b(;)830 b Fr(\(2.2\))758 3820 y(and)53 b(hence)h Fp(H)1676 3845 y Fo(!)1825 3820 y Fr(is)g(de\014ned)g(as)f(a)h(semi-b)5 b(ounded)55 b(self-adjoin)-5 b(t)54 b(op)5 b(erator)52 b(for)h Fl(P)6 b Fr(-a.e.)82 b Fp(!)6 b Fr(,)54 b(with)2825 4108 y Fp(H)2963 4133 y Fo(!)3105 4108 y Fn(\025)46 b(\000)p Fr(\002)3538 4133 y Fm(2)3751 4108 y Fr(for)55 b Fl(P)6 b Fr(-a.e.)83 b Fp(!)34 b(:)1726 b Fr(\(2.3\))758 4396 y(Moreo)-5 b(v)g(er,)71 b Fp(H)1687 4421 y Fo(!)1850 4396 y Fr(is)e(a)e(random)i(op)5 b(erator,)70 b(i.e.,)h(the)c(mappings) i Fp(!)k Fn(!)67 b Fp(f)18 b Fr(\()p Fp(H)5631 4421 y Fo(!)5727 4396 y Fr(\))67 b(are)g(strongly)758 4595 y(measurable)46 b(for)f(all)h(b)5 b(ounded)46 b(measurable)g(functions)g(on)g Fl(R)21 b Fr(.)80 b(Th)-5 b(us)47 b(there)d(exists)i(a)f(nonran-)758 4795 y(dom)55 b(set)g(\006)g(suc)-5 b(h)56 b(that)e Fp(\033)6 b Fr(\()p Fp(H)2597 4820 y Fo(!)2692 4795 y Fr(\))46 b(=)g(\006)55 b(with)g(probabilit)-5 b(y)55 b(one,)h(and)f(that)f(the)h (decomp)5 b(osition)758 4994 y(of)60 b Fp(\033)6 b Fr(\()p Fp(H)1256 5019 y Fo(!)1352 4994 y Fr(\))61 b(in)-5 b(to)61 b(pure)g(p)5 b(oin)-5 b(t)62 b(sp)5 b(ectrum,)62 b(absolutely)g(con)-5 b(tin)g(uous)63 b(sp)5 b(ectrum,)62 b(and)g(singular)758 5193 y(con)-5 b(tin)g(uous)68 b(sp)5 b(ectrum)67 b(is)g(also)g(indep)5 b(enden)-5 b(t)68 b(of)e(the)g(c)-5 b(hoice)67 b(of)f Fp(!)73 b Fr(with)67 b(probabilit)-5 b(y)67 b(one.)758 5393 y(\(See)54 b([KM)q(]and)i(the)f(discussion)i(in)f([GK3)q(].\))957 5592 y(In)70 b(this)g(article)f(w)-5 b(e)70 b(are)f(concerned)h(with)f (lo)5 b(calization,)73 b(from)d(b)5 b(oth)69 b(the)g(sp)5 b(ectral)70 b(and)758 5791 y(dynamical)56 b(p)5 b(oin)-5 b(t)55 b(of)g(views.)758 6101 y Fv(De\014nition)62 b(2.1.)84 b Ff(The)67 b(r)-8 b(andom)66 b(Schr\177)-85 b(odinger)68 b(op)-8 b(er)g(ator)66 b Fp(H)4715 6126 y Fo(!)4878 6101 y Ff(exhibits)h(exp)-8 b(onential)68 b(lo)-8 b(c)g(al-)758 6300 y(ization)64 b(in)h(the)f(op)-8 b(en)65 b(interval)h Fp(I)77 b Ff(if)64 b(it)h(has)f(pur)-8 b(e)64 b(p)-8 b(oint)64 b(sp)-8 b(e)g(ctrum)64 b(in)h Fp(I)13 b Ff(,)65 b(and)g(for)f Fl(P)5 b Ff(-)q(a.e.)100 b Fp(!)758 6499 y Ff(the)59 b(eigenfunctions)j(of)d Fp(H)2461 6524 y Fo(!)2617 6499 y Ff(with)h(eigenvalue)h(in)f Fp(I)73 b Ff(de)-8 b(c)g(ay)58 b(exp)-8 b(onential)8 b(ly)61 b(in)f(the)g Fp(L)6169 6439 y Fm(2)6243 6499 y Ff(-sense.)758 6699 y(The)69 b(exp)-8 b(onential)70 b(lo)-8 b(c)g(alization)70 b(r)-8 b(e)g(gion)69 b Fr(\006)3483 6724 y Fm(EL)3732 6699 y Ff(for)g(the)h(r)-8 b(andom)68 b(Schr\177)-85 b(odinger)71 b(op)-8 b(er)g(ator)68 b Fp(H)6639 6724 y Fo(!)758 6898 y Ff(is)63 b(the)i(set)f(of)g Fp(E)f Fn(2)54 b Fr(\006)64 b Ff(for)f(which)i(ther)-8 b(e)64 b(exists)g(some)g(op)-8 b(en)64 b(interval)h Fp(I)i Fn(3)54 b Fp(E)73 b Ff(such)64 b(that)g Fp(H)6639 6923 y Fo(!)758 7097 y Ff(exhibits)c(exp)-8 b(onential)60 b(lo)-8 b(c)g(alization)60 b(in)f Fp(I)13 b Ff(.)758 7407 y Fv(De\014nition)62 b(2.2.)84 b Ff(The)43 b(r)-8 b(andom)42 b(Schr\177)-85 b(odinger)43 b(op)-8 b(er)g(ator)42 b Fp(H)4618 7432 y Fo(!)4756 7407 y Ff(exhibits)h(str)-8 b(ong)43 b(HS-dynamic)-8 b(al)758 7606 y(lo)g(c)g(alization)58 b(in)g(the)h(op)-8 b(en)58 b(interval)i Fp(I)71 b Ff(if)58 b(for)g(al)8 b(l)59 b Fr(0)46 b Fn(\024)g(X)71 b(2)46 b Fp(C)4677 7546 y Fj(1)4875 7606 y Ff(with)59 b(supp)-8 b(ort)57 b(in)i Fp(I)71 b Ff(we)59 b(have)1722 7985 y Fl(E)1852 7751 y Fk(\032)1985 7985 y Fr(sup)1999 8125 y Fo(t)p Fj(2)p Fg(R)2263 7794 y Fk(\015)2263 7894 y(\015)2263 7993 y(\015)2355 7985 y Fn(h)p Fp(X)13 b Fn(i)2655 7857 y Fh(n)p 2655 7875 74 6 v 2663 7941 a Fi(2)2756 7985 y Fr(e)2830 7916 y Fj(\000)p Fo(itH)3140 7933 y Fh(!)3233 7985 y Fn(X)24 b Fr(\()p Fp(H)3578 8010 y Fo(!)3674 7985 y Fr(\))p Fp(\037)3843 8010 y Fm(0)3917 7794 y Fk(\015)3917 7894 y(\015)3917 7993 y(\015)4010 7835 y Fm(2)4010 8101 y(2)4084 7751 y Fk(\033)4255 7985 y Fp(<)46 b Fn(1)138 b Ff(for)60 b(al)8 b(l)60 b Fp(n)46 b Fn(\025)g Fr(0)56 b Fp(;)651 b Fr(\(2.4\))758 8380 y Ff(wher)-8 b(e)91 b Fn(h)p Fp(X)13 b Fn(i)89 b Ff(denotes)j(the)f(op)-8 b(er)g(ator)89 b(given)j(by)f (multiplic)-8 b(ation)91 b(by)g(the)g(function)g Fn(h)p Fp(x)p Fn(i)102 b Fr(=)758 8452 y Fk(p)p 924 8452 548 7 v 142 x Fr(1)37 b(+)g Fn(j)p Fp(x)p Fn(j)1397 8546 y Fm(2)1471 8594 y Ff(,)72 b(and)e Fp(\037)2031 8619 y Fo(x)2184 8594 y Ff(is)g(the)g(char)-8 b(acteristic)70 b(function)h(of)e(the)h(cub)-8 b(e)70 b(of)g(side)g Fr(1)f Ff(c)-8 b(enter)g(e)g(d)70 b(at)758 8802 y Fp(x)52 b Fn(2)h Fl(R)1180 8742 y Fo(d)1266 8802 y Ff(.)88 b(The)63 b(str)-8 b(ong)63 b(insulator)h(r)-8 b(e)g(gion)63 b Fr(\006)3570 8827 y Fm(SI)3763 8802 y Ff(for)g(the)g(r)-8 b(andom)62 b(Schr\177)-85 b(odinger)65 b(op)-8 b(er)g(ator)62 b Fp(H)6639 8827 y Fo(!)758 9001 y Ff(is)h(the)i(set)f(of)g Fp(E)f Fn(2)54 b Fr(\006)64 b Ff(for)f(which)i(ther)-8 b(e)64 b(exists)g(some)g(op)-8 b(en)64 b(interval)h Fp(I)i Fn(3)54 b Fp(E)73 b Ff(such)64 b(that)g Fp(H)6639 9026 y Fo(!)758 9201 y Ff(exhibits)c(str)-8 b(ong)59 b(HS-dynamic)-8 b(al)60 b(lo)-8 b(c)g(alization)60 b(in)f Fp(I)13 b Ff(.)957 9510 y Fr(T)-14 b(o)75 b(discuss)i(criteria)d(for)h(lo)5 b(calization,)80 b(w)-5 b(e)75 b(need)g(to)g(consider)g(the)g (restriction)g(of)g(the)758 9710 y(random)68 b(Sc)-5 b(hr\177)-83 b(odinger)68 b(op)5 b(erator)66 b Fp(H)3119 9735 y Fo(!)3282 9710 y Fr(to)h(a)f(\014nite)h(b)5 b(o)-5 b(x.)109 b(Throughout)67 b(this)h(pap)5 b(er)67 b(w)-5 b(e)67 b(use)758 9909 y(the)55 b(sup)h(norm)g(in)g Fl(R)2088 9849 y Fo(d)2174 9909 y Fr(:)2403 10197 y Fn(j)p Fp(x)p Fn(j)47 b Fr(=)f Fn(j)p Fp(x)p Fn(j)2999 10222 y Fj(1)3186 10197 y Fr(=)g(max)p Fn(fj)p Fp(x)3894 10222 y Fo(i)3950 10197 y Fn(j)p Fp(;)74 b(i)46 b Fr(=)g(1)p Fp(;)28 b(:)g(:)g(:)55 b(;)28 b(d)p Fn(g)p Fp(:)758 10485 y Fr(By)54 b(\003)1133 10510 y Fo(L)1233 10485 y Fr(\()p Fp(x)p Fr(\))g(w)-5 b(e)56 b(denote)f(the)g(op)5 b(en)55 b(b)5 b(o)-5 b(x)55 b(\(or)g(cub)5 b(e\))55 b(of)g(side)h Fp(L)46 b(>)g Fr(0:)2532 10782 y(\003)2647 10807 y Fo(L)2746 10782 y Fr(\()p Fp(x)p Fr(\))g(=)3192 10647 y Fk(\010)3289 10782 y Fp(y)52 b Fn(2)46 b Fl(R)3690 10713 y Fo(d)3776 10782 y Fr(;)111 b Fn(j)p Fp(y)44 b Fn(\000)37 b Fp(x)p Fn(j)46 b Fp(<)4652 10716 y Fo(L)p 4652 10743 92 7 v 4664 10839 a Fm(2)4763 10647 y Fk(\011)4887 10782 y Fp(;)1461 b Fr(\(2.5\))p eop %%Page: 4 4 4 3 bop 758 861 a Fm(4)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fr(and)44 b(b)-5 b(y)p 1288 1126 116 7 v 44 w(\003)1403 1284 y Fo(L)1502 1259 y Fr(\()p Fp(x)p Fr(\))43 b(the)g(closed)h(b)5 b(o)-5 b(x.)70 b(The)44 b(c)-5 b(haracteristic)43 b(function)h(of)f(a)h (set)f(\003)k Fn(\032)f Fl(R)5878 1199 y Fo(d)6008 1259 y Fr(is)e(denoted)758 1458 y(b)-5 b(y)55 b Fp(\037)1092 1483 y Fm(\003)1191 1458 y Fr(;)h(w)-5 b(e)55 b(set)2550 1768 y Fp(\037)2654 1793 y Fo(x;L)2914 1768 y Fr(=)46 b Fp(\037)3193 1798 y Fm(\003)3284 1815 y Fh(L)3371 1798 y Fm(\()p Fo(x)p Fm(\))3586 1768 y Fp(;)120 b Fr(with)93 b Fp(\037)4272 1793 y Fo(x)4401 1768 y Fr(=)46 b Fp(\037)4680 1793 y Fo(x;)p Fm(1)4897 1768 y Fp(:)1451 b Fr(\(2.6\))758 2095 y Ff(In)69 b(this)g(article)i(we)e(wil)8 b(l)71 b(take)f(b)-8 b(oxes)69 b(c)-8 b(enter)g(e)g(d)70 b(at)f(sites)h Fp(x)63 b Fn(2)h Fl(Z)4907 2034 y Fo(d)5062 2095 y Ff(with)70 b(side)f Fp(L)64 b Fn(2)f Fr(2)p Fl(N)26 b Ff(.)115 b Fr(F)-14 b(or)758 2294 y(suc)-5 b(h)56 b(a)g(b)5 b(o)-5 b(x)55 b(w)-5 b(e)55 b(set)1211 2603 y(\000)1315 2628 y Fo(x;L)1575 2603 y Fr(=)1895 2446 y Fk(X)1750 2810 y Fo(y)t Fj(2)p Fm(\007)2012 2827 y Fh(L)2101 2810 y Fm(\()p Fo(x)p Fm(\))2308 2603 y Fp(\037)2412 2628 y Fo(y)2519 2603 y Fp(;)166 b Fr(where)92 b(\007)3377 2628 y Fo(L)3477 2603 y Fr(\()p Fp(x)p Fr(\))45 b(=)3922 2469 y Fk(\010)4019 2603 y Fp(y)52 b Fn(2)46 b Fl(Z)4411 2535 y Fo(d)4497 2603 y Fr(;)84 b Fn(j)p Fp(y)43 b Fn(\000)37 b Fp(x)p Fn(j)46 b Fr(=)5345 2538 y Fo(L)p 5345 2565 92 7 v 5357 2661 a Fm(2)5493 2603 y Fn(\000)37 b Fr(1)5742 2469 y Fk(\011)5895 2603 y Fp(:)453 b Fr(\(2.7\))957 3097 y(The)53 b(\014nite)h(v)-5 b(olume)54 b(random)g(Sc)-5 b(hr\177)-83 b(odinger)55 b(op)5 b(erator)53 b Fp(H)4610 3122 y Fo(!)t(;x;L)5005 3097 y Fr(is)h(de\014ned)g(as)g(the)f(restric-) 758 3297 y(tion)70 b(of)g Fp(H)1456 3322 y Fo(!)1552 3297 y Fr(,)75 b(either)70 b(to)g(the)g(op)5 b(en)70 b(b)5 b(o)-5 b(x)71 b(\003)3543 3322 y Fo(L)3642 3297 y Fr(\()p Fp(x)p Fr(\))f(with)g(Diric)-5 b(hlet)71 b(b)5 b(oundary)70 b(condition,)75 b(or)758 3501 y(to)56 b(the)h(closed)h(b)5 b(o)-5 b(x)p 2062 3368 116 7 v 57 w(\003)2177 3526 y Fo(L)2276 3501 y Fr(\()p Fp(x)p Fr(\))57 b(with)g(p)5 b(erio)g(dic)56 b(b)5 b(oundary)58 b(condition.)79 b(\(W)-14 b(e)57 b(consisten)-5 b(tly)57 b(w)-5 b(ork)758 3701 y(with)65 b(either)g(Diric)-5 b(hlet)65 b(or)h(p)5 b(erio)g(dic)65 b(b)5 b(oundary)66 b(condition.\))104 b(T)-14 b(o)66 b(see)f(that)g Fp(H)5870 3726 y Fo(!)t(;x;L)6277 3701 y Fr(is)h(w)-5 b(ell)758 3900 y(de\014ned)48 b(as)g(a)g(semi-b)5 b(ounded)49 b(self-adjoin)-5 b(t)49 b(op)5 b(erator)47 b(on)h(L)4534 3840 y Fm(2)4608 3900 y Fr(\(\003)4788 3925 y Fo(L)4887 3900 y Fr(\()p Fp(x)p Fr(\))p Fp(;)28 b Fr(d)p Fp(y)6 b Fr(\),)49 b(note)e(that)g(if)h Fn(r)6521 3925 y Fo(x;L)758 4099 y Fr(is)62 b(the)f(gradien)-5 b(t)62 b(op)5 b(erator)61 b(restricted)g(to)g(either)g(to)g(the)g(op)5 b(en)61 b(b)5 b(o)-5 b(x)62 b(\003)5336 4124 y Fo(L)5435 4099 y Fr(\()p Fp(x)p Fr(\))f(with)g(Diric)-5 b(hlet)758 4304 y(b)5 b(oundary)53 b(condition,)i(or)e(to)g(the)g(closed)h(b)5 b(o)-5 b(x)p 3767 4171 V 53 w(\003)3882 4329 y Fo(L)3982 4304 y Fr(\()p Fp(x)p Fr(\))52 b(with)h(p)5 b(erio)g(dic)53 b(b)5 b(oundary)54 b(condition,)758 4503 y(then)h(it)g(follo)-5 b(ws)56 b(from)g(\(2.2\))e(that)h(for)g(all)h Fp( )c Fn(2)46 b(D)5 b Fr(\()p Fn(r)4134 4528 y Fo(x;L)4348 4503 y Fr(\))55 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)1808 4678 y Fk(\014)1808 4777 y(\014)1808 4877 y(\014)1864 4684 y(D)1965 4869 y Fp( )6 b(;)28 b(V)2287 4800 y Fm(\(2\))2250 4910 y Fo(!)2465 4869 y Fp( )2579 4684 y Fk(E)2681 4678 y(\014)2681 4777 y(\014)2681 4877 y(\014)2782 4869 y Fn(\024)46 b Fr(\002)3086 4894 y Fm(1)3161 4869 y Fn(kr)3382 4894 y Fo(x;L)3596 4869 y Fp( )6 b Fn(k)3793 4800 y Fm(2)3905 4869 y Fr(+)37 b(\002)4200 4894 y Fm(2)4275 4869 y Fn(k)p Fp( )6 b Fn(k)4555 4800 y Fm(2)4768 4869 y Fr(for)55 b Fl(P)5 b Fr(-)q(a.e.)83 b Fp(!)34 b(:)709 b Fr(\(2.8\))758 5228 y(\(F)-14 b(or)72 b(Diric)-5 b(hlet)72 b(b)5 b(oundary)73 b(condition)f(\(2.8\))g(follo)-5 b(ws)73 b(immediately)f(from)g (\(2.2\))g(with)g(the)758 5427 y(same)67 b(\002)1314 5452 y Fm(1)1456 5427 y Fr(and)g(\002)1919 5452 y Fm(2)2060 5427 y Fr(as)h(in)f(\(2.2\).)107 b(F)-14 b(or)68 b(p)5 b(erio)g(dic)66 b(b)5 b(oundary)67 b(condition)g(\(2.8\))f(follo)-5 b(ws)68 b(from)758 5626 y(\(2.2\))63 b(b)-5 b(y)65 b(using)h(a)e(smo)5 b(oth)65 b(partition)f(of)g(the)g(iden)-5 b(tit)g(y)65 b(on)f(the)g(torus,)j(with)e(the)f(same)h(\002)6661 5651 y Fm(1)758 5826 y Fr(but)e(with)f(\002)1584 5851 y Fm(2)1722 5826 y Fr(enlarged)h(b)-5 b(y)63 b(a)g(\014nite)g(constan)-5 b(t)64 b(dep)5 b(ending)63 b(only)g(on)g(the)g(dimension)h Fp(d)p Fr(,)i(so)758 6025 y(w)-5 b(e)54 b(can)h(mo)5 b(dify)54 b(\002)1991 6050 y Fm(2)2120 6025 y Fr(in)g(\(2.2\))g(so)h (\(2.2\))e(and)i(\(2.8\))f(hold)h(with)f(the)g(same)h(\002)5598 6050 y Fm(1)5727 6025 y Fr(and)g(\002)6178 6050 y Fm(2)6307 6025 y Fr(for)f(all)758 6224 y(b)5 b(o)-5 b(xes)55 b(\003)1330 6249 y Fo(L)1430 6224 y Fr(\()p Fp(x)p Fr(\).\))72 b(W)-14 b(e)55 b(write)g Fp(R)2675 6249 y Fo(!)t(;x;L)3017 6224 y Fr(\()p Fp(z)7 b Fr(\))45 b(=)i(\()p Fp(H)3655 6249 y Fo(!)t(;x;L)4033 6224 y Fn(\000)37 b Fp(z)7 b Fr(\))4348 6164 y Fj(\000)p Fm(1)4581 6224 y Fr(for)55 b(the)g(resolv)-5 b(en)g(t.)957 6423 y(A)60 b(random)h(Sc)-5 b(hr\177)-83 b(odinger)62 b(op)5 b(erator)59 b(satis\014es)j(all)f(the)f(requiremen) -5 b(ts)61 b(for)f(the)g(b)5 b(o)g(otstrap)758 6623 y(m)-5 b(ultiscale)53 b(analysis)h(of)e([GK1])h(with)f(the)g(p)5 b(ossible)54 b(exception)d(of)i(a)f(W)-14 b(egner)52 b(estimate,)h(i.e.,)758 6822 y(it)60 b(satis\014es)i(Assumptions)g (SLI,)f(EDI,)f(IAD,)h(NE,)f(and)h(SGEE)g(of)f([GK1)q(])h(in)g(an)-5 b(y)61 b(b)5 b(ounded)758 7021 y(in)-5 b(terv)c(al)55 b([GK3,)h(Theorem)f(A.1].)74 b(W)-14 b(e)55 b(refer)g(to)g([GK1,)h (GK3])g(for)f(a)g(discussion)j(of)d(all)g(these)758 7220 y(assumptions.)128 b(F)-14 b(or)74 b(our)f(purp)5 b(oses)74 b([GK3,)k(Theorem)73 b(A.1])f(can)h(b)5 b(e)73 b(restated)f(as)h(follo) -5 b(ws.)758 7420 y(\(Note)54 b(that)g(\(2.12\))h(b)5 b(elo)-5 b(w)55 b(is)h(giv)-5 b(en)56 b(in)g([GK3,)g(Eq.)73 b(\(A.7\)].