Content-Type: multipart/mixed; boundary="-------------0312111027136" This is a multi-part message in MIME format. ---------------0312111027136 Content-Type: text/plain; name="03-536.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-536.comments" This paper replaces mp_arc 02-450. A slightly shorter version of this paper will appear in: Markov Processes and Related Fields. ---------------0312111027136 Content-Type: text/plain; name="03-536.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-536.keywords" random Schroedinger operators ---------------0312111027136 Content-Type: application/postscript; name="pastur.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pastur.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: pastur.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Italic Times-Bold Helvetica Courier %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips pastur %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.12.11:1721 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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2033 y(Poincar)5 b(\264)-33 b(e)20 b(gauge)e(are)i(de\002ned)g(by)f Fx(A)1361 2045 y Fv(j)1397 2033 y FB(\()p Fx(x)p FB(\))24 b(:=)1652 2000 y Ft(1)p 1652 2014 34 4 v 1652 2062 a(2)1709 1971 y Fr(P)1797 1991 y Fv(d)1797 2058 y(i)p Ft(=1)1922 2033 y Fx(x)1969 2045 y Fv(i)1997 2033 y Fx(B)2060 2045 y Fv(ij)2140 2033 y FC(for)19 b(all)i Fx(x)j FB(=)e(\()p Fx(x)2598 2045 y Ft(1)2636 2033 y Fx(;)14 b(:)g(:)g(:)g(;)g(x)2868 2045 y Fv(d)2907 2033 y FB(\))23 b Fu(2)h Fw(R)3111 2003 y Fv(d)3149 2033 y FC(.)428 2133 y(Choosing)14 b(physical)h(units)g (such)g(that)h(the)f(mass)h(and)f(electric)h(char)o(ge)d(of)j(the)f (particle)g(as)h(well)g(as)g(Planck')-5 b(s)278 2232 y(constant)28 b(\(di)n(vided)e(by)h FB(2)p Fx(\031)s FC(\))h(are)f(all)i(equal)e(to)h(one,)h(the)f(Schr)7 b(\250)-35 b(odinger)25 b(operator)h(for)h(the)h(quantum)e(particle)278 2332 y(subject)g(to)g(a)g Fy(r)m(ealization)g Fx(V)1145 2302 y Ft(\()p Fv(!)r Ft(\))1278 2332 y FB(:)34 b Fw(R)1405 2302 y Fv(d)1477 2332 y Fu(!)g Fw(R)p 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FC(with)f(a)g(random)f(potential)g Fx(V)278 3863 y FC(and)j(a)h (constant)e(magnetic)g(\002eld,)43 b(corresponding)34 b(to)39 b(the)f(v)o(ector)f(potential)g Fx(A)p FC(,)43 b(we)c(mean)f(the)g(f)o(amily)278 3963 y FB(\012)338 3975 y Ft(0)399 3963 y Fu(3)23 b Fx(!)j Fu(7!)d Fx(H)7 b FB(\()p Fx(A;)14 b(V)935 3933 y Ft(\()p Fv(!)r Ft(\))1035 3963 y FB(\))21 b FC(of)f(Schr)7 b(\250)-35 b(odinger)17 b(operators)i(gi)n(v)o(en)g(by)h(\(1.1\).)428 4221 y(In)g(this)g(surv)o (e)o(y)e(we)j(basically)e(focus)h(on)f(tw)o(o)h(e)o(xamples)f(of)h (random)e(potentials)h(in)i(the)e(sense)i(of)e(De\002ni-)278 4320 y(tion)f(1.1,)f(namely)g(Gaussian)h(and)f(non-ne)o(gati)n(v)o(e)d (Poissonian)j(ones.)24 b(Both)18 b(are)f(rather)g(popular)f(in)i(the)g (physics)278 4420 y(literature)i([)p Fz(152)n(,)h(22)o(,)f(125)o(,)g (100)o(,)g(73)o(,)h(45)o FC(],)f(see)h(also)f([)p Fz(88)o FC(])g(and)g(references)f(therein.)428 4578 y Fz(De\002nition)26 b(1.3.)43 b FC(By)27 b(a)f Fy(Gaussian)f(r)o(andom)f(potential)h 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b(of)h(this)h(\002eld)g(which)e(is)j(jointly)d(measurable.)24 b(Referring)18 b(to)h(a)h(Gaussian)f(random)f(potential,)g(we)i(will) 278 5234 y(tacitly)j(assume)g(that)g(only)e(this)j(v)o(ersion)d(will)i (be)g(dealt)g(with.)32 b(By)23 b(the)g(Bochner)n(-Khintchine)c(theorem) j(there)278 5334 y(is)h(a)f(one-to-one)c(correspondence)g(between)j (Gaussian)g(random)e(potentials)i(and)g(\002nite)h(positi)n(v)o(e)e (\(and)h(e)n(v)o(en\))p eop %%Page: 4 4 4 3 bop 278 233 a FI(4)831 b(HAJO)19 b(LESCHKE,)e(PETER)g(M)1860 220 y(\250)1849 233 y(ULLER,)f(AND)j(SIMONE)g(W)-7 b(ARZEL)278 425 y FC(Borel)17 b(measures)g(on)f Fw(R)977 395 y Fv(d)1016 425 y FC(.)24 b(A)17 b(simple)g(suf)n(\002cient)g(criterion)e(ensuring) h Fw(R)2374 395 y Fv(d)2412 425 y FC(-er)o(godicity)f(is)i(the)g (mixing)f(condition)278 525 y FB(lim)394 540 y Fq(j)p Fv(x)p Fq(j!1)621 525 y Fx(C)6 b FB(\()p Fx(x)p FB(\))25 b(=)d(0)p FC(.)j(Furthermore,)18 b(the)i(e)o(xplicit)g(formula)278 744 y(\(1.2\))211 b Fw(E)715 676 y 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b(5)278 425 y FC(position)28 b(representation\))d FB(\002\()p Fx(E)k Fu(\000)24 b Fx(H)7 b FB(\()p Fx(A;)14 b(V)1651 395 y Ft(\()p Fv(!)r Ft(\))1751 425 y FB(\)\)\()p Fx(x;)g(y)s FB(\))p FC(,)32 b(which)27 b(is)i(a)g(jointly)e(continuous)f(function)h (of)g Fx(x)p FC(,)278 525 y Fx(y)f Fu(2)e Fw(R)494 495 y Fv(d)553 525 y FC(and)19 b(a)i Fw(P)p FC(-inte)o(grable)d(function)h (of)h Fx(!)25 b Fu(2)f FB(\012)1799 537 y Ft(0)1836 525 y FC(,)c(see)h([)p Fz(126)n(,)g(28)o(,)f(29)o FC(].)26 b(This)20 b(justi\002es)428 668 y Fz(De\002nition)c(1.5.)32 b FC(The)15 b Fy(inte)m(gr)o(ated)f(density)h(of)h(states)g FC(is)g(the)g(function)d Fx(N)32 b FB(:)23 b Fx(E)28 b Fu(7!)c Fx(N)9 b FB(\()p Fx(E)c FB(\))16 b FC(de\002ned)e(through)278 768 y(the)21 b(e)o(xpectation)d(v)n(alue)278 942 y(\(1.6\))182 b Fx(N)9 b FB(\()p Fx(E)c FB(\))24 b(:=)e Fw(E)1040 875 y Fr(\002)1075 942 y FB(\002)1140 875 y Fr(\000)1178 942 y Fx(E)h Fu(\000)18 b Fx(H)7 b FB(\()p Fx(A;)14 b(V)19 b FB(\))1651 875 y Fr(\001)1689 942 y FB(\()p Fx(x;)14 b(x)p FB(\))1884 875 y Fr(\003)1943 942 y FB(=)2017 829 y Fr(Z)2063 1018 y Ft(\012)2115 942 y Fw(P)p FB(\(d)p Fx(!)s FB(\))37 b(\002)2461 850 y Fr(\020)2510 942 y Fx(E)24 b Fu(\000)18 b Fx(H)7 b FB(\()p Fx(A;)14 b(V)2952 908 y Ft(\()p Fv(!)r Ft(\))3052 942 y FB(\))3084 850 y Fr(\021)3148 942 y FB(\()p Fx(x;)g(x)p FB(\))p Fx(:)428 1220 y FC(Thanks)21 b(to)h(the)g(unitary)f(in)m(v)n(ariance)f(of)i(the) g(kinetic-ener)o(gy)c(operator)j Fx(H)7 b FB(\()p Fx(A;)14 b FB(0\))22 b FC(under)f(so-called)g(mag-)278 1320 y(netic)g (translations)f([)p Fz(150)n FC(])h(and)f(to)h(the)g Fw(R)1486 1290 y Fv(d)1524 1320 y FC(-homogeneity)c(of)k Fx(V)e FC(,)i Fx(N)30 b FC(is)21 b(independent)d(of)j(the)f(chosen)g Fx(x)k Fu(2)h Fw(R)3492 1290 y Fv(d)3530 1320 y FC(.)278 1419 y(Moreo)o(v)o(er)m(,)17 b Fx(N)30 b FC(is)21 b(non-ne)o(gati)n(v)o (e,)16 b(non-decreasing)h(and)j(left-continuous.)428 1519 y(There)j(are)h(some)g(other)g(uni)n(v)o(ersally)e(v)n(alid)i 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b(who)278 824 y(w)o(as)32 b(the)e(\002rst)h(to)f (de)n(v)o(elop)f(a)i(quantitati)n(v)o(e)d(theory)h([)p Fz(97)o(,)h(98)o FC(])h(in)f(case)h Fx(A)42 b FB(=)g(0)p FC(.)55 b(Lifshits')30 b(ar)o(guments)e(can)278 923 y(be)23 b(summarized)d(in)j(terms)f(of)g(the)g(so-called)g Fy (optimal-\003uctuation)c(ideolo)o(gy)j FC([)p Fz(64)o(,)h(153)o(,)g(99) o FC(])h(\(see)f(also)h([)p Fz(125)n(,)278 1023 y(100)o(,)16 b(118)o FC(]\))f(according)f(to)i(which)f(the)h(lo)n(w-ener)o(gy)c(f)o (all-of)n(f)j(of)g Fx(N)25 b FC(near)16 b(the)f(almost-sure)g (ground-state)e(ener)o(gy)278 1123 y Fx(E)339 1135 y Ft(0)400 1123 y Fu(2)23 b FB([)p Fu(\0001)p Fx(;)14 b Fu(1)p FB([)21 b FC(of)f Fx(H)7 b FB(\()p Fx(A;)14 b(V)19 b FB(\))i FC(is)g(\(uni)n(v)o(ersally\))d(gi)n(v)o(en)h(by)g(the)i (formula)794 1302 y FB(log)14 b Fx(N)9 b FB(\()p Fx(E)c FB(\))24 b Fu(\030)33 b FB(inf)1232 1353 y