This is a multi-part message in MIME format. ---------------0109241027225 Content-Type: text/plain; name="01-335.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-335.keywords" random Schr\"odinger operators, Anderson model, localization ---------------0109241027225 Content-Type: application/postscript; name="india.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="india.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.76 (MiKTeX 1.11) Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: india.dvi %%CreationDate: Mon Apr 30 11:22:38 2001 %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: DVIPS.EXE india %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2001.04.30:1112 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 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Fm(H)1367 1442 y Fl(P)1423 1430 y Fr(\()p Fm(!)s Fr(\))c(=)f Fk(\000)1751 1374 y Fm(d)1794 1344 y Fn(2)p 1727 1411 128 4 v 1727 1487 a Fm(dx)1817 1463 y Fn(2)1883 1430 y Fr(+)1966 1352 y Fj(X)2015 1528 y Fl(i)2100 1430 y Fm(f)9 b Fr(\()p Fm(x)19 b Fk(\000)f Fm(X)2400 1442 y Fl(i)2427 1430 y Fr(\()p Fm(!)s Fr(\)\))p Fm(:)-2145 b Fr(\(2.2\))456 1644 y(The)30 b Fm(X)698 1656 y Fl(i)726 1644 y Fr(\()p Fm(!)s Fr(\))g(are)g(the)g(p)r(oin)n(ts) h(of)f(a)g(P)n(oisson)f(pro)r(cess)g(on)h Fg(R)36 b Fr(with)31 b(constan)n(t)f(densit)n(y)g Fm(\013)e(>)g Fr(0.)456 1744 y(They)22 b(can)g(almost)g(surely)f(b)r(e)i(measurably)e(lab)r (eled)h(b)n(y)h Fm(i)f Fk(2)i Fg(Z)7 b Fk(n)h(f)o Fr(0)p Fk(g)16 b Fr(suc)n(h)22 b(that)g Fm(:)14 b(:)g(:)23 b(<)g(X)3268 1756 y Fi(\000)p Fn(1)3380 1744 y Fm(<)456 1844 y Fr(0)f Fm(<)h(X)677 1856 y Fn(1)737 1844 y Fm(<)g(X)894 1856 y Fn(2)954 1844 y Fm(<)f(:)14 b(:)g(:)42 b Fr(and)27 b(ha)n(v)n(e)f(the)i(prop)r(ert)n(y)f(that)h(for)f(ev)n(ery)f(Borel)h (set)g Fm(B)g Fk(\032)c Fg(R)1224 2023 y(P)p Fr(\(#)p Fk(f)p Fm(i)f Fr(:)h Fm(X)1585 2035 y Fl(i)1635 2023 y Fk(2)h Fm(B)t Fk(g)e Fr(=)h Fm(n)p Fr(\))g(=)2136 1967 y(\()p Fm(\013)p Fk(j)p Fm(B)t Fk(j)p Fr(\))2366 1937 y Fl(n)p 2136 2004 277 4 v 2238 2080 a Fm(n)p Fr(!)2422 2023 y Fm(e)2461 1989 y Fi(\000)p Fl(\013)p Fi(j)p Fl(B)s Fi(j)2653 2023 y Fm(;)456 2188 y Fr(where)d Fk(j)5 b(\001)g(j)20 b Fr(is)g(Leb)r(esgue)g(measure.)34 b(If)21 b Fm(B)1709 2200 y Fn(1)1750 2188 y Fk(\\)5 b Fm(B)1873 2200 y Fn(2)1934 2188 y Fr(=)22 b Fk(;)p Fr(,)g(then)f(the)g(random)f(v)-5 b(ariables)19 b(#)p Fk(f)p Fm(i)j Fr(:)i Fm(X)3339 2200 y Fl(i)3389 2188 y Fk(2)456 2288 y Fm(B)519 2300 y Fn(1)556 2288 y Fk(g)32 b Fr(and)g(#)p Fk(f)p Fm(i)e Fr(:)h Fm(X)1089 2300 y Fl(i)1147 2288 y Fk(2)g Fm(B)1296 2300 y Fn(2)1333 2288 y Fk(g)h Fr(are)f(indep)r(enden)n(t.)52 b(If)33 b Fm(X)2224 2300 y Fn(0)2291 2288 y Fr(:=)e(0)h(\(but)h Fs(not)f Fr(a)g(P)n(oisson)e(p)r(oin)n(t\),)456 2387 y(then)k(the)g(random)g(distances)f Fm(p)1516 2399 y Fl(n)1595 2387 y Fr(:=)g Fm(X)1785 2399 y Fl(n)p Fn(+1)1937 2387 y Fk(\000)22 b Fm(X)2093 2399 y Fl(n)2138 2387 y Fr(,)36 b Fm(n)d Fk(2)h Fg(Z)p Fr(,)29 b(are)k(i.i.d.)i(with)f (distribution)456 2487 y(densit)n(y)27 b Fm(\013e)834 2457 y Fi(\000)p Fl(\013t)958 2487 y Fr(.)555 2586 y(Thinking)i(of)f (the)i Fm(p)1196 2598 y Fl(n)1269 2586 y Fr(rather)e(than)h(the)g Fm(X)1928 2598 y Fl(i)1984 2586 y Fr(as)f(the)h(de\014ning)g (parameters,)e(w)n(e)h(therefore)456 2686 y(ha)n(v)n(e)c Fs(smo)l(othness)h Fr(of)g(the)h(mo)r(del)f(in)h(the)g(parameters.)34 b(Ho)n(w)n(ev)n(er,)24 b(the)h(mo)r(del)h(is)f Fs(not)i(mono-)456 2786 y(tonic)f Fr(in)g(the)h Fm(p)941 2798 y Fl(n)1012 2786 y Fr(\(or)e(the)i Fm(X)1355 2798 y Fl(i)1382 2786 y Fr(\):)36 b(Change)26 b(of)g(the)g(parameters)e(mak)n(es)h(the)i (single)e(site)h(bumps)456 2885 y(mo)n(v)n(e)h(left)j(or)e(righ)n(t)g (rather)g(than)h(up)h(or)e(do)n(wn.)40 b(A)30 b(sign)e(restriction)g (on)h Fm(f)38 b Fr(do)r(esn't)29 b(c)n(hange)456 2985 y(this)e(and)h(th)n(us)g(w)n(e)f(only)g(assume)g(that)h Fm(f)j Fk(6)p Fr(=)23 b(0)k(is)h(b)r(ounded)g(and)f(compactly)g(supp)r (orted.)555 3085 y(An)j(additional)f(feature)g(of)g(the)h(P)n(oisson)e (mo)r(del)h(is)h(that)f(dense)h(clusters)e(of)i(single)f(sites)456 3184 y(as)23 b(w)n(ell)g(as)g(large)g(gaps)g(b)r(et)n(w)n(een)g(sites)h (o)r(ccur.)35 b(T)-7 b(ec)n(hnically)g(,)24 b(this)g(is)g(sometimes)f (helpful)i(and)456 3284 y(a)i(n)n(uisance)g(at)g(other)g(times.)456 3502 y Fu(\(iii\))33 b(The)h(Displacemen)m(t)e(Mo)s(del:)39 b Fr(In)30 b(sev)n(eral)e(w)n(a)n(ys)g(in)n(termediate)h(b)r(et)n(w)n (een)h Fm(H)3227 3514 y Fl(A)3311 3502 y Fr(and)456 3601 y Fm(H)525 3613 y Fl(P)608 3601 y Fr(is)d(the)h(displacemen)n(t)f(mo)r (del)1222 3786 y Fm(H)1291 3798 y Fl(D)1351 3786 y Fr(\()p Fm(!)s Fr(\))c(=)g Fk(\000)1679 3730 y Fm(d)1722 3700 y Fn(2)p 1655 3767 128 4 v 1655 3843 a Fm(dx)1745 3819 y Fn(2)1812 3786 y Fr(+)1897 3708 y Fj(X)1895 3886 y Fl(n)p Fi(2)p Fh(Z)2033 3786 y Fm(f)9 b Fr(\()p Fm(x)19 b Fk(\000)f Fm(n)g Fk(\000)g Fm(d)2458 3798 y Fl(n)2504 3786 y Fr(\()p Fm(!)s Fr(\)\))p Fm(:)-2222 b Fr(\(2.3\))456 4002 y(The)18 b(random)g(displacemen)n(ts)g Fm(d)1476 4014 y Fl(n)1540 4002 y Fr(are)g(i.i.d.)h(random)f(v)-5 b(ariables,)19 b(whic)n(h)g(can)f(b)r(e)h(though)n(t)g(of)f(as)456 4101 y(mo)r(deling)h(temp)r(erature)h(\015uctuations)g(in)g(a)f (crystal.)34 b(Th)n(us)19 b(it)i(is)e(reasonable)f(to)i(assume)f(that) 456 4201 y(they)29 b(ha)n(v)n(e)f(an)g(a.c.)h(distribution)g(with)h(b)r (ounded)f(densit)n(y)g(supp)r(orted)g(in)g([)p Fk(\000)p Fm(d)3001 4213 y Fr(max)3159 4201 y Fm(;)14 b(d)3239 4213 y Fr(max)3398 4201 y(],)456 4300 y(i.e.)24 b(that)h(the)g(mo)r (del)f(is)h Fs(smo)l(oth)g Fr(in)f(the)h Fm(d)1780 4312 y Fl(n)1825 4300 y Fr(.)36 b(The)25 b(single)f(site)g(p)r(oten)n(tial)h Fm(f)33 b Fr(is)24 b(assumed)g(to)g(b)r(e)456 4400 y(non-zero,)e(b)r (ounded)i(and)f(supp)r(orted)h(in)f([)p Fk(\000)p Fm(s;)14 b(s)p Fr(])23 b(with)h Fm(s)10 b Fr(+)g Fm(d)2384 4412 y Fr(max)2566 4400 y Fk(\024)22 b Fr(1)p Fm(=)p Fr(2,)h(whic)n(h)h (guaran)n(tees)456 4500 y(non-o)n(v)n(erlapping)g(adjacen)n(t)j(sites.) 555 4599 y Fm(H)624 4611 y Fl(D)713 4599 y Fr(is)h Fs(non-monotonic)h Fr(in)f(the)h(displacemen)n(ts)f Fm(d)2185 4611 y Fl(n)2230 4599 y Fr(,)g(similar)g(to)g(the)h(P)n(oisson)d(mo)r(del.)39 b(It)456 4699 y(di\013ers)23 b(from)f Fm(H)959 4711 y Fl(P)1038 4699 y Fr(b)n(y)h(the)g(more)f(crystalline)h(single)f(site)h (structure.)35 b(Neither)23 b(large)f(gaps)g(nor)456 4799 y(dense)27 b(clusters)g(o)r(ccur.)456 5016 y Fu(\(iv\))f (Inde\014nite)f(Anderson:)33 b Fr(Monotonicit)n(y)22 b(of)g(the)h(Anderson)e(mo)r(del)i Fm(H)2927 5028 y Fl(A)3003 5016 y Fr(is)f(due)h(to)f(the)456 5116 y(w)n(a)n(y)j(in)i(whic)n(h)f (the)h(random)f(parameters)f(en)n(ter,)h(but)i(also)d(to)i(the)g (sign-de\014niteness)f(of)g(the)456 5216 y(single)g(site)h(p)r(oten)n (tial)f Fm(f)9 b Fr(.)36 b(If)28 b(w)n(e)e(drop)g(the)h(assumption)f Fm(f)32 b Fk(\025)22 b Fr(0)27 b(\(or,)f(essen)n(tially)g(equiv)-5 b(alen)n(t,)p eop %%Page: 4 4 4 3 bop 456 226 a Fn(4)1303 b(G.)29 b(STOLZ)456 425 y Fm(f)j Fk(\024)24 b Fr(0\),)k(but)h(k)n(eep)f(all)g(other)f (assumptions)h(from)g(\(i\),)h(one)f(gets)f(a)h(non-monotonic)f(v)n (ersion)456 525 y(of)g(the)h(Anderson)f(mo)r(del.)37 b(This)28 b(is)f(due)h(to)g(p)r(ossible)f(sign)g(c)n(hanges)f(of)i Fm(f)9 b Fr(.)555 624 y(T)-7 b(o)28 b(distinguish)h(this)f(mo)r(del)h (from)f(the)h(monotonic)e(random)h(op)r(erator)e Fm(H)2967 636 y Fl(A)3021 624 y Fr(\()p Fm(!)s Fr(\),)j(w)n(e)f(will)456 724 y(denote)c(it)h(b)n(y)f Fm(H)981 736 y Fl(I)5 b(A)1069 724 y Fr(\()p Fm(!)s Fr(\),)25 b(the)g(inde\014nite)g(Anderson)f(mo)r (del.)36 b(W)-7 b(e)25 b(don't)f(c)n(hange)g(the)g(assump-)456 824 y(tions)j(on)g Fm(q)811 836 y Fl(n)884 824 y Fr(and)h(th)n(us)f (still)h(ha)n(v)n(e)f(smo)r(othness.)456 1041 y Fu(\(v\))i(Singular)f (Random)g(P)m(arameters:)35 b Fr(Not)25 b(just)g(for)g(the)g(sak)n(e)f (of)h(generalit)n(y)-7 b(,)24 b(but)i(also)456 1141 y(for)34 b(ph)n(ysical)h(reasons)e(it)i(is)h(desirable)e(to)h(understand)g(the)g (Anderson)g(mo)r(del)g Fm(H)3156 1153 y Fl(A)3245 1141 y Fr(if)h(the)456 1241 y(distributions)g(of)g(the)h Fm(q)1242 1253 y Fl(n)1323 1241 y Fr(are)f(singular.)61 b(An)37 b(extreme)f(case)f(is)i(the)f(Bernoulli-Anderson)456 1340 y(mo)r(del)31 b(where)h(the)g Fm(q)1135 1352 y Fl(n)1212 1340 y Fr(tak)n(e)f(only)g(t)n(w)n(o)g(v)-5 b(alues,)32 b(i.e.)g Fg(P)p Fr(\()p Fm(q)2277 1352 y Fl(n)2351 1340 y Fr(=)e Fm(a)p Fr(\))g(=)f Fm(p)h Fk(2)g Fr(\(0)p Fm(;)14 b Fr(1\))31 b(and)h Fg(P)p Fr(\()p Fm(q)3306 1352 y Fl(n)3380 1340 y Fr(=)456 1440 y Fm(b)p Fr(\))23 b(=)f(1)8 b Fk(\000)g Fm(p)p Fr(.)34 b(Lo)r(oking)21 b(at)h(this)g(and)g(other)g(discrete)g (distributions)g(is)g(motiv)-5 b(ated)22 b(b)n(y)g(thinking)456 1540 y(of)36 b(the)i Fm(q)749 1552 y Fl(n)831 1540 y Fr(as)e(the)i(n)n(uclear)d(c)n(harge)h(at)g(site)h Fm(n)g Fr(in)g(an)g(allo)n(y)f(consisting)g(of)h(\014nitely)g(man)n(y)456 1639 y(comp)r(onen)n(ts.)555 1739 y(W)-7 b(e)21 b(will)g(write)g Fm(H)1116 1751 y Fl(S)s(A)1214 1739 y Fr(\()p Fm(!)s Fr(\))g(\(singular)f(Anderson\))g(when)h(referring)e(to)i(the)g (Anderson)f(mo)r(del)456 1838 y(with)31 b(general,)e(p)r(ossibly)h (singular,)g(distributions)h(of)f(the)h(i.i.d.)g(random)f(v)-5 b(ariables)29 b Fm(q)3239 1850 y Fl(n)3284 1838 y Fr(,)i(i.e.)456 1938 y(it)c(will)h(only)e(b)r(e)i(assumed)f(that)g(the)h(supp)r(ort)f (of)g(their)g(distribution)g(con)n(tains)f(at)h(least)g(t)n(w)n(o)456 2038 y(p)r(oin)n(ts.)36 b(F)-7 b(or)27 b(simplicit)n(y)-7 b(,)28 b(w)n(e)f(will)h(still)g(assume)f(that)h(the)g(supp)r(ort)f(is)h (b)r(ounded.)555 2137 y(Th)n(us)c Fm(H)831 2149 y Fl(S)s(A)953 2137 y Fr(is)g(in)h(general)d(singular)h(in)h(its)h(random)e (parameters,)g(i.e.)h(lac)n(ks)f(smo)r(othness.)456 2237 y(Requiring)18 b(a)g(de\014nite)h(sign)f(for)g Fm(f)28 b Fr(w)n(ould)18 b(still)h(lea)n(v)n(e)e(us)i(with)g(monotonicit)n(y)-7 b(.)33 b(But)19 b(it)g(will)g(turn)456 2337 y(out)26 b(that)h(due)g(to)g(the)g(lac)n(k)f(of)h(smo)r(othness)f(w)n(e)g(will)h (not)g(b)r(e)g(able)f(to)h(exploit)f(monotonicit)n(y)456 2436 y(an)n(y)31 b(more.)49 b(One)31 b(w)n(a)n(y)g(to)h(understand)f (this)h(is)g(to)g(lo)r(ok)f(at)h(the)g(restrictions)e(of)i Fm(H)3165 2448 y Fl(A)3219 2436 y Fr(\()p Fm(!)s Fr(\))h(to)456 2536 y(a)f(\014xed)h(\014nite)h(in)n(terv)-5 b(al.)52 b(If)34 b Fm(f)40 b Fk(\025)32 b Fr(0)h(and)f(for)h(a)f (con\014guration)g Fm(!)k Fr(the)d(restricted)f(op)r(erator)456 2635 y(has)g(an)h(eigen)n(v)-5 b(alue)32 b(at)g Fm(E)1299 2647 y Fn(0)1337 2635 y Fr(,)i(then)g(b)n(y)e(c)n(ho)r(osing)g(a)g (con\014guration)g Fm(!)2690 2605 y Fi(0)2746 2635 y Fr(with)h(sligh)n(tly)f(larger)456 2735 y(couplings)22 b Fm(q)853 2747 y Fl(n)899 2735 y Fr(\()p Fm(!)986 2705 y Fi(0)1009 2735 y