\))758 7742 y Fv(Theorem)63 b(2.3.)84 b Ff(L)-8 b(et)60 b Fp(H)2374 7767 y Fo(!)2531 7742 y Ff(b)-8 b(e)60 b(a)g(r)-8 b(andom)60 b(Schr\177)-85 b(odinger)62 b(op)-8 b(er)g(ator.)78 b(Then)60 b Fp(H)5703 7767 y Fo(!)5860 7742 y Ff(satis\014es)h(As-)758 7941 y(sumptions)66 b(SLI,)i(EDI,)f (IAD,)g(NE,)g(and)g(SGEE)g(of)g Fr([GK1)q(])p Ff(.)99 b(In)67 b(p)-8 b(articular,)69 b(Assumptions)758 8140 y(SLI)60 b(and)f(NE)h(hold)f(in)h(the)f(folowing)i(form:)970 8395 y Fv(SLI:)83 b Ff(Given)68 b Fp(L;)28 b(`)2147 8334 y Fj(0)2194 8395 y Fp(;)g(`)2337 8334 y Fj(00)2480 8395 y Fn(2)58 b Fr(2)p Fl(N)26 b Ff(,)77 b Fp(x;)28 b(y)6 b(;)28 b(y)3388 8334 y Fj(0)3493 8395 y Fn(2)58 b Fl(Z)3764 8334 y Fo(d)3917 8395 y Ff(with)66 b Fr(\003)4399 8420 y Fo(`)4455 8386 y Fe(00)4545 8395 y Fr(\()p Fp(y)6 b Fr(\))58 b Fn(\032)h Fr(\003)5123 8420 y Fo(`)5179 8386 y Fe(0)5223 8420 y Fj(\000)p Fm(3)5402 8395 y Fr(\()p Fp(y)5554 8334 y Fj(0)5600 8395 y Fr(\))66 b Ff(and)g Fr(\003)6175 8420 y Fo(`)6231 8386 y Fe(0)6285 8395 y Fr(\()p Fp(y)6437 8334 y Fj(0)6483 8395 y Fr(\))58 b Fn(\032)1169 8594 y Fr(\003)1284 8619 y 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Fr(0)p Fn(g)36 b Fr(+)h(\002)4485 10246 y Fm(2)4597 10221 y Fr(+)g(50)p Fp(d)47 b(:)1203 b Fr(\(2.11\))970 10530 y Fv(NE:)84 b Ff(F)-13 b(or)60 b Fl(P)5 b Ff(-)q(a.e.)85 b Fp(!)66 b Ff(and)59 b(al)8 b(l)61 b Fp(E)55 b Fn(2)46 b Fl(R)90 b Ff(we)60 b(have)1417 10840 y Fr(tr)1602 10705 y Fk(\010)1699 10840 y Fp(\037)1803 10870 y Fm(\()p Fj(\0001)p Fo(;E)6 b Fm(\))2294 10840 y Fr(\()p Fp(H)2497 10865 y Fo(!)t(;x;L)2838 10840 y Fr(\))2903 10705 y Fk(\011)3046 10840 y Fn(\024)46 b Fp(C)3340 10865 y Fo(E)3452 10840 y Fp(L)3565 10771 y Fo(d)3781 10840 y Ff(for)59 b(al)8 b(l)60 b Fp(x)46 b Fn(2)g Fl(Z)4683 10780 y Fo(d)4829 10840 y Ff(and)59 b Fp(L)46 b Fn(2)g Fr(2)p Fl(N)63 b Fp(;)567 b Fr(\(2.12\))p eop %%Page: 5 5 5 4 bop 2654 861 a Fm(HIGH)58 b(DISORDER)h(LOCALIZA)-11 b(TION)1830 b(5)1169 1259 y Ff(with)2571 1696 y Fp(C)2690 1721 y Fo(E)2848 1696 y Fr(=)46 b Fp(c)3095 1721 y Fo(d)3200 1462 y Fk(\022)3343 1584 y Fr(max)p Fn(f)p Fp(E)g Fr(+)37 b(\002)4199 1609 y Fm(2)4301 1584 y Fp(;)28 b Fr(0)p Fn(g)p 3343 1658 1199 7 v 3697 1810 a Fr(1)37 b Fn(\000)g Fr(\002)4112 1835 y Fm(1)4561 1462 y Fk(\023)4703 1452 y Fh(d)p 4703 1470 62 6 v 4705 1537 a Fi(2)4848 1696 y Fp(;)1417 b Fr(\(2.13\))1169 2104 y Fp(c)1241 2129 y Fo(d)1378 2104 y Ff(b)-8 b(eing)60 b(a)f(\014nite)i(c)-8 b(onstant)60 b(dep)-8 b(ending)60 b(on)g Fp(d)g Ff(only.)957 2431 y Fr(Giv)-5 b(en)56 b(a)f(b)5 b(ounded)56 b(in)-5 b(terv)c(al)55 b Fp(I)69 b Fr(w)-5 b(e)55 b(set)2044 2749 y Fp(\015)2130 2774 y Fo(I)2252 2749 y Fr(=)d(sup)2427 2888 y Fo(E)6 b Fj(2)p Fo(I)2716 2749 y Fp(\015)2802 2774 y Fo(E)2960 2749 y Fp(;)167 b(C)3292 2774 y Fo(I)3414 2749 y Fr(=)51 b(sup)3589 2888 y Fo(E)6 b Fj(2)p Fo(I)3878 2749 y Fp(C)3997 2774 y Fo(E)4155 2749 y Fr(=)46 b Fp(C)4449 2774 y Fm(sup)4648 2805 y Fh(E)t Fe(2)p Fh(I)4904 2774 y Fo(E)5071 2749 y Fp(:)1194 b Fr(\(2.14\))758 3164 y(Note)59 b(that)h(Assumption)i(EDI)f(also)g(holds)h(at)e(an)h(energy)f Fp(E)70 b Fr(with)60 b(the)h(same)g(constan)-5 b(t)61 b Fp(\015)6623 3189 y Fo(E)758 3364 y Fr(as)56 b(in)f(\(2.10\))g(and)h (\(2.11\))o(.)957 3563 y(In)g(this)g(article)g(a)g(W)-14 b(egner)55 b(estimate)h(in)g(an)g(op)5 b(en)56 b(in)-5 b(terv)c(al)55 b(\(Assumption)i(W)f(in)g([GK1)q(]\))758 3762 y(will)f(b)5 b(e)55 b(an)h(explicit)f(h)-5 b(yp)5 b(othesis.)758 4089 y Fv(De\014nition)62 b(2.4.)84 b Ff(The)49 b(r)-8 b(andom)48 b(Schr\177)-85 b(odinger)50 b(op)-8 b(er)g(ator)48 b Fp(H)4643 4114 y Fo(!)4788 4089 y Ff(satis\014es)h(a)f(We)-8 b(gner)50 b(estimate)758 4288 y(in)59 b(an)h(op)-8 b(en)59 b(interval)i Fn(I)72 b Ff(if)60 b(for)f(some)g Fp(b)47 b Fn(\025)f Fr(1)59 b Ff(ther)-8 b(e)60 b(exists)g(a)f(c)-8 b(onstant)60 b Fp(Q)5452 4313 y Fj(I)5543 4288 y Ff(,)f(such)h(that)2330 4606 y Fl(P)34 b Fn(f)9 b Fr(dist\()p Fp(\033)d Fr(\()p Fp(H)3180 4631 y Fo(!)t(;x;L)3520 4606 y Fr(\))p Fp(;)28 b(E)10 b Fr(\))45 b Fn(\024)h Fp(\021)6 b Fn(g)46 b(\024)h Fp(Q)4601 4631 y Fj(I)4692 4606 y Fp(\021)6 b(L)4893 4537 y Fo(bd)5089 4606 y Fp(;)1176 b Fr(\(2.15\))758 4930 y Ff(for)68 b(al)8 b(l)70 b Fp(E)i Fn(2)63 b(I)12 b Ff(,)72 b Fr(0)63 b Fp(<)g(\021)69 b Fn(\024)63 b Fr(1)p Ff(,)71 b Fp(x)63 b Fn(2)g Fl(Z)3193 4869 y Fo(d)3279 4930 y Ff(,)71 b(and)e Fp(L)63 b Fn(2)g Fr(2)p Fl(N)26 b Ff(.)113 b(The)69 b(We)-8 b(gner)69 b(r)-8 b(e)g(gion)69 b Fr(\006)6032 4955 y Fm(W)6244 4930 y Ff(for)g(the)758 5129 y(r)-8 b(andom)65 b(Schr\177)-85 b(odinger)68 b(op)-8 b(er)g(ator)66 b Fp(H)3096 5154 y Fo(!)3258 5129 y Ff(is)h(the)g(set)f (of)h Fp(E)h Fn(2)58 b Fr(\006)67 b Ff(for)f(which)h(ther)-8 b(e)67 b(exists)f(some)758 5328 y(op)-8 b(en)59 b(interval)i Fp(I)e Fn(3)46 b Fp(E)68 b Ff(such)60 b(that)g Fp(H)3093 5353 y Fo(!)3248 5328 y Ff(satis\014es)g(a)f(We)-8 b(gner)60 b(estimate)g(in)g Fp(I)13 b Ff(.)957 5655 y Fr(W)-14 b(egner)67 b(estimates)h(ha)-5 b(v)g(e)67 b(b)5 b(een)67 b(pro)-5 b(v)g(en)69 b(for)e(a)g(large)g(v)-9 b(ariet)k(y)67 b(of)g(random)h(Sc)-5 b(hr\177)-83 b(odinger)758 5854 y(op)5 b(erators)67 b(\(e.g.,)k([W)-14 b(e)o(,)68 b(HM,)g(CKM,)g(CH1)o (,)g(Klo1,)g(CHM,)f(Ki)q(,)g(KSS1)q(,)h(St,)f(CHN)o(,)h(CHKN)o(,)758 6053 y(HK)o(]\),)42 b(under)e(some)f(assumptions)i(on)e(the)g(random)g (p)5 b(oten)-5 b(tials.)69 b(Usually)39 b Fp(b)46 b Fr(=)h(1)39 b(or)f(2.)69 b(In)39 b(this)758 6253 y(pap)5 b(er,)55 b(w)-5 b(e)56 b(shall)g(use)g(\(2.15\))f(as)h(stated,)f(the)g(mo)5 b(di\014cations)57 b(in)f(our)f(metho)5 b(ds)56 b(required)f(for)758 6452 y(the)c(other)h(forms)g(of)g(\(2.15\))f(b)5 b(eing)52 b(ob)-5 b(vious.)74 b(\(F)-14 b(or)52 b(a)g(discussion)j(of)c(p)5 b(ossible)53 b(mo)5 b(di\014cations)758 6651 y(see)55 b([GK1)q(,)g(Remark)h(2.4].\))957 6851 y(W)-14 b(e)48 b(will)h(lo)5 b(ok)48 b(for)h(lo)5 b(calization)48 b(b)-5 b(y)49 b(studying)g(the)g(deca)-5 b(y)48 b(of)h(the)f(\014nite)g(v)-5 b(olume)50 b(resolv)-5 b(en)g(t)758 7050 y(from)55 b(the)g(cen)-5 b(ter)55 b(of)g(a)h(b)5 b(o)-5 b(x)55 b(\003)2691 7075 y Fo(L)2791 7050 y Fr(\()p Fp(x)p Fr(\))f(to)h(its)g(b)5 b(oundary)56 b(as)g(measured)g(b)-5 b(y)2877 7368 y Fn(k)p Fr(\000)3064 7393 y Fo(x;L)3278 7368 y Fp(R)3404 7393 y Fo(!)t(;x;L)3745 7368 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)4112 7398 y Fo(x;L=)p Fm(3)4459 7368 y Fn(k)28 b Fp(:)1695 b Fr(\(2.16\))758 7686 y(W)-14 b(e)55 b(use)h(the)e(con)-5 b(v)g(en)g(tion)57 b(that)d Fn(k)p Fr(\000)2993 7711 y Fo(x;L)3207 7686 y Fp(R)3333 7711 y Fo(!)t(;x;L)3675 7686 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)4042 7716 y Fo(x;L=)p Fm(3)4388 7686 y Fn(k)47 b Fr(=)f Fn(1)55 b Fr(if)g Fp(E)h Fn(2)45 b Fp(\033)6 b Fr(\()p Fp(H)5705 7711 y Fo(!)t(;x;L)6046 7686 y Fr(\).)957 7887 y(W)-14 b(e)62 b(start)h(with)f(t)-5 b(w)g(o)63 b(deterministic)g(\(i.e.,)i (for)d(a)h(giv)-5 b(en)63 b Fp(!)6 b Fr(,)65 b(whic)-5 b(h)63 b(is)g(omitted)g(from)g(the)758 8086 y(notation\))54 b(de\014nitions.)758 8413 y Fv(De\014nition)62 b(2.5.)84 b Ff(Given)70 b Fp(\022)d(>)62 b Fr(0)p Ff(,)71 b Fp(E)g Fn(2)62 b Fl(R)22 b Ff(,)79 b Fp(x)62 b Fn(2)g Fl(Z)4104 8352 y Fo(d)4190 8413 y Ff(,)71 b(and)d Fp(L)62 b Fn(2)g Fr(6)p Fl(N)26 b Ff(,)80 b(we)69 b(say)e(that)i(the)f(b)-8 b(ox)758 8612 y Fr(\003)873 8637 y Fo(L)972 8612 y Fr(\()p Fp(x)p Fr(\))59 b Ff(is)g Fr(\()p Fp(\022)5 b(;)28 b(E)10 b Fr(\))p Ff(-suitable)59 b(if)2700 9008 y Fn(k)p Fr(\000)2887 9033 y Fo(x;L)3101 9008 y Fp(R)3227 9033 y Fo(x;L)3441 9008 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)3808 9038 y Fo(x;L=)p Fm(3)4155 9008 y Fn(k)46 b(\024)4532 8896 y Fr(1)p 4479 8970 189 7 v 4479 9122 a Fp(L)4592 9074 y Fo(\022)4747 9008 y Fp(:)1518 b Fr(\(2.17\))758 9407 y Fv(De\014nition)62 b(2.6.)84 b Ff(Given)68 b Fp(m)57 b(>)h Fr(0)p Ff(,)67 b Fp(E)g Fn(2)58 b Fl(R)22 b Ff(,)76 b Fp(x)58 b Fn(2)f Fl(Z)4132 9347 y Fo(d)4218 9407 y Ff(,)67 b(and)f Fp(L)58 b Fn(2)f Fr(6)p Fl(N)26 b Ff(,)77 b(we)66 b(say)f(that)h(the)g(b)-8 b(ox)758 9606 y Fr(\003)873 9631 y Fo(L)972 9606 y Fr(\()p Fp(x)p Fr(\))59 b Ff(is)g Fr(\()p Fp(m;)28 b(E)10 b Fr(\))p Ff(-r)-8 b(e)g(gular)58 b(if)2632 9953 y Fn(k)p Fr(\000)2819 9978 y Fo(x;L)3033 9953 y Fp(R)3159 9978 y Fo(x;L)3374 9953 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)3741 9983 y Fo(x;L=)p Fm(3)4087 9953 y Fn(k)46 b(\024)g Fr(e)4465 9884 y Fj(\000)p Fo(m)4707 9840 y Fh(L)p 4707 9858 80 6 v 4718 9924 a Fi(2)4814 9953 y Fp(:)1451 b Fr(\(2.18\))957 10280 y(W)-14 b(e)52 b(de\014ne)h(the)f(m)-5 b(ultiscale)53 b(analysis)h(region)e(\006)4014 10305 y Fm(MSA)4360 10280 y Fn(\032)46 b Fr(\006)52 b(b)-5 b(y)53 b(requiring)g(the)f(conclusions)758 10479 y(of)j(the)g(b)5 b(o)g(otstrap)55 b(m)-5 b(ultiscale)56 b(analysis)g([GK1)q(,)g(Theorem) f(3.4].)74 b(W)-14 b(e)55 b(use)h(the)f(notation)2610 10797 y([)p Fp(K)12 b Fr(])2855 10822 y Fm(6)p Fg(N)3060 10797 y Fr(=)46 b(max)p Fn(f)p Fp(L)g Fn(2)g Fr(6)p Fl(N)26 b Fr(;)83 b Fp(L)46 b Fn(\024)h Fp(K)12 b Fn(g)p Fp(:)1428 b Fr(\(2.19\))p eop %%Page: 6 6 6 5 bop 758 861 a Fm(6)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fv(De\014nition)k(2.7.)84 b Ff(The)58 b(multisc)-8 b(ale)58 b(analysis)f(r)-8 b(e)g(gion)57 b Fr(\006)4379 1284 y Fm(MSA)4735 1259 y Ff(for)g(the)g(r)-8 b(andom)57 b(Schr\177)-85 b(odinger)758 1458 y(op)-8 b(er)g(ator)59 b Fp(H)1550 1483 y Fo(!)1706 1458 y Ff(is)i(the)f(set)h (of)f Fp(E)d Fn(2)48 b Fr(\006)3077 1483 y Fm(W)3281 1458 y Ff(for)60 b(which)g(ther)-8 b(e)61 b(exists)g(some)f(op)-8 b(en)60 b(interval)i Fp(I)f Fn(3)47 b Fp(E)758 1658 y Ff(such)63 b(that,)g(given)i(any)d Fp(\020)12 b Ff(,)64 b Fr(0)52 b Fp(<)h(\020)64 b(<)53 b Fr(1)p Ff(,)63 b(and)g Fp(\013)q Ff(,)h Fr(1)52 b Fp(<)h(\013)g(<)g(\020)4615 1597 y Fj(\000)p Fm(1)4793 1658 y Ff(,)63 b(ther)-8 b(e)64 b(is)e(a)h(length)h(sc)-8 b(ale)64 b Fp(L)6661 1683 y Fm(0)758 1857 y Ff(and)59 b(a)g(mass)g Fp(m)1786 1882 y Fo(\020)1908 1857 y Fp(>)47 b Fr(0)p Ff(,)59 b(so)g(if)g(we)i(set)e Fp(L)3268 1882 y Fo(k)s Fm(+1)3564 1857 y Fr(=)47 b([)p Fp(L)3899 1797 y Fo(\013)3899 1904 y(k)3993 1857 y Fr(])4039 1882 y Fm(6)p Fg(N)4198 1857 y Ff(,)59 b Fp(k)52 b Fr(=)46 b(0)p Fp(;)28 b Fr(1)p Fp(;)g(:)g(:)g(:)f Ff(,)60 b(we)g(have)2524 2186 y Fl(P)70 b Fn(f)p Fp(R)29 b Fr(\()p Fp(m)3130 2211 y Fo(\020)3206 2186 y Fp(;)f(L)3393 2211 y Fo(k)3474 2186 y Fp(;)g(I)13 b(;)28 b(x;)g(y)6 b Fr(\))p Fn(g)46 b(\025)g Fr(1)37 b Fn(\000)g Fr(e)4693 2118 y Fj(\000)p Fo(L)4888 2061 y Fh(\020)4888 2158 y(k)6311 2186 y Fr(\(2.20\))758 2474 y Ff(for)59 b(al)8 b(l)60 b Fp(k)52 b Fr(=)47 b(0)p Fp(;)28 b Fr(1)p Fp(;)g(:)g(:)g(:)f Ff(,)59 b(and)g Fp(x;)28 b(y)53 b Fn(2)45 b Fl(Z)3102 2414 y Fo(d)3248 2474 y Ff(with)60 b Fn(j)p Fp(x)37 b Fn(\000)g Fp(y)6 b Fn(j)46 b Fp(>)g(L)4420 2499 y Fo(k)4539 2474 y Fr(+)37 b Fp(\045)p Ff(,)59 b(wher)-8 b(e)1332 2756 y Fp(R)q Fr(\()p Fp(m;)28 b(L;)g(I)13 b(;)28 b(x;)g(y)6 b Fr(\))45 b(=)3725 b(\(2.21\))1664 3005 y Fn(f)p Ff(for)59 b(every)47 b Fp(E)2571 2937 y Fj(0)2663 3005 y Fn(2)f Fp(I)13 b(;)87 b Ff(either)60 b Fr(\003)3627 3030 y Fo(L)3727 3005 y Fr(\()p Fp(x)p Fr(\))e Ff(or)i Fr(\003)4340 3030 y Fo(L)4439 3005 y Fr(\()p Fp(y)6 b Fr(\))59 b Ff(is)g Fr(\()p Fp(m;)28 b(E)5311 2945 y Fj(0)5356 3005 y Fr(\))p Ff(-r)-8 b(e)g(gular)p Fn(g)60 b Fp(:)957 3312 y Fr(On)48 b(\006)1346 3337 y Fm(MSA)1692 3312 y Fr(w)-5 b(e)47 b(ha)-5 b(v)g(e)48 b(all)g(desired)g(prop)5 b(erties)47 b(of)g(lo)5 b(calization,)49 b(including)g(exp)5 b(onen)-5 b(tial)47 b(lo-)758 3511 y(calization)52 b(and)h(strong)f(Hilb)5 b(ert-Sc)-5 b(hmidt)53 b(dynamical)g(lo)5 b(calization)52 b([GK1].)73 b(In)53 b(particular,)3023 3793 y(\006)3143 3818 y Fm(MSA)3488 3793 y Fn(\032)46 b Fr(\006)3783 3818 y Fm(EL)3999 3793 y Fn(\\)37 b Fr(\006)4267 3818 y Fm(SI)4424 3793 y Fp(:)1841 b Fr(\(2.22\))758 4075 y(On)55 b(the)g(other)g(hand,)i(w)-5 b(e)55 b(pro)-5 b(v)g(ed)56 b(in)g([GK3])g(that)3027 4357 y(\006)3147 4382 y Fm(SI)3313 4357 y Fn(\\)37 b Fr(\006)3581 4382 y Fm(W)3771 4357 y Fn(\032)46 b Fr(\006)4066 4382 y Fm(MSA)4393 4357 y Fp(;)1872 b Fr(\(2.23\))758 4639 y(and)56 b(hence)f(w)-5 b(e)55 b(ha)-5 b(v)g(e:)758 4946 y Fv(Theorem)63 b(2.8.)84 b Ff(L)-8 b(et)59 b Fp(H)2373 4971 y Fo(!)2529 4946 y Ff(b)-8 b(e)59 b(a)h(r)-8 b(andom)58 b(Schr\177)-85 b(odinger)61 b(op)-8 b(er)g(ator.)76 b(Then)2339 5228 y Fr(\006)2459 5253 y Fm(MSA)2804 5228 y Fr(=)46 b(\006)3099 5253 y Fm(SI)3266 5228 y Fn(\\)36 b Fr(\006)3533 5253 y Fm(W)3723 5228 y Fr(=)46 b(\006)4018 5253 y Fm(EL)4235 5228 y Fn(\\)36 b Fr(\006)4502 5253 y Fm(SI)4669 5228 y Fn(\\)h Fr(\006)4937 5253 y Fm(W)5108 5228 y Fp(:)1157 b Fr(\(2.24\))957 5534 y(In)45 b([GK4)q(])g(w)-5 b(e)46 b(ga)-5 b(v)g(e)45 b(explicit)g(\014nite)g(v)-5 b(olume)46 b(criteria)f(for)g Fp(E)55 b Fn(2)46 b Fr(\006)5046 5559 y Fm(MSA)5345 5534 y Fr(.)71 b(The)45 b(\014rst)g(criterion)758 5733 y(w)-5 b(orks)56 b(for)f(a)g Ff(pr)-8 b(escrib)g(e)g(d)72 b Fr(v)-9 b(alue)55 b(of)g(the)g(initial)h(length)f(scale)h Fp(L)4805 5758 y Fm(0)4935 5733 y Fr([GK4,)g(Theorem)f(2.4].)758 6040 y Fv(Theorem)63 b(2.9.)84 b Ff(L)-8 b(et)53 b Fp(H)2367 6065 y Fo(!)2517 6040 y Ff(b)-8 b(e)54 b(a)g(r)-8 b(andom)53 b(Schr\177)-85 b(odinger)55 b(op)-8 b(er)g(ator.)73 b(Fix)54 b(a)g(length)h(sc)-8 b(ale)54 b Fp(L)6504 6065 y Fm(0)6625 6040 y Fn(2)758 6239 y Fr(6)p Fl(N)26 b Ff(,)81 b Fp(L)1197 6264 y Fm(0)1335 6239 y Fn(\025)65 b Fr(max)28 b Fn(f)p Fr(6)p Fp(;)g Fr(3)p Fp(\045;)g Fn(g)55 b Fp(:)69 b Ff(L)-8 b(et)70 b Fp(E)3022 6264 y Fm(0)3160 6239 y Fn(2)65 b Fr(\006)3456 6264 y Fm(W)3599 6239 y Ff(,)72 b(with)e Fp(H)4231 6264 y Fo(!)4397 6239 y Ff(satisfying)f(a)h(We)-8 b(gner)70 b(estimate)g(in)758 6438 y(the)59 b(op)-8 b(en)60 b(b)-8 b(ounde)g(d)59 b(interval)i Fn(I)e(3)46 b Fp(E)3094 6463 y Fm(0)3168 6438 y Ff(,)59 b(and)h(supp)-8 b(ose)1444 6811 y Fl(P)1564 6577 y Fk(\032)1697 6811 y Fp(D)1834 6836 y Fj(I)1926 6811 y Fp(L)2039 6712 y Fr(\()2123 6655 y Fi(5)p 2123 6673 57 6 v 2123 6740 a(3)2200 6700 y Fm(+)p Fo(b)2360 6712 y Fr(\))2425 6700 y Fo(d)2039 6855 y Fm(0)2502 6811 y Fn(k)p Fr(\000)2689 6836 y Fm(0)p Fo(;L)2885 6853 y Fi(0)2959 6811 y Fp(R)3085 6836 y Fo(!)t(;)p Fm(0)p Fo(;L)3408 6853 y Fi(0)3482 6811 y Fr(\()p Fp(E)3670 6836 y Fm(0)3743 6811 y Fr(\))p Fp(\037)3912 6841 y Fm(0)p Fo(;L)4108 6858 y Fi(0)4173 6841 y Fo(=)p