Fv(\034)7 b(>)p Ft(0)1368 1210 y Fr(\020)1432 1302 y Fx(\034)i(E)24 b FB(+)80 b(sup)1604 1386 y Fv( )r Fq(2C)1734 1361 y 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FB(\()p Fx(T)7 b(;)18 b FB(\026)-46 b Fx(n)p FB(\))38 b Fu(\030)278 4248 y Fr(R)334 4269 y Fv(E)383 4277 y Fo(F)318 4345 y Fq(\0001)454 4315 y FB(d)p Fx(E)19 b(D)r FB(\()p Fx(E)5 b FB(\))p Fx(E)26 b Fu(\000)20 b Fx(T)1014 4285 y Ft(2)1050 4315 y Fx(D)r FB(\()p Fx(E)1214 4327 y Fv(F)1270 4315 y FB(\))14 b Fx(\031)1366 4285 y Ft(2)1404 4315 y Fx(=)p FB(6)22 b FC(as)i Fx(T)39 b Fu(#)28 b FB(0)p FC(,)23 b(where)g(the)g Fy(F)-6 b(ermi)23 b(ener)m(gy)f Fx(E)2726 4327 y Fv(F)2810 4315 y Fu(2)28 b Fw(R)c FC(is)f(the)g(solution)g(of) 278 4422 y(the)k(equation)d Fx(N)9 b FB(\()p Fx(E)888 4434 y Fv(F)944 4422 y FB(\))34 b(=)k(\026)-46 b Fx(n)p FC(.)43 b(Gi)n(v)o(en)25 b(\(1.7\),)h(a)h(suf)n(\002cient)e(condition)g (for)g(the)i(e)o(xistence)e(of)h(the)g(Lebesgue-)278 4522 y(Stieltjes)e(inte)o(gral)d(in)h(\(1.10\))f(is)i(the)f (\002niteness)h Fw(E)14 b FB([)o(exp)g(\()p Fu(\000)p Fx(\034)9 b(V)20 b FB(\(0\)\)])27 b Fx(<)f Fu(1)d FC(for)f(all)g Fx(\034)37 b Fu(2)27 b FB([0)p Fx(;)14 b Fu(1)p FB([)p 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b(Accordingly)17 b(these)i(eigenfunctions)e(are)i(delocalized)f(o)o(v)o (er)g(the)h(whole)f(solid,)278 5234 y(hence)f(not)g(square-inte)o (grable,)d(and)j(the)h(whole)e(ener)o(gy)g(spectrum)g(is)i(only)f (absolutely)f(continuous.)22 b(That)17 b(the)278 5334 y(spectrum)h(is)i(absolutely)e(continuous)f(e)n(v)o(en)h(in)h(the)g(lo) n(w-ener)o(gy)d(re)o(gime,)i(may)g(be)h(vie)n(wed)f(as)i(a)g (consequence)p eop %%Page: 7 7 7 6 bop 1025 234 a FI(RANDOM)18 b(SCHR)1465 221 y(\250)1454 234 y(ODINGER)g(OPERA)-6 b(T)o(ORS)18 b(FOR)h(AMORPHOUS)g(SOLIDS)718 b(7)278 425 y FC(of)25 b(the)g(tunnelling)e(ef)n(fect.)39 b(According)23 b(to)i(classical)h(mechanics)e(the)h(electron)f(w)o (ould)g(be)h(localized)g(in)g(one)278 525 y(of)g(the)g(identical)g (atomic)f(potential)g(wells)i(making)e(up)g(the)h(crystal.)39 b(Since)25 b(e)n(v)o(en)f(small)i(dif)n(ferences)d(in)i(the)278 624 y(potential)30 b(wells)h(may)e(suppress)h(\223quantum)e (coherence\224)g(and)i(hence)f(tunnelling,)i(localized)e(states)j(gi)n (v)o(en)278 724 y(by)23 b(square-inte)o(grable)c(ener)o(gy)i (eigenfunctions)f(associated)j(with)f(pure-point)f(spectrum)g(should)h (emer)o(ge)f(in)278 824 y(disordered)j(solids)h(at)h(least)g(at)g(lo)n (w)f(ener)o(gies.)39 b(In)25 b(particular)m(,)g(this)g(should)g(be)g (true)g(for)f(amorphous)f(solids.)278 923 y(As)g(we)f(will)h(see)f(in)g (Theorem)e(3.7)h(and)h(Theorem)e(3.9)h(belo)n(w)-5 b(,)21 b(localization)f(at)j(lo)n(w)f(ener)o(gies)e(can)i(indeed)e(be)278 1023 y(pro)o(v)o(en)e(in)j(case)f(of)g(certain)g(Gaussian)g(random)e (potentials)i(for)f(arbitrary)g Fx(d)k Fu(2)h Fw(N)p FC(.)428 1123 y(Since)32 b(by)f(their)h(v)o(ery)f(nature,)j(localized)d (states)i(are)f(not)f(capable)h(of)f(contrib)n(uting)f(to)i (macroscopic)278 1222 y(char)o(ge)20 b(transport,)f(at)i(a)h(certain)e (\(non-random\))c(ener)o(gy)j(a)j Fy(mobility)e(edg)o(e)g FC(is)i(e)o(xpected)d(to)i(occur)f([)p Fz(109)n FC(])h(in)g(an)278 1322 y(amorphous)f(solid)j(which)f(separates)g(localized)g(states)h(in) 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b(precisely)-5 b(,)18 b(one)i(requires)f(the)h(\002niteness)278 2775 y(\(1.11\))880 b FB(sup)1361 2844 y Fv(t)p Fq(2)p Fp(R)1513 2662 y Fr(Z)1559 2850 y Fp(R)1613 2834 y Fo(d)1639 2775 y FB(d)p Fx(x)1755 2704 y Fr(\014)1755 2754 y(\014)1783 2775 y Fx( )1840 2732 y Ft(\()p Fv(!)r Ft(\))1837 2799 y Fv(t;I)1940 2775 y FB(\()p Fx(x)p FB(\))2051 2704 y Fr(\014)2051 2754 y(\014)2080 2725 y Ft(2)2131 2775 y Fu(j)p Fx(x)p Fu(j)2224 2740 y Fv(q)2298 2775 y Fx(<)23 b Fu(1)278 3070 y FC(of)16 b(the)g Fx(q)s FC(th)g(absolute)f(spatial)h(moment)e(of)i Fx( )1560 3027 y Ft(\()p Fv(!)r Ft(\))1557 3094 y Fv(t;I)1683 3070 y FB(:=)22 b(e)1830 3040 y Fq(\000)p Ft(i)p Fv(tH)t Ft(\()p Fv(A;V)2134 3015 y Fn(\()p Fo(!)q Fn(\))2221 3040 y Ft(\))2252 3055 y Fx(\037)2304 3082 y Fv(I)2342 3070 y FB(\()p Fx(H)7 b FB(\()p Fx(A;)14 b(V)2648 3040 y Ft(\()p Fv(!)r Ft(\))2748 3070 y FB(\)\))p Fx( )2866 3082 y Ft(0)2920 3070 y FC(for)h(all)i(times)f Fx(t)23 b Fu(2)g Fw(R)p FC(,)278 3184 y(all)g Fx( )438 3196 y Ft(0)502 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y(quantum)24 b(Hall)i(ef)n(fect,)g(are)f (belie)n(v)o(ed)f(to)h(be)h(microscopically)d(e)o(xplainable)g(in)j (terms)f(of)g(a)h(system)f(of)h(non-)278 4836 y(interacting)19 b(electrically)g(char)o(ged)f(fermions)h(in)h(the)g(Euclidean)e(plane)i Fw(R)2471 4805 y Ft(2)2528 4836 y FC(subject)g(to)g(a)g(perpendicular)d (con-)278 4935 y(stant)24 b(magnetic)f(\002eld)g(of)g(strength)g Fx(B)33 b FB(:=)28 b Fx(B)1619 4947 y Ft(12)1719 4935 y Fx(>)g FB(0)c FC(and)e(a)i(random)e(potential)g([)p Fz(73)o(,)i(149)n FC(].)35 b(The)23 b(underlying)278 5035 y(one-particle)e(Schr)7 b(\250)-35 b(odinger)19 b(operator)i(is)i(kno)n(wn)e(as)i(the)f Fy(r)o(andom)f(Landau)f (Hamiltonian)h FC(which)h(acts)g(on)g(the)278 5134 y(Hilbert)f(space)g FB(L)793 5104 y Ft(2)830 5134 y FB(\()p Fw(R)932 5104 y Ft(2)969 5134 y FB(\))p FC(.)28 b(Apart)21 b(from)f(numerous)f (theoretical)h(and)g(numerical)g(studies)h(in)g(the)g(physics)f(liter)n (-)278 5234 y(ature,)26 b(there)e(are)h(no)n(w)o(adays)f(quite)g(a)i 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y Fu(\000)18 b FB(1)1309 1349 y Fr(\001)1347 1366 y Fv(l)1372 1416 y FC(,)j Fx(\030)27 b Fu(2)d Fw(R)p FC(,)c(is)h(the)f Fx(l)r FC(th)g(Le)o(gendre)e(polynomial)g([)p Fz(62)n FC(].)428 1522 y(Ne)o(glecting)d(ef)n(fects)i(of)g(le)n(v)o(el)f (mixing)g(by)g(only)g(dealing)g(with)h(the)g(sequence)f(of)h (restricted)f(random)f(oper)n(-)278 1622 y(ators)k Fu(f)o Fx(P)553 1634 y Fv(l)579 1622 y Fx(V)g(P)699 1634 y Fv(l)725 1622 y Fu(g)766 1647 y Fv(l)811 1622 y FC(is)g(a)g(simplifying)e (approximation)e(which)j(is)h(often)f(made.)23 b(The)18 b(interest)h(in)f(these)h(operators)278 1721 y(relates)26 b(to)f(spectral)g(localization)f([)p Fz(65)o(,)h(38)o(,)h(39)o(,)f(40)o (,)h(119)n(,)f(120)o FC(])g(and)g(to)g(properties)f(of)h(their)g (\(restricted\))e(in-)278 1821 y(te)o(grated)f(density)h(of)g(states)h Fx(R)1196 1833 y Fv(l)1222 1821 y FC(,)g(see)g(for)e(e)o(xample)g([)p Fz(146)n(,)i(23)o(,)f(83)o(,)h(19)o(,)f(18)o(,)h(11)o(,)f(124)o(,)g(25) o(,)h(21)o(,)f(70)o FC(].)35 b(From)22 b(the)278 1921 y(physical)j(point)g(of)h(vie)n(w)f(most)h(interesting)f(is)h(the)g (restriction)f(to)h(the)g(lo)n(west)g(Landau)e(band)h(\()p Fx(l)34 b FB(=)f(0)p FC(\).)42 b(F)o(or)278 2020 y(strong)20 b(enough)e(magnetic)h(\002elds)i(all)f(fermions)f(may)h(be)g (accommodated)e(in)i(the)g(lo)n(west)h(band)e(without)g(con-)278 2120 y(\003icting)j(with)h(P)o(auli')-5 b(s)22 b(e)o(xclusion)f (principle,)g(since)h(the)g(speci\002c)h(de)o(generac)o(y)c(increases)j (with)g(the)g(magnetic)278 2220 y(\002eld.)52 b(The)29 b(contrib)n(ution)d(of)j FB(2)p Fx(\031)s(R)1345 2232 y Ft(0)1382 2220 y FB(\()p Fx(E)1475 2232 y Fv(F)1556 2220 y Fu(\000)24 b Fx(B)t(=)p FB(2\))p Fx(=B)33 b FC(to)c(the)g(sum)g (of)g(the)g(series)h(in)f(\(1.15\))e(at)i(the)g(Fermi)278 2319 