Fr(\),)j(the)f(eigen)n(v)-5 b(alue)23 b(can)g(b)r(e)h(made)f(to)h(mo)n(v)n(e)e(a)n(w)n(a)n(y)g(from)h Fm(E)2860 2747 y Fn(0)2898 2735 y Fr(.)36 b(If)24 b(the)g Fm(q)3212 2747 y Fl(n)3281 2735 y Fr(ha)n(v)n(e)456 2835 y(discrete)e(distribution,)i(then)g(the)f(nearest)f(p)r(ossible)h (larger)e(v)-5 b(alues)23 b(of)g Fm(q)2742 2847 y Fl(n)2787 2835 y Fr(\()p Fm(!)2874 2805 y Fi(0)2897 2835 y Fr(\))h(migh)n(t)f (cause)f(a)456 2934 y(lo)n(w)n(er)i(lying)i(eigen)n(v)-5 b(alue)25 b(to)h(mo)n(v)n(e)f(up)i(to)f Fm(E)1863 2946 y Fn(0)1901 2934 y Fr(,)g(whic)n(h)g(diminishes)h(the)f(tec)n(hnical)g (adv)-5 b(an)n(tage)456 3034 y(of)29 b(monotonicit)n(y)-7 b(.)43 b(Therefore)28 b(our)h(assumptions)g(on)g Fm(f)39 b Fr(for)29 b Fm(H)2484 3046 y Fl(S)s(A)2612 3034 y Fr(will)h(b)r(e)g (the)g(same)f(as)g(for)456 3134 y Fm(H)525 3146 y Fl(I)5 b(A)613 3134 y Fr(.)555 3233 y(Belo)n(w,)25 b(w)n(e)g(will)h(o)r (ccasionally)e(discuss)h(sp)r(eci\014c)g(prop)r(erties)g(of)g(the)h (Bernoulli-Anderson)456 3333 y(mo)r(del,)h(denoting)h(it)g(b)n(y)f Fm(H)1334 3345 y Fl(B)s(A)1441 3333 y Fr(.)555 3432 y(W)-7 b(e)40 b(ma)n(y)e(also)h(in)n(tro)r(duce)f(a)h(singular)f(v)n(ersion)g (of)h(the)h(displacemen)n(t)f(mo)r(del)g Fm(H)3257 3444 y Fl(D)3357 3432 y Fr(b)n(y)456 3532 y(dropping)34 b(the)i(absolute)f (con)n(tin)n(uit)n(y)f(assumption)h(in)h(\(iii\))g(and)f(only)g (requiring)f(that)i(the)456 3632 y(supp)r(ort)g(of)h(the)g (distribution)g(of)f(the)i Fm(d)1789 3644 y Fl(n)1871 3632 y Fr(con)n(tains)d(at)i(least)f(t)n(w)n(o)g(p)r(oin)n(ts.)64 b(This)37 b(will)g(b)r(e)456 3731 y(referred)26 b(to)h(as)g Fm(H)1035 3743 y Fl(S)s(D)1140 3731 y Fr(.)456 3949 y Fu(\(vi\))37 b(Anderson)f(with)h(p)s(erio)s(dic)e(bac)m(kground:)47 b Fr(A)32 b(generalized)f(Anderson)g(mo)r(del)i(is)456 4049 y(obtained)i(if)h(in)g(addition)g(to)f(the)h(random)f(p)r(oten)n (tial)h(a)f(p)r(erio)r(dic)h(bac)n(kground)e(p)r(oten)n(tial)456 4148 y Fm(V)504 4160 y Fr(p)r(er)654 4148 y(is)27 b(in)n(tro)r(duced:) 1036 4420 y Fm(H)1105 4432 y Fl(P)9 b Fn(+)p Fl(A)1261 4420 y Fr(\()p Fm(!)s Fr(\))23 b(=)g Fk(\000)1589 4364 y Fm(d)1632 4334 y Fn(2)p 1565 4401 128 4 v 1565 4477 a Fm(dx)1655 4453 y Fn(2)1722 4420 y Fr(+)18 b Fm(V)1853 4432 y Fr(p)r(er)1975 4420 y(\()p Fm(x)p Fr(\))i(+)2191 4341 y Fj(X)2189 4520 y Fl(n)p Fi(2)p Fh(Z)2327 4420 y Fm(q)2364 4432 y Fl(n)2409 4420 y Fr(\()p Fm(!)s Fr(\))p Fm(f)9 b Fr(\()p Fm(x)19 b Fk(\000)f Fm(n)p Fr(\))p Fm(:)-2408 b Fr(\(2.4\))456 4714 y(Here)20 b Fm(V)693 4726 y Fr(p)r(er)836 4714 y(is)g(b)r(ounded)h(with)g Fm(V)1474 4726 y Fr(p)r(er)1596 4714 y(\()p Fm(x)t Fr(+)t(1\))j(=)f Fm(V)1982 4726 y Fr(p)r(er)2104 4714 y(\()p Fm(x)p Fr(\),)g(while)e Fm(q)2508 4726 y Fl(n)2573 4714 y Fr(and)g Fm(f)29 b Fr(satisfy)20 b(the)h(general)456 4814 y(assumptions)26 b(from)i(\(v\).)555 4914 y(Studying)d(this)f(mo)r(del)g(is)g(partly)g(motiv)-5 b(ated)24 b(b)n(y)g(applications,)g(where)g(it)g(can)g(b)r(e)g(used)h (to)456 5013 y(mo)r(del)31 b(a)f(crystal)g(with)h(random)f(impurities.) 46 b(There)31 b(is)f(also)g(a)g(mathematical)h(reason)e(for)456 5113 y(in)n(tro)r(ducing)j Fm(H)973 5125 y Fl(P)9 b Fn(+)p Fl(A)1129 5113 y Fr(:)47 b(Starting)32 b(from)g(a)g(singular)f (Anderson)h(mo)r(del)h Fm(H)2819 5125 y Fl(S)s(A)2948 5113 y Fr(=)e Fk(\000)p Fm(d)3152 5083 y Fn(2)3189 5113 y Fm(=dx)3321 5083 y Fn(2)3380 5113 y Fr(+)456 5153 y Fj(P)557 5216 y Fm(q)594 5228 y Fl(n)639 5216 y Fm(f)9 b Fr(\()p Fm(x)21 b Fk(\000)f Fm(n)p Fr(\),)32 b(w)n(e)e(ma)n(y)g(pic)n (k)g(an)n(y)g(n)n(um)n(b)r(er)g Fm(q)2002 5185 y Fn(\(0\))2122 5216 y Fr(in)h(the)g(supp)r(ort)f(of)h(the)g(distribution)g(of)p eop %%Page: 5 5 5 4 bop 1469 226 a Fn(LOCALIZA)-5 b(TION)30 b(STRA)-5 b(TEGIES)979 b(5)456 429 y Fm(q)493 441 y Fl(n)538 429 y Fr(,)27 b(de\014ne)h Fm(V)876 441 y Fr(p)r(er)998 429 y(\()p Fm(x)p Fr(\))c(=)1221 367 y Fj(P)1322 429 y Fm(q)1362 399 y Fn(\(0\))1452 429 y Fm(f)9 b Fr(\()p Fm(x)18 b Fk(\000)f Fm(n)p Fr(\))28 b(and)f Fm(q)1992 386 y Fn(\(0\))1989 439 y Fl(n)2104 429 y Fr(=)c Fm(q)2229 441 y Fl(n)2292 429 y Fk(\000)17 b Fm(q)2414 399 y Fn(\(0\))2531 429 y Fr(to)27 b(get)g Fm(H)2839 441 y Fl(S)s(A)2960 429 y Fr(=)c Fk(\000)p Fm(d)3156 399 y Fn(2)3193 429 y Fm(=dx)3325 399 y Fn(2)3380 429 y Fr(+)456 552 y Fm(V)504 564 y Fr(p)r(er)626 552 y(\()p Fm(x)p Fr(\))d(+)840 490 y Fj(P)941 552 y Fm(q)981 509 y Fn(\(0\))978 562 y Fl(n)1070 552 y Fm(f)9 b Fr(\()p Fm(x)19 b Fk(\000)f Fm(n)p Fr(\).)555 652 y(A)n(t)23 b(the)g(cost)g(of)f(the)i(new)e Fm(V)1416 664 y Fr(p)r(er)1562 652 y(w)n(e)g(ha)n(v)n(e)g(reac)n(hed)f(the)i(adv)-5 b(an)n(tage)21 b(that)i(the)h(new)e(random)456 776 y(couplings)27 b Fm(q)861 733 y Fn(\(0\))858 785 y Fl(n)978 776 y Fr(ha)n(v)n(e)g(0)g (in)i(the)f(supp)r(ort)g(of)g(their)g(distribution.)38 b(T)-7 b(o)28 b(some)f(exten)n(t)h(this)g(will)456 875 y(allo)n(w)33 b(to)i(treat)g Fm(H)1066 887 y Fl(S)s(A)1199 875 y Fr(as)f(a)h(p)r(erturbation)f(of)h Fk(\000)p Fm(d)2089 845 y Fn(2)2126 875 y Fm(=dx)2258 845 y Fn(2)2319 875 y Fr(+)23 b Fm(V)2455 887 y Fr(p)r(er)2577 875 y(.)59 b(In)35 b(particular,)h(w)n(e)e(will)456 975 y(outline)d(in)g(Section)g 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y(;)-1956 b Fr(\(2.5\))456 2978 y(a)28 b(random)h(op)r(erator)e(deriv)n(ed)h(from)h(the)h(one-dimensional)e(w) n(a)n(v)n(e)f(equation.)41 b(The)29 b(random)456 3078 y(co)r(e\016cien)n(t)j Fm(a)p Fr(\()p Fm(x;)14 b(!)s Fr(\))33 b(describ)r(es)e(prop)r(erties)h(of)g(an)g(inhomogeneous)f (medium,)j(suc)n(h)e(as)g(the)456 3177 y(recipro)r(cals)22 b(of)i(the)h(mass)e(densit)n(y)-7 b(,)25 b(compressibilit)n(y)-7 b(,)24 b(dielectricit)n(y)-7 b(,)24 b(etc.,)i(dep)r(ending)e(on)g(the) 456 3277 y(t)n(yp)r(e)j(of)h(w)n(a)n(v)n(es)e(to)h(b)r(e)h(studied.)555 3377 y(Sp)r(eci\014c)20 b(c)n(hoices)e(of)h Fm(a)p Fr(\()p Fm(x;)14 b(!)s Fr(\))20 b(allo)n(w)e(us)h(to)g(in)n(tro)r(duce)f (randomness)g(of)h(Anderson,)h(P)n(oisson,)456 3476 y(or)28 b(displacemen)n(t)h(t)n(yp)r(e.)43 b(The)30 b(corresp)r(onding)d(op)r (erators)g(will)j(b)r(e)g(denoted)g(b)n(y)f Fm(H)3142 3446 y Fl(W)3135 3499 y(A)3217 3476 y Fr(,)h Fm(H)3346 3446 y Fl(W)3339 3499 y(P)3421 3476 y Fr(,)456 3577 y Fm(H)532 3547 y Fl(W)525 3600 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503 y Fi(1)3012 554 y Fn(0)3089 533 y Fr(\()p Fg(R)p Fr(\).)41 b(Note)456 633 y(that)28 b(this)h(do)r(es)f(not)g(require)g(smo)r(othness)f(of)i Fm(a)p Fr(\()p Fm(x;)14 b(!)s Fr(\))29 b(and)f(therefore)f Fm(C)2806 603 y Fi(1)2800 653 y Fn(0)2877 633 y Fr(\()p Fg(R)p Fr(\))35 b(ma)n(y)28 b(not)g(b)r(e)456 732 y(in)f(the)h(op)r (erator)e(domain)h(of)h Fm(H)1496 702 y Fl(W)1571 732 y Fr(.)37 b(In)28 b(particular,)e Fm(a)p Fr(\()p Fm(x;)14 b(!)s Fr(\))29 b(ma)n(y)d(b)r(e)i(a)g(step)f(function.)555 832 y(All)d(random)e(op)r(erators)f Fm(H)7 b Fr(\()p Fm(!)s Fr(\))23 b(from)g(Chapter)g(2)g(are)f(ergo)r(dic)g(\(sometimes)h (called)g(metri-)456 932 y(cally)h(transitiv)n(e)h(or)f(co)n(v)-5 b(arian)n(t\),)24 b(i.e.)i(there)f(is)g(an)g(ergo)r(dic)f(dynamical)h (system)g Fk(f)p Fm(T)3110 944 y Fl(y)3149 932 y Fk(g)g Fr(on)g(the)456 1031 y(underlying)31 b(probabilit)n(y)f(space)h(\()p Fm(y)i Fk(2)d Fg(R)38 b Fr(for)31 b(the)h(P)n(oisson)d(mo)r(del,)k Fm(y)g Fk(2)d Fg(Z)25 b Fr(for)31 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(ergo)r(dic)g(op)r(erators)g(are)h(due)h(to)456 2472 y(Kotani,)j(e.g.)d([21,)h(22)o(].)76 b(T)-7 b(o)41 b(co)n(v)n(er)e(all) i(our)f(mo)r(dels,)k(w)n(e)c(also)g(need)h(the)g(extensions)g(of)456 2572 y(Kotani)27 b(theory)h(to)h Fg(Z)o Fr(-er)o(go)q(dic)23 b(op)r(erators)k(b)n(y)i(Kirsc)n(h)e([17)o(])i(and)f(to)h(op)r(erators) e(of)h(t)n(yp)r(e)h Fm(H)3369 2541 y Fl(W)456 2671 y Fr(b)n(y)e(Minami)h([24)o(,)f(25)o(].)555 2771 y(Assuming)19 b(in)g(eac)n(h)f(case)h(that)g(the)g(single)g(site)g(p)r(oten)n(tial)g (is)f(compactly)h(supp)r(orted)g(mak)n(es)456 2870 y(all)26 b(our)f(mo)r(dels)h(non-deterministic)g(in)h(Kotani's)e(sense.)36 b(By)26 b(one)g(of)g(the)g(results)g(of)g(Kotani)456 2970 y(theory)e(this)i(implies)f(that)h(the)f(Ly)n(apuno)n(v)f(exp)r (onen)n(t)h Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))25 b(is)h(strictly)f(p)r (ositiv)n(e)f(for)h(almost)456 3070 y(ev)n(ery)h(energy)g Fm(E)5 b Fr(.)37 b(Recall)28 b(that)f(the)h(Ly)n(apuno)n(v)e(exp)r (onen)n(t)i(is)f(de\014ned)h(b)n(y)1267 3239 y Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))23 b(=)76 b(lim)1556 3289 y Fl(x)p Fi(!\0061)1828 3183 y Fr(1)p 1802 3220 94 4 v 1802 3296 a Fk(j)p Fm(x)p Fk(j)1905 3239 y Fg(E)9 b Fr(\(log)20 b Fk(k)p Fm(T)12 b Fr(\()p Fm(x;)i(E)5 b(;)14 b(!)s Fr(\))p Fk(k)p Fr(\))p Fm(;)-2164 b Fr(\(3.3\))456 3422 y(where)34 b Fm(T)12 b Fr(\()p Fm(x;)i(E)5 b(;)14 b(!)s Fr(\))35 b(is)g(the)g(transfer)g(matrix)f(from)h(0)g(to)g Fm(x)g Fr(of)g Fm(H)7 b Fr(\()p Fm(!)s Fr(\))35 b(at)g Fm(E)5 b Fr(,)38 b(and)d Fg(E)49 b Fr(is)35 b(the)456 3522 y(exp)r(ectation)27 b(with)h(resp)r(ect)f(to)h Fm(!)s Fr(.)555 3621 y(By)33 b(another)f(general)g(result)g(this)i(is)f(equiv)-5 b(alen)n(t)32 b(to)h(the)h(fact)f(that)g(\006)2855 3633 y Fl(ac)2957 3621 y Fr(=)f Fk(;)p Fr(,)i(i.e.)f(that)456 3721 y Fm(H)7 b Fr(\()p Fm(!)s Fr(\))27 b(almost)g(surely)g(has)g(no)g(a.c.)h(sp)r (ectrum.)555 3821 y(In)i(order)e(to)h(get)g(results)g(whic)n(h)h(go)e (b)r(ey)n(ond)h(these)h(general)e(facts,)i(it)g(seems)f(to)g(b)r(e)h (nec-)456 3920 y(essary)25 b(to)i(use)g(more)g(sp)r(eci\014c)g (assumptions.)36 b(In)27 b(particular,)f(all)h(the)h(w)n(ork)e(on)h(lo) r(calization)456 4020 y(prop)r(erties)f(has)h(b)r(een)h(done)g(for)f (sp)r(eci\014c)g(mo)r(dels.)555 4119 y(W)-7 b(e)22 b(will)g(follo)n(w)g (the)g(con)n(v)n(en)n(tion)e(that)i Fs(lo)l(c)l(alization)i Fr(of)e(a)f(random)g(op)r(erator)f Fm(H)7 b Fr(\()p Fm(!)s Fr(\))22 b(means)456 4219 y(almost)d(surely)h(pure)f(p)r(oin)n(t)i(sp)r (ectrum,)h(i.e.)e(that)h(\006)2082 4231 y Fl(ac)2174 4219 y Fr(=)i(\006)2322 4231 y Fl(sc)2410 4219 y Fr(=)g Fk(;)p Fr(.)34 b Fs(Exp)l(onential)23 b(lo)l(c)l(alization)456 4319 y Fr(means)e(in)h(addition)f(that)h(all)f(eigenfunctions)h(of)f Fm(H)7 b Fr(\()p Fm(!)s Fr(\))22 b(deca)n(y)e(exp)r(onen)n(tially)-7 b(,)23 b(again)d(almost)456 4418 y(surely)26 b(in)i Fm(!)s Fr(.)555 4518 y(In)g(Section)g(3)f(w)n(e)g(will)h(discuss)f(the)h(v)-5 b(arious)26 b(metho)r(ds)i(whic)n(h)g(en)n(ter)f(the)h(pro)r(of)f(of) 456 4668 y Fu(Theorem)j(1.)41 b Fs(A)n(l)t(l)26 b(the)f(r)l(andom)i (Schr\177)-42 b(odinger)27 b(op)l(er)l(ators)f(de\014ne)l(d)g(in)g(Se)l (ction)g(2\(i\))g(to)f(2\(vi\))456 4767 y(ar)l(e)30 b(exp)l(onential)t (ly)h(lo)l(c)l(alize)l(d.)555 4917 y Fr(Under)h(suitable)f(assumptions) f(this)i(also)e(holds)h(for)g(the)g(op)r(erators)e(of)j(t)n(yp)r(e)f Fm(H)3169 4887 y Fl(W)3276 