Fm(3)4316 6811 y Fn(k)46 b Fp(<)g Fr(1)4703 6577 y Fk(\033)4874 6811 y Fn(\025)g Fr(1)37 b Fn(\000)5456 6745 y Fm(2)p 5355 6772 269 7 v 5355 6870 a(368)5553 6837 y Fh(d)5671 6811 y Fp(;)594 b Fr(\(2.25\))758 7183 y Ff(with)2196 7465 y Fp(D)2333 7490 y Fj(I)2471 7465 y Fr(=)46 b(39)2812 7397 y Fm(\(3+)p Fo(b)p Fm(\))p Fo(d)3247 7465 y Fr(max)3584 7331 y Fk(\010)3681 7465 y Fr(16)37 b Fn(\001)g Fr(60)4133 7397 y Fo(d)4210 7465 y Fp(Q)4341 7490 y Fj(I)4433 7465 y Fp(;)28 b Fr(1)4590 7331 y Fk(\011)4714 7465 y Fp(\015)4809 7397 y Fm(2)4800 7506 y Fj(I)4919 7465 y Fp(:)1346 b Fr(\(2.26\))758 7747 y Ff(Then)59 b Fp(E)1313 7772 y Fm(0)1434 7747 y Fn(2)45 b Fr(\006)1710 7772 y Fm(MSA)2010 7747 y Ff(.)957 8054 y Fr(The)66 b(second)h(criterion)f(is)h(for)f Ff(lar)-8 b(ge)66 b Fr(initial)h(length)f(scale)h Fp(L)4913 8079 y Fm(0)5053 8054 y Fr(and)g Ff(we)-8 b(ak)67 b Fr(initial)f(deca) -5 b(y:)758 8253 y(an)g(y)53 b(rate)f(that)h(is)h(faster)e(than)h(the)g (v)-5 b(olume)54 b(\(if)e Fp(b)46 b Fr(=)h(1)53 b(in)g(W)-14 b(egner\))52 b(or)i(the)e(v)-5 b(olume)54 b(squared)758 8452 y(\(if)67 b Fp(b)g Fr(=)g(2\))g(is)i(allo)-5 b(w)g(ed)69 b([GK4,)i(Theorem)d(2.5].)112 b(It)67 b(pro)-5 b(vides)69 b(a)f(precise)g(estimate)g(on)g(ho)-5 b(w)758 8652 y(large)57 b Fp(L)1279 8677 y Fm(0)1411 8652 y Fr(has)i(to)e(b)5 b(e,)58 b(dep)5 b(ending)58 b(on)g(the)f(parameters)h Fp(Q)4504 8677 y Fj(I)4653 8652 y Fr(and)g Fp(\015)5064 8677 y Fj(I)5213 8652 y Fr(of)f(the)g(mo)5 b(del,)59 b(and)f(on)758 8851 y(the)d(prescrib)5 b(ed)55 b(rate)g(of)g(deca)-5 b(y)56 b(of)f(the)g(resolv)-5 b(en)g(t.)758 9157 y Fv(Theorem)63 b(2.10.)84 b Ff(L)-8 b(et)70 b Fp(H)2479 9182 y Fo(!)2645 9157 y Ff(b)-8 b(e)70 b(a)g(r)-8 b(andom)69 b(Schr\177)-85 b(odinger)71 b(op)-8 b(er)g(ator.)106 b(L)-8 b(et)70 b Fp(E)5732 9182 y Fm(0)5871 9157 y Fn(2)65 b Fr(\006)6167 9182 y Fm(W)6311 9157 y Ff(,)72 b(with)758 9356 y Fp(H)896 9381 y Fo(!)1055 9356 y Ff(satis\014ng)63 b(a)g(We)-8 b(gner)64 b(estimate)g(in)f(the)g(op)-8 b(en)63 b(b)-8 b(ounde)g(d)63 b(interval)h Fn(I)h(3)53 b Fp(E)5704 9381 y Fm(0)5778 9356 y Ff(.)86 b(Fix)64 b Fp(s)52 b(>)h(bd)p Ff(,)758 9556 y(and)59 b(set)1789 9903 y Fn(L)45 b Fr(=)h(max)2461 9669 y Fk(\032)2586 9903 y Fr(3)p Fp(\045;)28 b Fr(42)p Fp(;)g Fr(3)3180 9719 y Fk(\020)3298 9838 y Fm(107)3496 9788 y Fh(d)p 3298 9865 V 3399 9960 a Fm(2)3586 9719 y Fk(\021)3807 9709 y Fi(2)p 3705 9727 260 6 v 3705 9793 a Fh(s)p Fe(\000)p Fh(bd)4021 9903 y Fp(;)4148 9838 y Fm(1)p 4114 9865 133 7 v 4114 9960 a(37)4294 9769 y Fk(\000)4371 9903 y Fr(16)36 b Fn(\001)h Fr(60)4822 9843 y Fo(d)4900 9903 y Fp(Q)5031 9928 y Fj(I)5122 9769 y Fk(\001)5320 9758 y Fi(2)p 5218 9776 260 6 v 5218 9843 a Fh(s)p Fe(\000)p Fh(bd)5506 9669 y Fk(\033)5658 9903 y Fp(:)607 b Fr(\(2.27\))758 10276 y Ff(Supp)-8 b(ose)59 b(that)g(for)h(some)f Fp(L)2532 10301 y Fm(0)2653 10276 y Fn(\025)46 b(L)p Ff(,)59 b Fp(L)3166 10301 y Fm(0)3286 10276 y Fn(2)46 b Fr(6)p Fl(N)35 b Ff(,)59 b(we)h(have)1341 10558 y Fl(P)1461 10423 y Fk(\010)1567 10558 y Fr(90)1733 10489 y Fo(d)1810 10558 y Fp(\015)1905 10489 y Fm(2)1896 10599 y Fj(I)1987 10558 y Fr(\(37)p Fp(L)2331 10583 y Fm(0)2405 10558 y Fr(\))2470 10489 y Fo(s)2541 10558 y Fn(k)p Fr(\000)2728 10583 y Fm(0)p Fo(;L)2924 10600 y Fi(0)2997 10558 y Fp(R)3123 10583 y Fo(!)t(;)p Fm(0)p Fo(;L)3446 10600 y Fi(0)3520 10558 y Fr(\()p Fp(E)3708 10583 y Fm(0)3782 10558 y Fr(\))p Fp(\037)3951 10588 y Fm(0)p Fo(;L)4147 10605 y Fi(0)4212 10588 y Fo(=)p Fm(3)4355 10558 y Fn(k)46 b Fp(<)g Fr(1)4742 10423 y Fk(\011)4885 10558 y Fn(\025)g Fr(1)37 b Fn(\000)5467 10492 y Fm(2)p 5366 10519 269 7 v 5366 10617 a(344)5564 10584 y Fh(d)5682 10558 y Fp(:)583 b Fr(\(2.28\))758 10840 y Ff(Then)59 b Fp(E)1313 10865 y Fm(0)1434 10840 y Fn(2)45 b Fr(\006)1710 10865 y Fm(MSA)2010 10840 y Ff(.)p eop %%Page: 7 7 7 6 bop 2654 861 a Fm(HIGH)58 b(DISORDER)h(LOCALIZA)-11 b(TION)1830 b(7)957 1259 y Fr(The)65 b(next)g(criterion)g(is)g(an)h (analog)f(of)g(Theorem)h(2.10)f(but)h(for)f(the)f(second)i(m)-5 b(ultiscale)758 1458 y(analysis)76 b(of)f(the)f(b)5 b(o)g(otstrap)74 b(sc)-5 b(heme)76 b(of)f([GK1)q(],)80 b(i.e.,)g(for)75 b(the)f(m)-5 b(ultiscale)76 b(analysis)g(with)758 1658 y(exp)5 b(onen)-5 b(tial)58 b(deca)-5 b(y)59 b(of)g(the)f(resolv)-5 b(en)g(t)60 b(and)f(p)5 b(olynomial)60 b(deca)-5 b(y)58 b(of)h(the)f(probabilities)i(as)g(in)758 1857 y(v)-5 b(on)54 b(Dreifus)g(and)h(Klein)f([vDK],)h(mo)5 b(di\014ed)54 b(as)h(in)f(Figotin)h(and)f(Klein)h([FK1,)g(Theorem)f(32])758 2056 y(to)j(allo)-5 b(w)58 b(the)f(mass)h(in)g(the)f(starting)h(h)-5 b(yp)5 b(otheses)58 b(to)f(decrease)g(as)h(the)f(initial)h(length)f (scale)758 2255 y(increases)f([GK4,)g(Theorem)f(2.6].)957 2455 y(Theorem)68 b(2.11)h(giv)-5 b(es)68 b(an)h(estimate)f(of)g(the)f (exp)5 b(onen)-5 b(tial)69 b(rate)e(of)h(deca)-5 b(y)68 b(of)g(the)g(eigen-)758 2654 y(functions)e(in)g(terms)f(of)h(the)f (constan)-5 b(ts)66 b(of)g(the)f(problems;)72 b(the)65 b(price)h(is)g(paid)g(in)g(stronger)758 2853 y(h)-5 b(yp)5 b(otheses)43 b(than)f(Theorem)g(2.10.)70 b(If)42 b(the)g(probabilit)-5 b(y)43 b(estimate)f(in)h(the)f(initial)h(length)f(scale)758 3052 y(is)68 b(su\016cien)-5 b(tly)69 b(go)5 b(o)g(d,)71 b(one)d(ma)-5 b(y)68 b(use)h(Theorem)f(2.11)g(to)g(start)f(the)h(b)5 b(o)g(otstrap)67 b(m)-5 b(ultiscale)758 3252 y(analysis,)46 b(instead)c(of)f(applying)i(\014rst)f(Theorem)g(2.9)g(or)g(Theorem)g (2.10)g(and)h(then)e(b)5 b(o)g(otstrap-)758 3451 y(ping)56 b(to)f(Theorem)g(2.11)h(to)f(obtain)g(the)g(exp)5 b(onen)-5 b(tial)55 b(rate)g(of)g(deca)-5 b(y)55 b(of)h(eigenfunctions.)758 3805 y Fv(Theorem)63 b(2.11.)84 b Ff(L)-8 b(et)70 b Fp(H)2479 3830 y Fo(!)2645 3805 y Ff(b)-8 b(e)70 b(a)g(r)-8 b(andom)69 b(Schr\177)-85 b(odinger)71 b(op)-8 b(er)g(ator.)106 b(L)-8 b(et)70 b Fp(E)5732 3830 y Fm(0)5871 3805 y Fn(2)65 b Fr(\006)6167 3830 y Fm(W)6311 3805 y Ff(,)72 b(with)758 4004 y Fp(H)896 4029 y Fo(!)1058 4004 y Ff(satis\014ng)67 b(a)f(We)-8 b(gner)67 b(estimate)g(in)f(the)h(op)-8 b(en)66 b(b)-8 b(ounde)g(d)66 b(interval)i Fn(I)j(3)58 b Fp(E)5748 4029 y Fm(0)5822 4004 y Ff(.)96 b(Fix)67 b Fp(p)58 b(>)g Fr(0)p Ff(,)758 4221 y Fp(\013)47 b Fr(=)f(1)37 b(+)1443 4140 y Fo(p=)p Fm(2)p 1392 4183 306 7 v 1392 4278 a Fo(p)p Fm(+2)p Fo(d)1717 4221 y Ff(,)60 b Fp(s)46 b(>)g Fr(2)p Fp(p)36 b Fr(+)h(\()p Fp(b)g Fr(+)g(2\))p Fp(d)p Ff(,)60 b(and)f Fp(\022)51 b(>)46 b Fr(4)p Fp(\013)q(s)p Ff(,)59 b(and)h(set)1174 4725 y Fn(L)45 b Fr(=)i(max)1846 4491 y Fk(\032)1971 4725 y Fr(3)p Fp(\045;)28 b Fr(417)2565 4609 y Fi(1)p 2483 4627 223 6 v 2483 4693 a Fh(\013)p Fe(\000)p Fi(1)2733 4725 y Fp(;)g Fr(dist\()p Fp(E)3263 4750 y Fm(0)3337 4725 y Fp(;)g Fl(R)22 b Fn(nI)12 b Fr(\))3772 4656 y Fj(\000)3905 4612 y Fi(1)p 3905 4630 57 6 v 3905 4696 a Fh(s)3989 4725 y Fp(;)28 b(Q)4392 4567 y Fi(1)p 4214 4585 411 6 v 4214 4652 a Fh(s)p Fe(\000)p Fh(p)p Fe(\000)p Fh(bd)4194 4774 y Fj(I)4653 4725 y Fp(;)1612 b Fr(\(2.29\))1646 5021 y Fk(\000)1722 5156 y Fr(16)37 b Fn(\001)g Fr(36)2174 5087 y Fo(d)2251 5156 y Fr(2)2334 5087 y Fo(s)p Fj(\000)p Fm(2)p Fo(p)2644 5156 y Fp(C)2763 5181 y Fj(I)2853 5156 y Fp(Q)2984 5181 y Fj(I)3076 5021 y Fk(\001)3576 5003 y Fi(1)p 3172 5021 865 6 v 3172 5089 a Fh(\013)p Fi(\()p Fh(s)p Fe(\000)p Fi(2)p Fh(p)p Fe(\000)p Fi(\()p Fh(b)p Fi(+2\))p Fh(d)p Fi(\))4093 5156 y Fp(;)4167 5021 y Fk(\000)4242 5156 y Fr(69)4408 5087 y Fo(d)4486 5156 y Fp(\015)4581 5087 y Fm(2)4572 5197 y Fj(I)4663 5021 y Fk(\001)4899 5003 y Fi(4)p 4759 5021 338 6 v 4759 5087 a Fh(\022)s Fe(\000)p Fi(4)p Fh(\013s)5152 5156 y Fp(;)5226 5021 y Fk(\000)5302 5156 y Fr(3)5385 5087 y Fo(d)5462 5156 y Fp(\015)5548 5181 y Fj(I)5639 5021 y Fk(\001)5826 5003 y Fi(16)p Fh(\013)p 5735 5021 372 6 v 5735 5089 a(\022)s Fi(\()p Fh(\013)p Fe(\000)p Fi(1\))6135 4971 y Fk(o)6273 5156 y Fp(:)758 5562 y Ff(If)59 b(for)g(some)h Fp(L)1735 5587 y Fm(0)1855 5562 y Fn(\025)46 b(L)p Ff(,)59 b Fp(L)2368 5587 y Fm(0)2489 5562 y Fn(2)46 b Fr(6)p Fl(N)35 b Ff(,)59 b(we)h(have)2286 5997 y Fl(P)34 b Fn(f)8 b Fr(\003)2613 6022 y Fo(L)2704 6039 y Fi(0)2778 5997 y Fr(\(0\))59 b Ff(is)g Fr(\()p Fp(\022)5 b(;)28 b(E)3573 6022 y Fm(0)3646 5997 y Fr(\))p Ff(-suitable)q Fn(g)46 b Fp(>)g Fr(1)37 b Fn(\000)4977 5885 y Fr(1)p 4923 5959 190 7 v 4923 6111 a Fp(L)5036 6031 y Fo(p)5036 6155 y Fm(0)5133 5997 y Fp(;)1132 b Fr(\(2.30\))758 6463 y Ff(then)61 b(ther)-8 b(e)61 b(exists)g(an)g(op)-8 b(en)60 b(interval)i Fp(I)g Fr(=)48 b Fp(I)13 b Fr(\()p Fp(\022)5 b(;)28 b(s;)g(L)4124 6488 y Fm(0)4197 6463 y Fr(\))p Ff(,)61 b(with)g Fp(E)4859 6488 y Fm(0)4981 6463 y Fn(2)48 b Fp(I)62 b Fn(\032)48 b(I)12 b Ff(,)62 b(such)e(that)h(if)g(we)758 6676 y(set)f Fp(m)1163 6701 y Fm(0)1283 6676 y Fr(=)46 b(2)p Fp(\022)1644 6602 y Fm(log)25 b Fo(L)1929 6619 y Fi(0)p 1644 6638 351 7 v 1741 6733 a Fo(L)1832 6750 y Fi(0)2014 6676 y Ff(,)59 b(and)h Fp(L)2560 6701 y Fo(k)s Fm(+1)2856 6676 y Fr(=)46 b([)p Fp(L)3190 6616 y Fo(\013)3190 6723 y(k)3285 6676 y Fr(])3331 6701 y Fm(6)p Fg(N)3489 6676 y Ff(,)60 b Fp(k)51 b Fr(=)c(0)p Fp(;)28 b Fr(1)p Fp(;)g(:)g(:)g(:)f Ff(,)59 b(we)h(have)1695 7153 y Fl(P)5 b Fn(f)p Fr(\003)1985 7178 y Fo(L)2077 7196 y Fh(k)2166 7153 y Fr(\(0\))92 b Ff(is)59 b Fr(\()2734 7085 y Fo(m)2852 7102 y Fi(0)p 2734 7115 183 7 v 2792 7210 a Fm(4)2936 7153 y Fp(;)28 b(E)10 b Fr(\))p Ff(-r)-8 b(e)g(gular)o Fn(g)46 b(\025)h Fr(1)37 b Fn(\000)4424 7040 y Fr(2)p 4369 7115 195 7 v 4369 7267 a Fp(L)4482 7187 y Fo(p)4482 7317 y(k)4768 7153 y Ff(for)59 b(al)8 b(l)93 b Fp(E)56 b Fn(2)45 b Fp(I)13 b(;)541 b Fr(\(2.31\))758 7615 y Ff(and)1160 8018 y Fl(P)1317 7884 y Fk(\002)1386 8018 y Fp(R)1541 7884 y Fk(\000)1637 7950 y Fo(m)1755 7967 y Fi(0)p 1637 7980 183 7 v 1695 8075 a Fm(4)1840 8018 y Fp(;)28 b(L)2027 8043 y Fo(k)2108 8018 y Fp(;)g(I)13 b(;)28 b(x;)g(y)2598 7884 y Fk(\001\003)2789 8018 y Fn(\025)47 b Fr(1)36 b Fn(\000)3357 7906 y Fr(4)p 3270 7980 256 7 v 3270 8155 a Fp(L)3383 8075 y Fm(2)p Fo(p)3383 8205 y(k)3592 8018 y Ff(for)46 b Fp(x;)28 b(y)53 b Fn(2)45 b Fl(Z)4405 7950 y Fo(d)4519 8018 y Fp(;)28 b Fn(j)p Fp(x)37 b Fn(\000)g Fp(y)6 b Fn(j)47 b Fp(>)f(L)5405 8043 y Fo(k)5523 8018 y Fr(+)37 b Fp(\045)60 b(;)430 b Fr(\(2.32\))758 8503 y Ff(for)59 b(al)8 b(l)60 b Fp(k)52 b Fr(=)47 b(0)p Fp(;)28 b Fr(1)p Fp(;)g(:)g(:)g(:)f Ff(,)59 b(wher)-8 b(e)1378 8869 y Fp(R)q Fr(\()p Fp(m;)28 b(L;)g(I)13 b(;)28 b(x;)g(y)6 b Fr(\))45 b(=)3679 b(\(2.33\))1711 9118 y Fn(f)p Ff(for)59 b(every)47 b Fp(E)55 b Fn(2)46 b Fp(I)13 b(;)87 b Ff(either)60 b Fr(\003)3627 9143 y Fo(L)3727 9118 y Fr(\()p Fp(x)p Fr(\))e Ff(or)i Fr(\003)4340 9143 y Fo(L)4439 9118 y Fr(\()p Fp(y)6 b Fr(\))59 b Ff(is)g Fr(\()p Fp(m;)28 b(E)10 b Fr(\))p Ff(-r)-8 b(e)g(gular)o Fn(g)59 b Fp(:)758 9484 y Ff(Mor)-8 b(e)g(over,)970 9767 y Fv(\(i\):)82 b Fp(I)50 b Fn(\\)37 b Fr(\006)46 b Fn(\032)g Fr(\006)2039 9792 y Fm(MSA)2366 9767 y Ff(.)970 9966 y Fv(\(ii\):)81 b Ff(F)-13 b(or)69 b(almost)f(every)h Fp(!)6 b Ff(,)71 b(an)e(eigenfunction)h Fp(')4288 9991 y Fo(!)t(;E)4596 9966 y Ff(of)e 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Fq(Anderson)61 b(Hamil)-11 b(tonians)957 1558 y Fr(In)68 b(this)g(section)f(w)-5 b(e)68 b(apply)f(our)h(explicit)f(criteria)g (to)g(Anderson)h(Hamiltonians)h(in)f(the)758 1757 y(con)-5 b(tin)g(uum.)80 b(These)57 b(are)g(the)g(most)g(studied)h(random)g(Sc) -5 b(hr\177)-83 b(odinger)58 b(op)5 b(erators)57 b(in)g(the)g(con-)758 1956 y(tin)-5 b(uum)56 b(\(e.g.,)f([HM,)h(CH1)o(,)g(Klo1,)f(CHM,)h(Ki,) f(KSS1)q(,)g(CHN)o(,)h(St,)f(Klo2,)h(GK4]\).)957 2156 y(In)f(this)h(article)f(an)g(Anderson)h(Hamiltonian)g(will)f(b)5 b(e)55 b(a)g(random)h(Sc)-5 b(hr\177)-83 b(odinger)57 b(op)5 b(erator)758 2355 y(of)55 b(the)g(form)2781 2645 y Fp(H)2919 2670 y Fo(\025;!)3179 2645 y Fr(=)46 b Fn(\000)p Fr(\001)38 b(+)f Fp(V)3922 2670 y Fm(p)t(er)4155 2645 y Fr(+)g Fp(\025V)4515 2670 y Fo(!)4638 2645 y Fp(;)1710 b Fr(\(3.1\))758 2935 y(where)970 3179 y Fv(\(a\):)83 b Fp(V)1444 3204 y Fm(p)t(er)1695 3179 y Fr(is)56 b(a)g(p)5 b(erio)g(dic)55 b(p)5 b(oten)-5 b(tial)55 b(with)g(p)5 b(erio)g(d)55 b(1)g(\(b)-5 b(y)55 b(rescaling)h(w)-5 b(e)55 b(can)h(alw)-5 b(a)g(ys)56 b(tak)-5 b(e)1169 3417 y(the)77 b(p)5 b(erio)g(d)76 b(to)h(b)5 b(e)76 b(one\),)82 b(suc)-5 b(h)79 b(that)d Fp(V)3804 3442 y Fm(p)t(er)4082 3417 y Fr(=)82 b Fp(V)4427 3331 y Fm(\(1\))4390 3436 y(p)t(er)4657 3417 y Fr(+)51 b Fp(V)4971 3331 y Fm(\(2\))4934 3436 y(p)t(er)5149 3417 y Fr(,)82 b(with)77 b Fp(V)5811 3331 y Fm(\()p Fo(i)p Fm(\))5774 3436 y(p)t(er)5970 3417 y Fr(,)83 b Fp(i)f Fr(=)g(1)p Fp(;)28 b Fr(2,)1169 3654 y(p)5 b(erio)g(dic)61 b(with)g(p)5 b(erio)g(d)61 b(one,)i(0)56 b Fn(\024)g Fp(V)3535 3568 y Fm(\(1\))3498 3674 y(p)t(er)3769 3654 y Fn(2)g Fr(L)4040 3594 y Fm(1)4040 3701 y(lo)t(c)4214 3654 y Fr(\()p Fl(R)4389 3594 y Fo(d)4476 3654 y Fp(;)28 b Fr(d)p Fp(x)p Fr(\),)62 b(and)g Fp(V)5373 3568 y Fm(\(2\))5336 3674 y(p)t(er)5612 3654 y Fr(relativ)-5 b(ely)61 b(form-)1169 3854 y(b)5 b(ounded)55 b(with)f(resp)5 b(ect)53 b(to)h Fn(\000)p Fr(\001)h(with)f(relativ)-5 b(e)53 b(b)5 b(ound)55 b Fp(<)46 b Fr(1:)74 b(there)53 b(are)h(nonnegativ)-5 b(e)1169 4053 y(constan)g(ts)56 b(\002)2033 4078 y Fm(p)t(er)p Fo(;)p Fm(1)2381 4053 y Fp(<)47 b Fr(1)55 b(and)h(\002)3147 4078 y Fm(p)t(er)p Fo(;)p Fm(2)3504 4053 y Fr(suc)-5 b(h)57 b(that)d(for)i(all)f Fp( )d Fn(2)46 b(D)5 b Fr(\()p Fn(r)p Fr(\))55 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)2179 4208 y Fk(\014)2179 