y(ener)o(gy)e Fx(E)593 2331 y Fv(F)678 2319 y FC(should)h(then)g (already)g(be)h(a)g(good)e(approximation)f(to)j Fx(N)9 b FB(\()p Fx(E)2521 2331 y Fv(F)2576 2319 y FB(\))p FC(,)32 b(since)d(the)g(ef)n(fects)f(of)h(higher)278 2426 y(Landau)18 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Fw(E)1535 3079 y Fr(\002)1570 3147 y Fu(j)p Fx(V)19 b FB(\(0\))p Fu(j)1789 3112 y Ft(2)p Fv(n)1867 3079 y Fr(\003)1925 3147 y Fu(\024)j FB(\(2)p Fx(n)p FB(\)!)14 b Fx(M)2295 3112 y Ft(2)p Fv(n)278 3305 y FC(for)29 b(all)g Fx(n)39 b Fu(2)g Fw(N)30 b FC(with)f(some)f(constant)g Fx(M)48 b(<)39 b Fu(1)p FC(.)51 b(While)29 b(\(1.20\))e(is)j (satis\002ed)g(for)e(all)h(Gaussian)g(random)278 3406 y(potentials)19 b(with)g Fx(M)31 b FB(=)988 3335 y Fr(p)p 1071 3335 V 71 x Fx(C)6 b FB(\(0\))20 b FC(\(see)f(\(1.2\)\),)e(its)j (v)n(alidity)e(for)h(non-ne)o(gati)n(v)o(e)14 b(Poissonian)19 b(random)e(potentials)278 3506 y(is)h(ensured)e([)p Fz(26)o FC(])h(by)g(the)g(additional)f(\(Lebesgue-essential\))e(boundedness)h (of)i(the)g(single-impurity)d(potential,)278 3605 y Fx(U)51 b Fu(2)42 b FB(L)535 3575 y Fq(1)605 3605 y FB(\()p Fw(R)707 3575 y Ft(2)744 3605 y FB(\))p FC(.)56 b(Moreo)o(v)o(er)m(,)30 b Fx(R)1292 3617 y Fv(l)1348 3605 y FC(in)h(\(1.16\))d(is)k(well)e 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FC(is)i(independent)d(of)h(the)i(chosen)e Fx(x)k Fu(2)h Fw(R)1934 3891 y Ft(2)1971 3921 y FC(.)278 4188 y FA(2.)46 b(Self-a)n(v)o(eraging)26 b(and)c(uniqueness)h(of)g(the)g(integrated)i (density)e(of)h(states)428 4337 y FC(Since)e(spatially)h(separated,)f (lar)o(ge)f(parts)i(of)f(a)h(macroscopic)d(sample)i(of)h(an)f (amorphous)e(solid)i(become)278 4437 y(decoupled)e(rather)h(f)o(ast)h (with)g(increasing)f(distance,)g(the)o(y)g(ef)n(fecti)n(v)o(ely)f (correspond)f(to)j(dif)n(ferent)e(realizations)278 4537 y(of)c(the)f(er)o(godic)f(random)g(potential)h(modelling)f(the)h (solid.)24 b(As)16 b(a)g(consequence,)e(it)i(should)f(mak)o(e)g(no)g (dif)n(ference)278 4636 y(whether)k(the)g(inte)o(grated)f(density)h(of) g(states)i Fx(N)28 b FC(is)21 b(de\002ned)d(as)i(an)g(ensemble)e(a)n(v) o(erage)h(or)g(as)h(a)g(spatial)f(a)n(v)o(erage)278 4736 y(for)h(a)h(gi)n(v)o(en)e(typical)g(realization.)428 4836 y(T)-7 b(o)28 b(specify)f(the)h(notion)f(of)g(a)i(spatial)f(a)n(v) o(erage)f(associated)g(with)h 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b(bounded)d(and)h(from)278 3183 y([)p Fz(36)o(,)30 b(67)o FC(])g(if)g Fx(V)50 b FC(is)30 b(bounded)e(from)g(belo)n(w)-5 b(.)53 b(F)o(or)30 b(random)e(potentials)h Fx(V)49 b FC(yielding)29 b(Schr)7 b(\250)-35 b(odinger)27 b(operators)278 3283 y Fx(H)7 b FB(\()p Fx(A;)14 b(V)20 b FB(\))25 b FC(which)e(are)i(almost)f(surely)f(unbounded)e(from)i(belo)n(w)-5 b(,)24 b(the)h(proof)d(of)i(\(2.2\))f(can)h(be)g(found)f(in)h([)p Fz(72)o(,)278 3382 y(29)o FC(],)d(see)f(also)h([)p Fz(118)n FC(,)g(Thm.)e(5.20])g(for)g Fx(A)k FB(=)g(0)p FC(.)428 3535 y Fz(Remark)f(2.2.)41 b FC(Similar)22 b(as)h(in)f(equilibrium)e (statistical-mechanics)i([)p Fz(123)n FC(])g(there)g(are)g(more)f (general)g(se-)278 3634 y(quences)i(of)g(re)o(gions)g(e)o(xpanding)d (to)k Fw(R)1458 3604 y Fv(d)1521 3634 y FC(than)f(concentric)f(open)g (cubes)h FB(\003)h FC(for)f(which)g(\(2.2\))f(is)j(true,)f(see)g(for) 278 3734 y(e)o(xample)g([)p Fz(118)n FC(,)j(p.)d(105],)h([)p Fz(30)o FC(,)h(p.)f(304])f(or)g([)p Fz(36)o FC(].)40 b(Moreo)o(v)o(er)m(,)23 b(the)i(con)m(v)o(er)o(gence)c(\(2.2\))i(holds) i(for)f(an)o(y)g(other)278 3834 y(boundary)j(condition)h FB(X)j FC(for)e(which)g(the)g(self-adjoint)g(operator)e Fx(H)2334 3846 y Ft(\003)p Fv(;)p Ft(X)2453 3834 y FB(\()p Fx(A;)14 b(V)2651 3804 y Ft(\()p Fv(!)r Ft(\))2751 3834 y FB(\))31 b FC(obe)o(ys)d(the)i(inequalities)278 3940 y Fx(H)347 3952 y Ft(\003)p Fv(;)p Ft(N)466 3940 y FB(\()p Fx(A;)14 b(V)664 3910 y Ft(\()p Fv(!)r Ft(\))764 3940 y FB(\))39 b Fu(\024)f Fx(H)1007 3952 y Ft(\003)p Fv(;)p Ft(X)1125 3940 y FB(\()p Fx(A;)14 b(V)1323 3910 y Ft(\()p Fv(!)r Ft(\))1423 3940 y FB(\))39 b Fu(\024)f Fx(H)1666 3952 y Ft(\003)p Fv(;)p Ft(D)1785 3940 y FB(\()p Fx(A;)14 b(V)1983 3910 y Ft(\()p Fv(!)r Ft(\))2083 3940 y FB(\))30 b FC(in)e(the)h(sense)g(of)f(\(sesquilinear\))f(forms.)49 b(The)278 4039 y(case)29 b(of)e(those)h(mix)o(ed)f(\(in)h(other)f(w)o (ords:)40 b(Robin\))27 b(boundary)e(conditions,)k(which)e(cannot)g(be)h (sandwiched)278 4139 y(between)20 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(\002eld.)29 b(In)22 b(particular)m(,)e(this)i(type)f(of)h(f)o(all-of)n (f)e(ensures)h(that)h Fx(H)7 b FB(\()p Fx(A;)14 b(V)19 b FB(\))p FC(,)j(although)e(the)278 5334 y(latter)d(is)h(almost)e (surely)g(unbounded)d(from)j(belo)n(w)-5 b(,)16 b(may)g(serv)o(e)g(as)h (the)f(one-particle)f(Schr)7 b(\250)-35 b(odinger)14 b(operator)h(of)p eop %%Page: 11 11 11 10 bop 1025 234 a FI(RANDOM)18 b(SCHR)1465 221 y(\250)1454 234 y(ODINGER)g(OPERA)-6 b(T)o(ORS)18 b(FOR)h(AMORPHOUS)g(SOLIDS)689 b(11)278 425 y FC(a)23 b(macroscopic)d(system)i(of)g(non-interacting)c (fermions)j(in)h(a)h(random)d(medium)g(with)j(well-de\002ned)d (speci\002c)278 525 y(free)g(ener)o(gy)e(\(1.10\))h(and)g(related)h (thermodynamic)d(quantities.)428 692 y Fz(Theor)o(em)24 b(3.1.)43 b Fy(Let)25 b Fx(V)44 b Fy(be)24 b(a)g(Gaussian)g(r)o(andom)f (potential)g(with)i(co)o(variance)e(function)g Fx(C)6 b Fy(.)38 b(Then)24 b(the)278 792 y(leading)19 b(low-ener)m(gy)g (fall-of)o(f)h(of)g(the)g(inte)m(gr)o(ated)f(density)h(of)g(states)h Fx(N)30 b Fy(is)21 b(Gaussian)e(in)h(the)h(sense)f(that)1332 1006 y FB(log)14 b Fx(N)9 b FB(\()p Fx(E)c FB(\))23 b Fu(\030)g(\000)1899 950 y Fx(E)1965 920 y Ft(2)p 1844 987 213 4 v 1844 1063 a FB(2)p Fx(C)6 b FB(\(0\))3120 1006 y(\()p Fx(E)29 b Fu(!)23 b(\0001)p FB(\))p Fx(:)-3273 b FC(\(3.1\))428 1339 y(Theorem)19 b(3.1)h(dates)g(back)g(to)h(P)o (astur)f([)p Fz(113)n(,)h(115)o FC(],)g(see)g(also)g([)p Fz(111)n(,)g(77)o FC(],)f(and)g([)p Fz(24)o(,)h(27)o(,)g(106)n(,)g(140) o FC(])f(for)g(the)278 1439 y(magnetic)j(case,)h(where)f(the)g(last)h (tw)o(o)g(references)e(e)n(v)o(en)g(allo)n(w)i(the)f(presence)f(of)i (certain)e Fy(r)o(andom)g FC(magnetic)278 1538 y(\002elds.)39 b(The)25 b(by)f(no)n(w)g(standard)f(w)o(ay)i(of)g(pro)o(ving)d (\(3.1\),)i(already)g(used)g(by)g(P)o(astur)m(,)h(is)h(in)f(the)f (spirit)h(of)g(\(1.9\))278 1638 y(\(setting)f(there)f Fx(E)803 1650 y Ft(0)871 1638 y FB(=)29 b Fu(\0001)p FC(\).)36 b(One)23 b(\002rst)i(determines)e(the)h(leading)e(asymptotic) h(beha)n(viour)f(of)h(the)h(Laplace-)278 1745 y(Stieltjes)h(transform,) 959 1724 y Fr(e)938 1745 y Fx(N)8 b FB(\()p Fx(\034)h FB(\))30 b(:=)1268 1678 y Fr(R)1307 1775 y Fp(R)1379 1745 y FB(d)p Fx(N)9 b FB(\()p Fx(E)c FB(\))29 b(exp)o(\()p Fu(\000)p Fx(\034)9 b(E)c FB(\))p FC(,)25 b Fx(\034)39 b(>)28 b FB(0)p FC(,)23 b(of)g Fx(N)32 b FC(as)24 b Fx(\034)38 b Fu(!)29 b(1)23 b FC(and)g(then)f(applies)h(an)278 1845 y(appropriate)18 b(T)-7 b(auberian)19 b(theorem)g([)p Fz(20)n FC(].)428 1944 y(The)34 b(Lifshits)h(tail)h(\(3.1\))d(in)i (case)h(of)e(Gaussian)h(random)e(potentials)h(is)i(highly)e(uni)n(v)o (ersal.)67 b(It)35 b(only)278 2044 y(depends)c(on)g(the)h(single-site)f (v)n(ariance)g Fx(C)6 b FB(\(0\))44 b(=)g Fw(E)1920 