4917 y Fr(from)456 5016 y(Section)c (2\(vii\),)h(see)f(the)h(remarks)e(in)i(Sections)f(3.5)g(and)g(3.6.)555 5116 y(More)19 b(recen)n(tly)-7 b(,)20 b(and)f(closer)f(to)h(what)g(is) g(relev)-5 b(an)n(t)19 b(in)g(ph)n(ysics,)h(atten)n(tion)f(has)g(b)r (een)h(shifted)456 5216 y(to)32 b(pro)n(ving)e Fs(dynamic)l(al)36 b(lo)l(c)l(alization)e Fr(of)e(random)f(op)r(erators.)48 b(W)-7 b(e)33 b(will)f(use)g(the)h(follo)n(wing)p eop %%Page: 7 7 7 6 bop 1469 226 a Fn(LOCALIZA)-5 b(TION)30 b(STRA)-5 b(TEGIES)979 b(7)456 425 y Fr(de\014nition,)26 b(whic)n(h)f(is)g(one)g (of)g(the)g(stronger)f(v)n(ersions)f(to)i(b)r(e)h(found)f(in)h(the)f (literature:)35 b Fm(H)7 b Fr(\()p Fm(!)s Fr(\))456 525 y(is)27 b(said)g(to)h(satisfy)f Fs(dynamic)l(al)k(lo)l(c)l(alization)f Fr(on)d(a)h(subset)f Fm(I)j Fk(\032)23 b Fg(R)33 b Fr(if)1183 720 y Fg(E)1246 603 y Fj(\032)1315 720 y Fr(sup)1322 787 y Fl(t>)p Fn(0)1453 720 y Fk(kj)p Fm(X)7 b Fk(j)1617 686 y Fl(p)1655 720 y Fm(e)1694 686 y Fi(\000)p Fl(itH)t Fn(\()p Fl(!)r Fn(\))1953 720 y Fm(P)2006 732 y Fl(I)2044 720 y Fr(\()p Fm(H)g Fr(\()p Fm(!)s Fr(\)\))p Fm(\037)2355 732 y Fl(K)2419 720 y Fk(k)2461 603 y Fj(\033)2546 720 y Fm(<)23 b Fk(1)-2261 b Fr(\(3.4\))456 916 y(for)28 b(ev)n(ery)h(compact)f(set)i Fm(K)35 b Fr(in)29 b Fg(R)36 b Fr(and)29 b(ev)n(ery)f Fm(p)e(>)g Fr(0.)42 b(Here)29 b Fm(P)2466 928 y Fl(I)2534 916 y Fr(is)g(the)h(sp)r(ectral)e(pro)5 b(jection)456 1016 y(on)n(to)35 b Fm(I)44 b Fr(and)37 b Fm(X)43 b Fr(the)37 b(op)r(erator)e(of)h(m)n(ultiplication)h(b)n(y)f (the)h(v)-5 b(ariable)36 b(in)h Fm(L)2915 986 y Fn(2)2952 1016 y Fr(\()p Fg(R)p Fr(\).)70 b(W)-7 b(e)37 b(will)456 1116 y(discuss)27 b(dynamical)g(lo)r(calization)f(for)h(the)h(mo)r (dels)g(from)f(Section)h(2)f(in)h(Section)f(3.6.)555 1215 y(All)40 b(self-adjoin)n(t)g(Hamiltonians)f Fm(H)46 b Fr(considered)39 b(in)h(this)g(pap)r(er)f(are)g(Sturm-Liouville)456 1315 y(op)r(erators)e(in)i(the)g(limit)h(p)r(oin)n(t)f(case)f(at)h(+)p Fk(1)f Fr(and)h Fk(\0001)p Fr(.)71 b(Th)n(us)38 b(their)h(sp)r(ectral)f (t)n(yp)r(e)i(is)456 1415 y(c)n(haracterized)31 b(b)n(y)i(their)h(W)-7 b(eyl-Titc)n(hmarsh)33 b(sp)r(ectral)g(measure)f Fm(\032)2645 1427 y Fl(H)2708 1415 y Fr(,)k(i.e.)d(the)h(trace)f(mea-)456 1514 y(sure)k(corresp)r(onding)f(to)i(the)g(standard)f(2)25 b Fk(\002)g Fr(2-matrix-v)-5 b(alued)36 b(sp)r(ectral)h(measure)g(for)h Fm(H)456 1614 y Fr([8)o(].)f(The)26 b(absolutely)g(con)n(tin)n(uous,)g (singular)f(con)n(tin)n(uous)h(and)g(p)r(oin)n(t)h(sp)r(ectra)f(of)g Fm(H)34 b Fr(are)25 b(the)456 1713 y(corresp)r(onding)g(parts)i(of)h (the)g(Leb)r(esgue)f(decomp)r(osition)g(of)g Fm(\032)2483 1725 y Fl(H)2546 1713 y Fr(.)456 1894 y(3.2.)46 b Fu(Sp)s(ectral)34 b(Av)m(eraging)g(in)f(Monotonic)f(Mo)s(dels.)39 b Fr(T)-7 b(o)29 b(see)f(ho)n(w)h(a)f(pro)r(of)g(of)h(lo)r(cal-)456 1993 y(ization)i(for)h(smo)r(oth,)h(monotonic)f(mo)r(dels)g(can)f(b)r (e)i(completed,)g(let)g(us)f(\014rst)g(fo)r(cus)g(on)g(the)456 2093 y(Anderson)27 b(mo)r(del)h Fm(H)1144 2105 y Fl(A)1226 2093 y Fr(under)g(the)h(assumptions)e(of)h(Section)g(2\(i\).)38 b(W)-7 b(e)28 b(essen)n(tially)f(outline)456 2193 y(the)h(argumen)n(t)e (in)i([23)o(],)g(where)f(more)g(details)g(can)g(b)r(e)h(found.)555 2292 y(The)35 b(quan)n(tit)n(y)g Fk(j)p Fm(x)p Fk(j)1166 2262 y Fi(\000)p Fn(1)1269 2292 y Fr(log)14 b Fk(k)p Fm(T)e Fr(\()p Fm(x;)i(E)5 b(;)14 b(!)s Fr(\))p Fk(k)34 b Fr(con)n(v)n(erges)e(to)j Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))35 b(not)g(only)g(in)g(exp)r(ectation)456 2392 y(but)29 b(also)f(almost)h(surely)f(as)h Fm(x)d Fk(!)g(\0061)p Fr(.)41 b(F)-7 b(ubini's)30 b(and)f(Osceledec's)f(Theorems)g(yield)h (that)456 2492 y(for)h(almost)g(ev)n(ery)f Fm(!)34 b Fr(there)c(exists)h Fm(S)5 b Fr(\()p Fm(!)s Fr(\))28 b Fk(\032)g Fg(R)37 b Fr(with)31 b Fk(j)p Fg(R)c Fk(n)20 b Fm(S)5 b Fr(\()p Fm(!)s Fr(\))p Fk(j)28 b Fr(=)g(0)i(suc)n(h)h(that)g (for)f(ev)n(ery)456 2591 y Fm(E)f Fk(2)24 b Fm(S)5 b Fr(\()p Fm(!)s Fr(\))28 b(solutions)f Fm(u)1223 2603 y Fn(+)1306 2591 y Fr(and)h Fm(u)1516 2603 y Fi(\000)1600 2591 y Fr(of)g Fm(H)7 b Fr(\()p Fm(!)s Fr(\))p Fm(u)24 b Fr(=)f Fm(E)5 b(u)28 b Fr(exist)g(whic)n(h)g(deca)n(y)f(exp)r(onen)n (tially)h(at)456 2691 y(+)p Fk(1)f Fr(and)g Fk(\0001)p Fr(,)h(resp)r(ectiv)n(ely)-7 b(.)36 b(All)28 b(other)f(solutions)g (increase)f(exp)r(onen)n(tially)h(at)g Fk(\0061)p Fr(.)555 2790 y(Let)33 b Fm(\032)752 2802 y Fl(!)831 2790 y Fr(=)e Fm(\032)970 2805 y Fl(H)1024 2813 y Ff(A)1071 2805 y Fn(\()p Fl(!)r Fn(\))1204 2790 y Fr(b)r(e)h(the)h(W)-7 b(eyl-Titc)n(hmarsh)32 b(sp)r(ectral)g(measures)f(of)h Fm(H)2976 2802 y Fl(A)3030 2790 y Fr(.)52 b(It)32 b(follo)n(ws)456 2890 y(from)c(either)g(Shnol's)h(Theorem)e(on)h(the)h(p)r(olynomial)f (b)r(oundedness)h(of)f(generalized)f(eigen-)456 2990 y(functions)i(or)f(from)h(Sub)r(ordinacy)g(Theory)f(\(e.g.)h([5,)h (Sect.2]\))f(that)h Fm(\032)2718 3002 y Fl(!)2795 2990 y Fr(restricted)e(to)h Fm(S)5 b Fr(\()p Fm(!)s Fr(\))456 3089 y(is)27 b(a)g(pure)h(p)r(oin)n(t)f(measure.)555 3189 y(The)37 b(set)g Fm(S)5 b Fr(\()p Fm(!)s Fr(\))37 b(is)g(indep)r(enden)n(t)h(of)f(the)g(v)-5 b(alue)37 b(of)g Fm(q)2280 3201 y Fn(0)2318 3189 y Fr(\()p Fm(!)s Fr(\),)i(since)e(a)g(c)n(hange)f(of)g Fm(q)3215 3201 y Fn(0)3290 3189 y Fr(only)456 3289 y(yields)31 b(a)h(lo)r(cal)f(c)n (hange)f(in)i(the)h(di\013eren)n(tial)e(equation,)i(not)e(e\013ecting)h (the)g(asymptotics)g(of)456 3388 y(solutions)20 b(at)h Fk(\0061)p Fr(.)34 b(This)21 b(is)g(crucial)g(since)g(it)g(allo)n(ws)f (to)h(use)g(the)g(follo)n(wing)f(result)h(on)g Fs(sp)l(e)l(ctr)l(al)456 3488 y(aver)l(aging)p Fr(,)28 b(whic)n(h)d(is)h(essen)n(tially)f(Lemma) h(2.2)f(of)h([23)o(])g(\(for)f(a)h(detailed)g(pro)r(of)f(in)h(the)h (stated)456 3587 y(form)g(see)g(also)g([30)o(,)g(Sect.5.3]\).)37 b(It)28 b(can)f(b)r(e)h(seen)g(as)e(a)i(v)n(ersion)e(of)h(\\Kotani's)f (tric)n(k".)456 3746 y Fu(Theorem)k(2.)41 b Fs(L)l(et)27 b Fm(f)37 b Fs(b)l(e)28 b(as)g(in)g(Se)l(ction)g(2\(i\),)h Fm(V)47 b Fs(a)28 b(r)l(e)l(al-value)l(d)h(p)l(otential)f(such)g(that)g Fm(H)3313 3758 y Fl(\025)3380 3746 y Fr(=)456 3845 y Fk(\000)p Fm(d)564 3815 y Fn(2)601 3845 y Fm(=dx)733 3815 y Fn(2)790 3845 y Fr(+)19 b Fm(V)38 b Fr(+)19 b Fm(\025f)40 b Fs(is)32 b(in)f(limit)h(p)l(oint)f(c)l(ase)h(at)f Fk(\0061)p Fs(,)h Fm(\032)2306 3857 y Fl(\025)2380 3845 y Fs(the)g(Weyl-Titchmarsh)h(sp)l(e)l(ctr)l(al)456 3945 y(me)l(asur)l(e)g(of)h Fm(H)951 3957 y Fl(\025)994 3945 y Fs(,)h(and)e Fk(\0001)c Fm(<)g(\025)1537 3957 y Fn(1)1604 3945 y Fm(<)g(\025)1746 3957 y Fn(2)1813 3945 y Fm(<)g Fk(1)p Fs(.)49 b(Then)34 b(the)g(me)l(asur)l(e)f Fm(\026)g Fs(de\014ne)l(d)g(for)h(Bor)l(el)456 4045 y(sets)29 b Fm(B)34 b Fs(by)1556 4252 y Fm(\026)p Fr(\()p Fm(B)t Fr(\))23 b(=)1848 4139 y Fj(Z)1931 4160 y Fl(\025)1970 4168 y Fe(2)1894 4328 y Fl(\025)1933 4336 y Fe(1)2021 4252 y Fm(\032)2064 4264 y Fl(\025)2107 4252 y Fr(\()p Fm(B)t Fr(\))14 b Fm(d\025)-1887 b Fr(\(3.5\))456 4452 y Fs(is)30 b(absolutely)g(c)l(ontinuous)f(on)h Fg(R)p Fs(.)555 4611 y Fr(The)36 b(pro)r(of)g(of)g(this)h(result)f(crucially)f (uses)h Fm(f)46 b Fk(\025)37 b Fr(0,)h(i.e.)e(monotonicit)n(y)f(of)i (the)f(mo)r(del.)456 4710 y(Smo)r(othness)22 b(comes)h(in)g(as)f(follo) n(ws:)34 b(W)-7 b(e)23 b(apply)g(Theorem)f(2)h(to)g(the)g(case)g(where) f Fm(\025)i Fr(=)e Fm(q)3250 4722 y Fn(0)3311 4710 y Fr(and)456 4810 y Fm(V)c Fr(\()p Fm(x)p Fr(\))25 b(=)745 4748 y Fj(P)833 4835 y Fl(n)p Fi(6)p Fn(=0)976 4810 y Fm(q)1013 4822 y Fl(n)1058 4810 y Fm(f)9 b Fr(\()p Fm(x)q Fk(\000)q Fm(n)p Fr(\).)34 b(Since)19 b Fk(j)p Fg(R)q Fk(n)q Fm(S)5 b Fr(\()p Fm(!)s Fr(\))p Fk(j)28 b Fr(=)23 b(0,)d(it)f(follo)n(ws)f(for)g(the)h Fm(q)2748 4822 y Fn(0)2786 4810 y Fr(-a)n(v)n(eraged)d(measure)456 4917 y Fm(\026)k Fr(that)g Fm(\026)p Fr(\()p Fg(R)5 b Fk(n)t Fm(S)g Fr(\()p Fm(!)s Fr(\)\))28 b(=)22 b(0.)34 b(This)20 b(implies)h(that)f Fm(\032)1978 4929 y Fl(!)2026 4917 y Fr(\()p Fg(R)5 b Fk(n)t Fm(S)g Fr(\()p Fm(!)s Fr(\)\))28 b(=)22 b(0)e(for)g(Leb)r(esgue-a.e.)e(v)-5 b(alue)21 b(of)456 5016 y Fm(q)493 5028 y Fn(0)555 5016 y Fr(\(let)26 b([)p Fm(\025)776 5028 y Fn(1)814 5016 y Fm(;)14 b(\025)899 5028 y Fn(2)936 5016 y Fr(])26 b(exhaust)e Fg(R)p Fr(\))q(.)42 b(Using)25 b(that)g Fm(q)1886 5028 y Fn(0)1924 5016 y Fr(\()p Fm(!)s Fr(\))g(has)g(absolutely)f(con)n(tin)n(uous)g (distribution)456 5116 y(w)n(e)k(get)h(that)h Fm(\032)944 5128 y Fl(!)992 5116 y Fr(\()p Fg(R)25 b Fk(n)19 b Fm(S)5 b Fr(\()p Fm(!)s Fr(\)\))26 b(=)f(0)k(for)f(a.e.)h Fm(q)1877 5128 y Fn(0)1914 5116 y Fr(\()p Fm(!)s Fr(\))h(and)f(th)n(us,)g(using)g (indep)r(endence,)h(for)f(a.e.)456 5216 y Fm(!)s Fr(.)p eop %%Page: 8 8 8 7 bop 456 226 a Fn(8)1303 b(G.)29 b(STOLZ)555 425 y Fr(Th)n(us,)c(kno)n(wing)f(that)h(the)g(restriction)f(of)h Fm(\032)1958 437 y Fl(!)2031 425 y Fr(to)g Fm(S)5 b Fr(\()p Fm(!)s Fr(\))24 b(is)h(pure)g(p)r(oin)n(t,)g(w)n(e)g(conclude)f(that) 456 525 y Fm(\032)499 537 y Fl(!)572 525 y Fr(almost)g(surely)g(is)h(a) f(pure)h(p)r(oin)n(t)g(measure,)g(whic)n(h)f(completes)h(the)g(pro)r (of)g(of)g(lo)r(calization)456 624 y(for)20 b Fm(H)645 636 y Fl(A)700 624 y Fr(.)34 b(In)22 b(fact,)h(the)f(pro)r(of)e(sho)n (ws)h(that)g Fm(\032)1833 636 y Fl(!)1903 624 y Fr(almost)g(surely)f (is)h(supp)r(orted)h(on)f(the)h(set)f(where)456 724 y(the)40 b(generalized)e(eigenfunctions)i(deca)n(y)f(exp)r(onen)n(tially)g(at)h (the)g(rate)f(of)h(the)g(Ly)n(apuno)n(v)456 824 y(exp)r(onen)n(t,)27 b(so)g(that)h(actually)f(exp)r(onen)n(tial)g(lo)r(calization)f(is)i (pro)n(v)n(en.)555 923 y(Muc)n(h)k(of)g(the)h(ab)r(o)n(v)n(e)e (strategy)f(also)h(w)n(orks)g(for)g(the)i(other)e(smo)r(oth)h(mo)r (dels)g(from)g(Sec-)456 1023 y(tion)g(2.)53 b(The)33 b(only)f(di\016cult)n(y)h(is)g(that)g(it)g(is)g(not)g(immediate)g(ho)n (w)f(to)h(\014nd)g(a)f(replacemen)n(t)456 1123 y(for)c(Theorem)h(2)f (in)i(non-monotonic)e(mo)r(dels.)41 b(One)29 b(exception)g(is)g(the)h (P)n(oisson)d(mo)r(del)i Fm(H)3389 1135 y Fl(P)456 1222 y Fr(at)e(p)r(ositiv)n(e)g(energies:)36 b(Indep)r(endence)28 b(of)f(the)h(random)f(distances)g Fm(d)2673 1234 y Fl(n)2745 1222 y Fr(allo)n(ws)g(to)g(adopt)g(the)456 1322 y(ab)r(o)n(v)n(e)k (strategy)g(for)h Fm(H)1223 1334 y Fl(A)1310 1322 y Fr(and)g(to)h (\014x)f(all)g Fm(d)1868 1334 y Fl(n)1946 1322 y Fr(for)g Fm(n)g Fk(6)p Fr(=)f(0)h(and)g(a)n(v)n(erage)e(sp)r(ectral)i(measures) 456 1421 y(o)n(v)n(er)25 b Fm(d)676 1433 y Fn(0)714 1421 y Fr(.)37 b(Essen)n(tially)-7 b(,)26 b(this)i(leads)f(to)h(the)g(follo) n(wing)e(one-parameter)f(mo)r(del)j(\(e.g.)g([31)o(]\):)555 1521 y(Let)d Fm(W)779 1533 y Fn(1)816 1521 y Fm(;)14 b(W)931 1533 y Fn(2)992 1521 y Fk(2)23 b Fm(L)1127 1491 y Fi(1)1127 1562 y Fr(lo)r(c)1235 1521 y(\()p