4307 y(\014)2179 4407 y(\014)2234 4214 y(D)2336 4399 y Fp( )6 b(;)28 b(V)2657 4330 y Fm(\(2\))2621 4440 y(p)t(er)2836 4399 y Fp( )2950 4214 y Fk(E)3051 4208 y(\014)3051 4307 y(\014)3051 4407 y(\014)3153 4399 y Fn(\024)46 b Fr(\002)3457 4424 y Fm(p)t(er)p Fo(;)p Fm(1)3759 4399 y Fn(kr)p Fp( )6 b Fn(k)4177 4330 y Fm(2)4289 4399 y Fr(+)37 b(\002)4584 4424 y Fm(p)t(er)p Fo(;)p Fm(2)4886 4399 y Fn(k)p Fp( )6 b Fn(k)5166 4330 y Fm(2)5268 4399 y Fp(:)1080 b Fr(\(3.2\))970 4738 y Fv(\(b\):)2803 5062 y Fp(V)2900 5087 y Fo(!)2996 5062 y Fr(\()p Fp(x)p Fr(\))45 b(=)3512 4904 y Fk(X)3442 5279 y Fo(i)p Fj(2)3599 5234 y Fi(1)p 3598 5252 58 6 v 3598 5319 a Fh(q)3676 5279 y Fg(Z)3761 5246 y Fh(d)3850 5062 y Fp(!)3953 5087 y Fo(i)4037 5062 y Fp(u)p Fr(\()p Fp(x)36 b Fn(\000)h Fp(i)p Fr(\))p Fp(;)1732 b Fr(\(3.3\))1169 5574 y(where)1282 5773 y Fv(\(b1\):)84 b Fp(q)52 b Fn(2)46 b Fl(N)26 b Fr(.)1282 5973 y Fv(\(b2\):)84 b Fp(!)76 b Fr(=)71 b Fn(f)p Fp(!)2333 5998 y Fo(i)2389 5973 y Fr(;)97 b Fp(i)47 b Fn(2)2812 5907 y Fm(1)p 2812 5934 67 7 v 2813 6030 a Fo(q)2899 5973 y Fl(Z)3000 5912 y Fo(d)3087 5973 y Fn(g)70 b Fr(a)g(family)g(of)f(indep)5 b(enden)-5 b(t,)74 b(iden)-5 b(tically)71 b(distributed)1481 6194 y(random)50 b(v)-9 b(ariables)49 b(taking)g(v)-9 b(alues)50 b(in)f(the)g(b)5 b(ounded)50 b(in)-5 b(terv)c(al)49 b([)p Fp(M)5701 6219 y Fj(\000)5813 6194 y Fp(;)28 b(M)6048 6219 y Fm(+)6158 6194 y Fr(],)51 b(whose)1481 6393 y(common)63 b(probabilit)-5 b(y)62 b(distribution)h Fp(\026)f Fr(has)h(a)e(b)5 b(ounded)63 b(densit)-5 b(y)62 b Fp(g)68 b Fr(with)62 b Fp(g)h(>)57 b Fr(0)1481 6592 y(a.e.)74 b(in)55 b([)p Fp(M)2204 6617 y Fj(\000)2317 6592 y Fp(;)28 b(M)2552 6617 y Fj(\000)2701 6592 y Fr(+)37 b Fp(a)p Fr(\))54 b(for)h(some)i Fp(a)45 b(>)i Fr(0.)1282 6791 y Fv(\(b3\):)84 b Fp(u)p Fr(\()p Fp(x)p Fr(\))53 b(is)i(a)f(real)h(v)-9 b(alued)54 b(measurable)h (function)f(with)h(compact)f(supp)5 b(ort,)55 b(sa)-5 b(y)1481 6991 y(supp)29 b Fp(u)51 b Fn(\032)f Fr(\003)2291 7016 y Fo(\045)2369 6991 y Fr(\(0\),)58 b(0)51 b Fn(\024)g Fp(u)g Fn(2)f Fr(L)3411 6930 y Fo(p)3487 6991 y Fr(\()p Fl(R)3663 6930 y Fo(d)3749 6991 y Fp(;)28 b Fr(d)p Fp(x)p Fr(\))58 b(with)g Fp(p)50 b Fn(2)h Fr([1)p Fp(;)28 b Fn(1)p Fr(])58 b(if)g Fp(d)51 b Fr(=)g(1)58 b(or)g Fp(M)6263 7016 y Fj(\000)6426 6991 y Fn(\025)51 b Fr(0,)1481 7201 y Fp(p)68 b Fn(2)g Fr(\()1897 7135 y Fo(d)p 1897 7162 70 7 v 1898 7258 a Fm(2)1986 7201 y Fp(;)28 b Fn(1)p Fr(])68 b(if)h Fp(d)g Fn(\025)g Fr(2)f(and)i Fp(M)3508 7226 y Fj(\000)3688 7201 y Fp(<)f Fr(0,)j(and)e(0)e Fp(<)h(U)4886 7226 y Fj(\000)5067 7201 y Fn(\024)f Fp(U)18 b Fr(\()p Fp(x)p Fr(\))69 b(where)f Fp(U)18 b Fr(\()p Fp(x)p Fr(\))68 b(=)1481 7276 y Fk(P)1656 7450 y Fo(i)p Fj(2)1813 7405 y Fi(1)p 1813 7423 58 6 v 1813 7490 a Fh(q)1890 7450 y Fg(Z)1975 7417 y Fh(d)2073 7400 y Fp(u)p Fr(\()p Fp(x)37 b Fn(\000)g Fp(i)p Fr(\).)970 7647 y Fv(\(c\):)83 b Fp(\025)46 b(>)h Fr(0)55 b(is)h(the)f(disorder)h(parameter.)957 7892 y Fp(H)1095 7917 y Fo(\025;!)1376 7892 y Fr(is)68 b(a)f(random)g(Sc)-5 b(hr\177)-83 b(odinger)68 b(op)5 b(erator)67 b(for)f(eac)-5 b(h)68 b Fp(\025)d(>)g Fr(0.)109 b(\()66 b(If)h Fp(M)5643 7917 y Fj(\000)5820 7892 y Fp(<)f Fr(0,)k Fp(V)6311 7917 y Fo(!)6474 7892 y Fr(is)d(a)758 8091 y(p)5 b(oten)-5 b(tial)60 b(in)g(Kato)g(class)h(for)f Fl(P)6 b Fr(-a.e.)97 b Fp(!)67 b Fr(\(see)60 b([Si,)i(Example)e(E,)h (p.)88 b(457]\).)h(Its)60 b(nonrandom)758 8290 y(sp)5 b(ectrum)73 b(will)h(b)5 b(e)73 b(denoted)g(b)-5 b(y)74 b(\006)3091 8315 y Fo(\025)3178 8290 y Fr(,)k(i.e.,)g(\006)3758 8315 y Fo(\025)3921 8290 y Fr(=)f Fp(\033)6 b Fr(\()p Fp(H)4431 8315 y Fo(\025;!)4644 8290 y Fr(\))73 b(for)g Fl(P)6 b Fr(-a.e.)137 b Fp(!)6 b Fr(.)129 b(W)-14 b(e)73 b(let)f Fp(E)6566 8230 y Fm(inf)6556 8337 y Fo(\025)758 8489 y Fr(denote)55 b(the)g(b)5 b(ottom)54 b(of)i(the)f(sp)5 b(ectrum,)55 b(i.e.,)3230 8779 y Fp(E)3363 8711 y Fm(inf)3353 8820 y Fo(\025)3578 8779 y Fr(=)46 b(inf)41 b(\006)4103 8804 y Fo(\025)4217 8779 y Fp(:)2131 b Fr(\(3.4\))758 9090 y Fv(Theorem)63 b(3.1.)84 b Ff(L)-8 b(et)59 b Fp(H)2373 9115 y Fo(\025;!)2647 9090 y Ff(b)-8 b(e)60 b(an)f(A)l(nderson)i (Hamiltonian.)77 b(Then)2960 9394 y Fp(E)3093 9326 y Fm(inf)3083 9435 y Fo(\025)3308 9394 y Fr(=)46 b(inf)40 b Fp(\033)6 b Fr(\()p Fp(H)4016 9419 y Fo(\025;M)4261 9436 y Fe(\000)4367 9394 y Fr(\))28 b Fp(;)1888 b Fr(\(3.5\))758 9684 y Ff(wher)-8 b(e)1839 9974 y Fp(H)1977 9999 y Fo(\025;M)2222 10016 y Fe(\000)2374 9974 y Fr(=)46 b Fp(H)2687 10015 y Fo(\025;)p Fj(f)p Fo(!)2957 10032 y Fh(i)3010 10015 y Fm(=)p Fo(M)3239 10032 y Fe(\000)3337 10015 y Fm(;)59 b Fo(i)p Fj(2)3591 9971 y Fi(1)p 3590 9989 V 3590 10055 a Fh(q)3668 10015 y Fg(Z)3753 9982 y Fh(d)3815 10015 y Fj(g)3937 9974 y Fr(=)47 b Fn(\000)p Fr(\001)37 b(+)g Fp(V)4680 9999 y Fm(p)t(er)4913 9974 y Fr(+)g Fp(\025M)5337 9999 y Fj(\000)5449 9974 y Fp(U)46 b(:)740 b Fr(\(3.6\))758 10294 y Ff(We)59 b(have)h(the)g(upp)-8 b(er)59 b(b)-8 b(ound)2029 10659 y Fp(E)2162 10591 y Fm(inf)2152 10700 y Fo(\025)2377 10659 y Fn(\024)2552 10433 y Fk(Z)2644 10811 y Fm(\003)2735 10828 y Fi(1)2800 10811 y Fm(\(0\))3006 10659 y Fp(V)3103 10684 y Fm(p)t(er)3299 10659 y Fr(\()p Fp(x)p Fr(\))28 b(d)p Fp(x)36 b Fr(+)h Fp(\025M)4199 10684 y Fj(\000)4338 10433 y Fk(Z)4431 10811 y Fm(\003)4522 10828 y Fi(1)4586 10811 y Fm(\(0\))4792 10659 y Fp(U)18 b Fr(\()p Fp(x)p Fr(\))28 b(d)p Fp(x)g(;)957 b Fr(\(3.7\))p eop %%Page: 9 9 9 8 bop 2654 861 a Fm(HIGH)58 b(DISORDER)h(LOCALIZA)-11 b(TION)1830 b(9)758 1259 y Ff(so,)59 b(in)g(p)-8 b(articular,)2038 1604 y Fp(E)2171 1535 y Fm(inf)2161 1645 y Fo(\025)2386 1604 y Fn(\024)2561 1378 y Fk(Z)2653 1755 y Fm(\003)2744 1772 y Fi(1)2809 1755 y Fm(\(0\))3015 1604 y Fp(V)3112 1629 y Fm(p)t(er)3308 1604 y Fr(\()p Fp(x)p Fr(\))28 b(d)p Fp(x)36 b Fr(+)h Fp(\025M)4208 1629 y Fj(\000)4320 1604 y Fp(U)4433 1629 y Fj(\000)4638 1604 y Ff(if)59 b Fp(M)4960 1629 y Fj(\000)5118 1604 y Fp(<)46 b Fr(0)28 b Ff(.)939 b Fr(\(3.8\))758 1986 y Ff(Mor)-8 b(e)g(over,)59 b(if)g Fp(M)1861 2011 y Fj(\000)2019 1986 y Fp(<)47 b Fr(0)59 b Ff(we)h(have)846 2111 y Fk(\014)846 2210 y(\014)846 2310 y(\014)902 2117 y(D)1003 2302 y Fp( )6 b(;)1191 2117 y Fk(\020)1290 2302 y Fp(V)1424 2233 y Fm(\(2\))1387 2343 y(p)t(er)1639 2302 y Fr(+)37 b Fp(\025M)2063 2327 y Fj(\000)2175 2302 y Fp(U)2306 2117 y Fk(\021)2433 2302 y Fp( )2547 2117 y Fk(E)2649 2111 y(\014)2649 2210 y(\014)2649 2310 y(\014)2751 2302 y Fn(\024)2946 2236 y Fm(1)p 2946 2263 67 7 v 2946 2359 a(2)3032 2302 y Fr(\(1)f(+)h(\002)3511 2327 y Fm(p)t(er)p Fo(;)p Fm(1)3813 2302 y Fr(\))p Fn(kr)p Fp( )6 b Fn(k)4296 2233 y Fm(2)4408 2302 y Fr(+)4574 2117 y Fk(\020)4673 2302 y Fr(\002)4802 2327 y Fm(p)t(er)p Fo(;)p Fm(2)5141 2302 y Fr(+)37 b Fp(\014)9 b(\025)5602 2174 y Fi(2)p Fh(p)p 5526 2201 270 6 v 5526 2268 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)5824 2117 y Fk(\021)5951 2302 y Fn(k)p Fp( )d Fn(k)6231 2233 y Fm(2)6394 2302 y Fr(\(3.9\))758 2621 y Ff(for)59 b(al)8 b(l)60 b Fp( )53 b Fn(2)46 b(D)5 b Fr(\()p Fn(r)p Fr(\))p Ff(,)58 b(wher)-8 b(e)1986 3050 y Fp(\014)55 b Fr(=)46 b(\()o Fp(C)2493 3075 y Fo(d;q)t(;\045;p)2919 3050 y Fn(j)p Fp(M)3126 3075 y Fj(\000)3238 3050 y Fn(j)28 b(k)p Fp(u)p Fn(k)3573 3075 y Fo(p)3650 3050 y Fr(\))3810 2913 y Fi(2)p Fh(p)p 3734 2940 V 3734 3007 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)4059 2816 y Fk(\022)4519 2938 y Fr(2)p 4202 3012 717 7 v 4202 3164 a(1)37 b Fn(\000)g Fr(\002)4617 3189 y Fm(p)t(er)p Fo(;)p Fm(1)4938 2816 y Fk(\023)5184 2806 y Fh(d)p 5081 2824 270 6 v 5081 2891 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)5433 3050 y Fp(;)832 b Fr(\(3.10\))758 3413 y Ff(with)58 b Fp(C)1236 3438 y Fo(d;q)t(;\045;p)1691 3413 y Ff(a)g(c)-8 b(onstant)59 b(dep)-8 b(ending)58 b(only)h(on)f(the)g(indic)-8 b(ate)g(d)58 b(p)-8 b(ar)g(ameters.)75 b(Thus)57 b(we)i(have)758 3613 y(the)g(lower)i(b)-8 b(ound)1981 3961 y Fp(E)2114 3892 y Fm(inf)2104 4002 y Fo(\025)2329 3961 y Fn(\025)2504 3727 y Fk(\032)2712 3839 y Fn(\000)p Fr(\002)2970 3864 y Fm(p)t(er)p Fo(;)p Fm(2)3309 3839 y Fr(+)37 b Fp(\025M)3733 3864 y Fj(\000)3845 3839 y Fp(U)3958 3864 y Fj(\000)4236 3839 y Ff(if)60 b Fp(M)4559 3864 y Fj(\000)4717 3839 y Fn(\025)46 b Fr(0)2712 4079 y Fn(\000)p Fr(\002)2970 4104 y Fm(p)t(er)p Fo(;)p Fm(2)3309 4079 y Fn(\000)37 b Fp(\014)9 b(\025)3770 3952 y Fi(2)p Fh(p)p 3694 3979 V 3694 4045 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)4236 4079 y Ff(if)60 b Fp(M)4559 4104 y Fj(\000)4717 4079 y Fp(<)46 b Fr(0)5134 3961 y Fp(:)1131 b Fr(\(3.11\))957 4368 y(W)-14 b(e)55 b(giv)-5 b(e)55 b(a)h(pro)5 b(of)55 b(for)g(Theorem)g(3.1)h(in)f(App)5 b(endix)56 b(A.)758 4665 y Fv(Theorem)63 b(3.2.)84 b Ff(L)-8 b(et)59 b Fp(H)2373 4690 y Fo(\025;!)2647 4665 y Ff(b)-8 b(e)60 b(an)f(A)l(nderson)i (Hamiltonian,)f(and)f(set)2077 4961 y Fp(L)2190 4986 y Fm(0)2310 4961 y Fr(=)47 b(min)p Fn(f)p Fp(L)g Fn(2)e Fr(6)p Fl(N)26 b Fr(;)83 b Fp(L)46 b Fn(\025)h Fr(max)p Fn(f)p Fr(3)p Fp(\045;)28 b Fr(3\(2)36 b(+)4887 4812 y Fn(p)p 5025 4812 87 7 v 149 x Fp(d)q Fr(\))p Fn(gg)28 b Fp(:)894 b Fr(\(3.12\))758 5230 y Ff(Then)57 b(ther)-8 b(e)56 b(exists)h Fp(\025)2147 5170 y Fj(\003)2223 5230 y Ff(,)g(dep)-8 b(ending)58 b(only)e(on)h Fp(d)p Ff(,)h Fp(U)4001 5255 y Fj(\000)4113 5230 y Ff(,)f Fn(k)p Fp(g)6 b Fn(k)4472 5255 y Fj(1)4613 5230 y Ff(,)57 b Fp(\045)p Ff(,)g Fp(p)p Ff(,)f Fn(k)p Fp(u)p Fn(k)5367 5255 y Fo(p)5444 5230 y Ff(,)h Fp(M)5713 5255 y Fj(\000)5825 5230 y Ff(,)g Fr(\002)6062 5255 y Fm(p)t(er)p Fo(;)p Fm(1)6364 5230 y Ff(,)g(and)758 5430 y Fr(\002)887 5455 y Fm(p)t(er)p Fo(;)p Fm(2)1189 5430 y Ff(,)i(such)h(that)f(for)g(any)g Fp(\025)46 b Fn(\025)h Fp(\025)3015 5369 y Fj(\003)3150 5430 y Ff(we)60 b(have)1747 5590 y Fk(\002)1817 5725 y Fp(E)1950 5656 y Fm(inf)1940 5766 y Fo(\025)2118 5725 y Fp(;)28 b(E)2325 5656 y Fm(inf)2315 5766 y Fo(\025)2531 5725 y Fr(+)37 b Fp(c\025)2866 5590 y Fk(\003)2972 5725 y Fn(\\)f Fr(\006)3239 5750 y Fo(\025)3373 5725 y Fn(\032)46 b Fr(\006)3668 5750 y Fm(MSA)4059 5725 y Ff(with)93 b Fp(c)46 b Fr(=)5099 5643 y Fo(U)5189 5660 y Fe(\000)p 4766 5687 855 7 v 4766 5784 a Fm(\(368)p Fo(L)5107 5801 y Fi(0)5172 5784 y Fm(\))5224 5751 y Fh(d)5294 5784 y Fj(k)p Fo(g)t Fj(k)5499 5801 y Fe(1)5699 5725 y Fp(:)566 b Fr(\(3.13\))758 6024 y Ff(Mor)-8 b(e)g(over)53 b(ther)-8 b(e)55 b(exists)f Fp(c)2412 5964 y Fj(0)2505 6024 y Fp(>)46 b Fr(0)p Ff(,)55 b(dep)-8 b(ending)55 b(on)f Fp(d)p Ff(,)h Fp(U)4173 6049 y Fj(\000)4286 6024 y Ff(,)g Fn(k)p Fp(g)6 b Fn(k)4643 6049 y Fj(1)4784 6024 y Ff(,)55 b(and)f Fp(\045)p Ff(,)h(but)f(not)h(on)f Fp(\025)46 b Fn(\025)g Fp(\025)6608 5964 y Fj(\003)6684 6024 y Ff(,)758 6224 y(such)72 b(that)g(if)g Fp(\025)d Fn(\025)g Fp(\025)2135 6163 y Fj(\003)2211 6224 y Ff(,)75 b(then)e(for)f(a.e.)114 b Fp(!)6 b Ff(,)75 b(if)d Fp(')3892 6249 y Fo(\025;!)t(;E)4321 6224 y Ff(is)g(an)g (eigenfunction)j(of)d Fp(H)6148 6249 y Fo(\025;!)6434 6224 y Ff(with)758 6429 y(eigenvalue)62 b Fp(E)55 b Fn(2)1895 6294 y Fk(\002)1964 6429 y Fp(E)2097 6369 y Fm(inf)2087 6476 y Fo(\025)2266 6429 y Fp(;)28 b(E)2473 6369 y Fm(inf)2463 6476 y Fo(\025)2678 6429 y Fr(+)37 b Fp(c\025)3013 6294 y Fk(\003)3082 6429 y Ff(,)60 b(then)2636 6810 y Fr(lim)28 b(inf)2656 6925 y Fj(j)p Fo(x)p Fj(j!1)3123 6810 y Fn(\000)3272 6698 y Fr(log)j Fn(k)p Fp(\037)3702 6723 y Fo(x)3785 6698 y Fp(')3894 6723 y Fo(!)t(;E)4133 6698 y Fn(k)p 3272 6772 944 7 v 3651 6924 a(j)p Fp(x)p Fn(j)4282 6810 y(\025)47 b Fp(c)4530 6742 y Fj(0)4576 6661 y Fn(p)p 4714 6661 97 7 v 149 x Fp(\025:)1454 b Fr(\(3.14\))758 7200 y Fv(Remark)62 b(3.3.)84 b Ff(In)59 b(the)f(we)-8 b(ak)58 b(disor)-8 b(der)57 b(r)-8 b(e)g(gime,)58 b(lo)-8 b(c)g(alization)57 b(of)h(A)l(nderson)g(Hamiltonians)758 7399 y(in)75 b(an)f(interval)i(of)f(size)g Fp(\025)f Ff(at)h(the)g(b)-8 b(ottom)74 b(of)g(the)h(sp)-8 b(e)g(ctrum)75 b(has)f(b)-8 b(e)g(en)75 b(pr)-8 b(ove)g(d)74 b(by)h(Klopp)758 7598 y Fr([Klo2])83 b Ff(using)g(Lifschitz)f(tails.)146 b(As)82 b(p)-8 b(ointe)g(d)82 b(out)h(in)f Fr([Klo2)q(,)k(p.)149 b(728])p Ff(,)89 b(in)82 b(this)g(c)-8 b(ase)83 b(the)758 7798 y(minimal)64 b(length)i(sc)-8 b(ale)65 b(for)f(the)g(initial)h (length)h(sc)-8 b(ale)65 b(estimate)g(gr)-8 b(ows)64 b(p)-8 b(olynomial)8 b(ly)64 b(in)6592 7732 y Fm(1)p 6586 7759 79 7 v 6586 7855 a Fo(\025)6684 7798 y Ff(,)758 8012 y(sinc)-8 b(e)60 b(the)h(c)-8 b(onstant)60 b(in)g(the)h(We)-8 b(gner)61 b(estimate)f(is)g(pr)-8 b(op)g(ortional)59 b(to)5206 7946 y Fm(1)p 5199 7973 V 5199 8069 a Fo(\025)5298 8012 y Ff(,)h(but,)h(b)-8 b(e)g(c)g(ause)60 b(of)g(the)758 8211 y(str)-8 b(ong)63 b(estimate)h(given)h(in)f Fr([Klo2,)d(Prop)5 b(osition)60 b(3.2])p Ff(,)65 b(the)f(usual)g(pr)-8 b(o)g(ofs)61 b(of)j(the)f(multisc)-8 b(ale)758 8410 y(analysis)61 b(may)h(b)-8 b(e)62 b(mo)-8 b(di\014e)g(d)61 b(to)h(yield)g(lo)-8 b(c)g(alization)62 b(in)g(this)g(situation.)84 b(The)-8 b(or)g(em)61 b(2.11)g(may)758 8610 y(b)-8 b(e)54 b(applie)-8 b(d)54 b(dir)-8 b(e)g(ctly)53 b(with)i(Klopp's)e(estimates)h(to)g(pr)-8 b(ove)54 b(lo)-8 b(c)g(alization)54 b(in)h(this)e(we)-8 b(ak)55 b(disor)-8 b(der)758 8809 y(r)g(e)g(gime.)758 9105 y(Pr)g(o)g(of)57 b(of)j(The)-8 b(or)g(em)58 b(3.2.)83 b Fr(W)-14 b(e)55 b(follo)-5 b(w)56 b(the)g(same)g(strategy)f(as)h(in)h ([GK1,)f(Pro)5 b(of)55 b(of)h(Theorem)758 9304 y(3.4],)63 b(but)f(taking)f(in)-5 b(to)62 b(accoun)-5 b(t)61 b(the)h(fact)e(that)h Fp(E)4041 9244 y Fm(inf)4031 