1977 y Fr(\002)1955 2044 y Fx(V)19 b FB(\(0\))2128 2014 y Ft(2)2165 1977 y Fr(\003)2244 2044 y Fx(>)44 b FB(0)p FC(,)34 b(b)n(ut)e(not)f(on)h(further)e(details)i(of)f(the)278 2144 y(co)o(v)n(ariance)38 b(function,)43 b(the)d(space)g(dimension)f (or)h(the)g(magnetic)f(\002eld.)84 b(As)41 b(physical)e(heuristics)h (and)278 2243 y(formula)45 b(\(1.9\))f(already)h(suggest,)52 b(the)46 b(\(non-ne)o(gati)n(v)o(e\))41 b(kinetic-ener)o(gy)i(operator) h Fx(H)7 b FB(\()p Fx(A;)14 b FB(0\))46 b FC(becomes)278 2343 y(irrele)n(v)n(ant)32 b(at)i(e)o(xtremely)e(ne)o(gati)n(v)o(e)g (ener)o(gies)g(and)h(the)g(tail)h(\(3.1\))e(is)j(purely)d(classical)i (in)g(the)f(sense)h(that)278 2442 y FB(log)15 b Fx(N)9 b FB(\()p Fx(E)c FB(\))23 b Fu(\030)g FB(log)14 b Fx(N)905 2454 y Fv(c)938 2442 y FB(\()p Fx(E)5 b FB(\))22 b FC(as)f Fx(E)28 b Fu(!)23 b(\0001)p FC(.)428 2642 y(In)d(contrast)g(to)g(the)g (uni)n(v)o(ersal)f(classical)j(Lifshits)e(tail)h(\(3.1\))e(of)h Fx(N)9 b FC(,)21 b(the)f(analogous)e(tail)j(of)f(the)h(restricted)278 2741 y(inte)o(grated)h(density)g(of)h(states)h Fx(R)1260 2753 y Fv(l)1310 2741 y FC(e)o(xhibits)e(non-uni)n(v)o(ersal)e(quantum) h(beha)n(viour)g(in)j(that)f(it)h(depends)e(on)g(the)278 2841 y(magnetic)e(\002eld)g(and)g(on)f(details)i(of)f(the)g(co)o(v)n (ariance)e(function.)428 3008 y Fz(Theor)o(em)23 b(3.2)f FC(\([)p Fz(25)n FC(]\))p Fz(.)43 b Fy(Let)24 b Fx(d)k FB(=)g(2)23 b Fy(and)f Fx(B)33 b(>)28 b FB(0)p Fy(.)33 b(Suppose)21 b(that)i Fx(V)43 b Fy(is)24 b(a)f(Gaussian)f(r)o(andom)f (potential)278 3108 y(with)h(co)o(variance)d(function)g Fx(C)6 b Fy(.)26 b(Mor)m(eo)o(ver)-9 b(,)20 b(let)h Fx(\033)1747 3078 y Ft(2)1744 3131 y Fv(l)1809 3108 y Fx(>)i FB(0)p Fy(,)d(see)h FC(\(1.17\))n Fy(.)26 b(Then)20 b(the)h(leading)e (low-ener)m(gy)g(fall-of)o(f)278 3207 y(of)i(the)f(r)m(estricted)h (inte)m(gr)o(ated)d(density)i(of)h(states)g Fx(R)1794 3219 y Fv(l)1840 3207 y Fy(is)g(Gaussian)e(in)i(the)f(sense)h(that)1366 3422 y FB(log)14 b Fx(R)1550 3434 y Fv(l)1576 3422 y FB(\()p Fx(E)5 b FB(\))24 b Fu(\030)e(\000)1905 3366 y Fx(E)1971 3336 y Ft(2)p 1892 3403 131 4 v 1892 3479 a FB(2\000)1986 3451 y Ft(2)1986 3504 y Fv(l)3120 3422 y FB(\()p Fx(E)29 b Fu(!)23 b(\0001)p FB(\))p Fx(:)-3273 b FC(\(3.2\))278 3635 y Fy(Her)m(e)21 b(the)f FC(f)o(all-of)n(f)f(ener) o(gy)f FB(\000)1134 3647 y Fv(l)1183 3635 y Fx(>)k FB(0)f Fy(is)g(given)e(by)i(a)f(solution)f(of)i(the)f(maximization)f(pr)l (oblem)278 3840 y FC(\(3.3\))536 b FB(\000)1027 3806 y Ft(2)1027 3861 y Fv(l)1087 3840 y FB(:=)119 b(sup)1170 3922 y Fv(')p Fq(2)p Fv(P)1301 3931 y Fo(l)1325 3922 y Fv(L)1371 3897 y Fn(2)1403 3922 y Ft(\()p Fp(R)1483 3897 y Fn(2)1516 3922 y Ft(\))1234 3989 y Fq(h)p Fv(';')p Fq(i)p Ft(=1)1556 3727 y Fr(Z)1602 3916 y Fp(R)1656 3899 y Fn(2)1651 3840 y FB(d)p Fx(x)1758 3727 y Fr(Z)1804 3916 y Fp(R)1858 3899 y Fn(2)1854 3840 y FB(d)p Fx(y)16 b Fu(j)p Fx(')p FB(\()p Fx(x)p FB(\))p Fu(j)2170 3798 y Ft(2)2221 3840 y Fx(C)6 b FB(\()p Fx(x)19 b Fu(\000)f Fx(y)s FB(\))28 b Fu(j)p Fx(')p FB(\()p Fx(y)s FB(\))p Fu(j)2780 3798 y Ft(2)2831 3840 y Fx(:)428 4253 y FC(A)i(proof)d(of)i (Theorem)f(3.2)h(follo)n(ws)f(the)i(lines)f(of)g(reasoning)f(in)h([)p Fz(25)o FC(],)j(which)c(amounts)h(to)g(establish)278 4352 y(the)j(appropriate)d(v)o(ersion)h(of)h(\(1.9\).)56 b(The)31 b(symmetry)f Fx(R)2024 4364 y Fv(l)2050 4352 y FB(\()p Fx(E)5 b FB(\))44 b(=)f(1)26 b Fu(\000)h Fx(R)2555 4364 y Fv(l)2580 4352 y FB(\()p Fu(\000)p Fx(E)5 b FB(\))p FC(,)35 b(for)30 b(all)i Fx(E)49 b Fu(2)44 b Fw(R)p FC(,)34 b(then)278 4452 y(immediately)24 b(gi)n(v)o(es)h(the)g(high-ener)o(gy)d (gro)n(wth)i FB(log)1850 4385 y Fr(\000)1888 4452 y FB(1)e Fu(\000)f Fx(R)2101 4464 y Fv(l)2127 4452 y FB(\()p Fx(E)5 b FB(\))2257 4385 y Fr(\001)2328 4452 y Fu(\030)32 b(\000)p Fx(E)2556 4422 y Ft(2)2593 4452 y Fx(=)p FB(\(2\000)2761 4422 y Ft(2)2761 4475 y Fv(l)2798 4452 y FB(\))26 b FC(as)g Fx(E)37 b Fu(!)c(1)p FC(.)40 b(F)o(or)25 b(the)278 4567 y(Gaussian)h(co)o(v)n(ariance)e(\(1.18\))g(a)j(maximizer)e(in)h (\(3.3\))e(is)j(gi)n(v)o(en)e(by)h Fx(')p FB(\()p Fx(x)p FB(\))35 b(=)2640 4496 y Fr(p)p 2723 4496 329 4 v 71 x Fx(B)t(=)p FB(\()p Fx(l)r FB(!)14 b(2)p Fx(\031)s FB(\))3065 4500 y Fr(\002)3099 4496 y(p)p 3182 4496 151 4 v 71 x Fx(B)t(=)p FB(2)f(\()p Fx(x)3425 4579 y Ft(1)3486 4567 y Fu(\000)278 4695 y FB(i)p Fx(x)348 4707 y Ft(2)386 4695 y FB(\))418 4627 y Fr(\003)453 4645 y Fv(l)492 4695 y FB(exp\()p Fu(\000)p Fx(B)18 b Fu(j)p Fx(x)p Fu(j)890 4664 y Ft(2)928 4695 y Fx(=)p FB(4\))i FC(and)g(the)g(squared)f(f)o (all-of)n(f)g(ener)o(gy)f(is)j(e)o(xplicitly)e(found)g(to)h(be)278 4869 y(\(3.4\))816 b FB(\000)1307 4835 y Ft(2)1307 4890 y Fv(l)1367 4869 y FB(=)1455 4802 y Fr(\002)1489 4869 y Fx(B)t(\025)1604 4835 y Ft(2)1642 4869 y Fx(=)p FB(\()p Fx(B)t(\025)1831 4835 y Ft(2)1887 4869 y FB(+)18 b(1\))2044 4802 y Fr(\003)2106 4869 y Fx(w)2165 4881 y Fv(l)2191 4869 y FB(\()p Fx(B)t(\025)2338 4835 y Ft(2)2395 4869 y FB(+)g(1\))p Fx(;)278 5035 y FC(see)24 b([)p Fz(25)n FC(])f(and)f(\(1.19\),)g(and)g(also)h([)p Fz(11)o FC(])f(for)h Fx(l)29 b FB(=)e(0)p FC(.)32 b(F)o(or)23 b(a)g(comparison)e(of)h(the)h (f)o(all-of)n(f)e(ener)o(gies)h(in)g(\(3.1\))g(and)278 5134 y(\(3.2\),)g(we)g(of)n(fer)g(the)g(chain)g(of)g(inequalities)f Fx(\033)1658 5104 y Ft(4)1655 5158 y Fv(l)1696 5134 y Fx(=C)6 b FB(\(0\))27 b Fu(\024)f FB(\000)2079 5104 y Ft(2)2079 5158 y Fv(l)2144 5134 y Fu(\024)g Fx(\033)2285 5104 y Ft(2)2282 5158 y Fv(l)2350 5134 y Fu(\024)g Fx(C)6 b FB(\(0\))24 b FC(which)e(is)h(actually)f(v)n(alid)f([)p Fz(25)o FC(])278 5234 y(for)d(the)h(co)o(v)n(ariance)d(function)h(of)h (a)g(general)g Fw(R)1663 5204 y Ft(2)1700 5234 y FC(-homogeneous)c (random)j(potential,)h(not)g(only)f(of)i(a)f(Gaussian)278 5334 y(one.)p eop %%Page: 12 12 12 11 bop 278 233 a FI(12)802 b(HAJO)19 b(LESCHKE,)e(PETER)g(M)1860 220 y(\250)1849 233 y(ULLER,)f(AND)j(SIMONE)g(W)-7 b(ARZEL)278 425 y Fz(3.2.)40 b(Existence)21 b(of)f(the)g(density)h(of)f(states)428 575 y FC(The)k(continuity)e(and)i(non-ne)o(gati)n(vity)d(of)j(the)g(co) o(v)n(ariance)e(function)h(of)h(a)g(Gaussian)h(random)d(potential)278 674 y(already)d(imply)f(that)i(the)f(corresponding)d(inte)o(grated)h (density)i(of)g(states)h Fx(N)29 b FC(is)20 b(locally)f(Lipschitz)g (continuous,)278 774 y(equi)n(v)n(alently)-5 b(,)15 b(that)i Fx(N)26 b FC(is)18 b(absolutely)d(continuous)g(on)i(an)o(y)f(bounded)e (interv)n(al)i(and)g(its)i(\(Lebesgue-\))c(deri)n(v)n(ati)n(v)o(e)278 874 y Fx(D)r FB(\()p Fx(E)5 b FB(\))24 b(=)f(d)p Fx(N)9 b FB(\()p Fx(E)c FB(\))p Fx(=)p FB(d)p Fx(E)g FC(,)21 b(the)f Fy(density)g(of)g(states)p FC(,)h(is)g(locally)f(bounded.)428 1027 y Fz(Theor)o(em)29 b(3.3)g FC(\([)p Fz(51)n(,)h(71)o FC(]\))p Fz(.)47 b Fy(Let)30 b Fx(V)49 b Fy(be)30 b(a)f(Gaussian)g(r)o (andom)f(potential)g(with)j(non-ne)m(gative)26 b(co)o(vari-)278 1127 y(ance)g(function)f Fx(C)6 b Fy(.)44 b(Then)25 b(the)i(inte)m(gr)o (ated)d(density)i(of)g(states)h Fx(N)36 b Fy(is)27 b(locally)e(Lipsc)o (hitz)i(continuous)d(and)h(the)278 1226 y(inequality)278 1412 y FC(\(3.5\))1623 1356 y FB(d)p Fx(N)9 b FB(\()p Fx(E)c FB(\))p 1623 1393 253 4 v 1693 1469 a(d)p Fx(E)1908 1412 y Fu(\024)23 b