Fg(R)p Fr(\))31 b(b)r(e)25 b(real-v)-5 b(alued,)23 b Fk(j)p Fm(W)2037 1533 y Fl(i)2066 1521 y Fr(\()p Fm(x)p Fr(\))p Fk(j)h Fr(=)e Fm(O)r Fr(\()p Fm(x)2455 1491 y Fn(2)2494 1521 y Fr(\))j(as)e Fk(j)p Fm(x)p Fk(j)h(!)f(1)p Fr(,)i Fm(i)e Fr(=)g(1)p Fm(;)14 b Fr(2,)24 b(and)456 1632 y(supp)14 b Fm(W)719 1644 y Fn(1)779 1632 y Fk(\032)23 b Fr([0)p Fm(;)14 b Fk(1)p Fr(\),)23 b(supp)14 b Fm(W)1393 1644 y Fn(2)1453 1632 y Fk(\032)23 b Fr(\()p Fk(\0001)p Fm(;)14 b Fr(0].)34 b(F)-7 b(or)22 b Fm(a)h(>)f Fr(0)f(let)i Fm(V)2404 1644 y Fl(a)2444 1632 y Fr(\()p Fm(x)p Fr(\))h(=)f Fm(W)2745 1644 y Fn(1)2783 1632 y Fr(\()p Fm(x)7 b Fk(\000)g Fm(a)p Fr(\))g(+)g Fm(W)3174 1644 y Fn(2)3210 1632 y Fr(\()p Fm(x)g Fr(+)g Fm(a)p Fr(\))456 1732 y(and)27 b Fm(\032)660 1744 y Fl(a)728 1732 y Fr(the)h(W)-7 b(eyl-Titc)n(hmarsh) 26 b(sp)r(ectral)h(measure)g(of)h Fm(H)2326 1744 y Fl(a)2389 1732 y Fr(=)22 b Fk(\000)p Fm(d)2584 1702 y Fn(2)2621 1732 y Fm(=dx)2753 1702 y Fn(2)2809 1732 y Fr(+)c Fm(V)2940 1744 y Fl(a)2981 1732 y Fr(\()p Fm(x)p Fr(\).)456 1882 y Fu(Theorem)30 b(3.)41 b Fs(F)-6 b(or)30 b(\014xe)l(d)f Fm(a)1367 1894 y Fn(2)1427 1882 y Fm(>)23 b(a)1559 1894 y Fn(1)1619 1882 y Fm(>)g Fr(0)29 b Fs(and)h(arbitr)l(ary)h(Bor)l(el)g (sets)e Fm(B)e Fk(\032)c Fg(R)36 b Fs(de\014ne)1550 2059 y Fm(\026)p Fr(\()p Fm(B)t Fr(\))23 b(=)1842 1946 y Fj(Z)1925 1967 y Fl(a)1961 1975 y Fe(2)1888 2135 y Fl(a)1924 2143 y Fe(1)2012 2059 y Fm(\032)2055 2071 y Fl(a)2095 2059 y Fr(\()p Fm(B)t Fr(\))14 b Fm(da:)456 2249 y Fs(Then)30 b(the)g(me)l(asur)l(e)f Fm(\026)h Fs(is)g(absolutely)h(c)l(ontinuous)e (on)h Fr(\(0)p Fm(;)14 b Fk(1)p Fr(\))p Fs(.)555 2399 y Fr(This)27 b(result)g(allo)n(ws)e(to)i(deduce)g(exp)r(onen)n(tial)f (lo)r(calization)g(of)h Fm(H)2645 2411 y Fl(P)2727 2399 y Fr(at)g(p)r(ositiv)n(e)g(energies.)456 2498 y(The)i(pro)r(of)g(of)g (Theorem)g(3)g(in)h([31)o(])g(is)f(based)g(on)g(a)g(monotonicit)n(y)g (prop)r(ert)n(y)f(of)i(the)f(family)456 2598 y Fm(H)525 2610 y Fl(a)565 2598 y Fr(.)46 b(In)31 b([5,)g(Sect.1])g(it)g(is)g (explained)f(ho)n(w)g(this)h(can)g(b)r(e)g(understo)r(o)r(d)f(in)h (terms)g(of)g(gro)n(wing)456 2698 y(phase)k(space)f(v)n(olume)h(at)g(p) r(ositiv)n(e)g(energy)g(as)f Fm(a)i Fr(increases.)59 b(T)-7 b(ec)n(hnically)g(,)36 b(monotonicit)n(y)456 2797 y(en)n(ters)25 b(the)i(pro)r(of)e(of)h(Theorem)g(3)f(in)i(the)f(form)g (of)g(monotonic)g(dep)r(endence)g(on)g Fm(a)g Fr(of)g(Pr)r(\177)-44 b(ufer)456 2897 y(phases)26 b(for)h(solutions)g(of)h Fm(H)1357 2909 y Fl(a)1397 2897 y Fm(u)23 b Fr(=)f Fm(E)5 b(u)p Fr(,)28 b Fm(E)g(>)22 b Fr(0,)28 b(see)f([31)o(].)555 2997 y(The)c(measure)f Fm(\026)h Fr(in)g(Theorem)f(3)h(is)g(not)g (necessarily)e(absolutely)h(con)n(tin)n(uous)g(at)h(negativ)n(e)456 3096 y(energies.)56 b(In)35 b(fact,)h(it)f(is)g(easy)e(to)i(construct)f (an)g(example)g(suc)n(h)g(that)h Fm(H)2884 3108 y Fl(a)2959 3096 y Fr(has)f(the)h(same)456 3196 y(eigen)n(v)-5 b(alue)20 b Fm(E)907 3208 y Fn(0)967 3196 y Fm(<)j Fr(0)d(for)h(all)f Fm(a)p Fr(,)j(see)d([5,)i(Sect.1].)35 b(Th)n(us)20 b Fm(\026)h Fr(also)f(gets)g(a)h(pure)g(p)r(oin)n(t)g(comp)r(onen)n(t.) 456 3295 y(Also,)29 b(Theorem)g(3)g(do)r(es)g(not)h(apply)f(to)g(the)h (situation)f(of)h(the)f(displacemen)n(t)h(mo)r(del,)g(since)456 3395 y(for)d Fm(H)652 3407 y Fl(D)740 3395 y Fr(the)h(displacemen)n (ts,)f(not)g(the)h(distances,)f(of)h(the)g(single)f(sites)g(are)g (indep)r(enden)n(t.)456 3545 y(3.3.)46 b Fu(Tw)m(o-P)m(arameter)e(Av)m (eraging)f(and)g(In)m(v)m(erse)h(Sp)s(ectral)f(Theory)-8 b(.)41 b Fr(Av)n(eraging)456 3645 y(the)30 b(W)-7 b(eyl-Titc)n(hmarsh) 29 b(sp)r(ectral)h(measures)f(of)h(a)g(random)f(op)r(erator)f(o)n(v)n (er)g(one)i(of)g(its)h(ran-)456 3745 y(dom)23 b(parameters)e(do)r(es)i (not)h(necessarily)d(lead)i(to)g(an)g(a.c.)g(measure)f(for)h(the)h (non-monotonic)456 3844 y(mo)r(dels)f(from)h(Section)g(2.)35 b(It)24 b(turns)g(out)g(that)g(sim)n(ultaneously)f(a)n(v)n(eraging)e(o) n(v)n(er)h(t)n(w)n(o)h(param-)456 3944 y(eters)31 b(can)g(b)r(e)h(used) f(to)g(o)n(v)n(ercome)f(this)h(problem)g(in)h(man)n(y)f(situations.)48 b(W)-7 b(e)32 b(will)g(describ)r(e)456 4044 y(this)25 b(here)g(for)f(the)i(displacemen)n(t)f(mo)r(del)g Fm(H)1865 4056 y Fl(D)1926 4044 y Fr(,)g(but)h(similar)f(ideas)f(also)g(yield)h (a)g(pro)r(of)g(of)g(ex-)456 4143 y(p)r(onen)n(tial)e(lo)r(calization)f (for)h(the)h(P)n(oisson)d(mo)r(del)j Fm(H)2121 4155 y Fl(P)2200 4143 y Fr(at)f(all,)h(including)f(negativ)n(e,)h(energies.) 555 4243 y(The)e(follo)n(wing)g(t)n(w)n(o-parameter)d(family)j(of)g(op) r(erators)f(arises)f(when)j(\014xing)e(all)h(but)h(t)n(w)n(o)f(of)456 4342 y(the)28 b Fm(d)642 4354 y Fl(n)716 4342 y Fr(in)g Fm(H)882 4354 y Fl(D)942 4342 y Fr(:)38 b(Let)28 b(0)c Fm(<)f(L)g(<)h Fr(1,)k Fm(q)e Fk(6)p Fr(=)e(0)j(b)r(ounded)i(and)f (real-v)-5 b(alued)27 b(with)h(supp)14 b Fm(q)27 b Fk(\032)c Fr([0)p Fm(;)14 b(L)p Fr(],)456 4442 y(and)27 b Fm(a;)14 b(b)23 b Fk(2)g Fr(\(0)p Fm(;)14 b Fr(1)k Fk(\000)g Fm(L)p Fr(\).)36 b(Let)1014 4727 y Fm(V)1062 4739 y Fl(a;b)1151 4727 y Fr(\()p Fm(x)p Fr(\))24 b(=)1374 4507 y Fj(8)1374 4581 y(>)1374 4606 y(>)1374 4631 y(<)1374 4781 y(>)1374 4805 y(>)1374 4830 y(:)1489 4576 y Fm(q)s Fr(\()p Fm(x)c Fk(\000)e Fm(a)p Fr(\))p Fm(;)218 b(a)23 b Fk(\024)f Fm(x)i Fk(\024)e Fm(a)d Fr(+)f Fm(L;)1489 4676 y(q)s Fr(\()p Fm(x)i Fk(\000)e Fr(1)g Fk(\000)g Fm(b)p Fr(\))p Fm(;)83 b Fr(1)18 b(+)g Fm(b)k Fk(\024)h Fm(x)g Fk(\024)g Fr(1)18 b(+)g Fm(b)g Fr(+)g Fm(L;)1489 4776 y Fr(0)p Fm(;)474 b Fr(elsewhere)26 b(in)i([0)p Fm(;)14 b Fr(2])o Fm(;)1489 4875 y(V)19 b Fr(\()p Fm(x)p Fr(\))p Fm(;)338 b(x)23 b Fk(62)h Fr([0)p Fm(;)14 b Fr(2])p Fm(;)456 5016 y Fr(where)37 b Fm(V)58 b Fr(is)38 b(a)g(\014xed)g(p)r(oten)n(tial)g (suc)n(h)g(that)h Fk(\000)p Fm(d)2055 4986 y Fn(2)2092 5016 y Fm(=dx)2224 4986 y Fn(2)2287 5016 y Fr(+)25 b Fm(V)57 b Fr(is)38 b(limit)h(p)r(oin)n(t)g(at)f(+)p Fk(1)g Fr(and)456 5116 y Fk(\0001)p Fr(.)49 b(By)32 b Fm(\032)854 5128 y Fl(a;b)975 5116 y Fr(w)n(e)g(denote)g(the)g(W)-7 b(eyl-Titc)n(hmarsh)31 b(sp)r(ectral)g(measures)g(of)h(the)g(op)r (erators)456 5216 y Fk(\000)p Fm(d)564 5185 y Fn(2)601 5216 y Fm(=dx)733 5185 y Fn(2)788 5216 y Fr(+)19 b Fm(V)920 5228 y Fl(a;b)1009 5216 y Fr(.)p eop %%Page: 9 9 9 8 bop 1469 226 a Fn(LOCALIZA)-5 b(TION)30 b(STRA)-5 b(TEGIES)979 b(9)456 425 y Fu(Theorem)30 b(4.)41 b Fs(Ther)l(e)f(is)f (a)g(discr)l(ete)h(subset)e Fm(M)48 b Fk(\032)39 b Fg(R)45 b Fs(such)39 b(that)g(for)g Fm(E)2884 437 y Fn(0)2961 425 y Fk(2)h Fg(R)31 b Fk(n)25 b Fm(M)47 b Fs(the)456 525 y(fol)t(lowing)30 b(holds:)39 b(F)-6 b(or)28 b(every)g Fm(a)1465 537 y Fn(0)1525 525 y Fk(2)c Fr(\(0)p Fm(;)14 b Fr(1)g Fk(\000)g Fm(L)p Fr(\))25 b Fs(and)k(for)f(every)g Fm(b)2507 537 y Fn(0)2567 525 y Fk(2)c Fr(\(0)p Fm(;)14 b Fr(1)g Fk(\000)g Fm(L)p Fr(\))g Fk(n)g Fm(M)9 b Fr(\()p Fm(a)3217 537 y Fn(0)3251 525 y Fm(;)14 b(E)3349 537 y Fn(0)3387 525 y Fr(\))p Fs(,)456 624 y(wher)l(e)26 b Fm(M)9 b Fr(\()p Fm(a)852 636 y Fn(0)889 624 y Fm(;)14 b(E)987 636 y Fn(0)1025 624 y Fr(\))26 b Fs(is)h(\014nite,)f(ther)l(e)g (exist)g Fm(")d(>)g Fr(0)i Fs(and)i Fm(\016)f(>)c Fr(0)k Fs(such)g(that)g(the)g(me)l(asur)l(e)g(de\014ne)l(d)456 724 y(by)1329 938 y Fm(\032)p Fr(\()p Fm(B)t Fr(\))e(=)1615 825 y Fj(Z)1698 845 y Fl(a)1734 853 y Fe(0)1766 845 y Fn(+)p Fl(\016)1661 1013 y(a)1697 1021 y Fe(0)1729 1013 y Fi(\000)p Fl(\016)1867 825 y Fj(Z)1950 845 y Fl(b)1979 853 y Fe(0)2012 845 y Fn(+)p Fl(\016)1914 1013 y(b)1943 1021 y Fe(0)1975 1013 y Fi(\000)p Fl(\016)2113 938 y Fm(\032)2156 950 y Fl(a;b)2245 938 y Fr(\()p Fm(B)t Fr(\))14 b Fm(db)g(da)456 1149 y Fs(is)30 b(absolutely)g(c)l(ontinuous)f(on)h Fr(\()p Fm(E)1554 1161 y Fn(0)1611 1149 y Fk(\000)18 b Fm(";)c(E)1831 1161 y Fn(0)1886 1149 y Fr(+)k Fm(")p Fr(\))p Fs(.)555 1311 y Fr(This)25 b(v)n(ersion)e(of)h(t)n(w)n (o-parameter)e(sp)r(ectral)i(a)n(v)n(eraging)d(still)k(allo)n(ws)e(to)i (complete)f(a)g(pro)r(of)456 1410 y(of)33 b(exp)r(onen)n(tial)h(lo)r (calization)e(for)i Fm(H)1657 1422 y Fl(D)1751 1410 y Fr(along)e(the)i(lines)g(of)g(the)g(general)e(argumen)n(ts)h(from)456 1510 y(Section)c(3.2,)g(see)g([5])g(for)g(details)h(and)f(a)g(pro)r(of) g(of)g(Theorem)g(4.)42 b(This)29 b(uses)g(that)h(the)g(\014nite)456 1609 y(set)k(whic)n(h)h(is)g(excluded)g(for)f(the)h(v)-5 b(alues)34 b(of)h Fm(b)1955 1621 y Fn(0)2027 1609 y Fr(has)f(zero)g (probabilit)n(y)g(since)g(the)h Fm(d)3200 1621 y Fl(n)3281 1609 y Fr(ha)n(v)n(e)456 1709 y(a.c.)g(distribution.)62 b(Also,)38 b(the)e(discrete)g(set)g(of)f(energies)g Fm(M)45 b Fr(where)35 b(a)n(v)n(eraging)e(do)r(es)i(not)456 1809 y(apply)27 b(can)h(not)g(carry)f(an)n(y)g(con)n(tin)n(uous)g(sp)r (ectrum.)39 b(In)28 b(fact,)h Fm(M)37 b Fr(almost)27 b(surely)g(carries)g(no)456 1908 y(sp)r(ectrum.)555 2008 y(The)e(crucial)f(asp)r(ect)g(of)h(the)g(w)n(ork)e(in)i([5])f(is)h(an)f (explicit)h(description)f(of)h Fm(M)33 b Fr(whic)n(h)25 b(allo)n(ws)456 2108 y(to)31 b(pro)n(v)n(e)f(its)i(discreteness.)48 b(F)-7 b(or)31 b(this)h(let)f Fm(q)k Fr(b)r(e)d(as)f(ab)r(o)n(v)n(e)f (and)i Fm(u)2597 2120 y Fn(0)2634 2108 y Fr(\()p Fk(\001)p Fm(;)14 b(\022)r(;)g(E)5 b(;)14 b(q)s Fr(\))32 b(the)g(solution)456 2207 y(of)j(the)i(initial)f(v)-5 b(alue)35 b(problem)h Fk(\000)p Fm(u)1625 2177 y Fi(0)o(0)1690 2207 y Fr(+)24 b Fm(q)s(u)36 b Fr(=)h Fm(E)5 b(u)p Fr(,)37 b Fm(u)p Fr(\(0\))g(=)f(sin)14 b Fm(\022)r Fr(,)38 b Fm(u)2737 2177 y Fi(0)2760 2207 y Fr(\(0\))f(=)f(cos)13 b Fm(\022)r Fr(.)62 b(Then)456 2307 y Fm(M)31 b Fr(=)23 b Fm(M)737 2319 y Fn(+)810 2307 y Fk([)c(f)p Fr(0)p Fk(g)e([)i Fm(M)1182 2319 y Fi(\000)1237 2307 y Fr(,)28 b(where)897 2464 y Fm(M)978 2476 y Fn(+)1116 2464 y Fr(:=)83 b Fk(f)p Fm(E)27 b(>)c Fr(0)g(:)g Fm(u)1664 2430 y Fi(0)1664 2485 y Fn(0)1701 2464 y Fr(\()p Fm(D)r(;)14 b(\022)r(;)g(E)5 b(;)14 b(q)s Fr(\))2094 2430 y Fn(2)2150 2464 y Fr(+)k Fm(E)5 b(u)2347 2476 y Fn(0)2384 2464 y Fr(\()p Fm(D)r(;)14 b(\022)r(;)g(E)5 b(;)14 b(q)s Fr(\))2777 2430 y Fn(2)2838 2464 y Fr(=)-2447 b(\(3.6\))1287 2590 y(cos)1398 2556 y Fn(2)1449 2590 y Fm(\022)21 b Fr(+)d Fm(E)h Fr(sin)1774 2555 y Fn(2)1825 2590 y Fm(\022)h Fr(for)28 b(all)f Fm(\022)e Fk(2)f Fg(R)6 b Fk(g)p Fm(;)897 2715 y(M)978 2727 y Fi(\000)1116 2715 y Fr(:=)83 b Fk(f)p Fm(E)27 b(<)c Fr(0)g(:)41 b(There)27 b(is)h(a)f(solution)g(of)h Fk(\000)p Fm(u)2549 2685 y Fi(0)o(0)2609 2715 y Fr(+)18 b Fm(q)s(u)k Fr(=)h Fm(E)5 b(u)-2548 b Fr(\(3.7\))1287 2887 y(suc)n(h)27 b(that)1655 2831 y Fm(u)1703 2801 y Fi(0)1725 2831 y Fr(\(0\))p 1655 2868 177 4 v 1667 2944 a Fm(u)p Fr(\(0\))1842 2887 y Fm(;)1888 2831 y(u)1936 2801 y Fi(0)1959 2831 y Fr(\()p Fm(D)r Fr(\))p 1888 2868 207 4 v 1900 2944 a Fm(u)p Fr(\()p Fm(D)r Fr(\))2128 2887 y Fk(2)c(f\000)2313 