9351 y Fo(\025)4266 9304 y Fn(!)c(\0061)k Fr(as)i Fp(\025)56 b Fn(!)g(1)62 b Fr(if)f Fp(M)5978 9329 y Fj(\000)6110 9256 y Fh(>)6110 9347 y(<)6220 9304 y Fr(0,)i(with)758 9513 y Fn(j)p Fp(E)937 9453 y Fm(inf)927 9560 y Fo(\025)1106 9513 y Fn(j)55 b Fr(gro)-5 b(wing)56 b(p)5 b(olynomially)56 b(in)g Fp(\025)p Fr(.)957 9712 y(The)38 b(\014nite)g(v)-5 b(olume)38 b(op)5 b(erators)38 b Fp(H)3098 9737 y Fo(\025;!)t(;x;L)3596 9712 y Fr(are)f(tak)-5 b(en)38 b(as)g(the)g(restriction)g(of)f Fp(H)5832 9737 y Fo(\025;!)6047 9712 y Fr(,)k(either)d(to)758 9923 y(the)51 b(op)5 b(en)52 b(b)5 b(o)-5 b(x)51 b(\003)1867 9948 y Fo(L)1967 9923 y Fr(\()p Fp(x)p Fr(\))f(with)i(Diric)-5 b(hlet)52 b(b)5 b(oundary)52 b(condition,)h(or)e(to)h(the)f(closed)h(b) 5 b(o)-5 b(x)p 6297 9790 116 7 v 52 w(\003)6412 9948 y Fo(L)6511 9923 y Fr(\()p Fp(x)p Fr(\))758 10122 y(with)62 b(p)5 b(erio)g(dic)62 b(b)5 b(oundary)63 b(condition.)95 b(The)62 b(random)h(p)5 b(oten)-5 b(tial)63 b Fp(V)5126 10147 y Fm(p)t(er)5363 10122 y Fr(+)42 b Fp(\025V)5728 10147 y Fo(!)5886 10122 y Fr(satis\014es)64 b(the)758 10322 y(\014nite)55 b(v)-5 b(olume)56 b(condition)g(\(2.8\))e(with,)i (if)f Fp(M)3617 10347 y Fj(\000)3775 10322 y Fn(\025)46 b Fr(0,)2917 10591 y(\002)3046 10616 y Fo(\025;)p Fm(1)3405 10591 y Fr(=)166 b(\002)3829 10616 y Fm(p)t(er)p Fo(;)p Fm(1)4159 10591 y Fp(;)2106 b Fr(\(3.15\))2917 10840 y(\002)3046 10865 y Fo(\025;)p Fm(2)3405 10840 y Fr(=)166 b(\002)3829 10865 y Fm(p)t(er)p Fo(;)p Fm(2)4168 10840 y Fr(+)37 b Fp(c)4406 10865 y Fo(d)4530 10840 y Fp(;)1735 b Fr(\(3.16\))p eop %%Page: 10 10 10 9 bop 758 861 a Fm(10)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fr(and,)e(if)f Fp(M)1440 1284 y Fj(\000)1598 1259 y Fp(<)46 b Fr(0,)56 b(in)g(view)e(of)68 b(\(3.9\),)2557 1574 y(\002)2686 1599 y Fo(\025;)p Fm(1)3045 1574 y Fr(=)3360 1509 y Fm(1)p 3360 1536 67 7 v 3360 1631 a(2)3446 1574 y Fr(\(1)37 b(+)g(\002)3926 1599 y Fm(p)t(er)p Fo(;)p Fm(1)4228 1574 y Fr(\))28 b Fp(;)1944 b Fr(\(3.17\))2557 1879 y(\002)2686 1904 y Fo(\025;)p Fm(2)3045 1879 y Fr(=)166 b(\002)3469 1904 y Fm(p)t(er)p Fo(;)p Fm(2)3808 1879 y Fr(+)37 b Fp(\014)9 b(\025)4269 1751 y Fi(2)p Fh(p)p 4194 1778 270 6 v 4194 1845 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)4528 1879 y Fr(+)37 b Fp(c)4766 1904 y Fo(d)4890 1879 y Fp(;)1375 b Fr(\(3.18\))758 2194 y(where)55 b Fp(c)1310 2219 y Fo(d)1442 2194 y Fr(is)h(a)f(constan)-5 b(t)55 b(dep)5 b(ending)56 b(only)f(on)g Fp(d)h Fr(\(see)e(the)h(remark)g(follo)-5 b(wing)58 b(\(2.8\))o(;)e Fp(c)6354 2219 y Fo(d)6477 2194 y Fr(=)46 b(0)758 2393 y(for)56 b(Diric)-5 b(hlet)58 b(b)5 b(oundary)57 b(coundition\).)79 b(F)-14 b(or)58 b(a)f(giv)-5 b(en)57 b(b)5 b(ounded)58 b(in)-5 b(terv)c(al)56 b Fn(I)12 b Fr(,)59 b(the)d(constan)-5 b(ts)758 2593 y Fp(\015)844 2618 y Fo(\025;)p Fj(I)1108 2593 y Fr(and)56 b Fp(C)1550 2618 y Fo(\025;)p Fj(I)1814 2593 y Fr(can)g(then)f(b)5 b(e)55 b(read)g(from)h(\(2.14\))o(,)g(\(2.10\))e(and)i(\(2.13\).)957 2792 y(The)68 b(W)-14 b(egner)67 b(estimate)h(\(2.15\))f(can)h(b)5 b(e)67 b(deriv)-5 b(ed)68 b(as)h(in)f([Ki,)j(Prop)5 b(osition)69 b(1])f(or)f([FK1)q(,)758 2991 y(Theorem)55 b(2.3])h(with)f Fp(b)46 b Fr(=)h(2,)55 b Fp(\021)2710 3016 y Fj(I)2848 2991 y Fr(=)46 b(1,)56 b(and)994 3467 y Fp(Q)1125 3492 y Fo(\025;)p Fj(I)1381 3467 y Fr(=)j(sup)1556 3607 y Fo(E)6 b Fj(2I)1860 3467 y Fp(Q)1991 3492 y Fo(\025;E)2249 3467 y Fp(;)74 b Fr(with)46 b Fp(Q)2869 3492 y Fo(\025;E)3146 3467 y Fr(=)3341 3355 y Fp(C)3460 3380 y Fo(d)3537 3355 y Fn(k)p Fp(g)6 b Fn(k)3788 3380 y Fj(1)p 3341 3429 588 7 v 3460 3581 a Fp(\025)28 b(U)3698 3606 y Fj(\000)3977 3233 y Fk(\022)4119 3355 y Fr(max)p Fn(f)p Fp(E)46 b Fr(+)37 b(\002)4975 3380 y Fo(\025;)p Fm(2)5205 3355 y Fr(+)g(1)28 b Fp(;)g Fr(0)p Fn(g)p 4119 3429 1603 7 v 4616 3581 a Fr(1)37 b Fn(\000)g Fr(\002)5031 3606 y Fo(\025;)p Fm(1)5741 3233 y Fk(\023)5883 3223 y Fh(d)p 5883 3241 62 6 v 5885 3307 a Fi(2)6001 3467 y Fp(;)264 b Fr(\(3.19\))758 3880 y(with)55 b Fp(C)1255 3905 y Fo(d)1387 3880 y Fr(a)h(constan)-5 b(t)55 b(dep)5 b(ending)56 b(only)g(on)f Fp(d)p Fr(.)957 4079 y(W)-14 b(e)65 b(no)-5 b(w)67 b(deriv)-5 b(e)66 b(an)g(elemen)-5 b(tary)66 b(probabilit)-5 b(y)66 b(estimate:)95 b(if)66 b Fp(\016)k(>)64 b Fr(0)i(and)g Fp(L)5916 4104 y Fm(0)6054 4079 y Fn(2)e Fr(2)p Fl(N)25 b Fr(,)78 b(w)-5 b(e)758 4278 y(ha)g(v)g(e)1803 4593 y Fl(P)34 b Fn(f)9 b Fp(!)2119 4618 y Fo(i)2220 4593 y Fp(>)47 b(M)2557 4618 y Fj(\000)2706 4593 y Fr(+)37 b Fp(\016)52 b Fr(for)j(all)47 b Fp(i)f Fn(2)g Fr(\003)3849 4618 y Fo(L)3940 4635 y Fi(0)4013 4593 y Fr(\(0\))p Fn(g)2002 b Fr(\(3.20\))1970 4842 y(=)166 b(1)37 b Fn(\000)g Fl(P)c Fn(f)9 b Fp(!)2866 4867 y Fo(i)2968 4842 y Fn(2)45 b Fr([)p Fp(M)3331 4867 y Fj(\000)3444 4842 y Fp(;)28 b(M)3679 4867 y Fj(\000)3828 4842 y Fr(+)37 b Fp(\016)6 b Fr(])46 b(for)55 b(some)47 b Fp(i)f Fn(2)g Fr(\003)5202 4867 y Fo(L)5293 4884 y Fi(0)5367 4842 y Fr(\(0\))p Fn(g)1970 5100 y(\025)166 b Fr(1)37 b Fn(\000)g Fp(\016)6 b Fn(k)p Fp(g)g Fn(k)2882 5125 y Fj(1)3023 5100 y Fp(L)3136 5031 y Fo(d)3136 5141 y Fm(0)3213 5100 y Fp(:)758 5415 y Fr(W)-14 b(e)55 b(\014x)g Fp(\016)61 b Fr(and)56 b Fp(L)1850 5440 y Fm(0)1925 5415 y Fr(,)f(and)h(set)2998 5730 y Fp(K)3139 5755 y Fo(\025)3393 5730 y Fr(=)3708 5665 y Fm(1)p 3708 5692 67 7 v 3708 5787 a(2)3794 5730 y Fp(\016)6 b(U)3987 5755 y Fj(\000)4099 5730 y Fp(\025)28 b(;)2041 b Fr(\(3.21\))3017 6003 y Fp(E)3140 6028 y Fo(\025)3393 6003 y Fr(=)166 b Fp(E)3821 5934 y Fm(inf)3811 6044 y Fo(\025)4027 6003 y Fr(+)37 b Fp(K)4334 6028 y Fo(\025)4448 6003 y Fp(:)1817 b Fr(\(3.22\))758 6318 y(It)55 b(follo)-5 b(ws)56 b(that)2871 6633 y Fl(P)6 b Fn(f)p Fp(A)3172 6658 y Fo(\025)3267 6633 y Fn(g)47 b(\025)f Fr(1)37 b Fn(\000)g Fp(\016)6 b Fn(k)p Fp(g)g Fn(k)4189 6658 y Fj(1)4330 6633 y Fp(L)4443 6565 y Fo(d)4443 6674 y Fm(0)4548 6633 y Fp(;)1717 b Fr(\(3.23\))758 6948 y(where)55 b Fp(A)1363 6973 y Fo(\025)1505 6948 y Fr(denotes)g(the)g(ev)-5 b(en)g(t)2322 7263 y Fp(A)2447 7288 y Fo(\025;L)2711 7263 y Fr(=)46 b Fn(f)p Fr(inf)41 b Fp(\033)6 b Fr(\()p Fp(H)3503 7288 y Fo(\025;!)t(;L)3839 7305 y Fi(0)3912 7263 y Fr(\))45 b Fn(\025)i Fp(E)4331 7195 y Fm(inf)4321 7305 y Fo(\025)4536 7263 y Fr(+)38 b(2)p Fp(K)4927 7288 y Fo(\025)5014 7263 y Fn(g)28 b Fp(:)1140 b Fr(\(3.24\))957 7579 y(W)-14 b(e)65 b(no)-5 b(w)66 b(use)h(Theorem)f(2.9.)105 b(T)-14 b(o)65 b(maximize)h Fp(\016)6 b Fr(,)69 b(w)-5 b(e)65 b(need)h(to)g(minimize)g Fp(L)5883 7604 y Fm(0)5958 7579 y Fr(;)71 b(hence)65 b(w)-5 b(e)758 7778 y(pic)g(k)62 b Fp(L)1228 7803 y Fm(0)1366 7778 y Fr(as)h(in)g(\(3.12\))f(and)h(c)-5 b(ho)5 b(ose)63 b Fp(\016)69 b Fr(b)-5 b(y)63 b(matc)-5 b(hing)63 b(the)f(righ)-5 b(t)63 b(hand)h(sides)f(of)76 b(\(3.23\))62 b(and)758 7977 y(\(2.25\))o(,)55 b(i.e.,)3114 8300 y Fp(\016)6 b Fn(k)p Fp(g)g Fn(k)3445 8325 y Fj(1)3586 8300 y Fp(L)3699 8232 y Fo(d)3699 8341 y Fm(0)3822 8300 y Fr(=)4118 8235 y Fm(2)p 4017 8262 269 7 v 4017 8360 a(368)4215 8326 y Fh(d)4333 8300 y Fp(:)1932 b Fr(\(3.25\))758 8616 y(If)55 b Fp(!)d Fn(2)46 b Fp(A)1361 8641 y Fo(\025)1448 8616 y Fr(,)55 b(it)g(follo)-5 b(ws)56 b(that)1958 8931 y(dist\()p Fp(E)10 b(;)28 b(\033)6 b Fr(\()p Fp(H)2802 8956 y Fo(\025;!)t(;L)3145 8931 y Fr(\)\))46 b Fn(\025)g Fp(K)3637 8956 y Fo(\025)3816 8931 y Fr(for)55 b(an)-5 b(y)93 b Fp(E)55 b Fn(2)46 b Fr([)p Fp(E)4935 8862 y Fm(inf)4925 8972 y Fo(\025)5104 8931 y Fp(;)28 b(E)5301 8956 y Fo(\025)5387 8931 y Fr(])g Fp(;)804 b Fr(\(3.26\))758 9246 y(so)64 b(w)-5 b(e)65 b(can)f(use)g(the)g(Com)-5 b(b)5 b(es-Thomas)66 b(estimate)e(to)g(get)g (the)f(deca)-5 b(y)64 b(of)g(the)g(resolv)-5 b(en)g(t)64 b(for)758 9445 y(an)-5 b(y)71 b Fp(E)81 b Fn(2)72 b Fr([)p Fp(E)1653 9385 y Fm(inf)1643 9492 y Fo(\025)1822 9445 y Fp(;)28 b(E)2019 9470 y Fo(\025)2106 9445 y Fr(].)120 b(The)71 b(exact)f(dep)5 b(endency)71 b(of)g(the)f(exp)5 b(onen)-5 b(tial)71 b(deca)-5 b(y)71 b(rate)f(in)i(the)758 9644 y(Com)-5 b(b)5 b(es-Thomas)49 b(estimate)e(in)g(terms)h(of)f(the)f (energy)h(parameter)g(and)h(the)e(distance)i(to)f(the)758 9844 y(sp)5 b(ectrum)69 b(is)h(crucial)f(in)g(our)h(argumen)-5 b(t,)73 b(since)d(w)-5 b(e)69 b(deal)g(with)g(large)g(energies)h(and)f (large)758 10043 y(distances)47 b(from)f(the)g(sp)5 b(ectrum.)71 b(Suc)-5 b(h)48 b(a)e(precise)h(estimate)f(is)h(pro)-5 b(vided)47 b(in)g([GK2,)h(Eq.)71 b(\(19\))758 10242 y(in)61 b(Theorem)g(1],)i(and)e(adapted)h(to)e(the)h(\014nite)g(v)-5 b(olume)61 b(case)g(in)h([GK1,)h(Pro)5 b(of)60 b(of)h(Theorem)758 10441 y(3.4]:)106 b(In)71 b(a)g(b)5 b(o)-5 b(x)71 b(\003)1994 10466 y Fo(L)2085 10483 y Fi(0)2159 10441 y Fr(\(0\),)j(with)d(either)g (Diric)-5 b(hlet)71 b(or)h(p)5 b(erio)g(dic)70 b(b)5 b(oundary)72 b(condition,)k(the)758 10641 y(same)81 b(estimates)g(as)h (in)f([GK2)q(,)87 b(Eq.)150 b(\(19\))80 b(in)h(Theorem)g(1])g(hold)h (for)f Fp(x;)28 b(y)94 b Fn(2)6090 10599 y Fr(\026)6074 10641 y(\003)6189 10666 y Fo(L)6280 10683 y Fi(0)6345 10666 y Fj(\000)p Fm(2)6523 10641 y Fr(\(0\))758 10840 y(\(i.e.,)59 b(at)f(a)g(distance)h Fn(\025)51 b Fr(1)58 b(from)h(the)f(b)5 b(oundary\))58 b(but)g(the)g(exp)5 b(onen)-5 b(tial)58 b(rates)h(of)f(deca)-5 b(y)58 b(get)p eop %%Page: 11 11 11 10 bop 2654 861 a Fm(HIGH)58 b(DISORDER)h(LOCALIZA)-11 b(TION)1764 b(11)758 1259 y Fr(divided)61 b(b)-5 b(y)61 b(1)41 b(+)g(2)1962 1118 y Fn(p)p 2100 1118 87 7 v 141 x Fp(dL)2299 1284 y Fm(0)2374 1259 y Fr(.)91 b(Th)-5 b(us,)63 b(if)e Fp(!)h Fn(2)55 b Fp(A)3598 1284 y Fo(\025)3745 1259 y Fr(and)62 b Fp(E)i Fn(2)56 b Fr([)p Fp(E)4607 1199 y Fm(inf)4597 1306 y Fo(\025)4775 1259 y Fp(;)28 b(E)4972 1284 y Fo(\025)5059 1259 y Fr(],)62 b(w)-5 b(e)62 b(ha)-5 b(v)g(e)61 b(that)f(for)h(an)-5 b(y)758 1477 y Fp(x;)28 b(y)52 b Fn(2)46 b Fl(Z)1318 1417 y Fo(d)1442 1477 y Fn(\\)36 b Fr(\003)1704 1502 y Fo(L)1795 1519 y Fi(0)1869 1477 y Fr(\(0\))45 b(=)h Fl(Z)2404 1417 y Fo(d)2527 1477 y Fn(\\)2691 1436 y Fr(\026)2675 1477 y(\003)2790 1502 y Fo(L)2881 1519 y Fi(0)2946 1502 y Fj(\000)p Fm(2)3124 1477 y Fr(\(0\))28 b(,)55 b(with)g Fn(j)p Fp(x)37 b Fn(\000)g Fp(y)6 b Fn(j)47 b(\025)4543 1336 y(p)p 4681 1336 V 141 x Fp(d)q Fr(,)1600 1821 y Fn(k)p Fp(\037)1787 1846 y Fo(x)1871 1821 y Fp(R)1997 1846 y Fo(\025;!)t(;L)2333 1863 y Fi(0)2406 1821 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)2773 1846 y Fo(y)2851 1821 y Fn(k)166 b(\024)3477 1756 y Fm(2)p 3415 1783 190 7 v 3415 1879 a Fo(K)3527 1897 y Fh(\025)3653 1821 y Fr(e)3727 1750 y Fj(\000)3831 1660 y(p)p 3939 1660 256 6 v 3939 1750 a Fm(2)p Fo(K)4117 1768 y Fh(\025)4218 1762 y Fr(\()4282 1750 y Fm(1+2)4516 1650 y Fj(p)p 4626 1650 70 6 v 4626 1750 a Fo(dL)4786 1767 y Fi(0)4851 1762 y Fr(\))4915 1678 y Fe(\000)p Fi(1)5070 1762 y Fr(\()5135 1750 y Fj(j)p Fo(x)p Fj(\000)p Fo(y)t Fj(j\000)5567 1650 y(p)p 5677 1650 V 5677 1750 a Fo(d)5746 1762 y Fr(\))5847 1821 y Fp(:)418 b Fr(\(3.27\))758 2125 y(Summing)65 b(o)-5 b(v)g(er)64 b(the)g(supp)5 b(ort)64 b(of)g(\000)3088 2150 y Fo(L)3179 2167 y Fi(0)3315 2125 y Fr(and)h(of)e Fp(\037)3948 2155 y Fo(L)4039 2172 y Fi(0)4104 2155 y Fo(=)p Fm(3)4247 2125 y Fr(,)j(noting)e(that)f(if)h Fp(x)c Fn(2)g Fr(\003)5854 2155 y Fo(L)5945 2172 y Fi(0)6010 2155 y Fo(=)p Fm(3)6153 2125 y Fr(\(0\),)65 b Fp(y)i Fn(2)758 2348 y Fr(\007)887 2373 y Fo(L)978 2390 y Fi(0)1051 2348 y Fr(\(0\))55 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b Fn(j)p Fp(x)37 b Fn(\000)g Fp(y)6 b Fn(j)46 b(\025)2665 2280 y Fo(L)2756 2297 y Fi(0)p 2665 2310 156 7 v 2709 2405 a Fm(3)2877 2348 y Fn(\000)37 b Fr(1,)56 b(and)g(using)i(\(3.12\))d(yields)1727 2708 y Fn(k)p Fr(\000)1914 2733 y Fo(L)2005 2750 y Fi(0)2078 2708 y Fp(R)2204 2733 y Fo(\025;!)t(;L)2540 2750 y Fi(0)2614 2708 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)2981 2738 y Fo(L)3072 2755 y Fi(0)3135 2738 y Fo(=)p Fm(3)3278 2708 y Fn(k)46 b(\024)3629 2642 y Fm(2)p Fo(d)p 3602 2669 190 7 v 3602 2765 a(K)3714 2783 y Fh(\025)3839 2708 y Fp(L)3952 2637 y Fm(2)p Fo(d)p Fj(\000)p Fm(1)3952 2752 y(0)4266 2708 y Fr(e)4340 2636 y Fj(\000)4444 2546 y(p)p 4553 2546 256 6 v 90 x Fm(2)p Fo(K)4731 2654 y Fh(\025)4831 2648 y Fr(\()4896 2636 y Fm(1+2)5130 2537 y Fj(p)p 5239 2537 70 6 v 99 x Fo(dL)5399 2653 y Fi(0)5464 2648 y Fr(\))5529 2565 y Fe(\000)p Fi(1)5719 2708 y Fp(:)546 b Fr(\(3.28\))957 3028 y(W)-14 b(e)68 b(will)h(sho)-5 b(w)69 b(that)f(\(2.25\))g(of)g (Theorem)g(2.9)h(is)g(satis\014ed)g(for)f(all)h Fp(E)77 b Fn(2)68 b Fr([)p Fp(E)5970 2967 y Fm(inf)5960 3075 y Fo(\025)6139 3028 y Fp(;)28 b(E)6336 3053 y Fo(\025)6422 3028 y Fr(])68 b(for)758 3227 y(large)60 b(disorder.)92 b(Let)59 b Fn(I)2281 3252 y Fo(\025)2424 3227 y Fr(=)c(\()p Fn(\000)p Fr(\002)2931 3252 y Fo(\025;)p Fm(2)3164 3227 y Fn(\000)41 b Fr(1)p Fp(;)28 b(E)3614 3252 y Fo(\025)3741 3227 y Fr(+)40 b Fp(K)4051 3252 y Fo(\025)4179 3227 y Fn(\000)h Fr(1\),)62 b(w)-5 b(e)60 b(ha)-5 b(v)g(e,)63 b(using)h(\(2.26\),)e(\(2.14\))o(,)758 3426 y(\(2.10\))o(,)55 b(\(3.7\),)g(\(3.15\)-\(3.18\))o(,)h(\(3.19\))o(,)g(\(3.21\))o(,)f(and) h(\(3.28\),)1092 3795 y Fp(D)1229 3820 y Fo(\025;)p Fj(I)1422 3838 y Fh(\025)1510 3795 y Fp(L)1623 3696 y Fr(\()1707 3639 y Fi(5)p 1707 3657 57 6 v 1707 3724 a(3)1784 3684 y Fm(+)p Fo(b)1944 3696 y Fr(\))2008 3684 y Fo(d)1623 3839 y Fm(0)2086 3795 y Fn(k)p Fr(\000)2273 3820 y Fo(L)2364 3837 y Fi(0)2437 3795 y Fp(R)2563 3820 y Fo(\025;!)t(;L)2899 3837 y Fi(0)2973 3795 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)3340 3825 