Fx(W)12 b FB(\()p Fx(E)5 b FB(\))278 1581 y Fy(holds)27 b(for)g(Lebesgue-almost)d(all)j(ener)m(gies)g Fx(E)40 b Fu(2)35 b Fw(R)27 b Fy(with)g(some)g(non-ne)m(gative)c Fx(W)47 b Fu(2)35 b FB(L)3000 1551 y Fq(1)3000 1605 y Ft(lo)r(c)3087 1581 y FB(\()p Fw(R)p FB(\))p Fy(,)29 b(whic)o(h)d(is)278 1681 y(independent)18 b(of)i(the)h(ma)o(gnetic)e (\002eld.)428 1834 y FC(A)i(simple,)f(b)n(ut)g(not)g(optimal)f(choice)g (for)h(the)g(Lipschitz)g(constant)f(is)j(gi)n(v)o(en)c(by)278 2034 y(\(3.6\))359 b Fx(W)12 b FB(\()p Fx(E)5 b FB(\))23 b(=)1129 1942 y Fr(\020)1178 2034 y Fx(r)1217 2000 y Fq(\000)p Ft(1)1326 2034 y FB(+)18 b(\(2)p Fx(\031)s(\034)9 b FB(\))1610 2000 y Fq(\000)p Ft(1)p Fv(=)p Ft(2)1767 1942 y Fr(\021)1817 1959 y Fv(d)1902 1973 y FB(exp)2029 1906 y Fr(\010)2078 1973 y Fx(\034)g(E)24 b FB(+)18 b Fx(\034)2336 1943 y Ft(2)2374 1973 y Fx(C)6 b FB(\(0\))2559 1906 y Fr(\002)2594 1973 y FB(1)17 b Fu(\000)i Fl({)2795 1943 y Ft(2)2792 1993 y Fv(r)2832 1973 y Fx(=)p FB(2)2916 1906 y Fr(\003)o(\011)p 1902 2015 1097 4 v 2224 2103 a Fl({)2279 2115 y Fv(r)2330 2032 y Fr(p)p 2413 2032 264 4 v 71 x FB(2)p Fx(\031)s(C)6 b FB(\(0\))3008 2034 y Fx(:)278 2240 y FC(Here)28 b Fx(r)n(;)14 b(\034)47 b Fu(2)p FB(]0)p Fx(;)14 b Fu(1)p FB([)28 b FC(are)f(arbitrary)g(e)o (xcept)f(that)i Fl({)1809 2252 y Fv(r)1883 2240 y FB(:=)37 b(inf)2109 2267 y Fq(j)p Fv(x)p Fq(j\024)p Fv(r)r(=)2306 2217 y Fq(p)p 2359 2217 35 3 v 2359 2267 a Fv(d)2425 2240 y Fx(C)6 b FB(\()p Fx(x)p FB(\))p Fx(=C)g FB(\(0\))38 b Fx(>)f FB(0)28 b FC(must)f(be)h(strictly)278 2346 y(positi)n(v)o(e.) 58 b(By)31 b(the)g(assumed)g(continuity)f(of)g(the)i(co)o(v)n(ariance)d (function,)j(the)f(latter)g(condition)f(is)i(ful\002lled)278 2446 y(at)27 b(least)g(for)f(all)h(suf)n(\002ciently)f(small)g Fx(r)r FC(.)45 b(Figure)26 b(1,)i(which)e(is)h(tak)o(en)f(from)f([)p Fz(71)o 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5315 a(\033)2943 5327 y Ft(0)2991 5258 y Fx(;)p eop %%Page: 14 14 14 13 bop 278 233 a FI(14)802 b(HAJO)19 b(LESCHKE,)e(PETER)g(M)1860 220 y(\250)1849 233 y(ULLER,)f(AND)j(SIMONE)g(W)-7 b(ARZEL)278 425 y FC(see)26 b(also)g([)p Fz(23)n(,)g(83)o(,)g(103)n FC(].)40 b(As)26 b(for)f(the)g(second)g(result,)h(it)g(is)g(kno)n(wn)e ([)p Fz(19)o(,)h(124)o FC(])g(that)g FB(d)p Fx(R)2924 437 y Fv(l)2950 425 y FB(\()p Fx(E)5 b FB(\))p Fx(=)p FB(d)p Fx(E)31 b FC(becomes)278 525 y(semi-elliptic,)278 733 y(\(3.11\))697 b FB(lim)1159 787 y Fv(l)p Fq(!1)1337 677 y FB(d)p Fx(R)1446 689 y Fv(l)1471 677 y FB(\()p Fx(E)5 b FB(\))p 1337 714 266 4 v 1413 790 a(d)p Fx(E)1635 733 y FB(=)1800 677 y(1)p 1733 714 177 4 v 1733 790 a(2)p Fx(\031)s(\033)1872 802 y Ft(0)1933 733 y FB(\002)2012 666 y Fr(\000)2049 733 y FB(4)18 b Fu(\000)g Fx(\021)2236 699 y Ft(2)2274 666 y Fr(\001)2339 656 y(p)p 2422 656 225 4 v 77 x FB(4)g Fu(\000)g Fx(\021)2609 709 y Ft(2)2647 733 y Fx(:)278 931 y 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1082 y(due)e(to)h(Anderson)d([)p Fz(9)p FC(].)34 b(The)23 b(multi-scale)g(analysis)g(in)m(v)n(ok)o(es)g(elements)g(from)f(K)m (olmogoro)o(v-Arnold-M)o(oser)278 1182 y(theory)j(for)g(coping)f(with)i (small)g(denominators)e(in)i(order)e(to)i(bound)e(resolv)o(ents)h(of)g (\002nite-v)n(olume)f(random)278 1281 y(Schr)7 b(\250)-35 b(odinger)28 b(operators)i(with)g(high)g(probabilities.)55 b(Its)31 b(applicability)f(to)g(the)h(present)f(situation)g(requires) 278 1381 y(additional)24 b(assumptions)g(on)h(the)g(co)o(v)n(ariance)e (function)h(be)o(yond)e(those)j(needed)f(for)h(the)g(W)-7 b(e)o(gner)24 b(estimate.)278 1480 y(The)d(technical)e(realization)h (of)g(the)g(proof)f(of)h(Theorem)f(3.7)h(is)h(patterned)e(after)h(the)h (\223\002x)o(ed-ener)o(gy\224)16 b(analysis)278 1580 y(of)21 b(v)n(on)f(Dreifus)g(and)h(Klein)f([)p Fz(43)o FC(])h(in)g(order)e(to)i(handle)f(the)h(long-ranged)c(correlations)j (of)g(the)h(\003uctuations)f(of)278 1680 y(Gaussian)25 b(random)d(potentials.)37 b(In)24 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FC(.)428 3000 y(As)26 b(a)g(consequence)e(of)h(the)h(unbounded)c(ne)o(gati)n(v)o (e)i(\003uctuations)g(of)i(Gaussian)f(random)f(potentials)h(the)278 3099 y(authors)17 b(of)h([)p Fz(52)n FC(])g(were)g(only)f(able)g(to)h (pro)o(v)o(e)e(algebraic)g(instead)i(of)f(e)o(xponential)f(decay)h(of)g (the)h(eigenfunctions)278 3199 y(corresponding)d(to)j(the)g(pure-point) d(spectrum.)23 b(This)18 b(technical)f(problem)f(should)h(be)h (mastered)f(by)h(e)o(xtending)278 3299 y(either)f(the)f(\224v)n (ariable-ener)o(gy\224)d(multi-scale)j(analysis)h([)p Fz(130)n(,)g(42)o(,)g(80)o FC(])g(or)f(the)h(po)n(werful)e(bootstrap)g (programme)278 3398 y(of)i(Germinet)e(and)i(Klein)f([)p Fz(57)o(,)h(59)o FC(])g(to)f(certain)g(Schr)7 b(\250)-35 b(odinger)14 b(operators)h(which)i(are)f(almost)h(surely)f(unbounded) 278 3498 y(from)k(belo)n(w)-5 b(.)27 b(Indeed,)19 b(by)h(doing)g(the)h (latter)m(,)g(Ueki)g([)p Fz(141)n FC(])g(succeeded)f(in)h(sho)n(wing)f (e)o(xponential)e(localization)278 3597 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16 15 bop 278 233 a FI(16)802 b(HAJO)19 b(LESCHKE,)e(PETER)g(M)1860 220 y(\250)1849 233 y(ULLER,)f(AND)j(SIMONE)g(W)-7 b(ARZEL)278 425 y FC(practically)16 b(no)g(further)f(assumptions)h(than)g(required) f(to)h(pro)o(v)o(e)f(spectral)h(localization.)23 b(It)17 b(w)o(ould)f(be)g(desirable)278 525 y(to)21 b(pro)o(v)o(e)d(some)i (sort)g(of)g(dynamical)f(localization)g(also)i(in)f(the)g(situation)g (of)g(Theorem)e(3.7.)278 724 y FA(4.)46 b(Results)24 b(in)e(case)i(of)g(P)n(oissonian)g(random)f(potentials)428 874 y FC(In)e(comparison)f(to)h(Gaussian)h(random)d(potentials,)j(less) g(is)h(kno)n(wn)d(about)g(re)o(gularity)g(properties)g(of)h(the)278 973 y(inte)o(grated)c(density)i(of)f(states)i Fx(N)28 b FC(and)18 b(localization)f(in)i(case)g(of)g(Poissonian)f(random)f (potentials)h(for)g(arbitrary)278 1073 y(space)j(dimension.)i(Most)d (results)h(concern)e(the)h(Lifshits)g(tail)h(of)f Fx(N)9 b FC(.)278 1245 y Fz(4.1.)40 b(Lifshits)22 b(tails)428 1394 y 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FB(\()p Fx(x)p FB(\))24 b(=)e(0)p Fy(,)325 2818 y FC(\()p Fs(D2)p FC(\))82 b Fx(U)56 b Fy(has)46 b(\(de\002nite\))e(alg)o(ebr)o(aic)i(decay)f(with)i(e)n(xponent)e Fx(\013)72 b Fu(2)p FB(])14 b Fx(d;)g(d)38 b FB(+)g(2)14 b([)46 b Fy(in)g(the)h(sense)g(that)569 2918 y FB(lim)684 2933 y Fq(j)p Fv(x)p Fq(j!1)912 2918 y Fu(j)p Fx(x)p Fu(j)1005 2888 y Fv(\013)1066 2918 y Fx(U)9 b FB(\()p Fx(x)p FB(\))24 b(=)f Fx(g)g Fy(with)e(some)f(constant)g Fx(g)25 b(>)e FB(0)p Fy(.)278 3055 y(Mor)m(eo)o(ver)-9 b(,)19 b(assume)g(that)f(the)h(ma)o(gnetic)f(\002eld)g(vanishes,)h FB(\()p Fx(B)2073 3067 y Fv(ij)2131 3055 y FB(\))24 b(=)e(0)p Fy(.)j(Then)18 b(the)h(leading)e(low-ener)m(gy)h(fall-of)o(f)278 3154 y(of)j(the)f(inte)m(gr)o(ated)e(density)i(of)h(states)g(r)m(eads) 670 3435 y FB(log)14 b Fx(N)9 b FB(\()p Fx(E)c FB(\))23 b Fu(\030)1108 3215 y Fr(8)1108 3290 y(>)1108 3315 y(>)1108 3340 y(<)1108 3489 y(>)1108 3514 y(>)1108 3539 y(:)1403 3334 y Fu(\000)p Fx(\045)1525 3242 y Fr(\020)1595 3278 y Fx(\024)1643 3290 y Fv(d)p 1584 3315 108 4 v 1584 3391 a FB(2)p Fx(E)1702 3242 y Fr(\021)1751 3260 y Fv(d=)p Ft(2)2203 3334 y