2812 y Fj(p)p 2395 2812 113 4 v 2395 2887 a Fk(j)p Fm(E)5 b Fk(j)q Fm(;)2545 2812 y Fj(p)p 2627 2812 V 2627 2887 a Fk(j)p Fm(E)g Fk(j)q(gg)p Fm(:)555 3092 y Fr(Discreteness)32 b(of)g Fm(M)1211 3104 y Fn(+)1298 3092 y Fr(in)g([0)p Fm(;)14 b Fk(1)p Fr(\))32 b(follo)n(ws)f(b)n(y)h(analytic)g(con)n(tin)n (uation)f(\(in)h Fm(E)5 b Fr(\))33 b(from)f(dis-)456 3192 y(creteness)39 b(of)h Fm(M)1008 3204 y Fi(\000)1103 3192 y Fr(in)h(\()p Fk(\0001)p Fm(;)14 b Fr(0].)74 b(The)40 b(latter)f(can)h(b)r(e)g(seen)g(to)g(follo)n(w)g(from)f(a)h(classical) 456 3291 y(result)i(of)h(Borg)e(in)i(in)n(v)n(erse)e(sp)r(ectral)h (theory:)67 b(Let)43 b Fm(Q)f Fr(b)r(e)h(the)g(p)r(oten)n(tial)g(on)f Fg(R)49 b Fr(found)456 3391 y(b)n(y)36 b Fm(L)p Fr(-p)r(erio)r(dic)f (extension)h(of)g Fm(q)s Fr(.)64 b(If)37 b Fm(M)1770 3403 y Fi(\000)1862 3391 y Fr(w)n(ere)e(not)i(discrete,)h(then)f(one)f (can)g(sho)n(w)f(that)456 3491 y Fm(\033)s Fr(\()p Fk(\000)p Fm(d)646 3461 y Fn(2)683 3491 y Fm(=dx)815 3461 y Fn(2)871 3491 y Fr(+)18 b Fm(Q)p Fr(\))23 b(=)g([0)p Fm(;)14 b Fk(1)p Fr(\),)27 b(see)h([5)o(].)37 b(Th)n(us)27 b(w)n(e)h(can)f(use) 456 3652 y Fu(Theorem)j(5)i Fr(\([3)o(]\))p Fu(.)42 b Fs(If)30 b Fm(Q)f Fs(is)h(p)l(erio)l(dic)i(and)f Fm(\033)s Fr(\()p Fk(\000)p Fm(d)2095 3622 y Fn(2)2132 3652 y Fm(=dx)2264 3622 y Fn(2)2320 3652 y Fr(+)18 b Fm(Q)p Fr(\))23 b(=)g([0)p Fm(;)14 b Fk(1)p Fr(\))p Fs(,)30 b(then)f Fm(Q)23 b Fr(=)g(0)p Fs(.)555 3814 y Fr(Since)28 b(w)n(e)f(ha)n(v)n(e)g(assumed)g Fm(q)f Fk(6)p Fr(=)c(0,)28 b(w)n(e)f(conclude)g(that)h Fm(M)2386 3826 y Fi(\000)2469 3814 y Fr(is)g(discrete.)456 4007 y(3.4.)46 b Fu(The)25 b(Inde\014nite)g(Anderson)f(Mo)s(del.)40 b Fr(It)22 b(turns)f(out)g(that)h(t)n(w)n(o-parameter)d(sp)r(ectral)456 4106 y(a)n(v)n(eraging)24 b(and)k(a)f(simple)h(fact)g(from)f(in)n(v)n (erse)f(sp)r(ectral)h(theory)g(can)g(also)g(b)r(e)h(used)g(to)f(pro)n (v)n(e)456 4206 y(exp)r(onen)n(tial)37 b(lo)r(calization)g(for)g(the)h (inde\014nite)h(Anderson)e(mo)r(del)h Fm(H)2735 4218 y Fl(I)5 b(A)2823 4206 y Fr(.)68 b(W)-7 b(e)38 b(sk)n(etc)n(h)f(the)456 4306 y(t)n(w)n(o)27 b(main)h(ingredien)n(ts)g(from)g(the)h(pro)r(of)e (in)i([32)o(],)g(whic)n(h)f(in)h(fact)f(allo)n(ws)f(the)i(presence)f (of)g(a)456 4405 y(p)r(erio)r(dic)i(bac)n(kground)e Fm(V)1275 4417 y Fr(p)r(er)1428 4405 y(as)i(in)h(\(2.4\))f(in)h Fm(H)2003 4417 y Fl(I)5 b(A)2091 4405 y Fr(,)31 b(but)g(needs)f(an)g (a.c.)g(comp)r(onen)n(t)g(in)h(the)456 4505 y(distribution)c(of)h Fm(q)1040 4517 y Fl(n)1085 4505 y Fr(.)555 4605 y(Fixing)g(all)f(but)h (t)n(w)n(o)f(of)h(the)g Fm(q)1514 4617 y Fl(n)1586 4605 y Fr(w)n(e)g(arriv)n(e)d(at)j(a)f(family)1290 4761 y Fm(H)1359 4773 y Fl(\025;\026)1486 4761 y Fr(=)22 b Fk(\000)p Fm(d)1681 4726 y Fn(2)1718 4761 y Fm(=dx)1850 4726 y Fn(2)1906 4761 y Fr(+)c Fm(V)37 b Fr(+)18 b Fm(\025W)2283 4773 y Fn(1)2340 4761 y Fr(+)g Fm(\025W)2549 4773 y Fn(2)2587 4761 y Fm(;)456 4917 y Fr(where)24 b Fm(V)44 b Fr(is)25 b(a)f(\014xed)h(bac)n(kground)e(p)r(oten)n(tial)i(suc)n(h)f(that)h(w)n (e)g(ha)n(v)n(e)e(limit)j(p)r(oin)n(t)f(case)f(at)h Fk(\0061)p Fr(,)456 5016 y(and)c Fm(W)689 5028 y Fn(1)750 5016 y Fk(6)p Fr(=)h(0)f(and)g Fm(W)1133 5028 y Fn(2)1194 5016 y Fk(6)p Fr(=)i(0)e(are)f(in)n(tegrable)g(p)r(oten)n(tials)i(of)f (disjoin)n(t,)i(compact)e(supp)r(orts,)h(but)456 5116 y(not)i(necessarily)f(of)h(\014xed)g(sign.)35 b(F)-7 b(or)24 b(their)g(sp)r(ectral)g(measures)f Fm(\032)2547 5128 y Fl(\025;\026)2675 5116 y Fr(w)n(e)h(ha)n(v)n(e)f(the)i(follo)n (wing)456 5216 y([32)o(,)i(Section)h(2])p eop %%Page: 10 10 10 9 bop 456 226 a Fn(10)1270 b(G.)29 b(STOLZ)456 425 y Fu(Theorem)h(6.)41 b Fs(L)l(et)32 b Fm(E)1184 437 y Fn(0)1251 425 y Fk(2)d Fg(R)39 b Fs(b)l(e)34 b(\014xe)l(d.)48 b(Then)33 b(ther)l(e)h(is)f(a)g(discr)l(ete)h(set)f Fm(M)38 b Fk(\032)28 b Fg(R)39 b Fs(such)33 b(that)456 525 y(for)c(every)f Fm(\025)852 537 y Fn(0)913 525 y Fk(2)c Fg(R)c Fk(n)15 b Fm(M)36 b Fs(ther)l(e)28 b(is)h(a)f(discr)l(ete)h(set)f Fm(N)9 b Fr(\()p Fm(\025)2190 537 y Fn(0)2228 525 y Fr(\))23 b Fk(\032)g Fg(R)34 b Fs(with)28 b(the)h(fol)t(lowing)h(pr)l(op)l (erty:)456 624 y(If)j Fm(\026)596 636 y Fn(0)664 624 y Fk(2)d Fg(R)d Fk(n)20 b Fm(N)9 b Fr(\()p Fm(\025)1048 636 y Fn(0)1086 624 y Fr(\))34 b Fs(then)f(ther)l(e)h(ar)l(e)f Fm(")d(>)f Fr(0)k Fs(and)h Fm(\016)f(>)c Fr(0)k Fs(such)h(that)f(the)h (Bor)l(el)g(me)l(asur)l(e)f Fm(\026)456 724 y Fs(de\014ne)l(d)d(by)1302 927 y Fm(\026)p Fr(\()p Fm(B)t Fr(\))24 b(=)1594 814 y Fj(Z)1677 834 y Fl(\025)1716 842 y Fe(0)1749 834 y Fn(+)p Fl(\016)1640 1002 y(\025)1679 1010 y Fe(0)1712 1002 y Fi(\000)p Fl(\016)1850 814 y Fj(Z)1933 834 y Fl(\026)1973 842 y Fe(0)2006 834 y Fn(+)p Fl(\016)1896 1002 y(\026)1936 1010 y Fe(0)1969 1002 y Fi(\000)p Fl(\016)2107 927 y Fm(\032)2150 939 y Fl(\025;\026)2254 927 y Fr(\()p Fm(B)t Fr(\))14 b Fm(d\026)g(d\025)456 1129 y Fs(is)30 b(absolutely)g(c)l (ontinuous)f(on)h Fr(\()p Fm(E)1554 1141 y Fn(0)1611 1129 y Fk(\000)18 b Fm(";)c(E)1831 1141 y Fn(0)1886 1129 y Fr(+)k Fm(")p Fr(\))p Fs(.)555 1285 y Fr(Note)25 b(that)g(in)g (Theorem)f(6,)h(as)f(opp)r(osed)g(to)g(Theorem)g(4,)h(there)g(is)f(no)h (exceptional)f(set)h(of)456 1384 y(energies.)35 b(Sp)r(ectral)28 b(a)n(v)n(eraging)c(is)k(p)r(ossible)f(in)h(a)f(neigh)n(b)r(orho)r(o)r (d)f(of)i(an)n(y)f(giv)n(en)g(energy)f Fm(E)3377 1396 y Fn(0)3414 1384 y Fr(.)555 1484 y(But)e(it)f(is)g(still)h(crucial)e (for)h(the)g(pro)r(of)g(of)g(lo)r(calization)f(that)i(the)f (exceptional)g(sets)g Fm(M)32 b Fr(and)456 1584 y Fm(N)9 b Fr(\()p Fm(\025)612 1596 y Fn(0)649 1584 y Fr(\))31 b(for)e(the)h(parameters)f Fm(\025)1467 1596 y Fn(0)1534 1584 y Fr(and)h Fm(\026)1748 1596 y Fn(0)1816 1584 y Fr(are)f(discrete.)43 b(This)30 b(is)g(deriv)n(ed)f(in)h([32)o(])g (from)g(the)456 1683 y(follo)n(wing)c(in)n(v)n(erse)g(sp)r(ectral)h (result:)456 1839 y Fu(Theorem)j(7.)41 b Fs(L)l(et)e Fm(q)44 b Fs(and)c Fm(w)j Fs(b)l(e)d(inte)l(gr)l(able)h(r)l(e)l(al)f(p) l(otentials)h(on)f(a)g(\014nite)g(interval)h Fr([)p Fm(a;)14 b(b)p Fr(])456 1938 y Fs(and)31 b(let)g Fm(T)785 1950 y Fl(\025)859 1938 y Fs(b)l(e)g(a)g(self-adjoint)i(op)l(er)l(ator)f(on) f Fm(L)1972 1908 y Fn(2)2009 1938 y Fr(\()p Fm(a;)14 b(b)p Fr(\))30 b Fs(de\014ne)l(d)i(by)f Fk(\000)p Fm(d)2719 1908 y Fn(2)2756 1938 y Fm(=dx)2888 1908 y Fn(2)2945 1938 y Fr(+)19 b Fm(q)j Fr(+)d Fm(\025w)34 b Fs(and)456 2038 y(arbitr)l(ary,)29 b Fm(\025)p Fs(-indep)l(endent,)g(b)l(oundary)f (c)l(onditions)h(at)e Fm(a)g Fs(and)h Fm(b)p Fs(.)38 b(If)28 b Fm(E)2710 2050 y Fn(0)2775 2038 y Fs(is)f(an)h(eigenvalue)h (of)456 2137 y Fm(T)505 2149 y Fl(\025)577 2137 y Fs(for)i(every)g Fm(\025)p Fs(,)f(then)g Fm(w)25 b Fr(=)e(0)p Fs(.)555 2293 y Fr(This)41 b(follo)n(ws)f(quite)h(easily)f(from)g(the)i(v)-5 b(ariational)39 b(principle)i(and)f(the)i(fact)f(that)g(no)456 2392 y(eigen)n(v)-5 b(alue)26 b(crossings)g(are)g(allo)n(w)n(ed)h(in)g (dimension)h(one.)456 2563 y(3.5.)46 b Fu(T)-8 b(o)s(ols)30 b(from)f(Scattering)h(Theory)-8 b(.)42 b Fr(It)27 b(is)f(p)r(ossible)g (to)g(replace)f(Borg's)g(result)h(from)456 2662 y(in)n(v)n(erse)d(sp)r (ectral)i(theory)f(b)n(y)h(to)r(ols)f(from)h(in)n(v)n(erse)f (scattering)g(theory)g(in)i(pro)r(ofs)e(of)h(lo)r(caliza-)456 2762 y(tion.)35 b(This)21 b(is)h(closer)f(to)g(ph)n(ysics)g(since)h(it) g(supp)r(orts)g(the)g(follo)n(wing)f(heuristics:)33 b(If)23 b(scattering)456 2861 y(at)28 b(eac)n(h)f(single)h(site)g(is)g (non-trivial)f(and)i(a)e(global)h(p)r(oten)n(tial)g(is)g(constructed)g (b)n(y)g(randomly)456 2961 y(disp)r(ersing)e(these)i(sites,)g(then)g (states)f(should)g(b)r(e)h(lo)r(calized.)555 3061 y(A)22 b(w)n(a)n(y)f(to)g(c)n(haracterize)f(\\non-trivial)g(scattering)g(at)i (single)f(sites")g(is)g(b)n(y)h(using)f(re\015ection)456 3160 y(and)28 b(transmission)f(co)r(e\016cien)n(ts.)38 b(F)-7 b(or)28 b(their)g(de\014nition)g(let)h Fm(f)37 b Fr(b)r(e)29 b(compactly)e(supp)r(orted)h(in)456 3260 y([)p Fk(\000)p Fr(1)p Fm(=)p Fr(2)p Fm(;)14 b Fr(1)p Fm(=)p Fr(2],)27 b Fm(k)j Fk(2)e Fg(C)41 b Fk(n)20 b(f)p Fr(0)p Fk(g)p Fr(,)29 b(and)h Fm(u)p Fr(\()p Fk(\001)p Fm(;)14 b(k)s Fr(\))30 b(the)h(Jost)e(solution)h(of)g Fk(\000)p Fm(u)2646 3230 y Fi(0)o(0)2707 3260 y Fr(+)20 b Fm(f)9 b(u)26 b Fr(=)h Fm(k)3054 3230 y Fn(2)3091 3260 y Fm(u)p Fr(,)k(i.e.)f(the)456 3360 y(solution)d(whic)n(h)g (satis\014es)1031 3534 y Fm(u)p Fr(\()p Fm(x;)14 b(k)s Fr(\))24 b(=)1384 3417 y Fj(\032)1488 3484 y Fm(e)1527 3453 y Fl(ik)q(x)2306 3484 y Fr(for)j Fm(x)d Fk(\024)e(\000)p Fr(1)p Fm(=)p Fr(2)n Fm(;)1488 3584 y(a)p Fr(\()p Fm(k)s Fr(\))p Fm(e)1681 3554 y Fl(ik)q(x)1801 3584 y Fr(+)c Fm(b)p Fr(\()p Fm(k)s Fr(\))p Fm(e)2069 3554 y Fi(\000)p Fl(ik)q(x)2306 3584 y Fr(for)27 b Fm(x)d Fk(\025)e Fr(1)p Fm(=)p Fr(2.)456 3534 y(\(3.8\))555 3730 y(This)h(c)n(haracterizes)e Fm(a)p Fr(\()p Fm(k)s Fr(\))i(and)g Fm(b)p Fr(\()p Fm(k)s Fr(\))g(uniquely)-7 b(,)24 b(whic)n(h)f(are)f(related)h(to)g(the)g (re\015ection)g(and)456 3829 y(transmission)j(co)r(e\016cien)n(ts)i (from)f(ph)n(ysics)h(b)n(y)f Fm(r)r Fr(\()p Fm(k)s Fr(\))e(=)e Fm(b)p Fr(\()p Fm(k)s Fr(\))p Fm(=a)p Fr(\()p Fm(k)s Fr(\))28 b(and)g Fm(t)p Fr(\()p Fm(k)s Fr(\))c(=)f(1)p Fm(=a)p Fr(\()p Fm(k)s Fr(\).)37 b(The)456 3929 y(co)r(e\016cien)n(ts) 21 b Fm(a)p Fr(\()p Fm(k)s Fr(\))h(and)g Fm(b)p Fr(\()p Fm(k)s Fr(\))f(are)g(analytic)g(in)h Fm(k)j Fr(and)c(for)g(real)g Fm(k)k Fr(satisfy)c Fk(j)p Fm(a)p Fr(\()p Fm(k)s Fr(\))p Fk(j)2924 3899 y Fn(2)2968 3929 y Fk(\000)7 b(j)p Fm(b)p Fr(\()p Fm(k)s Fr(\))p Fk(j)3232 3899 y Fn(2)3292 3929 y Fr(=)23 b(1.)456 4029 y(One)k(has)456 4184 y Fu(Theorem)j(8.)41 b Fs(If)30 b Fk(\000)p Fm(u)1177 4154 y Fi(0)o(0)1237 4184 y Fr(+)18 b Fm(f)9 b(u)22 b Fr(=)h Fm(k)1574 4154 y Fn(2)1611 4184 y Fm(u)29 b Fs(is)h(r)l(e\015e)l(ctionless,)h(i.e.)g (if)g Fm(b)p Fr(\()p Fm(k)s Fr(\))23 b Fk(\021)f Fr(0)p Fs(,)30 b(then)g Fm(f)h Fk(\021)23 b Fr(0)p Fs(.)555 4339 y Fr(This)28 b(follo)n(ws)f(from)h(in)n(v)n(erse)e(scattering)h (theory)-7 b(,)28 b(e.g.)f([12)o(],)i(where)e(all)h(re\015ectionless)f (p)r(o-)456 4439 y(ten)n(tials)32 b(\(solitons\))g(ha)n(v)n(e)g(b)r (een)h(c)n(haracterized,)f(and)h(none)f(of)h(them)g(are)f(compactly)g (sup-)456 4539 y(p)r(orted.)j(One)25 b(can)f(also)g(deduce)h(Theorem)f (8)g(from)h(Borg's)e(Theorem)h(since)h(it)g(is)g(seen)f(quite)456 4638 y(easily)31 b(that)i Fm(b)p Fr(\()p Fm(k)s Fr(\))e Fk(\021)g Fr(0)h(implies)h(that)g(the)g(p)r(oten)n(tial)f(found)h(b)n (y)f(p)r(erio)r(dically)g(extending)h Fm(f)456 4738 y Fr(has)27 b(sp)r(ectrum)g([0)p Fm(;)14 b Fk(1)p Fr(\).)555 4837 y(A)25 b(pro)r(of)f(of)h(exp)r(onen)n(tial)f(lo)r(calization)f (for)h(the)h(displacemen)n(t)f(mo)r(del)h Fm(H)2897 4849 y Fl(D)2982 4837 y Fr(can)f(b)r(e)h(based)456 