y Fo(L)3431 3842 y Fi(0)3494 3825 y Fo(=)p Fm(3)3636 3795 y Fn(k)2592 b Fr(\(3.29\))1731 4116 y(=)166 b(39)2192 4047 y Fm(5)p Fo(d)2363 4116 y Fr(max)2700 3981 y Fk(\010)2797 4116 y Fr(16)37 b Fn(\001)g Fr(60)3249 4047 y Fo(d)3326 4116 y Fp(Q)3457 4141 y Fo(\025;)p Fj(I)3650 4159 y Fh(\025)3737 4116 y Fp(;)28 b Fr(1)3894 3981 y Fk(\011)4019 4116 y Fp(\015)4114 4047 y Fm(2)4105 4157 y Fo(\025;)p Fj(I)4298 4175 y Fh(\025)4384 4116 y Fp(L)4517 3974 y Fi(11)p 4517 3992 114 6 v 4546 4059 a(3)4650 4019 y Fo(d)4497 4160 y Fm(0)4728 4116 y Fn(k)p Fr(\000)4915 4141 y Fo(L)5006 4158 y Fi(0)5079 4116 y Fp(R)5205 4141 y Fo(\025;!)t(;;L)5580 4158 y Fi(0)5654 4116 y Fr(\()p Fp(E)10 b Fr(\))p Fp(\037)6021 4146 y Fo(L)6112 4163 y Fi(0)6175 4146 y Fo(=)p Fm(3)6317 4116 y Fn(k)1731 4425 y(\024)166 b Fp(c)2098 4450 y Fm(1)2172 4425 y Fp(\025)2269 4356 y Fo(\034)2352 4425 y Fr(e)2426 4356 y Fj(\000)p Fo(c)2589 4373 y Fi(2)2654 4257 y Fj(p)p 2763 4257 79 6 v 99 x Fo(\025)6311 4425 y Fr(\(3.30\))1731 4674 y Fp(<)g Fr(1)28 b Fp(;)4128 b Fr(\(3.31\))758 4961 y(with)1619 5359 y Fp(\034)64 b Fr(=)1932 5075 y Fk(\()2148 5211 y Fr(max)2485 5077 y Fk(\010)2602 5146 y Fo(d)p 2602 5173 70 7 v 2603 5268 a Fm(2)2728 5211 y Fn(\000)37 b Fr(1)p Fp(;)28 b Fr(0)3134 5077 y Fk(\011)4608 5211 y Fr(if)55 b Fp(M)4921 5236 y Fj(\000)5079 5211 y Fn(\025)47 b Fr(0)2148 5462 y(max)2485 5278 y Fk(n)2702 5388 y Fm(2)p Fo(p)p 2616 5424 308 7 v 2616 5519 a Fm(2)p Fo(p)p Fj(\000)p Fo(d)2963 5396 y(d)p 2963 5423 70 7 v 2964 5519 a Fm(2)3089 5462 y Fn(\000)37 b Fr(1)p Fp(;)28 b Fr(0)3495 5278 y Fk(o)3642 5462 y Fr(+)3915 5388 y Fm(2)p Fo(p)p 3828 5424 308 7 v 3828 5519 a Fm(2)p Fo(p)p Fj(\000)p Fo(d)4193 5462 y Fn(\000)37 b Fr(1)166 b(if)55 b Fp(M)4921 5487 y Fj(\000)5079 5462 y Fp(<)47 b Fr(0)5496 5359 y Fp(;)769 b Fr(\(3.32\))758 5798 y(where)58 b(\(3.30\))g(holds)i(for)e Fp(!)g Fn(2)52 b Fp(A)2868 5823 y Fo(\025)3013 5798 y Fr(with)58 b Fp(c)3466 5823 y Fm(1)3599 5798 y Fr(and)h Fp(c)3997 5823 y Fm(2)4130 5798 y Fr(constan)-5 b(ts)60 b(dep)5 b(ending)59 b(only)g(on)g Fp(d)p Fr(,)h Fp(U)6577 5823 y Fj(\000)6689 5798 y Fr(,)758 5998 y Fn(k)p Fp(g)6 b Fn(k)1009 6023 y Fj(1)1150 5998 y Fr(,)49 b Fp(\045)p Fr(,)g Fp(p)p Fr(,)g Fn(k)p Fp(u)p Fn(k)1866 6023 y Fo(p)1943 5998 y Fr(,)h Fp(M)2200 6023 y Fj(\000)2312 5998 y Fr(,)f(\002)2536 6023 y Fm(p)t(er)p Fo(;)p Fm(1)2838 5998 y Fr(,)h(and)e(\002)3378 6023 y Fm(p)t(er)p Fo(;)p Fm(2)3728 5998 y Fr(\(note)f(that)g(w)-5 b(e)48 b Ff(\014xe)-8 b(d)65 b Fp(L)5244 6023 y Fm(0)5366 5998 y Fr(and)49 b Fp(\016)k Fr(in)c(\(3.12\))e(and)758 6197 y(\(3.25\))o(\).)102 b(W)-14 b(e)65 b(conclude)g(that)f(there)g (exists)h Fp(\025)3751 6137 y Fj(\003)3828 6197 y Fr(,)i(dep)5 b(ending)65 b(only)g(on)g Fp(d)p Fr(,)j Fp(U)5675 6222 y Fj(\000)5788 6197 y Fr(,)f Fn(k)p Fp(g)6 b Fn(k)6152 6222 y Fj(1)6293 6197 y Fr(,)67 b Fp(\045)p Fr(,)h Fp(p)p Fr(,)758 6396 y Fn(k)p Fp(u)p Fn(k)1019 6421 y Fo(p)1096 6396 y Fr(,)55 b Fp(M)1358 6421 y Fj(\000)1470 6396 y Fr(,)h(\002)1701 6421 y Fm(p)t(er)p Fo(;)p Fm(1)2003 6396 y Fr(,)f(and)h(\002)2556 6421 y Fm(p)t(er)p Fo(;)p Fm(2)2858 6396 y Fr(,)f(suc)-5 b(h)57 b(that)e(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b(\(3.31\))e(for)i(all)f Fp(\025)46 b(>)g(\025)5700 6336 y Fj(\003)5832 6396 y Fr(and)56 b Fp(!)c Fn(2)46 b Fp(A)6592 6421 y Fo(\025)6679 6396 y Fr(.)957 6601 y(Th)-5 b(us)78 b(condition)f(\(2.25\))f(holds)h(for)g Fp(\025)k(>)g(\025)3866 6541 y Fj(\003)4019 6601 y Fr(and)c Fp(E)91 b Fn(2)81 b Fr([)p Fp(E)4948 6541 y Fm(inf)4938 6648 y Fo(\025)5117 6601 y Fp(;)28 b(E)5314 6626 y Fo(\025)5400 6601 y Fr(])77 b(b)-5 b(y)79 b(\(3.23\))o(,)j(hence)758 6806 y(Theorem)55 b(2.9)h(implies)g(that)f([)p Fp(E)2829 6746 y Fm(inf)2819 6853 y Fo(\025)2998 6806 y Fp(;)28 b(E)3195 6831 y Fo(\025)3281 6806 y Fr(])38 b Fn(\\)e Fr(\006)3632 6831 y Fo(\025)3765 6806 y Fn(\032)46 b Fr(\006)4060 6831 y Fm(MSA)4359 6806 y Fr(.)957 7005 y(It)54 b(remains)h(to)e(pro)-5 b(v)g(e)55 b(the)f(estimate)f(\(3.14\)) h(on)g(the)g(the)f(rate)h(of)g(the)f(exp)5 b(onen)-5 b(tial)54 b(deca)-5 b(y)758 7204 y(of)46 b(the)g(eigenfunctions)h(with) f(energies)g(in)h([)p Fp(E)3643 7144 y Fm(inf)3633 7251 y Fo(\025)3812 7204 y Fp(;)28 b(E)4009 7229 y Fo(\025)4096 7204 y Fr(].)71 b(T)-14 b(o)46 b(do)h(so)g(w)-5 b(e)46 b(use)h(the)f(criterion)g(giv)-5 b(en)758 7404 y(in)55 b(Theorem)h(2.11.)74 b(W)-14 b(e)55 b(start)g(b)-5 b(y)56 b(de\014ning)g Fp(\022)3688 7429 y Fo(\025)3830 7404 y Fr(b)-5 b(y)2361 7738 y Fp(L)2474 7664 y Fj(\000)p Fo(\022)2641 7682 y Fh(\025)2474 7783 y Fm(0)2773 7738 y Fr(=)2996 7673 y Fm(2)p Fo(d)p 2969 7700 190 7 v 2969 7796 a(K)3081 7814 y Fh(\025)3206 7738 y Fp(L)3319 7667 y Fm(2)p Fo(d)p Fj(\000)p Fm(1)3319 7783 y(0)3632 7738 y Fr(e)3706 7666 y Fj(\000)3810 7577 y(p)p 3919 7577 256 6 v 89 x Fm(2)p Fo(K)4097 7684 y Fh(\025)4198 7678 y Fr(\()4262 7666 y Fm(1+2)4496 7567 y Fj(p)p 4606 7567 70 6 v 4606 7666 a Fo(dL)4766 7683 y Fi(0)4831 7678 y Fr(\))4895 7595 y Fe(\000)p Fi(1)5086 7738 y Fp(:)1179 b Fr(\(3.33\))758 8042 y(Then)65 b(\(3.28\))f(sa)-5 b(ys)66 b(that)e(for)g Fp(!)k Fn(2)62 b Fp(A)3153 8067 y Fo(\025)3304 8042 y Fr(the)j(b)5 b(o)-5 b(x)64 b(\003)4042 8067 y Fo(L)4133 8084 y Fi(0)4207 8042 y Fr(\(0\))g(is)h(\()p Fp(\022)4803 8067 y Fo(\025)4890 8042 y Fp(;)28 b(E)10 b Fr(\)-suitable)64 b(for)h(an)-5 b(y)65 b Fp(E)71 b Fn(2)758 8241 y Fr([)p Fp(E)937 8181 y Fm(inf)927 8288 y Fo(\025)1106 8241 y Fp(;)28 b(E)1303 8266 y Fo(\025)1389 8241 y Fr(].)74 b(Moreo)-5 b(v)g(er,)56 b(it)f(follo)-5 b(ws)56 b(from)g(\(3.33\))e(that)2713 8555 y Fp(\022)2791 8580 y Fo(\025)2924 8555 y Fn(\025)46 b Fp(c)3171 8580 y Fm(0)3246 8406 y Fn(p)p 3384 8406 97 7 v 149 x Fp(\025)138 b Fr(for)55 b(all)139 b Fp(\025)46 b Fn(\025)g Fp(\025)4602 8487 y Fj(\003)4706 8555 y Fp(;)1559 b Fr(\(3.34\))758 8842 y(where)62 b Fp(c)1317 8867 y Fm(0)1454 8842 y Fr(is)h(some)g (constan)-5 b(t)63 b(and)g Fp(\025)3155 8782 y Fj(\003)3294 8842 y Fr(is)g(tak)-5 b(en)62 b(large)g(enough)i(\(b)5 b(oth)61 b(dep)5 b(ending)63 b(only)g(on)758 9041 y Fp(d)p Fr(,)56 b Fp(U)1059 9066 y Fj(\000)1171 9041 y Fr(,)f Fn(k)p Fp(g)6 b Fn(k)1523 9066 y Fj(1)1664 9041 y Fr(,)56 b(and)g Fp(\045)p Fr(\).)957 9241 y(As)j(in)g([GK1,)h(Pro)5 b(of)58 b(of)g(Theorem)h(3.4],)h(w)-5 b(e)59 b(b)5 b(o)g(otstrap)58 b(from)h([GK4,)h(Theorem)f(2.7])g(\(of)758 9440 y(whic)-5 b(h)48 b(Theorem)g(2.9)g(is)g(a)g(sp)5 b(ecial)48 b(case\))f(to)h (Theorem)g(2.11.)71 b(It)47 b(follo)-5 b(ws,)51 b(as)d(in)g([GK4,)i (Pro)5 b(of)758 9639 y(of)47 b(Theorem)h(3.1],)h(that)e(if)g(w)-5 b(e)48 b(set)f Fp(L)3083 9664 y Fo(k)3211 9639 y Fr(=)f(39)3552 9579 y Fo(k)3634 9639 y Fp(L)3747 9664 y Fm(0)3821 9639 y Fr(,)k Fp(k)h Fr(=)c(0)p Fp(;)28 b Fr(1)p Fp(;)g Fr(2)p Fp(;)g(:)g(:)g(:)f Fr(,)49 b(where)e Fp(L)5602 9664 y Fm(0)5724 9639 y Fr(is)h(as)g(in)g(\(3.12\))o(,)758 9838 y(then)55 b(for)g(all)h Fp(\025)46 b(>)g(\025)2036 9778 y Fj(\003)2167 9838 y Fr(w)-5 b(e)56 b(ha)-5 b(v)g(e)2227 10168 y Fl(P)5 b Fn(f)p Fr(\003)2518 10193 y Fo(L)2609 10211 y Fh(k)2699 10168 y Fr(\(0\))91 b(is)56 b(\()p Fp(\022)3313 10193 y Fo(\025)3400 10168 y Fp(;)28 b(E)10 b Fr(\)-suitable)o Fn(g)46 b(\025)g Fr(1)37 b Fn(\000)4967 10056 y Fr(1)p 4899 10130 218 7 v 4899 10345 a Fp(L)5032 10204 y Fi(5)p 5032 10222 57 6 v 5032 10288 a(3)5012 10395 y Fo(k)5192 10168 y Fp(;)1073 b Fr(\(3.35\))758 10641 y(for)58 b(all)g Fp(E)i Fn(2)50 b Fr([)p Fp(E)1771 10580 y Fm(inf)1761 10688 y Fo(\025)1940 10641 y Fp(;)28 b(E)2137 10666 y Fo(\025)2223 10641 y Fr(])58 b(and)h Fp(k)d Fn(\025)51 b(K)r Fr(,)59 b(where)f Fn(K)52 b Fr(=)f Fn(K)r Fr(\()p Fp(d;)28 b(\045)p Fr(\))50 b Fp(<)g Fn(1)59 b Fr(is)f(a)g(constan)-5 b(t)59 b(dep)5 b(ending)758 10840 y(only)55 b(on)h Fp(d)f Fr(and)h Fp(\045)p Fr(.)74 b(A)55 b(k)-5 b(ey)55 b(fact)f(is)i(that)f Fn(K)i Fr(do)5 b(es)56 b(not)f(dep)5 b(end)55 b(on)h Fp(\025)p Fr(.)p eop %%Page: 12 12 12 11 bop 758 861 a Fm(12)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)957 1259 y Fr(W)-14 b(e)50 b(no)-5 b(w)52 b(feed)e(\(3.35\))h(in)-5 b(to)51 b(the)g(h)-5 b(yp)5 b(otheses)51 b(of)g(Theorem)g(2.11.)73 b(W)-14 b(e)50 b(ha)-5 b(v)g(e)52 b(already)e(\014xed)758 1458 y Fp(p)58 b Fr(=)1108 1393 y Fm(5)p 1108 1420 67 7 v 1108 1516 a(3)1194 1458 y Fr(,)66 b(and)d(hence)g Fp(\013)h Fr(is)g(\014xed.)97 b(On)63 b(the)g(other)g(hand)h(for)e(eac) -5 b(h)64 b Fp(\025)f Fr(w)-5 b(e)63 b(ha)-5 b(v)g(e)63 b Fp(\022)h Fr(=)59 b Fp(\022)6235 1483 y Fo(\025)6385 1458 y Fr(as)64 b(in)758 1658 y(\(3.33\))o(,)69 b(and)e Fp(Q)1762 1683 y Fo(\025;)p Fj(I)1955 1701 y Fh(\025)2042 1658 y Fr(,)j Fp(C)2277 1683 y Fo(\025;)p Fj(I)2470 1701 y Fh(\025)2556 1658 y Fr(,)g(and)d Fp(\015)3092 1683 y Fo(\025;)p Fj(I)3285 1701 y Fh(\025)3438 1658 y Fr(gro)-5 b(w)67 b(p)5 b(olynomially)67 b(with)f Fp(\025)p Fr(.)107 b(T)-14 b(o)67 b(con)-5 b(trol)67 b Fn(L)6434 1683 y Fo(\025)6587 1658 y Fr(as)758 1857 y(giv)-5 b(en)44 b(in)g(\(2.29\))f (with)g(all)h(this)g(dep)5 b(endence)44 b(in)g Fp(\025)p Fr(,)i(w)-5 b(e)44 b(pic)-5 b(k)44 b(the)f(remaining)i(parameter,)h Fp(s)p Fr(,)g(to)758 2056 y(also)g(dep)5 b(end)45 b(on)h Fp(\025)f Fr(b)-5 b(y)46 b Fp(s)2309 2081 y Fo(\025)2442 2056 y Fr(=)g(log)30 b Fp(\025)p Fr(.)71 b(By)45 b(taking)g Fp(\025)3923 1996 y Fj(\003)4044 2056 y Fr(su\016cien)-5 b(tly)47 b(large,)g(as)f(b)5 b(efore)45 b(dep)5 b(ending)758 2255 y(only)56 b(on)h Fp(d)p Fr(,)g Fp(U)1657 2280 y Fj(\000)1770 2255 y Fr(,)g Fn(k)p Fp(g)6 b Fn(k)2124 2280 y Fj(1)2265 2255 y Fr(,)57 b Fp(\045)p Fr(,)f Fp(p)p Fr(,)h Fn(k)p Fp(u)p Fn(k)3004 2280 y Fo(p)3081 2255 y Fr(,)g Fp(M)3345 2280 y Fj(\000)3457 2255 y Fr(,)g(\002)3689 2280 y Fm(p)t(er)p Fo(;)p Fm(1)3991 2255 y Fr(,)f(and)h(\002)4546 2280 y Fm(p)t(er)p Fo(;)p Fm(2)4848 2255 y Fr(,)g(and)g(using)j (\(3.34\),)d(w)-5 b(e)56 b(can)758 2455 y(also)70 b(guaran)-5 b(tee)69 b Fp(\022)1958 2480 y Fo(\025)2114 2455 y Fp(>)h Fr(4)p Fp(\013)q(s)2581 2480 y Fo(\025)2737 2455 y Fr(and)g Fp(s)3152 2480 y Fo(\025)3308 2455 y Fp(>)f Fr(2)p Fp(p)46 b Fr(+)g(4)p Fp(d)70 b Fr(for)f(all)h Fp(\025)f(>)g(\025)5107 2394 y Fj(\003)5184 2455 y Fr(.)115 b(It)69 b(follo)-5 b(ws)70 b(from)g(the)758 2654 y(explicit)54 b(form)i(of)68 b(\(2.29\))55 b(that)2472 2944 y Fn(L)2587 2969 y Fj(1)2773 2944 y Fr(=)87 b(sup)2948 3085 y Fo(\025>\025)3210 3051 y Fe(\003)3549 2944 y Fr(sup)3307 3102 y Fo(E)6 b Fj(2)p Fm([)p Fo(E)3640 3062 y Fi(inf)3634 3142 y Fh(\025)3789 3102 y Fo(;E)3925 3120 y Fh(\025)4004 3102 y Fm(])4069 2944 y Fn(L)4184 2969 y Fo(\025)4271 2944 y Fr(\()p Fp(E)k Fr(\))44 b Fp(<)j Fn(1)28 b Fp(;)1317 b Fr(\(3.36\))758 3406 y(where)59 b Fn(L)1357 3431 y Fo(\025)1443 3406 y Fr(\()p Fp(E)10 b Fr(\))58 b(is)i(giv)-5 b(en)59 b(b)-5 b(y)62 b(\(2.29\))d(with)g Fp(E)3598 3431 y Fm(0)3725 3406 y Fr(=)53 b Fp(E)10 b Fr(,)59 b Fn(I)65 b Fr(=)53 b Fn(I)4572 3431 y Fo(\025)4660 3406 y Fr(,)60 b Fp(p)52 b Fr(=)5104 3341 y Fm(5)p 5104 3368 V 5104 3464 a(3)5190 3406 y Fr(,)61 b Fp(\022)c Fr(=)c Fp(\022)5692 3431 y Fo(\025)5779 3406 y Fr(,)60 b(and)g Fp(s)52 b Fr(=)h Fp(s)6602 3431 y Fo(\025)6689 3406 y Fr(.)758 3606 y(Note)h(that)h Fn(L)1634 3631 y Fj(1)1829 3606 y Fr(dep)5 b(ends)56 b(only)g(on)f Fp(d)p Fr(,)h Fp(U)3367 3631 y Fj(\000)3480 3606 y Fr(,)f Fn(k)p Fp(g)6 b Fn(k)3832 3631 y Fj(1)3973 3606 y Fr(,)55 b Fp(\045)p Fr(,)h Fp(p)p Fr(,)f Fn(k)p Fp(u)p Fn(k)4708 3631 y Fo(p)4785 3606 y Fr(,)g Fp(M)5047 3631 y Fj(\000)5159 3606 y Fr(,)h(\002)5390 3631 y Fm(p)t(er)p Fo(;)p Fm(1)5692 3606 y Fr(,)f(and)h(\002)6245 3631 y Fm(p)t(er)p Fo(;)p Fm(2)6547 3606 y Fr(.)957 3811 y(W)-14 b(e)54 b(no)-5 b(w)54 b(\014x)g Fp(\024)g Fr(to)g(b)5 b(e)53 b(that)h(smallest)h Fp(k)d Fn(\025)46 b(K)56 b Fr(suc)-5 b(h)55 b(that)e Fp(L)4732 3836 y Fo(k)4860 3811 y Fr(=)47 b(39)5202 3750 y Fo(k)5283 3811 y Fp(L)5396 3836 y Fm(0)5517 3811 y Fn(\025)f(L)5807 3836 y Fj(1)5947 3811 y Fr(.)74 b(It)53 b(follo)-5 b(ws)758 4010 y(from)74 b(Theorem)g(2.11)g(that)g(if)f Fp(\025)k(>)h(\025)3285 3950 y Fj(\003)3361 4010 y Fr(,)h(then)73 b(for)h(almost)h(ev)-5 b(ery)73 b Fp(!)6 b Fr(,)79 b(the)74 b(eigenfunctions)758 4209 y Fp(')867 4234 y Fo(\025;!)t(;E)1279 4209 y Fr(with)55 b(energy)g Fp(E)h Fn(2)45 b Fr([)p Fp(E)2702 4149 y Fm(inf)2692 4256 y Fo(\025)2871 4209 y Fp(;)28 b(E)3068 4234 y Fo(\025)3155 4209 y Fr(])55 b(deca)-5 b(y)56 b(exp)5 b(onen)-5 b(tially)54 b(\(in)i(the)f Fp(L)5399 4149 y Fm(2)5473 4209 y Fr(-sense\))h(with) 2393 4601 y(lim)28 b(inf)2414 4716 y Fj(j)p Fo(x)p Fj(j!1)2881 4601 y Fn(\000)3030 4489 y Fr(log)j Fn(k)p Fp(\037)3460 4514 y Fo(x)3543 4489 y Fp(')3652 4514 y Fo(\025;!)t(;E)4009 4489 y Fn(k)p 3030 4563 1062 7 v 3468 4715 a(j)p Fp(x)p Fn(j)4158 4601 y(\025)4354 4489 y Fp(\022)4432 4514 y Fo(\025)4546 4489 y Fr(log)f Fp(L)4901 4514 y Fo(\024)p 4354 4563 634 7 v 4529 4715 a Fr(2)p Fp(L)4725 4740 y Fo(\024)5053 4601 y Fp(:)1212 b Fr(\(3.37\))758 4994 y(so)56 b(\(3.14\))e(follo)-5 b(ws)56 b(from)g(\(3.34\))o(.)p 6614 4994 7 113 v 6621 4888 100 7 v 6621 4994 V 6720 4994 7 113 v 957 5323 a(.)2256 5722 y Fq(Appendix)61 b Fr(A.)91 b Fq(Pr)l(oof)63 b(of)f(Theorem)f(3.1)957 6021 y Fr(The)55 b(iden)-5 b(tit)g(y)58 b(\(3.5\))d(follo)-5 b(ws)56 b(from)g(ergo)5 b(dicit)-5 b(y)55 b(and)h(h)-5 b(yp)5 b(othesis)56 b(\(b2\),)f(as)h(in)g([KM,)g(The-)758 6220 y(orem)f(3].)74 b(Note)55 b(that)1237 6543 y Fp(H)1375 6568 y Fo(\025;!)1636 6543 y Fr(=)46 b Fp(H)1949 6568 y Fo(\025;M)2194 6585 y