Fy(for)e(the)f(decay)45 b FC(\()p Fs(D1)p FC(\))1223 3556 y Fu(\000)p Fx(C)1347 3568 y Fv(d)1386 3556 y FB(\()p Fx(\013;)14 b(\045)p FB(\))1597 3464 y Fr(\020)1669 3500 y Fx(g)p 1657 3537 67 4 v 1657 3613 a(E)1733 3464 y Fr(\021)1783 3481 y Fv(d=)p Ft(\()p Fv(\013)p Fq(\000)p Fv(d)p Ft(\))2203 3556 y Fy(for)21 b(the)f(decay)45 b FC(\()p Fs(D2)p FC(\))2886 3215 y Fr(9)2886 3290 y(>)2886 3315 y(>)2886 3340 y(=)2886 3489 y(>)2886 3514 y(>)2886 3539 y(;)3268 3435 y FB(\()p Fx(E)28 b Fu(#)23 b FB(0\))p Fx(:)-3273 b FC(\(4.1\))278 3720 y Fy(Her)m(e)31 b Fx(\024)520 3732 y Fv(d)590 3720 y Fy(is)g(the)g(lowest)g(eig)o(en)m(value)e(of)h (the)h(ne)m(gative)e(Laplacian)g FB(2)p Fx(H)7 b FB(\(0)p Fx(;)14 b FB(0\))p Fy(,)32 b(when)e(Diric)o(hlet)h(r)m(estricted)278 3820 y(to)42 b(a)g(ball)f(in)h Fw(R)817 3789 y Fv(d)897 3820 y Fy(of)g(unit)f(volume)o(.)88 b(Mor)m(eo)o(ver)-9 b(,)46 b(we)d(have)e(intr)l(oduced)f(the)h(constant)g Fx(C)3164 3832 y Fv(d)3203 3820 y FB(\()p Fx(\013;)14 b(\045)p FB(\))63 b(:=)288 3934 y Fv(\013)p Fq(\000)p Fv(d)p 288 3948 130 4 v 336 3996 a(d)442 3875 y Fr(h)481 3967 y Fx(\045)552 3934 y Fv(d)p 548 3948 44 4 v 548 3996 a(\013)697 3934 y(\031)738 3909 y Fo(d=)p Fn(2)p 625 3948 279 4 v 625 3996 a Ft(\000\(1+)p Fv(d=)p Ft(2\))927 3967 y FB(\000)979 3900 y Fr(\000)1027 3934 y Fv(\013)p Fq(\000)p Fv(d)p 1027 3948 130 4 v 1070 3996 a(\013)1167 3900 y Fr(\001)1205 3875 y(i)1244 3892 y Fv(\013=)p Ft(\()p Fv(\013)p Fq(\000)p Fv(d)p Ft(\))1507 3967 y Fy(.)428 4147 y Fz(Remarks)20 b(4.2.)170 b FC(\(i\))82 b(One)22 b(has,)g(for)f(e)o(xample,)g Fx(\024)2034 4159 y Ft(1)2097 4147 y FB(=)k Fx(\031)2237 4117 y Ft(2)2275 4147 y FC(,)d Fx(\024)2366 4159 y Ft(2)2429 4147 y FB(=)k Fx(\031)s(a)2614 4117 y Ft(2)2614 4168 y(0)2651 4147 y FC(,)d(where)e Fx(a)2964 4159 y Ft(0)3027 4147 y FB(=)26 b(2)p Fx(:)p FB(4048)14 b Fx(:)g(:)g(:)33 b FC(is)278 4250 y(the)17 b(smallest)f(positi)n(v)o(e)g(zero)f(of)h(the)g(zeroth)f(Bessel)i (function)e(of)h(the)g(\002rst)h(kind)e([)p Fz(62)n FC(],)i(and)f Fx(\024)2951 4262 y Ft(3)3011 4250 y FB(=)23 b Fx(\031)3149 4220 y Ft(2)3186 4250 y FB(\(4)p Fx(\031)s(=)p FB(3\))3426 4220 y Ft(2)p Fv(=)p Ft(3)3530 4250 y FC(.)385 4368 y(\(ii\))82 b(Con)m(vincing)28 b(ar)o(guments)f(for)i(the)h(v)n(alidity)f(of)h (Lifshits')f(result)h(\(4.1\))f(for)g(the)h(decay)f(\()p Fs(D1)p FC(\))g(were)278 4468 y(also)f(gi)n(v)o(en)e(in)h([)p Fz(54)o(,)g(102)n FC(].)46 b(An)27 b(alternati)n(v)o(e)f(\(rigorous\))f (proof)g(of)i(the)g(underlying)e(long-time)g(asymptotics)278 4567 y(is)30 b(due)e(to)h(Sznitman)e(who)i(in)m(v)o(ented)d(a)j (coarse-graining)d(scheme)i(called)g(the)h Fy(method)e(of)i(enlar)m(g)o (ement)e(of)278 4667 y(obstacles)20 b FC([)p Fz(137)o FC(].)26 b(More)19 b(elementary)g(proofs)g(which)h(rely)g(on)g (Dirichlet-Neumann)e(brack)o(eting)h(were)h(found)278 4767 y(in)i([)p Fz(79)o(,)g(132)o FC(],)g(b)n(ut)f(for)g(the)h(price)f (of)h(obtaining)e(only)h(the)g(so-called)g Fy(Lifshits)i(e)n(xponent)f FC(\(here:)27 b Fx(d=)p FB(2)p FC(\))21 b(and)h(not)278 4866 y(the)f(other)e(constants)h(in)g(\(4.1\))f(for)g(the)i(decay)e(\() p Fs(D1)p FC(\).)361 4985 y(\(iii\))83 b(As)20 b(an)g(aside,)g(we)g (note)f(that)h(\(4.1\))e(for)h(the)h(decay)f(\()p Fs(D1)p FC(\))g(with)h Fx(d)j FB(=)g(1)d FC(remains)f(v)n(alid)g(in)h(the)g (limiting)278 5084 y(case)g(of)f(Poissonian)g(point)f(impurities,)h Fx(U)9 b FB(\()p Fx(x)p FB(\))23 b(=)g Fx(u)1814 5096 y Ft(0)1865 5084 y Fx(\016)s FB(\()p Fx(x)p FB(\))p FC(,)d Fx(u)2105 5096 y Ft(0)2165 5084 y Fx(>)j FB(0)p FC(;)d(see)f([)p Fz(44)o(,)h(63)o(,)f(85)o FC(])h(and)e([)p Fz(118)n FC(,)i(Thm.)f (6.7].)428 5234 y(F)o(or)24 b(\(non-ne)o(gati)n(v)o(e\))19 b(single-impurity)j(potentials)h Fx(U)34 b FC(with)24 b(rapid)f(decay)g(\()p Fs(D1)p FC(\),)i(the)f(Lifshits)h(tail)f(is)h (in-)278 5334 y(sensiti)n(v)o(e)32 b(to)h(the)f(details)h(of)e(the)i (decay)e(of)h Fx(U)41 b FC(and)32 b(is)h(dominated)d(by)i(the)g (quantum)f(kinetic)g(ener)o(gy)-5 b(.)59 b(It)p eop %%Page: 17 17 17 16 bop 1025 234 a FI(RANDOM)18 b(SCHR)1465 221 y(\250)1454 234 y(ODINGER)g(OPERA)-6 b(T)o(ORS)18 b(FOR)h(AMORPHOUS)g(SOLIDS)689 b(17)278 425 y FC(has)32 b(therefore)e(a)j(quantum)c(character)-5 b(.)59 b(Ov)o(er)31 b(against)g(this,)k(if)d Fx(U)53 b Fu(\025)44 b FB(0)32 b FC(has)g(the)g(slo)n(w)g(decay)f(\()p Fs(D2)p FC(\),)j(the)278 525 y(Lifshits)e(tail)h(sensiti)n(v)o(ely)e (depends)f(on)h(the)h(details)g(of)g(this)g(decay)-5 b(.)58 b(It)32 b(is)h(classical)f(in)g(character)e(in)i(that)278 624 y FB(log)15 b Fx(N)9 b FB(\()p Fx(E)c FB(\))33 b Fu(\030)f FB(log)14 b Fx(N)924 636 y Fv(c)957 624 y FB(\()p Fx(E)5 b FB(\))27 b FC(as)f Fx(E)38 b Fu(#)32 b FB(0)p FC(.)41 b(Therefore,)24 b(the)h(character)f(of)h(the)h(tail)g(\(4.1\))e (changes)g(from)h(purely)278 724 y(quantum)d(to)i(purely)f(classical,)i (when)f(the)f(decay)g(changes)g(from)g(\()p Fs(D1)p FC(\))h(to)g(\()p Fs(D2)p FC(\).)36 b(The)23 b(Lifshits)h(tail)h(for)e(the)278 824 y(borderline)18 b(case)j(of)f(algebraic)f(decay)h(with)g(e)o (xponent)e Fx(\013)24 b FB(=)e Fx(d)d FB(+)f(2)i FC(seems)h(to)g(be)f (open.)k(In)c(vie)n(w)g(of)g(\(1.9\))f(we)278 923 y(ha)n(v)o(e)h(the)g (follo)n(wing)428 1075 y Fz(Conjectur)o(e)f(4.3.)40 b FC(If)20 b FB(lim)1201 1090 y Fq(j)p Fv(x)p Fq(j!1)1428 1075 y Fu(j)p Fx(x)p Fu(j)1521 1045 y Fv(d)p Ft(+2)1659 1075 y Fx(U)9 b FB(\()p Fx(x)p FB(\))23 b(=)g Fx(g)g FC(with)e(some)f(constant)f Fx(g)26 b(>)d FB(0)p FC(,)d(then)671 1312 y FB(log)14 b Fx(N)9 b FB(\()p Fx(E)c FB(\))24 b Fu(\030)e(\000)1188 1195 y Fr(\024)1232 1312 y Fx(\045)1321 1254 y Fn(2)p 1285 1263 102 3 v 1285 1296 a Fo(d)p Fn(+2)1414 1220 y Fr(\020)1484 1256 y Fx(\024)1532 1268 y Fv(d)p 1474 1293 108 4 v 1474 1369 a FB(2)p Fx(E)1591 1220 y Fr(\021)1686 1215 y Fo(d)p 1651 1224 102 3 v 1651 1257 a(d)p Fn(+2)1785 1312 y FB(+)1868 1245 y Fr(\002)1903 1312 y Fx(C)1962 1324 y Fv(d)2001 1312 y FB(\()p Fx(d)d FB(+)f(2)p Fx(;)c(\045)p FB(\))2332 1245 y Fr(\003)2412 1240 y Fn(2)p 2376 1249 V 2376 1282 a Fo(d)p Fn(+2)2505 1220 y Fr(\020)2577 1256 y Fx(g)p 2565 1293 67 4 v 2565 1369 a(E)2641 1220 y Fr(\021)2736 1215 y Fo(d)p 2700 1224 102 3 v 2700 1257 a(d)p Fn(+2)2816 1195 y Fr(\025)2870 1181 y Fo(d)p Fn(+2)p 2870 1194 V 2907 1226 a(2)3268 1312 y FB(\()p Fx(E)28 b Fu(#)23 b FB(0\))p Fx(:)-3273 b FC(\(4.2\))428 1508 y(This)20 b(tail)h(e)o(xhibits)f(a)g(mix)o(ed)f (quantum/classical)g(character)-5 b(.)428 1707 y(A)19 b(similar)g(transition)f(from)f(a)i(purely)e(quantum)g(to)i(a)g(purely) e(classical)i(Lifshits)g(tail)h(can)e(be)g(observ)o(ed)f(in)278 1807 y(case)g(of)e(the)h(random)e(Landau)g(Hamiltonian)h(with)h(non-ne) o(gati)n(v)o(e)c(Poissonian)j(potential.)23 b(Ho)n(we)n(v)o(er)m(,)14 b(since)i(the)278 1906 y(Landau)22 b(Hamiltonian)g(possesses)h (ground-state)e(\(w)o(a)n(v)o(e\))h(functions)f(with)j(Gaussian)e (decay)-5 b(,)23 b(the)f(borderline)278 2006 y(decay)e(of)g Fx(U)29 b FC(turns)20 b(out)g(to)g(be)g(Gaussian)g(and)g(not)g (algebraic)f(\(with)h(e)o(xponent)e Fx(d)h FB(+)f(2)k(=)h(4)p FC(\).)428 2158 y Fz(Theor)o(em)f(4.4)f FC(\([)p Fz(26)n(,)i(47)o(,)f (68)o(,)g(69)o(,)h(48)o FC(]\))p Fz(.)41 b Fy(Let)23 b Fx(d)j FB(=)g(2)c Fy(and)g