4937 y(on)33 b(Theorem)g(8)h(b)n(y)f(observing)f(that)i(the)h(sets)e Fm(M)2094 4949 y Fn(+)2183 4937 y Fr(and)g Fm(M)2431 4949 y Fi(\000)2521 4937 y Fr(in)h(\(3.6\))f(and)h(\(3.7\))g(can)f(b)r (e)456 5037 y(c)n(haracterized)25 b(as)i(follo)n(ws:)1330 5182 y Fm(M)1411 5194 y Fn(+)1488 5182 y Fr(=)c Fk(f)p Fm(k)1664 5147 y Fn(2)1724 5182 y Fm(>)f Fr(0)h(:)41 b Fm(k)26 b(>)d Fr(0)p Fm(;)32 b(b)p Fr(\()p Fm(k)s Fr(\))23 b(=)f(0)p Fk(g)p Fm(;)-2101 b Fr(\(3.9\))p eop %%Page: 11 11 11 10 bop 1469 226 a Fn(LOCALIZA)-5 b(TION)30 b(STRA)-5 b(TEGIES)946 b(11)1001 425 y Fm(M)1082 437 y Fi(\000)1161 425 y Fr(=)22 b Fk(f\000)p Fm(\013)1408 391 y Fn(2)1468 425 y Fr(:)42 b Fm(\013)23 b(>)g Fr(0)p Fm(;)32 b(b)p Fr(\()p Fm(i\013)p Fr(\))p Fm(b)p Fr(\()p Fk(\000)p Fm(i\013)p Fr(\))p Fm(a)p Fr(\()p Fm(i\013)p Fr(\))p Fm(a)p Fr(\()p Fk(\000)p Fm(i\013)p Fr(\))23 b(=)g(0)p Fk(g)p Fm(;)-2430 b Fr(\(3.10\))456 581 y(see)39 b([29)o(].)72 b(F)-7 b(or)39 b Fm(f)51 b Fk(6)p Fr(=)43 b(0)c(one)g(has)f(that)i Fm(b)p Fr(\()p Fm(k)s Fr(\))g(and)f Fm(a)p Fr(\()p Fm(k)s Fr(\))h(are)e (analytic)h(and)g(don't)h(v)-5 b(anish)456 681 y(iden)n(tically)e(.)57 b(Th)n(us)34 b(they)h(ha)n(v)n(e)e(discrete)h(ro)r(ots,)h(yielding)f (discreteness)g(of)g Fm(M)3027 693 y Fn(+)3116 681 y Fr(and)h Fm(M)3366 693 y Fi(\000)3421 681 y Fr(.)456 781 y(Therefore)26 b(Theorem)h(4)g(applies)g(and)h(the)g(pro)r(of)f(of) g(lo)r(calization)g(is)g(concluded)h(as)e(b)r(efore.)555 880 y(F)-7 b(or)36 b(the)h(Sc)n(hr\177)-42 b(odinger-t)n(yp)r(e)35 b(mo)r(del)i Fm(H)1841 892 y Fl(D)1901 880 y Fr(,)i(the)f(ab)r(o)n(v)n (e)d(ideas)h(are)g(mainly)g(a)h(v)-5 b(arian)n(t)36 b(of)456 980 y(the)30 b(previous)f(pro)r(of,)h(but)h(they)f(also)f(can)h(b)r(e)h (applied)f(to)g(pro)n(v)n(e)e(exp)r(onen)n(tial)i(lo)r(calization)456 1080 y(for)k Fm(H)666 1049 y Fl(W)659 1102 y(D)742 1080 y Fr(,)j(the)f(displacemen)n(t)f(mo)r(del)g(for)g(classical)f(w)n(a)n (v)n(es)f([29)o(].)60 b(The)35 b(main)h(di\016cult)n(y)f(in)456 1179 y(extending)26 b(the)g(pro)r(of)f(to)h(this)h(mo)r(del)f(is)g(to)g (\014nd)g(an)g(analogue)e(of)i(Theorem)f(8.)36 b(F)-7 b(or)26 b(this)g(let)456 1279 y(a)31 b(function)i Fm(f)39 b Fk(2)31 b Fm(L)1082 1249 y Fn(1)1151 1279 y Fr(b)r(e)i(supp)r(orted)f (in)g([)p Fk(\000)p Fr(1)p Fm(=)p Fr(2)p Fm(;)14 b Fr(1)p Fm(=)p Fr(2])29 b(and)j(suc)n(h)g(that)g(1)21 b(+)g Fm(f)39 b(>)31 b Fr(0.)50 b(F)-7 b(or)31 b(the)456 1378 y(single)c(site)g (equation)1545 1530 y Fk(\000)p Fr(\(\(1)18 b(+)g Fm(f)9 b Fr(\))1899 1495 y Fi(\000)p Fn(1)1988 1530 y Fm(u)2036 1495 y Fi(0)2059 1530 y Fr(\))2091 1495 y Fi(0)2137 1530 y Fr(=)23 b Fm(k)2271 1495 y Fn(2)2308 1530 y Fm(u)-1900 b Fr(\(3.11\))456 1686 y(one)27 b(can)g(de\014ne)h Fm(a)p Fr(\()p Fm(k)s Fr(\))g(and)f Fm(b)p Fr(\()p Fm(k)s Fr(\))h(as)f(b)r (efore)g(b)n(y)h(\(3.8\).)36 b(One)27 b(indeed)h(has)f([29)o(])456 1845 y Fu(Theorem)j(9.)41 b Fs(If)30 b Fk(\000)p Fr(\(\(1)18 b(+)g Fm(f)9 b Fr(\))1418 1815 y Fi(\000)p Fn(1)1507 1845 y Fm(u)1555 1815 y Fi(0)1578 1845 y Fr(\))1610 1815 y Fi(0)1657 1845 y Fr(=)22 b Fm(k)1790 1815 y Fn(2)1827 1845 y Fm(u)30 b Fs(is)g(r)l(e\015e)l(ctionless,)g(then)g Fm(f)h Fk(\021)23 b Fr(0)p Fs(.)555 2004 y Fr(If)35 b Fm(f)44 b Fr(is)34 b(t)n(wice)h(di\013eren)n(tiable,)h(then)f(\(3.11\)) f(can)g(b)r(e)h(transformed)f(in)n(to)g(a)g(Sc)n(hr\177)-42 b(odinger)456 2103 y(equation)29 b(b)n(y)g(a)g(Liouville-Green)f (transformation,)h(whic)n(h)g(can)g(b)r(e)h(used)g(to)f(reduce)g(Theo-) 456 2203 y(rem)f(9)g(to)g(Theorem)g(8.)39 b(F)-7 b(or)28 b(non-smo)r(oth)f Fm(f)38 b Fr(the)28 b(result)h(is)f(deep)r(er.)39 b(It)29 b(w)n(as)f(pro)n(v)n(ed)f(in)h([29)o(])456 2303 y(b)n(y)34 b(using)h(v)-5 b(arious)34 b(facts)h(ab)r(out)g(Herglotz)f (function)i(represen)n(tations)d(and)i(the)h(follo)n(wing)456 2402 y(result)27 b(on)g(the)h(asymptotics)f(of)h Fm(m)p Fr(-functions:)456 2561 y Fu(Theorem)i(10.)41 b Fs(L)l(et)j Fm(m)p Fr(\()p Fm(z)t(;)14 b(x)p Fr(\))46 b Fs(b)l(e)f(the)h (Weyl-Titchmarsh)h Fm(m)p Fs(-function)e(of)h(the)f(op)l(er)l(ator)456 2663 y Fk(\000)549 2631 y Fl(d)p 530 2645 73 4 v 530 2692 a(dx)612 2663 y Fm(p)683 2631 y Fl(d)p 664 2645 V 664 2692 a(dx)785 2663 y Fs(on)40 b Fr(\()p Fm(x;)14 b Fk(1)p Fr(\))40 b Fs(with)g(Dirichlet-b)l(oundary)h(c)l(ondition)f (at)g Fm(x)p Fs(,)i(wher)l(e)e Fm(p)g(>)g Fr(0)p Fs(,)i Fr(1)p Fm(=p)d Fk(2)456 2768 y Fm(L)513 2738 y Fn(1)513 2791 y Fl(loc)600 2768 y Fr(\()p Fg(R)p Fr(\))d Fs(and)31 b Fm(p)18 b Fk(\000)g Fr(1)29 b Fs(is)h(c)l(omp)l(actly)h(supp)l(orte)l (d.)39 b(If)30 b Fm(x)g Fs(is)h(a)f(L)l(eb)l(esgue-p)l(oint)g(for)g Fr(1)p Fm(=p)p Fs(,)f(then)1388 2929 y Fm(m)p Fr(\()p Fk(\000)p Fm(\013)1611 2895 y Fn(2)1649 2929 y Fm(;)14 b(x)p Fr(\))23 b(=)g Fk(\000)p Fm(\013p)p Fr(\()p Fm(x)p Fr(\))2147 2895 y Fn(1)p Fl(=)p Fn(2)2271 2929 y Fr(+)18 b Fm(o)p Fr(\()p Fm(\013)p Fr(\))456 3080 y Fs(for)30 b(r)l(e)l(al)g Fm(\013)24 b Fk(!)f Fr(+)p Fk(1)p Fs(.)555 3239 y Fr(An)38 b(elemen)n(tary)e(pro)r(of)h(of)g(this)g(fact)h(is)f (giv)n(en)f(in)i([29)o(].)66 b(Muc)n(h)37 b(more)f(general)g(results) 456 3339 y(whic)n(h)27 b(apply)g(to)h(our)f(situation)g(are)g(con)n (tained)g(in)h([1)o(],)g(see)f(also)g([28)o(].)456 3522 y(3.6.)46 b Fu(Singular)40 b(Random)f(P)m(arameters.)i Fr(W)-7 b(e)35 b(ha)n(v)n(e)f(demonstrated)g(that)i(the)f(metho)r(d)456 3622 y(of)28 b(sp)r(ectral)g(a)n(v)n(eraging)e(can)i(b)r(e)h(made)g(to) f(w)n(ork)g(in)h(a)f(n)n(um)n(b)r(er)g(of)h(non-monotonic)f(mo)r(dels.) 456 3721 y(But)h(it)g(is)f(not)h(p)r(ossible)f(to)h(extend)f(this)h (metho)r(d)h(to)e(mo)r(dels)h(with)g(singularly)e(distributed)456 3821 y(random)20 b(parameters,)h(in)h(particular)e(the)i(singular)e (Anderson)h(mo)r(del)h Fm(H)2791 3833 y Fl(S)s(A)2889 3821 y Fr(.)35 b(In)22 b(an)f(attempt)456 3920 y(to)g(pro)n(v)n(e)e(lo) r(calization)i(this)g(do)r(oms)g(m)n(uc)n(h)g(of)g(the)h(basic)f (strategy)f(describ)r(ed)h(in)g(Sections)g(3.1)456 4020 y(and)k(3.2)f(to)i(failure.)35 b(F)-7 b(or)25 b(example,)h(Kotani's)e (theory)g(guaran)n(tees)g(p)r(ositivit)n(y)h(of)g Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))26 b(only)456 4120 y(for)37 b(a.e.)g Fm(E)45 b Fk(2)40 b Fg(R)p Fr(.)74 b(The)38 b(set)f(where)h Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))40 b(=)f(0)f(migh)n(t)f(b)r(e)h (uncoun)n(table)g(and)f(therefore,)456 4219 y(without)i(sp)r(ectral)f (a)n(v)n(eraging)d(at)k(hand,)j(supp)r(ort)c(singular)f(con)n(tin)n (uous)h(sp)r(ectrum)h(with)456 4319 y(p)r(ositiv)n(e)27 b(probabilit)n(y)-7 b(.)555 4419 y(Th)n(us)33 b(it)g(b)r(ecomes)f(imp)r (ortan)n(t)h(to)f(ha)n(v)n(e)g(stronger)f(results)h(on)h(p)r(ositivit)n (y)f(of)h(Ly)n(apuno)n(v)456 4518 y(exp)r(onen)n(ts.)59 b(Since)35 b(w)n(e)g(assume)f(that)i(supp)14 b Fm(f)44 b Fk(\032)35 b Fr([)p Fk(\000)p Fr(1)p Fm(=)p Fr(2)p Fm(;)14 b Fr(1)p Fm(=)p Fr(2])32 b(the)j(transfer)g(matrices)f(for)456 4618 y Fm(H)525 4630 y Fl(S)s(A)623 4618 y Fr(\()p Fm(!)s Fr(\))39 b(from)f Fm(n)26 b Fk(\000)g Fr(1)p Fm(=)p Fr(2)37 b(to)i Fm(n)26 b Fr(+)f(1)p Fm(=)p Fr(2)38 b(are)g(i.i.d.)h(unimo)r (dular)g(random)f(matrices.)70 b(This)456 4717 y(allo)n(ws)32 b(us)j(to)f(use)g(F)r(\177)-44 b(ursten)n(b)r(erg's)33 b(Theorem,)i(see)f(e.g.)g([4)o(],)i(as)e(a)g(p)r(o)n(w)n(erful)f(to)r (ol)h(to)g(pro)n(v)n(e)456 4817 y(p)r(ositivit)n(y)c(of)g(Ly)n(apuno)n (v)e(exp)r(onen)n(ts.)44 b(This)30 b(has)g(frequen)n(tly)g(b)r(een)g (exploited)g(for)g(discrete)456 4917 y(mo)r(dels,)25 b(compare)e([26)o(,)i(Sect.14.A].)f(In)g(particular)f(it)i(sho)n(ws)e (that)i Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))23 b Fm(>)g Fr(0)h(for)f Fs(al)t(l)j Fm(E)i Fk(2)23 b Fg(R)456 5016 y Fr(in)32 b(the)h(discrete)f(one-dimensional)g(Bernoulli-Anderson)e (mo)r(del,)k(whic)n(h)f(has)f(serv)n(ed)f(as)h(a)456 5116 y(starting)20 b(p)r(oin)n(t)h(in)g(a)g(pro)r(of)f(of)h(lo)r (calization)f(for)g(this)h(mo)r(del)g(in)h([6)o(].)35 b(Based)20 b(on)h(F)r(\177)-44 b(ursten)n(b)r(erg's)456 5216 y(Theorem)26 b(one)i(can)f(sho)n(w)f(the)i(follo)n(wing)f(result)g (for)g(the)h(con)n(tin)n(uum)g(case:)p eop %%Page: 12 12 12 11 bop 456 226 a Fn(12)1270 b(G.)29 b(STOLZ)456 425 y Fu(Theorem)h(11.)41 b Fs(L)l(et)25 b Fm(H)1233 437 y Fl(P)9 b Fn(+)p Fl(A)1415 425 y Fs(b)l(e)25 b(the)h(singular)g(A)n (nderson)g(mo)l(del)h(with)f(p)l(erio)l(dic)i(b)l(ackgr)l(ound)456 525 y(as)35 b(intr)l(o)l(duc)l(e)l(d)h(in)f(Se)l(ctions)h(2\(v\))f(and) h(\(vi\).)57 b(Then)36 b(ther)l(e)f(exists)g(a)h(discr)l(ete)g(set)f Fm(M)44 b Fs(such)456 624 y(that)29 b Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))24 b Fm(>)e Fr(0)29 b Fs(for)i(every)g Fm(E)d Fk(2)23 b Fg(R)i Fk(n)18 b Fm(M)9 b Fs(.)555 770 y Fr(In)34 b(fact,)i(the)e(set)g Fm(M)43 b Fr(of)34 b(critical)f(energies)g(where) g Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))34 b(ma)n(y)f(v)-5 b(anish,)36 b(can)d(b)r(e)i(c)n(harac-)456 870 y(terized)e(explicitly)g (in)h(terms)f(of)g(re\015ection)g(and)g(transmission)f(co)r(e\016cien)n (ts)h(at)g(the)h(single)456 970 y(site)h(p)r(oten)n(tials.)58 b(While)36 b(this)f(is)g(done)f(in)h([9])g(in)g(the)h(con)n(text)e(of)h (scattering)f(at)h(p)r(erio)r(dic)456 1069 y(bac)n(kground,)28 b(w)n(e)h(only)g(state)g(the)h(result)f(here)g(for)g(the)h(sp)r(ecial)f (case)g Fm(V)2774 1081 y Fr(p)r(er)2922 1069 y(=)d(0,)k(where)f(w)n(e) 456 1169 y(can)f(w)n(ork)f(with)j(the)f(re\015ection)f(and)g (transmission)g(co)r(e\016cien)n(ts)g(as)g(de\014ned)h(in)g(\(3.8\).)40 b(It)30 b(is)456 1269 y(also)e(su\016cien)n(t)i(to)g(fo)r(cus)g(on)g (the)g(Bernoulli)f(case,)h(since)f(b)n(y)h(the)g(nature)g(of)g(F)r (\177)-44 b(ursten)n(b)r(erg's)456 1368 y(Theorem)26 b(the)i(critical)f(set)h(b)r(ecomes)f(smaller)g(as)g(the)h(supp)r(ort)f (of)h Fm(q)2657 1380 y Fl(n)2730 1368 y Fr(increases.)456 1514 y Fu(Theorem)i(12.)41 b Fs(L)l(et)30 b Fm(H)1238 1526 y Fl(B)s(A)1345 1514 y Fr(\()p Fm(!)s Fr(\))24 b(=)g Fk(\000)p Fm(d)1685 1484 y Fn(2)1722 1514 y Fm(=dx)1854 1484 y Fn(2)1911 1514 y Fr(+)1994 1452 y Fj(P)2082 1539 y Fl(n)2141 1514 y Fm(q)2178 1526 y Fl(n)2223 1514 y Fr(\()p Fm(!)s Fr(\))p Fm(f)9 b Fr(\()p Fm(x)20 b Fk(\000)e Fm(n)p Fr(\))31 b Fs(with)g Fm(f)i Fk(6)p Fr(=)24 b(0)p Fs(,)31 b(supp)14 b Fm(f)33 b Fk(\032)456 1614 y Fr([)p Fk(\000)p Fr(1)p Fm(=)p Fr(2)p Fm(;)14 b Fr(1)p Fm(=)p Fr(2])23 b Fs(and)28 b(i.i.d.)i(r)l(andom)e(variables)h Fm(q)1914 1626 y Fl(n)1986 1614 y Fs(taking)f(only)g(the)f(values)h Fr(0)e Fs(and)i Fr(1)p Fs(.)37 b(Then)28 b(the)456 1713 y(critic)l(al)j(set)e Fm(M)38 b Fs(in)30 b(The)l(or)l(em)h(11)f(c)l(an) g(b)l(e)g(chosen)h(as)1288 1848 y Fm(M)h Fr(=)22 b Fm(M)1569 1860 y Fn(+)1642 1848 y Fk([)d Fm(M)1797 1860 y Fi(\000)1871 1848 y Fk([)g(f)p Fr(\()p Fm(n\031)s(=)p Fr(2\))2235 1814 y Fn(2)2295 