Fe(\000)2337 6543 y Fr(+)37 b Fp(\025)2699 6386 y Fk(X)2628 6761 y Fo(i)p Fj(2)2785 6716 y Fi(1)p 2784 6734 58 6 v 2784 6801 a Fh(q)2862 6761 y Fg(Z)2947 6727 y Fh(d)3009 6543 y Fr(\()p Fp(!)3177 6568 y Fo(i)3269 6543 y Fn(\000)g Fp(M)3596 6568 y Fj(\000)3709 6543 y Fr(\))28 b Fp(u)p Fr(\()p Fp(x)35 b Fn(\000)i Fp(i)p Fr(\))92 b(with)g Fp(!)4990 6568 y Fo(i)5083 6543 y Fn(\000)37 b Fp(M)5410 6568 y Fj(\000)5568 6543 y Fn(\025)47 b Fr(0)p Fp(:)479 b Fr(\(A.1\))957 7082 y(T)-14 b(o)79 b(obtain)h(the)f(upp)5 b(er)80 b(b)5 b(ound)80 b(\(3.7\))o(,)85 b(let)79 b Fp(')86 b Fn(2)g Fp(C)4371 7022 y Fj(1)4359 7123 y Fo(c)4511 7082 y Fr(\()p Fl(R)4687 7022 y Fo(d)4773 7082 y Fr(\))79 b(with)g Fn(k)p Fp(')p Fn(k)5594 7107 y Fm(2)5755 7082 y Fr(=)86 b(1,)f(and)80 b(set)758 7281 y Fp(')867 7306 y Fo(n)957 7281 y Fr(\()p Fp(x)p Fr(\))45 b(=)1524 7216 y Fm(1)p 1422 7243 270 7 v 1422 7345 a Fo(n)1504 7312 y Fh(d=)p Fi(2)1711 7281 y Fp(')p Fr(\()1912 7216 y Fm(1)p 1904 7243 83 7 v 1904 7338 a Fo(n)2006 7281 y Fp(x)p Fr(\))55 b(for)g Fp(n)46 b Fn(2)g Fl(N)26 b Fr(;)64 b(note)55 b Fn(k)p Fp(')3560 7306 y Fo(n)3650 7281 y Fn(k)3733 7306 y Fm(2)3854 7281 y Fr(=)46 b(1.)74 b(It)55 b(follo)-5 b(ws)56 b(from)g(\(3.5\))f(that)1505 7602 y Fp(E)1638 7533 y Fm(inf)1628 7643 y Fo(\025)1853 7602 y Fn(\024)46 b(h)p Fp(')2202 7627 y Fo(n)2291 7602 y Fp(;)28 b(H)2503 7627 y Fo(\025;M)2748 7644 y Fe(\000)2855 7602 y Fp(')2964 7627 y Fo(n)3054 7602 y Fn(i)3233 b Fr(\(A.2\))1837 7950 y(=)2078 7837 y(1)p 2032 7911 175 7 v 2032 8064 a Fp(n)2132 8016 y Fm(2)2226 7950 y Fn(h)p Fp(';)28 b Fn(\000)p Fr(\001)p Fp(')p Fn(i)36 b Fr(+)3117 7724 y Fk(Z)3209 8101 y Fg(R)3286 8068 y Fh(d)3399 7950 y Fr(\()o Fp(V)3560 7975 y Fm(p)t(er)3756 7950 y Fr(\()p Fp(nx)p Fr(\))g(+)h Fp(\025M)4541 7975 y Fj(\000)4653 7950 y Fp(U)18 b Fr(\()p Fp(nx)p Fr(\)\))26 b Fn(j)p Fp(')p Fr(\()p Fp(x)p Fr(\))p Fn(j)5626 7881 y Fm(2)5727 7950 y Fr(d)p Fp(x)i(:)758 8333 y Fr(Letting)54 b Fp(n)46 b Fn(!)g(1)55 b Fr(w)-5 b(e)56 b(get)f(\(3.7\))o(.)957 8532 y(W)-14 b(e)52 b(no)-5 b(w)53 b(turn)f(to)g(\(3.9\))g(and)g(the)g (lo)-5 b(w)g(er)53 b(b)5 b(ound)53 b(\(3.11\).)72 b(The)53 b(lo)-5 b(w)g(er)53 b(b)5 b(ound)53 b(for)f Fp(M)6319 8557 y Fj(\000)6477 8532 y Fn(\025)46 b Fr(0)758 8732 y(is)60 b(ob)-5 b(vious.)87 b(F)-14 b(or)60 b Fp(M)2069 8757 y Fj(\000)2234 8732 y Fp(<)53 b Fr(0)60 b(it)f(follo)-5 b(ws)60 b(from)f(\(3.9\),)h(whic)-5 b(h)60 b(will)g(follo)-5 b(w)60 b(from)f(the)h(follo)-5 b(wing)758 8931 y(lemma.)758 9242 y Fv(Lemma)63 b(A.1.)84 b Ff(L)-8 b(et)75 b Fp(W)97 b Ff(b)-8 b(e)76 b(a)e(r)-8 b(e)g(al)75 b(value)-8 b(d)75 b(me)-8 b(asur)g(able)75 b(function)h(on)f Fl(R)5546 9182 y Fo(d)5707 9242 y Ff(such)g(that)g(that)758 9441 y Fp(W)69 b Fn(2)45 b Fr(L)1244 9381 y Fo(p)1244 9482 y(u)1331 9441 y Fr(\()p Fl(R)1507 9381 y Fo(d)1593 9441 y Fp(;)28 b Fr(d)p Fp(x)p Fr(\))p Ff(,)59 b(i.e.,)2646 9731 y Fn(k)p Fp(W)23 b Fn(k)2992 9756 y Fo(p;u)3233 9731 y Fr(=)77 b(sup)3408 9882 y Fo(x)p Fj(2)p Fg(Z)3657 9849 y Fh(d)3748 9731 y Fn(k)p Fp(\037)3935 9756 y Fo(x)4018 9731 y Fp(W)23 b Fn(k)4281 9756 y Fo(p)4404 9731 y Fp(<)46 b Fn(1)28 b Fp(;)1533 b Fr(\(A.3\))758 10161 y Ff(with)59 b Fp(p)46 b Fn(2)g Fr(\()1490 10096 y Fo(d)p 1489 10123 70 7 v 1490 10218 a Fm(2)1578 10161 y Fp(;)28 b Fn(1)p Fr(])60 b Ff(if)f Fp(d)47 b Fn(\025)f Fr(2)p Ff(,)59 b(and)h Fp(p)45 b Fn(2)h Fr([1)p Fp(;)28 b Fn(1)p Fr(])60 b Ff(if)f Fp(d)46 b Fr(=)h(1)p Ff(.)76 b(Then)60 b(for)f(every)h Fp(m)46 b(>)g Fr(0)60 b Ff(we)g(have)984 10502 y Fn(jh)o Fp( )6 b(;)28 b(W)23 b( )6 b Fn(ij)46 b(\024)g Fp(C)2027 10527 y Fo(d;p)2240 10502 y Fp(m)2386 10428 y Fj(\000)p Fm(2+)2678 10383 y Fh(d)p 2678 10401 62 6 v 2678 10468 a(p)2767 10502 y Fn(k)p Fp(W)23 b Fn(k)3113 10527 y Fo(p;u)3336 10368 y Fk(\000)3412 10502 y Fn(kr)p Fp( )6 b Fn(k)3830 10434 y Fm(2)3942 10502 y Fr(+)37 b Fp(m)4254 10434 y Fm(2)4328 10502 y Fn(k)p Fp( )6 b Fn(k)4608 10434 y Fm(2)4683 10368 y Fk(\001)4832 10502 y Ff(for)60 b(al)8 b(l)60 b Fp( )53 b Fn(2)45 b(D)5 b Fr(\()p Fn(r)p Fr(\))p Fp(;)253 b Fr(\(A.4\))758 10792 y Ff(wher)-8 b(e)60 b Fp(C)1346 10817 y Fo(d;p)1590 10792 y Ff(is)f(a)g(\014nite)i(c)-8 b(onstant)60 b(dep)-8 b(ending)60 b(only)g(on)g Fp(d)f Ff(and)h Fp(p)p Ff(.)p eop %%Page: 13 13 13 12 bop 2654 861 a Fm(HIGH)58 b(DISORDER)h(LOCALIZA)-11 b(TION)1764 b(13)758 1259 y Ff(Pr)-8 b(o)g(of.)81 b Fr(It)55 b(su\016ces)h(to)f(pro)-5 b(v)g(e)1670 1473 y Fk(\000)1746 1607 y Fn(\000)p Fr(\001)38 b(+)f Fp(m)2363 1539 y Fm(2)2437 1473 y Fk(\001)2513 1505 y Fj(\000)2637 1460 y Fi(1)p 2637 1478 57 6 v 2637 1545 a(2)2749 1607 y Fn(j)p Fp(W)23 b Fn(j)3049 1473 y Fk(\000)3125 1607 y Fn(\000)p Fr(\001)38 b(+)f Fp(m)3742 1539 y Fm(2)3816 1473 y Fk(\001)3892 1505 y Fj(\000)4016 1460 y Fi(1)p 4016 1478 V 4016 1545 a(2)4146 1607 y Fn(\024)47 b Fp(C)4441 1632 y Fo(d;p)4653 1607 y Fp(m)4799 1533 y Fj(\000)p Fm(2+)5091 1488 y Fh(d)p 5091 1506 62 6 v 5091 1573 a(p)5181 1607 y Fn(k)p Fp(W)23 b Fn(k)5527 1632 y Fo(p;u)5749 1607 y Fp(;)557 b Fr(\(A.5\))758 1905 y(or,)55 b(equiv)-9 b(alen)k(tly)-14 b(,)1984 2230 y Fn(j)p Fp(W)23 b Fn(j)2276 2117 y Fi(1)p 2276 2135 57 6 v 2276 2201 a(2)2388 2095 y Fk(\000)2465 2230 y Fn(\000)p Fr(\001)37 b(+)g Fp(m)3081 2161 y Fm(2)3155 2095 y Fk(\001)3231 2127 y Fj(\000)p Fm(1)3437 2230 y Fn(j)p Fp(W)23 b Fn(j)3729 2117 y Fi(1)p 3729 2135 V 3729 2201 a(2)3860 2230 y Fn(\024)47 b Fp(C)4155 2255 y Fo(d;p)4367 2230 y Fp(m)4513 2156 y Fj(\000)p Fm(2+)4805 2111 y Fh(d)p 4805 2129 62 6 v 4805 2196 a(p)4894 2230 y Fn(k)p Fp(W)23 b Fn(k)5240 2255 y Fo(p;u)5463 2230 y Fp(:)843 b Fr(\(A.6\))957 2578 y(W)-14 b(e)36 b(recall)h(\(e.g.,)k ([GJ,)g(Prop)5 b(osition)37 b(7.2.1]\))g(that)g(for)f Fp(m)46 b(>)h Fr(0)36 b(the)h(resolv)-5 b(en)g(t)5773 2444 y Fk(\000)5849 2578 y Fn(\000)p Fr(\001)38 b(+)f Fp(m)6466 2518 y Fm(2)6540 2444 y Fk(\001)6616 2478 y Fj(\000)p Fm(1)758 2777 y Fr(has)56 b(an)f(in)-5 b(tegral)56 b(k)-5 b(ernel)55 b Fp(K)2519 2802 y Fo(m)2646 2777 y Fr(\()p Fp(x)36 b Fn(\000)h Fp(y)6 b Fr(\),)55 b(suc)-5 b(h)57 b(that)2850 3083 y Fp(K)2991 3108 y Fo(m)3117 3083 y Fr(\()p Fp(x)p Fr(\))45 b(=)i Fp(m)3709 3015 y Fo(d)p Fj(\000)p Fm(2)3956 3083 y Fp(K)4097 3108 y Fm(1)4171 3083 y Fr(\()p Fp(mx)p Fr(\))28 b Fp(;)1736 b Fr(\(A.7\))758 3381 y(and)1787 3958 y(0)46 b Fp(<)h(K)2233 3983 y Fm(1)2307 3958 y Fr(\()p Fp(x)p Fr(\))e Fn(\024)2753 3518 y Fk(8)2753 3667 y(>)2753 3717 y(>)2753 3767 y(<)2753 4066 y(>)2753 4115 y(>)2753 4165 y(:)2983 3644 y Fp(C)3102 3669 y Fm(1)3204 3644 y Fr(e)3278 3583 y Fj(\000j)p Fo(x)p Fj(j)4626 3644 y Fr(if)55 b Fp(d)47 b Fr(=)f(1)2983 3851 y Fp(C)3102 3876 y Fm(2)3232 3851 y Fr(\()o Fn(j)p Fr(ln)29 b Fn(j)p Fp(x)p Fn(jj)37 b Fr(+)g(1\))27 b(e)4194 3791 y Fj(\000j)p Fo(x)p Fj(j)4626 3851 y Fr(if)55 b Fp(d)47 b Fr(=)f(2)2983 4058 y Fp(C)3102 4083 y Fm(3)3204 4058 y Fn(j)p Fp(x)p Fn(j)3391 3998 y Fj(\000)p Fm(1)3570 4058 y Fr(e)3644 3998 y Fj(\000j)p Fo(x)p Fj(j)4626 4058 y Fr(if)55 b Fp(d)47 b Fr(=)f(3)2983 4276 y Fp(C)3102 4301 y Fo(d)3207 4276 y Fn(j)p Fp(x)p Fn(j)3394 4216 y Fj(\000)p Fo(d)p Fm(+2)3771 4276 y Fr(e)3845 4216 y Fj(\000)3969 4171 y Fi(1)p 3969 4189 57 6 v 3969 4256 a(2)4045 4216 y Fj(j)p Fo(x)p Fj(j)4626 4276 y Fr(if)55 b Fp(d)47 b Fn(\025)f Fr(4)5280 3958 y Fp(;)1026 b Fr(\(A.8\))758 4597 y(where)63 b Fp(C)1365 4622 y Fo(d)1506 4597 y Fr(is)i(a)f(constan)-5 b(t)64 b(dep)5 b(ending)64 b(only)g(on)g(the)g(dimension.)101 b(If)64 b Fp(p)59 b(>)5623 4532 y Fo(d)p 5623 4559 70 7 v 5624 4655 a Fm(2)5776 4597 y Fr(if)k Fp(d)e Fn(\025)g Fr(2,)66 b(and)758 4797 y Fp(p)45 b Fn(\025)i Fr(1)55 b(if)g Fp(d)47 b Fr(=)f(1,)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)2659 5120 y Fn(k)p Fp(K)2883 5145 y Fo(m)3009 5120 y Fn(k)3185 5099 y Fh(p)p 3112 5126 208 6 v 3112 5193 a(p)p Fe(\000)p Fi(1)3394 5120 y Fn(\024)46 b Fp(C)3688 5145 y Fo(d;p)3901 5120 y Fp(m)4047 5046 y Fj(\000)p Fm(2+)4339 5002 y Fh(d)p 4339 5020 62 6 v 4339 5086 a(p)4456 5120 y Fp(;)1850 b Fr(\(A.9\))758 5439 y(where)55 b Fp(C)1357 5464 y Fo(d;p)1588 5439 y Fp(<)46 b Fn(1)56 b Fr(dep)5 b(ends)56 b(only)f(on)h Fp(d)f Fr(and)h Fp(p)957 5644 y Fr(If)f Fp( )d Fn(2)46 b Fr(L)1544 5584 y Fm(2)1618 5644 y Fr(\()p Fl(R)1794 5584 y Fo(d)1880 5644 y Fp(;)28 b Fr(d)p Fp(x)p Fr(\),)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e,)912 5808 y Fk(D)1014 5992 y Fp( )6 b(;)28 b Fn(j)p Fp(W)23 b Fn(j)1494 5879 y Fi(1)p 1494 5897 57 6 v 1494 5963 a(2)1606 5858 y Fk(\000)1682 5992 y Fn(\000)p Fr(\001)38 b(+)f Fp(m)2299 5923 y Fm(2)2373 5858 y Fk(\001)2449 5890 y Fj(\000)p Fm(1)2655 5992 y Fn(j)p Fp(W)23 b Fn(j)2947 5879 y Fi(1)p 2947 5897 V 2947 5963 a(2)3032 5992 y Fp( )3146 5808 y Fk(E)3294 5992 y Fr(=)3469 5808 y Fk(D)3570 5992 y Fp( )6 b(;)28 b Fn(j)p Fp(W)23 b Fn(j)4050 5879 y Fi(1)p 4050 5897 V 4050 5963 a(2)4135 5992 y Fp(K)4276 6017 y Fo(m)4439 5992 y Fn(\003)37 b(j)p Fp(W)23 b Fn(j)4851 5879 y Fi(1)p 4851 5897 V 4851 5963 a(2)4936 5992 y Fp( )5050 5808 y Fk(E)6269 5992 y Fr(\(A.10\))2444 6357 y Fn(\024)2711 6200 y Fk(X)2620 6567 y Fo(x;y)t Fj(2)p Fg(Z)2979 6534 y Fh(d)3070 6166 y Fk(\014)3070 6266 y(\014)3070 6366 y(\014)3126 6173 y(D)3227 6357 y Fp( )6 b(;)28 b Fn(j)p Fp(W)23 b Fn(j)3707 6244 y Fi(1)p 3707 6262 V 3707 6329 a(2)3792 6357 y Fp(\037)3896 6382 y Fo(x)3979 6357 y Fp(K)4120 6382 y Fo(m)4283 6357 y Fn(\003)37 b Fp(\037)4507 6382 y Fo(y)4587 6357 y Fn(j)p Fp(W)23 b Fn(j)4879 6244 y Fi(1)p 4879 6262 V 4879 6329 a(2)4964 6357 y Fp( )5078 6173 y Fk(E)5179 6166 y(\014)5179 6266 y(\014)5179 6366 y(\014)2444 6847 y Fn(\024)2711 6689 y Fk(X)2620 7057 y Fo(x;y)t Fj(2)p Fg(Z)2979 7024 y Fh(d)3070 6847 y Fn(k)p Fp(W)g Fn(k)3416 6872 y Fo(p;u)3611 6847 y Fn(k)p Fp(K)3835 6872 y Fo(m)3961 6847 y Fn(k)4137 6826 y Fh(p)p 4064 6853 208 6 v 4064 6919 a(p)p Fe(\000)p Fi(1)4300 6847 y Fn(k)p Fp(\037)4487 6872 y Fo(x)4571 6847 y Fp( )6 b Fn(k)4768 6872 y Fm(2)4842 6847 y Fn(k)p Fp(\037)5029 6872 y Fo(y)5109 6847 y Fp( )g Fn(k)5306 6872 y Fm(2)6269 6847 y Fr(\(A.11\))2444 7306 y Fn(\024)47 b(k)p Fp(W)23 b Fn(k)2966 7331 y Fo(p;u)3160 7306 y Fn(k)p Fp(K)3384 7331 y Fo(m)3510 7306 y Fn(k)3687 7285 y Fh(p)p 3613 7312 V 3613 7378 a(p)p Fe(\000)p Fi(1)3849 7306 y Fn(k)p Fp( )6 b Fn(k)4129 7238 y Fm(2)4129 7347 y(2)4232 7306 y Fp(;)1991 b Fr(\(A.12\))758 7634 y(where)71 b(w)-5 b(e)72 b(used)h(Holder's)f(and)h(Y)-14 b(oung's)73 b(inequalities)f(to) f(get)h(\(A.11\))n(,)77 b(and)72 b(the)g(Cauc)-5 b(h)g(y)758 7833 y(inequalit)g(y)55 b(to)g(obtain)g(\(A.12\))o(.)957 8033 y(The)g(estimate)g(\(A.6\))g(follo)-5 b(ws.)74 b(from)56 b(\(A.12\))e(and)i(\(A.9\))o(.)p 6614 8033 7 113 v 6621 7927 100 7 v 6621 8033 V 6720 8033 7 113 v 957 8373 a(W)-14 b(e)43 b(can)g(no)-5 b(w)43 b(pro)-5 b(v)g(e)44 b(\(3.9\))o(.)70 b(Note)42 b(that)g(there)h(exists)g(a)g(\014nite)g(constan)-5 b(t)43 b Fp(c)5611 8398 y Fo(d;q)t(;\045)5901 8373 y Fr(,)j(dep)5 b(ending)758 8572 y(only)55 b(on)h Fp(d)f Fr(and)h Fp(\045)p Fr(,)g(suc)-5 b(h)56 b(that)2445 8870 y Fn(k)p Fp(U)18 b Fn(k)2742 8895 y Fo(p;u)2983 8870 y Fr(=)47 b Fn(k)p Fp(\037)3346 8895 y Fm(0)3448 8870 y Fp(U)18 b Fn(k)3662 8895 y Fo(p)3785 8870 y Fn(\024)46 b Fp(c)4032 8895 y Fo(d;q)t(;\045)4323 8870 y Fn(k)p Fp(u)p Fn(k)4584 8895 y Fo(p)4706 8870 y Fp(<)h Fn(1)1221 b Fr(\(A.13\))758 9168 y(It)55 b(th)-5 b(us)56 b(follo)-5 b(ws)56 b(from)g(\(A.4\))e(that)g(for)i(all)f Fp(m)46 b(>)g Fr(0)56 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)1084 9515 y Fn(jh)p Fp( )6 b(;)28 b(\025M)1641 9540 y Fj(\000)1753 9515 y Fp(U)18 b( )6 b Fn(ij)46 b(\024)g Fp(\013)q(\025)2562 9331 y Fk(\020)2661 9515 y Fp(m)2807 9441 y Fj(\000)p Fm(2+)3099 9397 y Fh(d)p 3099 9415 62 6 v 3099 9481 a(p)3188 9515 y Fn(kr)p Fp( )6 b Fn(k)3606 9447 y Fm(2)3718 9515 y Fr(+)37 b Fp(m)4050 9397 y Fh(d)p 4050 9415 V 4050 9481 a(p)4140 9515 y Fn(k)p Fp( )6 b Fn(k)4420 9447 y Fm(2)4494 9331 y Fk(\021)4667 9515 y Fr(for)55 b(all)h Fp( )c Fn(2)46 b(D)5 b Fr(\()p Fn(r)p Fr(\))o Fp(;)354 b Fr(\(A.14\))758 9854 y(with)2819 10152 y Fp(\013)47 b Fr(=)g Fp(c)3220 10177 y Fo(d;q)t(;\045)3537 10152 y Fp(C)3656 10177 y Fo(d;p)3869 10152 y Fn(j)p Fp(M)4076 10177 y Fj(\000)4188 10152 y Fn(j)28 b(k)p Fp(u)p Fn(k)4523 10177 y Fo(p)4628 10152 y Fp(:)1595 b Fr(\(A.15\))758 10450 y(T)-14 b(o)55 b(obtain)h(\(3.9\))o(,)f(w)-5 b(e)56 b(c)-5 b(ho)5 b(ose)56 b Fp(m)f Fr(dep)5 b(ending)56 b(on)f Fp(\025)h Fr(b)-5 b(y)2744 10783 y Fp(\013)q(\025m)3094 10709 y Fj(\000)p Fm(2+)3385 10664 y Fh(d)p 3385 10682 V 3385 10749 a(p)3521 10783 y Fr(=)3716 10717 y Fm(1)p 3716 10744 67 7 v 3716 10840 a(2)3802 10783 y Fr(\(1)36 b Fn(\000)h Fr(\002)4281 10808 y Fm(p)t(er)p Fo(;)p Fm(1)4583 10783 y Fr(\))28 b Fp(;)1547 b Fr(\(A.16\))p eop %%Page: 14 14 14 13 bop 758 861 a Fm(14)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fr(i.e.,)2839 1697 y Fp(m)46 b Fr(=)3206 1463 y Fk(\022)3563 1584 y Fr(2)p Fp(\013)q(\025)p 3348 1659 717 7 v 3348 1811 a Fr(1)37 b Fn(\000)g Fr(\002)3763 1836 y Fm(p)t(er)p Fo(;)p Fm(1)4085 1463 y Fk(\023)4331 1443 y Fh(p)p 4227 1470 270 6 v 4227 1537 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)4580 1697 y Fp(;)1643 b Fr(\(A.17\))758 2090 y(and)56 b(w)-5 b(e)55 b(obtain)970 2389 y Fn(jh)o Fp( )6 b(;)28 b(\025M)1526 2414 y Fj(\000)1638 2389 y Fp(U)18 b( )6 b Fn(ij)47 b(\024)2236 2323 y Fm(1)p 2236 2350 67 7 v 2236 2446 a(2)2322 2389 y Fr(\(1)36 b Fn(\000)h Fr(\002)2801 2414 y Fm(p)t(er)p Fo(;)p Fm(1)3103 2389 