Fx(B)30 b(>)c FB(0)p Fy(.)k(Suppose)20 b(that)i Fx(V)42 b Fy(is)23 b(a)f(non-ne)m(gative)278 2257 y(P)-7 b(oissonian)17 b(r)o(andom)g(potential)g(with)i(mean)e (concentr)o(ation)f Fx(\045)23 b(>)g FB(0)p Fy(.)h(Furthermor)m(e)o(,) 18 b(assume)g(that)g(the)g(single-)278 2357 y(impurity)j(potential)e Fx(U)31 b Fu(\025)23 b FB(0)d Fy(has)g(one)g(of)g(the)g(following)g (thr)m(ee)g(decays)g(at)g(in\002nity:)325 2493 y FC(\()p Fs(D3)p FC(\))82 b Fx(U)g Fy(is)74 b(compactly)d(supported)g(or)i(has)g (super)n(-Gaussian)e(decay)h(in)h(the)f(sense)i(that)569 2593 y FB(lim)14 b(sup)823 2613 y Fq(j)p Fv(x)p Fq(j!1)1051 2593 y Fu(j)p Fx(x)p Fu(j)1144 2563 y Fq(\000)p Ft(2)1247 2593 y FB(log)g Fx(U)9 b FB(\()p Fx(x)p FB(\))24 b(=)f Fu(\0001)p Fy(.)325 2740 y FC(\()p Fs(D4)p FC(\))82 b Fx(U)36 b Fy(has)26 b(\(de\002nite\))e(Gaussian)h(decay)h(in)g(the)g (sense)h(that)f FB(lim)14 b(sup)2527 2760 y Fq(j)p Fv(x)p Fq(j!1)2754 2740 y Fu(j)p Fx(x)p Fu(j)2847 2710 y Fq(\000)p Ft(2)2951 2740 y FB(log)g Fx(U)9 b FB(\()p Fx(x)p FB(\)=)35 b Fu(\000)p Fx(\025)3462 2710 y Fq(\000)p Ft(2)569 2841 y Fy(with)21 b(some)f(length)f Fx(\025)24 b(>)f FB(0)p Fy(.)325 2977 y FC(\()p Fs(D5)p FC(\))82 b Fx(U)29 b Fy(has)20 b(sub-Gaussian)e(decay)i(in)g(the)g(sense)g(that)g FB(lim)14 b(sup)2294 2997 y Fq(j)p Fv(x)p Fq(j!1)2522 2977 y Fu(j)p Fx(x)p Fu(j)2615 2947 y Fq(\000)p Ft(2)2718 2977 y FB(log)g Fx(U)9 b FB(\()p Fx(x)p FB(\))24 b(=)f(0)p Fy(.)i(Mor)m(eo)o(ver)-9 b(,)569 3078 y(the)20 b(decay)g(of)g Fx(U)29 b Fy(is)22 b(inte)m(gr)o(able)c(and)h(r)m(e)m(gular)h(in)g(the) g(sense)h(of)f FC([)p Fz(69)o FC(,)g(Def.)g(3.5])p Fy(.)278 3214 y(Then)g(the)g(leading)f(low-ener)m(gy)g(fall-of)o(f)h(of)g(the)g 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3452 y(>)2931 3477 y(=)2931 3627 y(>)2931 3652 y(>)2931 3677 y(>)2931 3702 y(>)2931 3726 y(>)2931 3751 y(;)3268 3573 y FB(\()p Fx(E)28 b Fu(#)23 b FB(0\))p Fx(:)-3273 b FC(\(4.3\))428 4025 y Fz(Remarks)20 b(4.5.)170 b FC(\(i\))82 b(F)o(or)34 b(super)n(-Gaussian)e(decay)h(\()p Fs(D3)p FC(\))h(and)g(Gaussian)g (decay)f(\()p Fs(D4)p FC(\))h(the)g(inte-)278 4125 y(grated)20 b(density)g(of)h(states)h(has)f(a)g(po)n(wer)n(-la)o(w)f(f)o(all-of)n (f)f(\(on)h(a)h(logarithmic)e(scale\).)27 b(The)21 b(corresponding)c(e) o(xpo-)278 4224 y(nent)25 b FB(2)p Fx(\031)s(\045=B)k FC(in)c(\(4.3\))f(for)g(the)h(decay)f(\()p Fs(D3)p FC(\))h(is)h(just)g (the)f(mean)f(number)f(of)i(impurities)f(in)h(a)h(disc)f(of)g(radius) 278 4260 y Fr(p)p 361 4260 151 4 v 71 x FB(2)p Fx(=B)t FC(.)41 b(T)-7 b(w)o(o)26 b(important)e(e)o(xamples)g(for)h(\()p Fs(D5)p FC(\))h(are)f(an)h(algebraic)e(decay)h(with)h(e)o(xponent)d Fx(\013)33 b Fu(2)p FB(])14 b(2)p Fx(;)g Fu(1)p FB([)26 b FC(\(see)278 4435 y(Theorem)20 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eop %%Page: 19 19 19 18 bop 1025 234 a FI(RANDOM)18 b(SCHR)1465 221 y(\250)1454 234 y(ODINGER)g(OPERA)-6 b(T)o(ORS)18 b(FOR)h(AMORPHOUS)g(SOLIDS)689 b(19)278 425 y FC(This)20 b(may)e(be)h(ascribed)g(to)g(the)g(ef)n (fecti)n(v)o(e)f(zero-\002eld)g(motion)g(of)g(the)i(particle)e (parallel)h(to)g(the)g(magnetic)f(\002eld,)278 525 y(which)28 b(dominates)e(the)i(lo)n(w-ener)o(gy)d(asymptotics.)47 b(Actually)-5 b(,)28 b(in)g([)p Fz(144)n FC(])g(some)n(what)f(more)g (detailed)g(infor)n(-)278 624 y(mation)g(on)h(the)f(f)o(all-of)n(fs)g (depending)e(on)j(the)f(actual)h(decay)f(can)g(be)h(found.)46 b(F)o(or)27 b(e)o(xample,)h(for)f(algebraic)278 724 y(decay)e(\()p Fs(D2)p FC(\))h(with)g Fx(d)33 b FB(=)g(3)p FC(,)27 b(that)f(is,)i (with)e(e)o(xponent)d Fx(\013)34 b Fu(2)p FB(])14 b(3)p Fx(;)g FB(5[)p FC(,)27 b(the)e(tail)i(coincides)e(with)g(the)h (correspond-)278 824 y(ing)h(one)f(for)g Fx(B)40 b FB(=)35 b(0)27 b FC(\(see)g(Theorem)e(4.1\))h(and)g(has)h(therefore)e(a)i 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1809 y FC(There)i(are)i(only)e(a)i (fe)n(w)f(rigorous)f(results)i(on)f(these)g(issues)i(for)d(Poissonian)h (random)f(potentials.)34 b(F)o(or)23 b(a)278 1908 y(special)e(class)g (of)g(non-ne)o(gati)n(v)o(e)16 b(single-impurity)i(potentials)i Fx(U)9 b FC(,)21 b(T)m(ip)f([)p Fz(139)n FC(])h(has)g(pro)o(v)o(en)d (that)i(the)h(inte)o(grated)278 2008 y(density)29 b(of)f(states)i Fx(N)38 b FC(is)30 b(absolutely)d(continuous)g(at)i(suf)n(\002ciently)f (high)g(ener)o(gies.)49 b(The)28 b(only)g(localization)278 2108 y(result)21 b(is)g(due)e(to)i(Stolz)f([)p Fz(134)n FC(].)26 b(It)20 b(concerns)f(the)h(case)h(of)f(one)f(space)h (dimension.)428 2287 y Fz(Theor)o(em)25 b(4.9)g FC(\([)p Fz(134)n FC(]\))p Fz(.)44 b Fy(Let)27 b Fx(d)33 b FB(=)g(1)26 b Fy(and)f(let)h Fx(V)45 b Fy(be)26 b(a)f(non-ne)m(gative)e(P)-7 b(oissonian)24 b(r)o(andom)h(potential.)278 2387 y(Mor)m(eo)o(ver)-9 b(,)25 b(let)g(the)g(single-impurity)e(potential)g Fx(U)39 b Fu(\025)31 b FB(0)24 b Fy(be)g(compactly)f(supported.)36 b(Then)24 b(the)h(almost-sur)m(e)278 2486 y(spectrum)c FB([0)p Fx(;)14 b Fu(1)p FB([)20 b Fy(of)g Fx(H)7 b FB(\(0)p Fx(;)14 b(V)19 b FB(\))i Fy(is)g(only)e(pur)m(e)h(point)g(with)h(e)n (xponentially)d(localized)h(eig)o(enfunctions.)428 2665 y FC(The)30 b(proof)g(b)n(uilds)g(on)h(techniques)e(which)i(are)f(a)n (v)n(ailable)h(only)f(for)g(one)g(dimension)g(and)g(are)h(nicely)278 2765 y(summarized)19 b(and)h(discussed)g(in)g(the)g(recent)g(surv)o(e)o (y)e([)p Fz(135)o FC(].)428 2944 y Fz(Remark)23 b(4.10.)42 b FC(T)-7 b(o)23 b(our)f(kno)n(wledge,)g(the)h(only)g(other)f(rigorous) f(w)o(orks)i([)p Fz(105)n(,)h(32)o FC(])f(which,)g(among)f(the)278 3044 y(rest,)e(deal)g(with)f(localization)g(proofs)f(for)h(Poissonian)g (random)f(potentials)h(in)g(multi-dimensional)e(situations)278 3144 y(ha)n(v)o(e)j(to)g(assume)h(an)f(additional)f(randomness)f(of)i (the)g(impurity)f(coupling-strengths.)278 3491 y FA(5.)46 b(Some)23 b(open)g(pr)n(oblems)428 3640 y FC(While)e(most)f(rigorous)e (w)o(orks)i(on)g(random)e(Schr)7 b(\250)-35 b(odinger)18 b(operators)g(concern)h Fw(Z)2831 3610 y Fv(d)2870 3640 y FC(-er)o(godic)f(random)g(po-)278 3740 y(tentials,)33 b(the)d(present)g(surv)o(e)o(y)e(has)i(focused)f(on)h Fw(R)1838 3710 y Fv(d)1876 3740 y FC(-er)o(godic)e(ones.)54 b(More)29 b(precisely)-5 b(,)31 b(for)f(Gaussian)g(and)278 3839 y(Poissonian)g(random)e(potentials,)k(rigorous)c(results)i(ha)n(v) o(e)f(been)h(presented)e(on)i(the)g(inte)o(grated)e(density)i(of)278 3939 y(states)g(and)d(Anderson)f(localization.)48 b(In)28 b(this)g(conte)o(xt,)h(a)f(lot)g(of)g(issues,)j(which)c(are)h (interesting)f(from)g(the)278 4039 y(\(theoretical-\))19 b(physics)g(point)g(of)h(vie)n(w)-5 b(,)20 b(are)g(still)h(unsolv)o (ed.)428 4138 y(One)h(major)g(open)g(problem)f(concerns)g(a)i(proof)e (of)h(Anderson)f(localization)h(in)h(case)g(of)f(\(non-ne)o(gati)n(v)o (e\))278 4238 y(Poissonian)32 b(random)d(potentials)j(for)f(arbitrary)f (space)h(dimension.)58 b(Another)31 b(problem)f(is)i(to)g(isolate)g (the)278 4337 y(weak)o(est)g(possible)f(conditions)e(for)i(an)g Fw(R)1567 4307 y Fv(d)1605 4337 y FC(-er)o(godic)e(random)g(potential)i (which)f(imply)h(continuity)e(of)i(the)278 4437 y(corresponding)26 b(inte)o(grated)h(density)h(of)g(states)i(if)f Fx(d)39 b Fu(\025)f FB(2)29 b FC(and)f Fx(A)39 b FB(=)g(0)p FC(.)50 