1848 y Fr(:)k Fm(n)g Fk(2)g Fg(Z)o Fk(g)p Fm(;)456 1982 y Fs(wher)l(e)30 b Fm(M)771 1994 y Fn(+)855 1982 y Fs(and)h Fm(M)1098 1994 y Fi(\000)1183 1982 y Fs(ar)l(e)f(the)g(sets)f(fr)l(om)i(\(3.9\))f(and)h(\(3.10\).)555 2128 y Fr(Th)n(us)38 b(discreteness)g(of)g Fm(M)47 b Fr(follo)n(ws)37 b(from)h(the)h(results)f(in)g(Section)h(3.5.)68 b(Theorem)37 b(12)456 2228 y(follo)n(ws)23 b(from)g(the)i(more)e (general)g(results)g(in)i([9)o(].)36 b(A)24 b(more)g(direct)g(pro)r(of) f(of)h(this)h(sp)r(ecial)e(case)456 2328 y(is)k(pro)n(vided)g(in)g ([10].)555 2427 y(In)36 b(the)g(general)f(case)g(of)g(Theorem)g(11,)i (i.e.)f(in)g(the)g(presence)f(of)h Fm(V)2792 2439 y Fr(p)r(er)2915 2427 y(,)i(one)d(needs)h(to)456 2527 y(rede\014ne)28 b Fm(a)h Fr(and)f Fm(b)h Fr(as)f(scattering)g(co)r(e\016cien)n(ts)g (relativ)n(e)g(to)h(the)g(p)r(erio)r(dic)f(bac)n(kground,)g(and)456 2626 y(can)f(then)i(again)e(describ)r(e)h(the)h(critical)e(set)h Fm(M)37 b Fr(in)29 b(terms)f(of)g(their)g(zeros.)38 b(In)28 b(particular,)f(it)456 2726 y(can)g(again)f(b)r(e)i(sho)n(wn)f(that)h (their)f(zeros)g(are)f(discrete,)h(e.g.)h([9)o(].)555 2826 y(While)38 b(the)h(discrete)e(set)h Fm(M)46 b Fr(can)37 b(not)h(supp)r(ort)f(singular)g(con)n(tin)n(uous)g(sp)r(ectrum,)j(the) 456 2925 y(argumen)n(t)32 b(from)g(Section)h(3.2)f(can)h(still)g(not)g (b)r(e)h(pushed)f(through)f(to)h(pro)n(v)n(e)f(lo)r(calization.)456 3025 y(By)22 b(the)h(w)n(a)n(y)f(in)h(whic)n(h)g(F)-7 b(ubini)24 b(en)n(ters)e(the)h(argumen)n(t)f(at)g(the)i(b)r(eginning)e (of)h(Section)g(3.2,)g(for)456 3125 y(\014xed)32 b Fm(!)j Fr(w)n(e)c(still)i(get)f(the)g(existence)g(of)g(exp)r(onen)n(tially)f (deca)n(ying)g(solutions)h(only)f(for)h(a.e.)456 3224 y Fm(E)5 b Fr(.)40 b(Instead,)29 b(one)f(can)g(use)h(Theorem)f(11)f(as) h(a)h(starting)f(p)r(oin)n(t)g(for)h(a)f(pro)r(of)g(of)h(lo)r (calization)456 3324 y(follo)n(wing)e(the)i(argumen)n(ts)f(for)g(the)h (discrete)g(case)e(in)i([6].)41 b(W)-7 b(e)29 b(refer)f(to)g([9])h(for) f(details)h(and)456 3423 y(only)f(men)n(tion)i(that)f(the)h(follo)n (wing)e(facts)h(can)g(b)r(e)h(established:)39 b(H\177)-42 b(older)29 b(con)n(tin)n(uit)n(y)g(of)g(the)456 3523 y(Ly)n(apuno)n(v)f(exp)r(onen)n(t)i(and)g(the)g(in)n(tegrated)f(densit) n(y)h(of)g(states,)g(a)f(W)-7 b(egner-t)n(yp)r(e)29 b(estimate,)456 3623 y(and)34 b(a)g(so-called)e(initial)j(length)f(scale)g(estimate)g (for)g(the)g(deca)n(y)g(of)g(resolv)n(en)n(t)f(k)n(ernels)g(on)456 3722 y(\014nite)38 b(in)n(terv)-5 b(als.)69 b(It)38 b(then)h(follo)n (ws)e(from)h(w)n(ell)g(established)g(facts,)j(based)d(on)g(m)n (ultiscale)456 3822 y(analysis)19 b(metho)r(ds,)k(that)e Fm(H)1356 3834 y Fl(P)9 b Fn(+)p Fl(A)1533 3822 y Fr(is)21 b(exp)r(onen)n(tially)f(lo)r(calized.)35 b(In)21 b(fact,)h(one)f(gets)f (dynamical)456 3922 y(lo)r(calization:)456 4068 y Fu(Theorem)30 b(13.)41 b Fs(The)32 b(singular)g(A)n(nderson)f(mo)l(del)h Fm(H)2199 4080 y Fl(P)9 b Fn(+)p Fl(A)2387 4068 y Fs(with)32 b(p)l(erio)l(dic)i(b)l(ackgr)l(ound)e(sat-)456 4167 y(is\014es)d (dynamic)l(al)i(lo)l(c)l(alization)g(in)f(the)f(sense)g(of)i(\(3.4\))f (for)g(c)l(omp)l(act)g(intervals)g Fm(I)g Fk(\032)23 b Fg(R)g Fk(n)17 b Fm(M)9 b Fs(,)456 4267 y(wher)l(e)30 b Fm(M)38 b Fs(is)30 b(the)g(discr)l(ete)h(critic)l(al)g(set)e(fr)l(om) h(The)l(or)l(em)h(11.)555 4413 y Fr(Critical)g(energies)g(with)h Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))30 b(=)f(0)j(do)f(indeed)h(exist)g(in) g(Bernoulli-Anderson)e(mo)r(dels.)456 4512 y(F)-7 b(or)33 b(example,)j(in)f(the)g(situation)f(of)g(Theorem)g(12)g(let)g Fm(E)40 b Fr(=)34 b Fm(k)2524 4482 y Fn(2)2595 4512 y Fm(>)g Fr(0)h(b)r(e)f(an)h(energy)e(suc)n(h)456 4612 y(that)f Fm(b)p Fr(\()p Fm(k)s Fr(\))f(=)f(0.)51 b(It)32 b(is)g(then)h(easily)e(seen)h(from)g(\(3.8\))g(that)h(all)e(solutions)h (of)g Fm(H)3050 4624 y Fl(B)s(A)3157 4612 y Fm(u)e Fr(=)h Fm(E)5 b(u)456 4712 y Fr(are)23 b(globally)g(b)r(ounded,)i(implying)f Fm(\015)5 b Fr(\()p Fm(E)g Fr(\))23 b(=)g(0.)35 b(F)-7 b(or)24 b(the)g(sp)r(ecial)g(case)g Fm(f)31 b Fr(=)23 b Fm(\025\037)2972 4727 y Fn([)p Fi(\000)p Fn(1)p Fl(=)p Fn(2)p Fl(;)p Fn(1)p Fl(=)p Fn(2])3310 4712 y Fr(this)456 4817 y(happ)r(ens)34 b(if)g Fm(E)39 b Fr(=)34 b(\()p Fm(n\031)s Fr(\))1233 4787 y Fn(2)1294 4817 y Fr(+)22 b Fm(\025)p Fr(,)36 b Fm(n)e Fk(2)g Fg(N)t Fr(.)63 b(Critical)33 b(energies)g(also)g(app)r(ear)g(in)h(the)h(so-called)456 4917 y(dimer)25 b(mo)r(del,)i(a)f(generalized)e(discrete)i(Anderson)f (mo)r(del,)i(where)e(dynamical)h(lo)r(calization)456 5016 y(a)n(w)n(a)n(y)f(from)i(the)h(critical)f(energies)g(has)g(b)r (een)h(sho)n(wn)f(in)g([2].)555 5116 y(The)21 b(restriction)f(in)g (Theorem)g(13)g(to)g(in)n(terv)-5 b(als)20 b Fm(I)30 b Fk(\032)23 b Fg(R)t Fk(n)t Fm(M)36 b Fr(is)20 b(crucial.)34 b(Numerical)20 b(results)456 5216 y(in)40 b(the)g(ph)n(ysics)f (literature,)k(e.g.)c([13)o(],)44 b(indicate)c(that)g(the)g(dimer)g (and)g(related)f(discrete)p eop %%Page: 13 13 13 12 bop 1469 226 a Fn(LOCALIZA)-5 b(TION)30 b(STRA)-5 b(TEGIES)946 b(13)456 425 y Fr(mo)r(dels)33 b(ha)n(v)n(e)f(p)r(ositiv)n (e)h(di\013usion)g(exp)r(onen)n(ts)g(if)h(w)n(a)n(v)n(e)e(pac)n(k)n (ets)g(con)n(tain)g(critical)h(energies)456 525 y(in)d(their)g(energy)g (supp)r(ort.)44 b(This)31 b(will)f(b)r(e)h(rigorously)d(established)i (for)g(discrete)f(as)h(w)n(ell)g(as)456 624 y(con)n(tin)n(uous)24 b(mo)r(dels)i(in)f([16)o(].)37 b(This)25 b(is)h(quite)f(remark)-5 b(able)24 b(since)i(it)g(is)f(the)h(\014rst)f(result)h(whic)n(h)456 724 y(establishes)j(at)h(least)g(sub)r(di\013usiv)n(e)h(w)n(a)n(v)n(e)e (transp)r(ort)g(for)h(some)f(t)n(yp)r(es)i(of)f(one-dimensional)456 824 y(random)c(op)r(erators.)555 923 y(Finally)-7 b(,)27 b(w)n(e)g(note)g(that)g(the)g(tec)n(hniques)g(whic)n(h)g(led)g(to)g (Theorem)f(13)g(can)g(b)r(e)i(applied)f(to)456 1023 y(sev)n(eral)h (other)h(mo)r(dels:)41 b(Results)30 b(for)f(singular)f(displacemen)n(t) i(mo)r(dels)f Fm(H)2855 1035 y Fl(S)s(D)2989 1023 y Fr(and)h(classical) 456 1123 y(w)n(a)n(v)n(e)36 b(mo)r(dels)h Fm(H)1034 1092 y Fl(W)1148 1123 y Fr(are)f(con)n(tained)h(in)h([28].)67 b(Generalized)37 b(discrete)g(Anderson)g(mo)r(dels,)456 1222 y(whic)n(h)27 b(include)h(the)g(dimer)g(mo)r(del,)f(will)h(b)r(e)g (considered)f(in)h([11)o(].)456 1393 y(3.7.)46 b Fu(Some)37 b(Op)s(en)h(Problems.)h Fr(W)-7 b(e)34 b(conclude)f(b)n(y)g(men)n (tioning)g(some)g(op)r(en)g(problems.)456 1492 y(Among)28 b(the)i(man)n(y)f(op)r(en)g(problems)f(in)i(the)f(\014eld)h(of)f (random)f(op)r(erators)g(and)h(their)g(appli-)456 1592 y(cations,)f(w)n(e)h(restrict)f(our)g(remarks)f(to)i(problems)f(whic)n (h)g(arise)g(directly)h(in)g(the)g(con)n(text)g(of)456 1692 y(results)e(presen)n(ted)g(in)h(this)f(pap)r(er.)555 1909 y(\(i\))37 b(In)g(Section)f(3.6)f(w)n(e)h(describ)r(ed)g(metho)r (ds)h(whic)n(h)f(allo)n(w)f(to)h(pro)n(v)n(e)f(lo)r(calization)g(for) 456 2009 y(one-dimensional)23 b(random)h(op)r(erators)f(with)j (singularly)e(distributed)h(random)f(parameters,)456 2109 y(in)f(particular)e(the)i(Bernoulli-Anderson)f(mo)r(del.)35 b(The)23 b(pro)r(of)f(uses)h(m)n(ultiscale)f(analysis)g(and)456 2208 y(therefore)d(the)i(en)n(tire)f(collection)g(of)h(to)r(ols)f(whic) n(h)g(en)n(ter)g(in)n(to)h(this)f(metho)r(d,)j(whic)n(h)e(w)n(as)e (origi-)456 2308 y(nally)h(dev)n(elop)r(ed)g(mainly)h(for)f(the)h(purp) r(ose)f(of)h(pro)n(ving)e(lo)r(calization)h(for)g(m)n(ulti-dimensional) 456 2408 y(Anderson)28 b(mo)r(dels.)42 b(After)29 b(establishing)g(p)r (ositivit)n(y)g(of)g(the)h(Ly)n(apuno)n(v)d(exp)r(onen)n(t)i(in)h(The-) 456 2507 y(orem)e(11,)i(it)g(seems)f(to)g(b)r(e)i(somewhat)d (inappropriate)h(that)g(one)h(has)f(to)g(em)n(bark)g(in)n(to)g(this)456 2607 y(sophisticated)k(mac)n(hinery)h(rather)f(than)h(b)r(eing)h(able)f (to)g(complete)g(the)h(pro)r(of)f(within)h(ar-)456 2706 y(gumen)n(ts)30 b(from)g(the)h(one-dimensional)e(theory)-7 b(.)46 b(A)31 b(more)e(direct)i(pro)r(of)f(of)h(lo)r(calization)e(for) 456 2806 y(one-dimensional)d(Bernoulli-Anderson)g(mo)r(dels)h(w)n(ould) g(b)r(e)h(desirable.)555 2906 y(Some)41 b(ideas)f(for)g(this)h(could)f (come)h(from)f(an)g(existing)h(second)f(metho)r(d)h(of)f(pro)r(of)h(of) 456 3005 y(lo)r(calization)28 b(for)h(the)h(one-dimensional)f(discrete) g(Bernoulli-Anderson)f(mo)r(del)i(whic)n(h)f(w)n(as)456 3105 y(giv)n(en)d(in)i([27)o(].)37 b(Ho)n(w)n(ev)n(er,)25 b(the)j(application)f(of)g(this)h(metho)r(d)g(to)f(con)n(tin)n(uum)h (mo)r(dels)f(seems)456 3205 y(to)g(yield)h(considerable)e (di\016culties.)555 3422 y(\(ii\))i(Throughout)e(this)h(pap)r(er)g(w)n (e)g(ha)n(v)n(e)f(assumed)g(that)i(the)f(single)g(site)g(p)r(oten)n (tials)g(ha)n(v)n(e)456 3522 y(compact)e(supp)r(ort,)h(most)f(of)h(the) g(time)g(ev)n(en)f(that)h(they)f(do)h(not)f(o)n(v)n(erlap.)35 b(F)-7 b(or)25 b(man)n(y)g(of)g(the)456 3622 y(tec)n(hniques)f(whic)n (h)h(w)n(ere)f(used)g(this)h(is)g(quite)g(crucial,)g(for)f(example)g (for)h(most)f(of)h(the)g(results)456 3721 y(on)h(sp)r(ectral)g(a)n(v)n (eraging,)e(or)h(the)i(results)f(based)g(on)g(F)r(\177)-44 b(ursten)n(b)r(erg's)26 b(Theorem.)36 b(While)27 b(most)456 3821 y(of)i(the)g(results)g(on)g(lo)r(calization)f(should)h(hold)g(for) f(non-compactly)g(supp)r(orted)h(single)g(sites)456 3920 y(under)24 b(suitable)g(deca)n(y)f(assumptions,)h(the)g(pro)r(ofs)g(do) f(not)i(easily)e(extend)h(to)g(this)h(situation.)456 4020 y(T)-7 b(ec)n(hnically)g(,)30 b(the)g(main)g(reason)f(for)h(the)g (di\016culties)h(is)f(that)g(long)g(range)f(correlations)e(are)456 4120 y(in)n(tro)r(duced)33 b(in)n(to)h(the)g(random)f(p)r(oten)n(tial.) 56 b(There)33 b(are)g(results)h(whic)n(h)g(sho)n(w)f(lo)r(calization) 456 4219 y(near)f(the)i(b)r(ottom)g(of)g(the)g(sp)r(ectrum)g(or,)g (more)f(generally)-7 b(,)33 b(near)g(band)h(edges,)g(for)f(m)n(ulti-) 456 4319 y(dimensional)24 b(Anderson)g(mo)r(dels)h(with)g (non-compactly)f(supp)r(orted)h(single)f(site)h(p)r(oten)n(tials,)456 4419 y(e.g.)30 b([18)o(].)46 b(While)31 b(these)f(results)g(include)h (dimension)g(one,)g(they)f(do)g(not)h(giv)n(e)f(lo)r(calization)456 4518 y(at)f(arbitrary)e(energy)h(in)h Fm(d)d Fr(=)f(1.)41 b(It)30 b(is)f(quite)g(surprising)f(that)h(there)g(do)r(es)g(not)g (seem)g(to)g(b)r(e)456 4618 y(a)f(pro)r(of)h(of)g(lo)r(calization)f(at) h(all)g(energies)f(for)h(Anderson)g(mo)r(dels)g Fm(H)2672 4630 y Fl(A)2755 4618 y Fr(from)g(\(2.1\))g(where)g Fm(f)456 4717 y Fr(is)e(in)h(a)f(class)g(of)g(not)h(necessarily)e(compactly)h (supp)r(orted)g(p)r(oten)n(tials.)555 4817 y(The)i(scattering)e (theoretic)i(argumen)n(ts)e(used)h(in)h(Section)g(3.5)e(indicate)i (that)g(some)f(non-)456 4917 y(trivial)d(di\016culties)h(ma)n(y)f (arise.)36 b(If)26 b Fm(f)34 b Fr(is)26 b(not)g(compactly)f(supp)r (orted,)h(then)h(the)f(structure)g(of)456 