y Fr(\))p Fn(kr)p Fp( )6 b Fn(k)3586 2320 y Fm(2)3698 2389 y Fr(+)37 b Fp(\014)9 b(\025)4159 2261 y Fi(2)p Fh(p)p 4083 2288 270 6 v 4083 2355 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)4381 2389 y Fn(k)p Fp( )d Fn(k)4661 2320 y Fm(2)4782 2389 y Fr(for)55 b(all)g Fp( )e Fn(2)45 b(D)5 b Fr(\()p Fn(r)p Fr(\))p Fp(;)239 b Fr(\(A.18\))758 2688 y(with)2648 3106 y Fp(\014)55 b Fr(=)46 b Fp(\013)3175 2978 y Fi(2)p Fh(p)p 3099 3005 V 3099 3072 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)3424 2872 y Fk(\022)3883 2994 y Fr(2)p 3567 3068 717 7 v 3567 3220 a(1)36 b Fn(\000)i Fr(\002)3982 3245 y Fm(p)t(er)p Fo(;)p Fm(1)4303 2872 y Fk(\023)4549 2862 y Fh(d)p 4446 2880 270 6 v 4446 2946 a Fi(2)p Fh(p)p Fe(\000)p Fh(d)4798 3106 y Fp(:)1425 b Fr(\(A.19\))957 3508 y(The)55 b(estimate)g(\(3.9\))g(no)-5 b(w)56 b(follo)-5 b(ws)56 b(from)g(\(3.2\))e(and)i(\(A.18\))o(.)p 6614 3508 7 113 v 6621 3403 100 7 v 6621 3508 V 6720 3508 7 113 v 3262 3932 a Fq(References)758 4198 y Fu([AM])259 b(Aizenman,)52 b(M.,)i(Molc)l(hano)l(v,)f(S.:)74 b(Lo)t(calization)47 b(at)52 b(large)e(disorder)h(and)i(extreme)e(energies:)71 b(an)1330 4364 y(elemen)l(tary)46 b(deriv)-8 b(ation.)45 b(Comm)l(un.)i(Math.)h(Ph)l(ys.)g Fd(157)p Fu(,)g(245-278)c(\(1993\)) 758 4530 y([ASFH])112 b(Aizenman,)45 b(M.,)h(Sc)l(henk)l(er,)i(J.,)f(F) -12 b(riedric)l(h,)45 b(R.)h(,)g(Hundertmark,)h(D.:)62 b(Finite-v)l(olume)43 b(criteria)h(for)1330 4696 y(Anderson)j(lo)t (calization.)42 b(Comm)l(un.)47 b(Math.)g(Ph)l(ys.)h Fd(224)p Fu(,)h(219-253)44 b(\(2001\))758 4862 y([AENSS])71 b(Aizenman,)46 b(M.,)h(Elgart,)f(A.,)h(Nab)t(ok)l(o,)f(S.,)i(Sc)l(henk) l(er,)h(J.,)f(Stolz,)e(G.:)62 b(In)48 b(preparation)758 5028 y([CKM])153 b(Carmona,)60 b(R.,)h(Klein,)f(A.,)h(Martinelli,)d (F.:)86 b(Anderson)58 b(lo)t(calization)53 b(for)58 b(Bernoulli)e(and)j (other)1330 5194 y(singular)45 b(p)t(oten)l(tials.)g(Comm)l(un.)i (Math.)g(Ph)l(ys.)h Fd(108)p Fu(,)g(41-66)e(\(1987\))758 5360 y([CH1])215 b(Com)l(b)t(es,)52 b(J.M.,)j(Hislop,)c(P)-12 b(.D.:)73 b(Lo)t(calization)48 b(for)j(some)g(con)l(tin)l(uous,)i (random)e(Hamiltonian)e(in)1330 5526 y(d-dimension.)d(J.)i(F)-12 b(unct.)48 b(Anal.)e Fd(124)p Fu(,)i(149-180)d(\(1994\))758 5692 y([CHN])180 b(Com)l(b)t(es,)43 b(J.M.,)h(Hislop,)d(P)-12 b(.D.,)44 b(Nak)-8 b(am)l(ura,)43 b(S.:)61 b(The)43 b Fc(L)4399 5646 y Fb(p)4471 5692 y Fu(-theory)g(of)f(the)g(sp)t(ectral)f (shift)g(function,)1330 5858 y(the)59 b(W)-12 b(egner)59 b(estimate)e(and)i(the)h(in)l(tegrated)d(densit)l(y)i(of)g(states)f (for)h(some)f(random)h(op)t(erators.)1330 6025 y(Comm)l(un.)47 b(Math.)g(Ph)l(ys.)h Fd(218)p Fu(,)h(113-130)44 b(\(2001\))758 6191 y([CHKN])70 b(Com)l(b)t(es,)62 b(J.M.,)h(Hislop,)e(P)-12 b(.D.,)63 b(Klopp,)f(F.,)h(Nak)-8 b(am)l(ura,)61 b(S.:)88 b(The)60 b(W)-12 b(egner)59 b(estimate)d(and)k(the)1330 6357 y(in)l(tegrated)45 b(densit)l(y)i(of)f(states)g(for)g(some)f (random)i(op)t(erators.)e(Pro)t(c.)h(Indean)h(Acad.)g(Sci.)f(\(Math.) 1330 6523 y(Sci.\))h Fd(112)p Fu(,)h(31-53)d(\(2002\))758 6689 y([CHM])157 b(Com)l(b)t(es,)62 b(J.M.,)j(Hislop,)c(P)-12 b(.D.,)64 b(Mourre,)f(E.:)89 b(Sp)t(ectral)59 b(a)l(v)l(eraging,)i(p)t (erturbation)f(of)f(singular)1330 6855 y(sp)t(ectra,)46 b(and)i(lo)t(calization.)42 b(T)-12 b(rans.)47 b(Amer.)f(Math.)h(So)t (c.)g Fd(348)p Fu(,)i(4883-4894)43 b(\(1996\))758 7021 y([CHT])184 b(Com)l(b)t(es,)56 b(J.M.,)i(Hislop,)c(P)-12 b(.D.,)58 b(Tip,)e(A.:)78 b(Band)55 b(edge)f(lo)t(calization)c(and)55 b(the)g(densit)l(y)f(of)h(states)1330 7187 y(for)50 b(acoustic)e(and)j (electromagnetic)46 b(w)l(a)l(v)l(es)k(in)f(random)h(media.)f(Ann.)i (Inst.)g(H.)f(P)l(oincare)e(Ph)l(ys.)1330 7353 y(Theor.)f Fd(70)h Fu(,)f(381-428)d(\(1999\))758 7519 y([DS])308 b(Damanik,)46 b(D.,)h(Stollmann,)f(P)-12 b(.:)64 b(Multi-scale)44 b(analysis)g(implies)h(strong)h(dynamical)f(lo)t(calization.)1330 7685 y(Geom.)h(F)-12 b(unct.)48 b(Anal.)e Fd(11)p Fu(,)i(11-29)d (\(2001\))758 7851 y([DRJLS])71 b(Del)49 b(Rio,)h(R.,)h(Jitomirsk)-8 b(a)l(y)l(a,)49 b(S.,)j(Last,)e(Y.,)i(Simon,)e(B.:)69 b(Op)t(erators)49 b(with)h(singular)e(con)l(tin)l(uous)1330 8017 y(sp)t(ectrum)36 b(IV:)h(Hausdor\013)g(dimensions,)g(rank)h(one)e (p)t(ertubations)h(and)g(lo)t(calization.)32 b(J.)38 b(d'Analyse)1330 8183 y(Math.)47 b Fd(69)p Fu(,)h(153-200)d(\(1996\).) 758 8349 y([vD])312 b(v)l(on)42 b(Dreifus,)g(H.:)60 b Ft(On)47 b(the)e(e\013e)-7 b(cts)46 b(of)f(r)-7 b(andomness)46 b(in)g(ferr)-7 b(omagnetic)43 b(mo)-7 b(dels)46 b(and)g(Schr\177)-73 b(odinger)1330 8515 y(op)-7 b(er)g(ators)p Fu(.)47 b(Ph.D.)g(thesis,)f (New)h(Y)-12 b(ork)47 b(Univ)l(ersit)l(y)f(\(1987\))758 8681 y([vDK])202 b(v)l(on)57 b(Dreifus,)i(H.,)g(Klein,)e(A.:)83 b(A)56 b(new)h(pro)t(of)f(of)h(lo)t(calization)51 b(in)56 b(the)h(Anderson)g(tigh)l(t)f(binding)1330 8847 y(mo)t(del.)46 b(Comm)l(un.)h(Math.)g(Ph)l(ys.)h Fd(124)p Fu(,)g(285-299)d(\(1989\)) 758 9013 y([vDK2])131 b(v)l(on)33 b(Dreifus,)h(H.,)h(Klein,)e(A.:)56 b(Lo)t(calization)28 b(for)j(random)h(Sc)l(hro)t(dinger)g(op)t(erators) e(with)h(correlated)1330 9179 y(p)t(oten)l(tials.)45 b(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fd(140)p Fu(,)g(133-147)d(\(1991\)) 758 9345 y([FK1])221 b(Figotin,)38 b(A.,)j(Klein,)d(A.:)59 b(Lo)t(calization)34 b(of)k(classical)c(w)l(a)l(v)l(es)k(I:)g(Acoustic) e(w)l(a)l(v)l(es.)j(Comm)l(un.)f(Math.)1330 9511 y(Ph)l(ys.)48 b Fd(180)p Fu(,)g(439-482)d(\(1996\).)758 9677 y([FK2])221 b(Figotin,)52 b(A.,)h(Klein,)f(A.:)73 b(Lo)t(calization)48 b(of)k(classical)47 b(w)l(a)l(v)l(es)52 b(I)t(I:)g(Electromagnetic)c(w) l(a)l(v)l(es.)k(Com-)1330 9844 y(m)l(un.)c(Math.)f(Ph)l(ys.)h Fd(184)p Fu(,)g(411-441)d(\(1997\).)758 10010 y([FMSS])117 b(F)-12 b(r\177)-71 b(ohlic)l(h,)44 b(J.,)i(Martinelli,)41 b(F.,)46 b(Scopp)t(ola,)d(E.,)j(Sp)t(encer,)f(T.:)62 b(Constructiv)l(e)43 b(pro)t(of)h(of)f(lo)t(calization)1330 10176 y(in)k(the)g(Anderson)g(tigh)l(t)f(binding)h(mo)t(del.)e(Comm)l (un.)i(Math.)g(Ph)l(ys.)h Fd(101)p Fu(,)h(21-46)c(\(1985\))758 10342 y([FS])324 b(F)-12 b(r\177)-71 b(ohlic)l(h,)58 b(J.,)i(Sp)t(encer,)g(T.:)83 b(Absence)56 b(of)g(di\013usion)g(with)g (Anderson)h(tigh)l(t)f(binding)g(mo)t(del)g(for)1330 10508 y(large)45 b(disorder)h(or)h(lo)l(w)f(energy.)h(Comm)l(un.)g (Math.)g(Ph)l(ys.)h Fd(88)p Fu(,)g(151-184)c(\(1983\))758 10674 y([FLM])185 b(Fisc)l(her,)35 b(W.,)i(Lesc)l(hk)l(e,)f(H.,)g(M)t (\177)-74 b(uller,)35 b(P)-12 b(.:)56 b(Sp)t(ectral)32 b(lo)t(calization)d(b)l(y)34 b(Gaussian)e(random)h(p)t(oten)l(tials) 1330 10840 y(in)47 b(m)l(ulti-dimensional)42 b(con)l(tin)l(uous)47 b(space.)f(J.)i(Statist.)e(Ph)l(ys.)h Fd(101)i Fu(935{985)44 b(\(2000\))p eop %%Page: 15 15 15 14 bop 2654 861 a Fm(HIGH)58 b(DISORDER)h(LOCALIZA)-11 b(TION)1764 b(15)758 1259 y Fu([Ge])320 b(Germinet,)73 b(F.:)106 b(Dynamical)66 b(lo)t(calization)d(I)t(I)69 b(with)f(an)g(application)e(to)i(the)h(almost)d(Mathieu)1330 1425 y(op)t(erator.)46 b(J.)i(Stat)e(Ph)l(ys.)i Fd(95)p Fu(,)g(273-286)d(\(1999\).)758 1591 y([GDB])175 b(Germinet,)46 b(F,)h(De)f(Bi)l(\022)-67 b(evre,)47 b(S.:)63 b(Dynamical)45 b(lo)t(calization)d(for)k(discrete)g(and)h(con)l(tin)l(uous)g(random) 1330 1757 y(Sc)l(hr\177)-71 b(odinger)47 b(op)t(erators.)e(Comm)l(un.)i (Math.)g(Ph)l(ys.)h Fd(194)p Fu(,)h(323-341)44 b(\(1998\))758 1923 y([GK1])202 b(Germinet,)71 b(F,)d(Klein,)j(A.:)103 b(Bo)t(otstrap)66 b(Multiscale)e(Analysis)i(and)i(Lo)t(calization)63 b(in)k(Random)1330 2089 y(Media.)46 b(Comm)l(un.)h(Math.)h(Ph)l(ys.)g Fd(222)p Fu(,)g(415-448)c(\(2001\).)758 2255 y([GK2])202 b(Germinet,)36 b(F,)g(Klein,)g(A.:)57 b(Op)t(erator)35 b(k)l(ernel)g(estimates)d(for)j(functions)g(of)h(generalized)c(Sc)l (hrdinger)1330 2421 y(op)t(erators.)46 b(Pro)t(c.)g(Amer.)g(Mat.)h(So)t (c.)g(\(2002\))758 2587 y([GK3])202 b(Germinet,)59 b(F,)e(Klein,)i(A.:) 83 b(A)57 b(c)l(haracterization)d(of)j(the)g(Anderson)h (metal-insulator)53 b(transp)t(ort)1330 2753 y(transition.)45 b(Submitted.)758 2919 y([GK4])202 b(Germinet,)70 b(F,)d(Klein,)j(A.:) 102 b(Explicit)64 b(\014nite)i(v)l(olume)g(criteria)e(for)i(lo)t (calization)c(in)k(con)l(tin)l(uous)1330 3086 y(random)47 b(media)e(and)j(applications.)c(Submitted)758 3252 y([GJ])311 b(Glimm,)45 b(J.,)j(Ja\013e,)f(A.:)63 b Ft(Quantum)50 b(Physics.)d Fu(New)f(Y)-12 b(ork:)63 b(Springer-V)-12 b(erlag,)45 b(1981)758 3418 y([HK])278 b(Hislop,)37 b(P)-12 b(.D.,)40 b(Klopp,)e(F.:)59 b(The)37 b(in)l(tegrated)f(densit)l(y)h(of) f(states)g(for)h(some)f(random)g(op)t(erators)g(with)1330 3584 y(nonsign)46 b(de\014nite)h(p)t(oten)l(tials.)e(J.)j(F)-12 b(unct.)48 b(Anal.)e(T)-12 b(o)47 b(app)t(ear)758 3750 y([HM])259 b(Holden,)46 b(H.,)h(Martinelli,)d(F.:)63 b(On)47 b(absence)g(of)f(di\013usion)g(near)h(the)g(b)t(ottom)f(of)g (the)h(sp)t(ectrum)g(for)1330 3916 y(a)g(random)f(Sc)l(hr\177)-71 b(odinger)47 b(op)t(erator.)f(Comm)l(un.)h(Math.)g(Ph)l(ys.)h Fd(93)p Fu(,)g(197-217)c(\(1984\))758 4082 y([Ki])344 b(Kirsc)l(h,)37 b(W.:)58 b(W)-12 b(egner)36 b(estimates)d(and)k(lo)t (calization)31 b(for)36 b(allo)l(y-t)l(yp)t(e)f(p)t(oten)l(tials.)f (Math.)i(Zeit.)g Fd(221)p Fu(,)1330 4248 y(507-512)45 b(\(1996\))758 4414 y([KM])255 b(W.)40 b(Kirsc)l(h)f(and)i(F.)f (Martinelli,)e(On)i(the)h(Sp)t(ectrum)f(of)f(Sc)l(hr\177)-71 b(odinger)40 b(Op)t(erators)f(with)g(a)h(Random)1330 4580 y(P)l(oten)l(tial.)45 b(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fd(85)p Fu(,)g(329-350)d(\(1982\).)758 4746 y([KSS1])157 b(Kirsc)l(h,)55 b(W.,)h(Stolz,)f(G.,)h(Stollman,)e(P)-12 b(.:)78 b(Lo)t(calization)50 b(for)j(random)h(p)t(erturbations)f(of)h (p)t(erio)t(dic)1330 4912 y(Sc)l(hr\177)-71 b(odinger)47 b(op)t(erators.)e(Random)i(Op)t(er.)g(Sto)t(c)l(hastic)e(Equations)h Fd(6)p Fu(,)i(241-268)c(\(1998\))758 5078 y([KSS2])157 b(Kirsc)l(h,)43 b(W.,)g(Stolz,)g(G.,)h(Stollman,)d(P)-12 b(.:)62 b(Anderson)42 b(lo)t(calization)c(for)k(random)g(Sc)l(hr\177) -71 b(odinger)43 b(op)t(er-)1330 5244 y(ators)j(with)g(long)g(range)g (in)l(teractions.)f(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fd(195)p Fu(,)h(495-507)44 b(\(1998\))758 5410 y([K])384 b(Klein,)33 b(A.:)56 b(Lo)t(calization)27 b(in)k(the)h(Anderson)g(mo)t(del)e(with)h (long)g(range)g(hopping.)h(Brazilian)d(Journal)1330 5576 y(of)47 b(Ph)l(ysics)f Fd(23)p Fu(,)i(363-371)d(\(1993\))758 5742 y([KK1])203 b(Klein,)53 b(A.,)h(Koines,)f(A.:)74 b(A)52 b(general)f(framew)l(ork)h(for)h(lo)t(calization)47 b(of)53 b(classical)48 b(w)l(a)l(v)l(es:)74 b(I.)53 b(Inho-)1330 5908 y(mogeneous)45 b(media)h(and)h(defect)g(eigenmo)t(des.)e(Math.)i (Ph)l(ys.)h(Anal.)e(Geom.)g Fd(4)p Fu(,)i(97-130)d(\(2001\))758 6074 y([KK2])203 b(Klein,)34 b(A.,)i(Koines,)f(A.:)56 b(A)32 b(general)g(framew)l(ork)g(for)h(lo)t(calization)28 b(of)33 b(classical)28 b(w)l(a)l(v)l(es:)56 b(I)t(I.)33 b(Random)1330 6240 y(media.)46 b(In)h(preparation.)758 6406 y([KLS])218 b(Klein.,)48 b(A.,)i(Lacroix,)e(J.,)i(Sp)t(eis,)f(A.:) 133 b(Lo)t(calization)45 b(for)j(the)h(Anderson)g(mo)t(del)e(on)i(a)g (strip)f(with)1330 6572 y(singular)d(p)t(oten)l(tials.)g(J.)j(F)-12 b(unct.)48 b(Anal.)e Fd(94)p Fu(,)i(135-155)c(\(1990\))758 6739 y([Klo1])202 b(Klopp,)49 b(F.:)66 b(Lo)t(calization)44 b(for)49 b(con)l(tin)l(uous)f(random)g(Sc)l(hr\177)-71 b(odinger)48 b(op)t(erators.)f(Comm)l(un.)i(Math.)1330 6905 y(Ph)l(ys.)f Fd(167)p Fu(,)g(553-569)d(\(1995\))758 7071 y([Klo2])202 b(Klopp)47 b(F.:)65 b(W)-12 b(eak)48 b(disorder)f(lo)t(calization)c(and)48 b(Lifshitz)f(tails:)62 b(con)l(tin)l(uous)47 b(Hamiltonians.)e(Ann.)1330 7237 y(I.H.P)-12 b(.)47 b Fd(3)p Fu(,)h(711-737)d(\(2002\))758 7403 y([Si])376 b(Simon,)47 b(B.:)63 b(Sc)l(hr\177)-71 b(odinger)46 b(semi-groups.)f(Bull.)h(Amer.)h(Math.)g(So)t(c.)g Fd(V)-14 b(ol.7)p Fu(,)48 b(447-526)d(\(1982\))758 7569 y([Sp])338 b(Sp)t(encer,)57 b(T.)d(:)78 b(Lo)t(calization)50 b(for)55 b(random)f(and)g(quasip)t(erio)t(dic)e(p)t(oten)l(tials.)h(J.) i(Stat.)f(Ph)l(ys.)h Fd(51)p Fu(,)1330 7735 y(1009-1019)44 b(\(1988\))758 7901 y([St])361 b(Stollmann,)46 b(P)-12 b(.:)63 b(Caugh)l(t)47 b(b)l(y)h(disorder.)e(Bound)i(States)e(in)h (Random)f(Media.)h(Birk)-8 b(a)t(\177)-74 b(user)45 b(2001.)758 8067 y([U])388 b(Ueki,)46 b(N.:)63 b(W)-12 b(egner)47 b(estimates)d(and)j(lo)t(calization)42 b(for)47 b(Gaussian)e(random)i (p)t(oten)l(tials.)e(Preprin)l(t)758 8233 y([W)-12 b(a])290 b(W)-12 b(ang,)35 b(W.-M.:)56 b(Lo)t(calization)27 b(and)32 b(univ)l(ersalit)l(y)e(of)h(P)l(oisson)f(statistics)e(for)j(the)h(m)l (ultidimensional)1330 8399 y(Anderson)47 b(mo)t(del)f(at)g(w)l(eak)h (disorder.)f(In)l(v)l(en)l(t.)i(Math.)g Fd(146)p Fu(,)g(365-398)d (\(2001\))758 8565 y([W)-12 b(e])298 b(W)-12 b(egner,)47 b(F.:)63 b(Bounds)48 b(on)f(the)g(densit)l(y)g(of)g(states)f(in)g (disordered)h(systems.)f(Z.)h(Ph)l(ys.)h(B)f Fd(44)p Fu(,)h(9-15)1330 8731 y(\(1981\))957 9042 y Fs(Universit)1601 9031 y(\023)1601 9042 y(e)k(de)f(Lille)g(1,)i(UMR)e(8524)i(CNRS,)g(UFR) e(de)g(Ma)-9 b(th)4661 9031 y(\023)4661 9042 y(ema)g(tiques,)54 b(F-59655)f(Villeneuve)758 9208 y(d'Ascq)d(C)1369 9197 y(\023)1369 9208 y(edex,)i(France)957 9374 y Ft(E-mail)e(addr)-7 b(ess)11 b Fu(:)62 b Fa(germinet@agat.univ-lille)o(1.)o(fr)957 9657 y Fs(University)44 b(of)g(Calif)m(ornia,)k(Ir)-12 b(vine,)47 b(Dep)-9 b(ar)g(tment)44 b(of)g(Ma)-9 b(thema)g(tics,)46 b(Ir)-12 b(vine,)46 b(CA)e(92697-3875,)758 9823 y(USA)957 9989 y Ft(E-mail)50 b(addr)-7 b(ess)11 b Fu(:)62 b Fa(aklein@uci.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0210160415450--