5016 y(its)31 b(re\015ection)g(co)r(e\016cien)n(t)g Fm(b)p Fr(\()p Fm(k)s Fr(\))h(ma)n(y)e(b)r(e)i(m)n(uc)n(h)f(more)g(complicated.)48 b Fm(f)40 b Fr(ma)n(y)30 b(b)r(e)i(a)f(soliton)456 5116 y(\()p Fm(b)p Fr(\()p Fm(k)s Fr(\))38 b Fk(\021)f Fr(0\))f(or,)i (probably)d(ev)n(en)h(w)n(orse,)g Fm(b)g Fr(ma)n(y)g(v)-5 b(anish)36 b(on)g(large)f(sets)h(without)h(v)-5 b(anish-)456 5216 y(ing)30 b(iden)n(tically)-7 b(.)47 b(This)31 b(ma)n(y)g(happ)r (en)g(in)g(situations)g(where)f Fm(f)40 b Fr(has)31 b(sub)r(exp)r(onen) n(tial)f(deca)n(y)p eop %%Page: 14 14 14 13 bop 456 226 a Fn(14)1270 b(G.)29 b(STOLZ)456 425 y Fr(and)j(th)n(us)g Fm(b)g Fr(is)g(not)h(real-analytic.)49 b(Another)32 b(di\016cult)n(y)g(is)g(the)h(non-linear)e(in)n(teraction) h(of)456 525 y(scattering-co)r(e\016cien)n(ts)f(at)i(m)n(ultiple)g(o)n (v)n(erlapping)e(sites.)53 b(One)33 b(has)g(to)f(ask)h(if)g(the)h(sets) f(of)456 624 y(critical)j(energies)f(whic)n(h)i(ha)n(v)n(e)f(app)r (eared)g(in)h(Sections)g(3.3,)h(3.5)e(and)h(3.6)f(could)h(b)r(ecome)456 724 y(signi\014can)n(tly)23 b(larger)e(and)j(giv)n(e)f(rise)g(to)h(con) n(tin)n(uous)f(sp)r(ectrum.)35 b(W)-7 b(e)25 b(feel)f(that)g(this)g (question)456 824 y(is)j(wide)h(op)r(en)f(and)h(do)f(not)h(w)n(an)n(t)f (to)g(predict)h(the)g(answ)n(er.)555 1041 y(\(iii\))k(While)f(w)n(e)f (ha)n(v)n(e)f(restricted)h(this)h(pap)r(er)f(to)g(the)h (one-dimensional)e(case,)i(it)g(should)456 1141 y(at)i(least)g(b)r(e)h (men)n(tioned)f(that)h(man)n(y)f(of)g(the)h(results)f(discussed)g(here) g(ha)n(v)n(e)f(not)i(y)n(et)f(b)r(een)456 1241 y(extended)22 b(to)h(higher)f(dimension.)35 b(There)22 b(are)f(results)h(for)g(the)h (m)n(ulti-dimensional)f(displace-)456 1340 y(men)n(t)30 b(mo)r(del)g([19)o(])f(and)h(for)f(the)h(inde\014nite)h(Anderson)e(mo)r (del)h([20)o(,)g(15)o(].)43 b(But)30 b(there)g(are)f(no)456 1440 y(results)37 b(y)n(et)h(on)g(lo)r(calization)f(for)h(the)h(m)n (ulti-dimensional)e(P)n(oisson)g(mo)r(del.)69 b(And)38 b(lo)r(cal-)456 1540 y(ization)28 b(for)h(the)h(m)n(ulti-dimensional)e (Bernoulli-Anderson)g(mo)r(del)h(curren)n(tly)f(seems)h(to)g(b)r(e)456 1639 y(quite)e(far)h(out)f(of)h(reac)n(h,)e(ev)n(en)h(in)h(the)g (discrete)f(case.)1708 1845 y Fp(References)491 1978 y Ft([1])35 b(C.)22 b(Bennewitz,)i(Sp)r(ectral)f(asymptotics)g(for)f (Sturm-Liouville)f(equations.)j Fo(Pr)l(o)l(c.)i(L)l(ondon)g(Math.)f (So)l(c.)601 2061 y Fd(59)e Ft(\(1989\),)i(294{338)491 2144 y([2])35 b(S.)25 b(De)i(Bi)n(\022)-33 b(evre)25 b(and)i(F.)f(Germinet,)f(Dynamical)h(Lo)r(calization)h(for)e(the)i (Random)f(Dimer)e(Sc)n(hr\177)-35 b(odinger)601 2227 y(Op)r(erator.)23 b Fo(J.)j(Stat.)f(Phys.)f Fd(98)f Ft(\(2000\),)i (1135{1148)491 2310 y([3])35 b(G.)29 b(Borg,)i(Uniqueness)g(theorems)e (in)h(the)g(sp)r(ectral)h(theory)f(of)g Fc(y)2427 2286 y Fb(00)2488 2310 y Ft(+)19 b(\()p Fc(\025)i Fa(\000)e Fc(q)r Ft(\()p Fc(x)p Ft(\)\))p Fc(y)34 b Ft(=)29 b(0.)h Fo(Pr)l(o)l(c.)i(11th)601 2393 y(Sc)l(andinavian)27 b(Congr)l(ess)f(of) f(Mathematicians)p Ft(,)f(Johan)g(Grundt)g(T)-6 b(an)n(ums)23 b(F)-6 b(orlag,)23 b(Oslo)g(\(1952\))i(276{)601 2476 y(287)491 2559 y([4])35 b(P)-6 b(.)27 b(Bougerol)h(and)g(J.)f(Lacroix,) h Fo(Pr)l(o)l(ducts)j(of)f(r)l(andom)h(matric)l(es)f(with)f(applic)l (ations)i(to)f(Schr\177)-36 b(odinger)601 2642 y(op)l(er)l(ators)p Ft(.)25 b(Birkh\177)-35 b(auser,)23 b(Boston{Stuttgart)k(\(1985\))491 2725 y([5])35 b(D.)d(Busc)n(hmann)g(and)h(G.)f(Stolz,)k(Tw)n (o-parameter)c(sp)r(ectral)h(a)n(v)n(eraging)g(and)g(lo)r(calization)g (for)f(non-)601 2808 y(monotonic)24 b(random)e(Sc)n(hr\177)-35 b(odinger)25 b(op)r(erators.)f Fo(T)-5 b(r)l(ans.)26 b(A)n(mer.)f(Math.)h(So)l(c.)e Fd(353)f Ft(\(2000\),)i(635{653)491 2891 y([6])35 b(R.)f(Carmona,)j(A.)d(Klein,)j(and)f(F.)e(Martinelli,)i (Anderson)g(lo)r(calization)g(for)e(Bernoulli)g(and)i(other)601 2974 y(singular)23 b(p)r(oten)n(tials.)h Fo(Commun.)j(Math.)f(Phys.)e Fd(108)e Ft(\(1987\),)j(41{66)491 3057 y([7])35 b(R.)27 b(Carmona)h(and)h(J.)f(Lacroix,)h Fo(Sp)l(e)l(ctr)l(al)j(the)l(ory)e (of)g(r)l(andom)i(Schr\177)-36 b(odinger)31 b(op)l(er)l(ators)p Ft(.)f(Birkh\177)-35 b(auser,)601 3140 y(Basel{Berlin)23 b(\(1990\))491 3223 y([8])35 b(E.A.)24 b(Co)r(ddington)j(and)g(N.)e (Levinson,)h Fo(The)l(ory)i(of)g(or)l(dinary)h(di\013er)l(ential)e(e)l (quations)p Ft(.)f(McGra)n(w-Hill,)601 3306 y(New)d(Y)-6 b(ork)24 b(\(1955\))491 3389 y([9])35 b(D.)24 b(Damanik,)f(R.)h(Sims)f (and)i(G.)g(Stolz,)g(Lo)r(calization)g(for)f(one-dimensional,)g(con)n (tin)n(uum,)h(Bernoulli-)601 3472 y(Anderson)f(mo)r(dels.)e(Preprin)n (t)h(2000,)h(a)n(v)l(ailable)h(at)f(mp-arc)e(00-404)456 3555 y([10])35 b(D.)26 b(Damanik,)h(R.)g(Sims)f(and)h(G.)h(Stolz,)g(Ly) n(apuno)n(v)h(exp)r(onen)n(ts)h(in)d(con)n(tin)n(uum)g (Bernoulli-Anderson)601 3638 y(mo)r(dels.)16 b(T)-6 b(o)18 b(app)r(ear)g(in:)27 b Fo(Pr)l(o)l(c)l(e)l(e)l(dings)c(of)d(the)f (Kovalevski-Symp)l(osium,)k(Op)l(er)l(ator)e(The)l(ory:)30 b(A)l(dvanc)l(es)601 3721 y(and)c(Applic)l(ation)p Ft(,)f(Birkh\177)-35 b(auser)456 3804 y([11])35 b(D.)23 b(Damanik,)f(R.)h(Sims)f(and)i(G.)g (Stolz,)g(in)f(preparation)456 3887 y([12])35 b(P)-6 b(.)28 b(Deift)g(and)g(E.)g(T)-6 b(rub)r(o)n(witz,)30 b(In)n(v)n(erse)f(Scattering)h(on)f(the)g(line.)e Fo(Commun.)k(Pur)l(e) g(Appl.)g(Math.)d Fd(32)601 3970 y Ft(\(1979\),)d(121{251)456 4053 y([13])35 b(D.H.)30 b(Dunlap,)k(H.-L.)c(W)-6 b(u)32 b(and)g(P)-6 b(.)31 b(W.)h(Phillips,)g(Absence)h(of)e(lo)r(calization)h (in)g(the)g(random-dimer)601 4136 y(mo)r(del.)22 b Fo(Phys.)k(R)l(ev.)f (L)l(ett.)e Fd(65)g Ft(\(1990\),)i(88{91)456 4219 y([14])35 b(I.)15 b(Herbst)h(and)h(J.)f(Ho)n(wland,)h(The)f(Stark)h(ladder)f(and) g(other)h(one-dimensional)e(external)i(\014eld)f(problems.)601 4302 y Fo(Commun.)26 b(Math.)g(Phys.)e Fd(80)f Ft(\(1981\),)i(23{42)456 4385 y([15])35 b(P)-6 b(.D.)23 b(Hislop)h(and)h(F.)f(Klopp,)g(The)h(in) n(tegrated)h(densit)n(y)f(of)g(states)g(for)f(some)g(random)f(op)r (erators)i(with)601 4468 y(nonsign)f(de\014nite)h(p)r(oten)n(tials.)f (Preprin)n(t)g(2001,)g(a)n(v)l(ailable)g(at)g(mp-arc)e(01-139)456 4551 y([16])35 b(S.)23 b(Jitomirsk)l(a)n(y)n(a,)g(H.)g(Sc)n(h)n (ulz-Baldes)h(and)g(G.)g(Stolz,)g(in)f(preparation)456 4634 y([17])35 b(W.)23 b(Kirsc)n(h,)f(On)h(a)g(class)g(of)g(random)f (Sc)n(hr\177)-35 b(odinger)24 b(op)r(erators.)f Fo(A)l(dv.)j(in)f (Appl.)h(Math.)d Fd(6)g Ft(\(1985\),)h(177{)601 4717 y(187)456 4800 y([18])35 b(W.)18 b(Kirsc)n(h,)h(P)-6 b(.)18 b(Stollmann,)h(G.)g(Stolz,)h(Anderson)f(Lo)r(calization)h(for)e (Random)h(Sc)n(hr\177)-35 b(odinger)19 b(Op)r(erators)601 4883 y(with)k(Long)i(Range)f(In)n(teractions.)h Fo(Commun.)i(Math.)f (Phys.)e Fd(195)e Ft(\(1998\),)k(495{507)456 4967 y([19])35 b(F.)c(Klopp,)i(Lo)r(calization)f(for)f(semiclassical)f(con)n(tin)n (uous)j(random)d(Sc)n(hr\177)-35 b(odinger)32 b(op)r(erators)g(I)r(I:)g (The)601 5050 y(random)22 b(displacemen)n(t)i(mo)r(del.)f Fo(Helv.)i(Phys.)h(A)l(cta)j Fd(66)23 b Ft(\(1993\),)i(810{841)456 5133 y([20])35 b(F.)23 b(Klopp,)g(Lo)r(calization)h(for)f(some)f(con)n (tin)n(uous)j(random)e(Sc)n(hr\177)-35 b(odinger)24 b(op)r(erators.)g Fo(Commun.)i(Math.)601 5216 y(Phys.)e Fd(167)e Ft(\(1995\),)j(553{569)p eop %%Page: 15 15 15 14 bop 1469 226 a Fn(LOCALIZA)-5 b(TION)30 b(STRA)-5 b(TEGIES)946 b(15)456 425 y Ft([21])35 b(S.)28 b(Kotani,)i(Ly)n(apuno)n (v)h(indices)d(determine)h(absolutely)h(con)n(tin)n(uous)g(sp)r(ectra)f (of)g(stationary)h(one)f(di-)601 508 y(mensional)i(Sc)n(hr\177)-35 b(odinger)32 b(op)r(erators.)g(In:)48 b Fo(Sto)l(chastic)34 b(A)n(nalysis)p Ft(,)g(ed.)e(K.)f(Ito,)k(pp.)d(225{247,)j(North)601 591 y(Holland,)23 b(Amsterdam)e(\(1984\))456 674 y([22])35 b(S.)24 b(Kotani,)h(Ly)n(apuno)n(v)i(exp)r(onen)n(ts)g(and)e(sp)r (ectra)h(for)e(one-)h(dimensional)e(random)h(Sc)n(hr\177)-35 b(odinger)26 b(op)r(er-)601 757 y(ators.)i(In:)42 b Fo(R)l(andom)32 b(Matric)l(es)e(and)h(their)f(Applic)l(ations)p Ft(,)i(eds.)c(J.E.)g (Cohen,)j(H.)d(Kesten)h(and)h(C.M.)601 840 y(Newman,)k(American)d (Mathematical)i(So)r(ciet)n(y)-6 b(,)37 b Fo(Contemp)l(or)l(ary)f (Math.)d Fd(50)p Ft(,)h(277{286,)i(Pro)n(vidence)601 923 y(\(1986\))456 1006 y([23])f(S.)21 b(Kotani)g(and)h(B.)f(Simon,)f (Lo)r(calization)j(in)d(general)i(one-dimensional)f(random)f(systems.)g Fo(Commun.)601 1089 y(Math.)25 b(Phys.)f Fd(112)f Ft(\(1987\))i (103{119)456 1172 y([24])35 b(N.)21 b(Minami,)f(An)j(Extension)g(of)f (Kotani's)g(Theorem)g(to)h(Random)f(Generalized)h(Sturm-Liouville)e (Op-)601 1255 y(erators.)i Fo(Commun.)k(Math.)f(Phys.)d Fd(103)g Ft(\(1986\))j(387{402)456 1338 y([25])35 b(N.)22 b(Minami,)f(An)i(extension)h(of)f(Kotani's)g(theorem)g(to)h(random)e (generalized)i(Sturm-Liouville)d(op)r(era-)601 1421 y(tors)h(I)r(I.)h (In:)31 b Fo(Sto)l(chastic)25 b(pr)l(o)l(c)l(esses)i(in)d(classic)l(al) i(and)g(quantum)f(systems)j Ft(411{419,)c Fo(L)l(e)l(ctur)l(e)h(Notes)f (in)601 1504 y(Physics)29 b Fd(262)p Ft(,)22 b(Springer,)h (Berlin-Heidelb)r(erg-New)f(Y)-6 b(ork)24 b(\(1986\))456 1588 y([26])35 b(L.)15 b(P)n(astur)h(and)h(A.)e(Figotin,)j Fo(Sp)l(e)l(ctr)l(a)h(of)g(R)l(andom)h(and)g(A)n(lmost-Perio)l(dic)f (Op)l(er)l(ators)p Ft(.)e(Springer)f(V)-6 b(erlag,)601 1671 y(Berlin-Heidelb)r(erg-New)22 b(Y)-6 b(ork)23 b(\(1992\))456 1754 y([27])35 b(C.)40 b(Sh)n(ubin,)46 b(R.)41 b(V)-6 b(akilian)41 b(and)g(T.)g(W)-6 b(ol\013,)46 b(Some)41 b(harmonic)f(analysis)h(questions)h(suggested)h(b)n(y)601 1837 y(Anderson-Bernoulli)22 b(mo)r(dels.)h Fo(Ge)l(om.)j(F)-5 b(unct.)25 b(A)n(nal.)f Fd(8)g Ft(\(1998\),)h(932{964)456 1920 y([28])35 b(R.)23 b(Sims,)e(PhD-Thesis,)i(Univ.)g(of)g(Alabama)g (at)h(Birmingham,)d(in)i(preparation)456 2003 y([29])35 b(R.)23 b(Sims)f(and)i(G.)f(Stolz,)h(Lo)r(calization)h(in)e(one)i (dimensional)d(random)h(media:)30 b(a)24 b(scattering)h(theoretic)601 2086 y(approac)n(h.)f Fo(Commun.)j(Math.)f(Phys.)e Fd(213)e Ft(\(2000\),)j(575{597)456 2169 y([30])35 b(G.)27 b(Stolz,)j(Sp)r (ectral)e(theory)h(of)f(Sc)n(hr\177)-35 b(odinger)28 b(op)r(erators)h(with)f(p)r(oten)n(tials)h(of)f(in\014nite)g(barriers)f (t)n(yp)r(e.)601 2252 y(Habilitationssc)n(hrift,)c(F)-6 b(rankfurt)23 b(Univ)n(ersit)n(y)h(\(1994\))456 2335 y([31])35 b(G.)26 b(Stolz,)i(Lo)r(calization)f(for)f(random)g(Sc)n (hr\177)-35 b(odinger)27 b(op)r(erators)g(with)f(P)n(oisson)h(p)r(oten) n(tial.)g Fo(A)n(nn.)h(Inst.)601 2418 y(Henri)d(Poinc)l(ar)n(\023)-35 b(e)31 b Fd(63)22 b Ft(\(1995\),)k(297{314)456 2501 y([32])35 b(G.)29 b(Stolz,)i(Non-monotonic)e(Random)g(Sc)n(hr\177)-35 b(odinger)29 b(Op)r(erators:)43 b(The)29 b(Anderson)h(Mo)r(del.)e Fo(J.)j(Math.)601 2584 y(A)n(nal.)26 b(Appl.)e Fd(248)f Ft(\(2000\),)i(173{183)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0109241027225--