b(In)29 b(particular)m(,)g(one)f(may)g(ask:)43 b(Is)278 4537 y Fw(R)348 4507 y Fv(d)387 4537 y FC(-er)o(godicity)17 b(already)i(enough?)428 4636 y(Most)f(striking)f(is)h(de\002nitely)f (the)g(f)o(act)h(that)g(there)f(is)h(not)g(a)g(single)f(non-zero)e Fw(R)2734 4606 y Fv(d)2773 4636 y FC(-)j(or)f Fw(Z)2965 4606 y Fv(d)3004 4636 y FC(-er)o(godic)e(random)278 4736 y(potential)i(for)f(which)g(the)h(e)o(xistence)f(of)h(an)g(absolutely)f (continuous)f(component)f(in)j(the)g(ener)o(gy)e(spectrum)h(has)278 4836 y(been)21 b(pro)o(v)o(en,)d(that)j(is,)h(spectral)f (delocalization)e(in)i(certain)g(ener)o(gy)e(re)o(gimes.)26 b(After)21 b(all,)h(physical)d(intuition,)278 4935 y(approximate)e (calculations)h(and)g(numerical)g(studies)h(suggest)f(the)h(occurrence) e(of)h(a)h(mobility)f(edge)g(if)i Fx(d)j Fu(\025)f FB(3)p FC(.)278 5035 y(If)j Fx(d)32 b FB(=)g(2)25 b FC(it)h(is)g(not)e(yet)h (rigorously)e(settled)i(whether)f(the)h(whole)g(ener)o(gy)e(spectrum)h (is)i(almost)f(surely)f(only)278 5134 y(pure)c(point)f(or)h(not.)428 5234 y(Last)d(b)n(ut)f(not)h(least,)g(one)f(can)h(hardly)e(claim)i(to)f (utterly)g(understand)e(electronic)i(properties)f(of)h(disordered)278 5334 y(solids)21 b(without)e(ha)n(ving)h(a)g(more)f(solid)i(foundation) c(of)j(their)g(transport)f(theory)-5 b(.)p eop %%Page: 20 20 20 19 bop 278 233 a FI(20)802 b(HAJO)19 b(LESCHKE,)e(PETER)g(M)1860 220 y(\250)1849 233 y(ULLER,)f(AND)j(SIMONE)g(W)-7 b(ARZEL)278 425 y FA(Ackno)o(wledgement)428 575 y FC(Our)20 b(thanks)g(go)g(to)h (Ale)o(xandra)d(W)-7 b(eichlein)21 b(for)f(helpful)f(remarks.)25 b(This)c(w)o(ork)f(w)o(as)h(partially)f(supported)278 674 y(by)h(the)g(Deutsche)g(F)o(orschungsgemeinschaft)16 b(\(DFG\))21 b(under)f(grant)g(no.)g(Le)h(330/12.)26 b(The)21 b(latter)g(is)h(a)f(project)278 774 y(within)27 b(the)g(DFG)g(Priority)f(Programme)e(SPP)k(1033)d(\223Interagierende)f (stochastische)i(Systeme)h(v)n(on)f(hoher)278 874 y(K)m(omple)o(xit)5 b(\250)-33 b(at\224.)53 b(Peter)30 b(M)7 b(\250)-35 b(uller)30 b(ackno)n(wledges)e(partial)i(\002nancial)f(support)g(of)g(the)h(DFG)h (under)e(grant)g(no.)278 973 y(Zi)21 b(209/6-1)d(and)i(SFB)h(602.)278 1299 y FA(Refer)n(ences)363 1432 y FD([1])34 b(R.)21 b(J.)g(Adler)m(,)i(The)e(geometry)i(of)f(random)g(\002elds,)g (Chichester:)j(W)m(ile)o(y)l(,)e(1981)f(\(A)f(major)i(re)n(vision,)g (co-authored)h(by)e(J.)f(T)-5 b(aylor)m(,)474 1515 y(will)18 b(appear)h(with)e(Birkh)t(\250)-26 b(auser)m(,)19 b(Boston\))363 1598 y([2])34 b(M.)16 b(Aizenman,)i Fh(Localization)i(at)d(weak)i (disor)n(der:)e(some)g(elementary)j(bounds)p FD(,)d(Re)n(v)l(.)g(Math.) g(Phys.)f Fg(6)h FD(\(1994\))h(1163\2261182)363 1681 y([3])34 b(M.)16 b(Aizenman,)i(G.)f(M.)f(Graf,)h Fh(Localization)i (bounds)f(for)f(an)g(electr)m(on)i(gas)p FD(,)e(J.)f(Phys.)g(A)g Fg(31)h FD(\(1998\))h(6783\2266806)363 1764 y([4])34 b(M.)24 b(Aizenman,)j(S.)d(Molchano)o(v)l(,)j Fh(Localization)h(at)d (lar)n(g)o(e)h(disor)n(der)f(and)h(at)f(e)o(xtr)n(eme)h(ener)n(gies:)h (an)e(elementary)j(derivation)p FD(,)474 1847 y(Commun.)16 b(Math.)i(Phys.)e Fg(157)h FD(\(1993\))h(245\226278)363 1930 y([5])34 b(M.)17 b(Aizenman,)j(J.)d(H.)g(Schenk)o(er)m(,)j(R.)e (M.)f(Friedrich,)j(D.)d(Hundertmark,)i Fh(Constructive)j(fr)o (actional-moment)g(criteria)e(for)e(local-)474 2013 y(ization)g(in)g(r) o(andom)f(oper)o(ator)o(s)p FD(,)h(Physica)g(A)e Fg(279)h FD(\(2000\))h(369\226377)363 2096 y([6])34 b(M.)20 b(Aizenman,)h(J.)f (H.)g(Schenk)o(er)m(,)i(R.)e(M.)g(Friedrich,)i(D.)e(Hundertmark,)i Fh(F)m(inite-volume)h(criteria)g(for)d(Ander)o(son)h(localization)p FD(,)474 2179 y(Commun.)16 b(Math.)i(Phys.)e Fg(224)h FD(\(2001\))h(219\226253)363 2262 y([7])34 b(M.)e(Aizenman,)i(A.)e (Elgart,)h(S.)f(Nabok)o(o,)i(J.)e(H.)g(Schenk)o(er)m(,)j(G.)d(Stolz,)h Fh(Moment)i(analysis)f(for)f(localization)j(in)d(r)o(andom)474 2345 y(Sc)o(hr)5 b(\250)-27 b(oding)o(er)19 b(oper)o(ator)o(s)p FD(,)f(preprint)h(math-ph/0308023)363 2428 y([8])34 b(W)-6 b(.)16 b(O.)g(Amrein,)h(Non-relati)n(vistic)22 b(quantum)c(dynamics,)g (Dordrecht:)h(Reidel,)g(1981)363 2511 y([9])34 b(P)-7 b(.)16 b(W)-6 b(.)16 b(Anderson,)h Fh(Absence)h(of)f(dif)o(fusion)i(in) e(certain)i(r)o(andom)e(lattices)p FD(,)i(Phys.)e(Re)n(v)l(.)g Fg(109)g FD(\(1958\))h(1492\2261505)330 2594 y([10])34 b(T)-5 b(.)16 b(Ando,)i(A.)e(B.)h(F)o(o)n(wler)m(,)i(F)-5 b(.)16 b(Stern,)i Fh(Electr)m(onic)i(pr)m(operties)f(of)f (two-dimensional)j(systems)p FD(,)d(Re)n(v)l(.)f(Mod.)h(Phys.)e Fg(54)i FD(\(1982\))g(437\226)474 2677 y(672)330 2760 y([11])34 b(W)-6 b(.)23 b(Apel,)h Fh(Asymptotic)i(density)g(of)e (states)h(for)f(a)g Fk(2)p Fj(D)i Fh(electr)m(on)g(system)f(in)f(a)g (str)m(ong)g(ma)o(gnetic)j(\002eld)p FD(,)d(J.)g(Phys.)f(C)h Fg(20)g FD(\(1987\))474 2843 y(L577\226L581)330 2926 y([12])34 b(J.)17 b(E.)h(A)-5 b(vron,)18 b(L.)f(Sadun,)i Fh(F)l(r)n(edholm)g(indices)h(and)f(the)g(phase)g(dia)o(gr)o(am)g(of)g (quantum)h(Hall)f(systems)p FD(,)f(J.)g(Math.)h(Phys.)e Fg(42)i FD(\(2001\))474 3009 y(1\22614)330 3092 y([13])34 b(J.-M.)15 b(Barbaroux,)k(J.)c(M.)h(Combes,)h(P)-7 b(.)15 b(D.)h(Hislop,)h Fh(Landau)g(Hamiltonians)i(with)f(unbounded)g(r)o (andom)f(potentials)p FD(,)i(Lett.)e(Math.)474 3175 y(Phys.)f Fg(40)h FD(\(1997\))h(335\226369)330 3258 y([14])34 b(J.)12 b(M.)g(Barbaroux,)j(J.)d(M.)h(Combes,)g(R.)f(Montcho,)j Fh(Remarks)e(on)g(the)h(r)n(elation)g(between)i(quantum)e(dynamics)g (and)g(fr)o(actal)h(spectr)o(a)p FD(,)474 3341 y(J.)h(Math.)h(Anal.)h (Appl.)f Fg(213)g FD(\(1997\))h(698\226722)330 3424 y([15])34 b(J.)25 b(M.)g(Barbaroux,)i(W)-6 b(.)24 b(Fischer)m(,)j(P)-7 b(.)25 b(M)6 b(\250)-28 b(uller)m(,)27 b Fh(Dynamical)g(pr)m(operties)h (of)e(r)o(andom)g(Sc)o(hr)5 b(\250)-27 b(oding)o(er)28 b(oper)o(ator)o(s)p FD(,)e(preprint)i(math-)474 3507 y(ph/9907002)330 3590 y([16])34 b(J.)22 b(Bellissard,)j(A.)d(v)n(an)h (Elst,)g(H.)f(Schulz-Baldes,)j Fh(The)f(non-commutative)i(g)o(eometry)f (of)e(the)h(quantum)g(Hall)f(ef)o(fect)p FD(,)i(J.)d(Math.)474 3673 y(Phys.)16 b Fg(35)h FD(\(1994\))h(5373\2265451)330 3756 y([17])34 b(M.)19 b(M.)h(Benderskij,)i(L.)d(A.)h(P)o(astur)m(,)h Fh(On)f(the)h(spectrum)g(of)g(the)g(one-dimensional)i(Sc)o(hr)5 b(\250)-27 b(oding)o(er)23 b(equation)f(with)g(a)e(r)o(andom)g(po-)474 3839 y(tential)p FD(,)f(Math.)e(USSR)g(Sb)m(.)g Fg(11)g FD(\(1970\))g(245\226256)i([Russian)f(original:)h(Mat.)e(Sb)m(.)g Fg(82)g FD(\(1970\))h(273\226284])330 3922 y([18])34 b(K.)16 b(A.)g(Benedict,)k Fh(The)d(fate)h(of)f(the)h(Lifshitz)g(tails) g(of)g(high)f(Landau)h(le)o(vels)p FD(,)g(Nucl.)g(Phys.)e(B)h Fg(280)g FD(\(1987\))h(549\226560)330 4005 y([19])34 b(K.)17 b(A.)g(Benedict,)j(J.)d(T)-5 b(.)17 b(Chalk)o(er)m(,)j Fh(An)d(e)o(xactly)k(solvable)f(model)e(of)h(the)f(disor)n(der)n(ed)h (two-dimensional)i(electr)m(on)f(gas)e(in)g(a)g(str)m(ong)474 4088 y(ma)o(gnetic)h(\002eld)p FD(,)f(J.)e(Phys.)g(C)h Fg(19)g FD(\(1986\))h(3587\2263604)330 4171 y([20])34 b(N.)17 b(H.)f(Bingham,)j(C.)e(M.)g(Goldie,)h(J.)f(L.)f(T)-5 b(eugels,)18 b(Re)o(gular)i(v)n(ariation,)g(paperback)g(edition)g(with) e(additions,)h(Cambridge:)h(Cam-)474 4254 y(bridge)e(UP)-7 b(,)16 b(1989)330 4337 y([21])34 b(M.)16 b(B)6 b(\250)-28 b(ohm,)16 b(K.)g(Broderix,)i(H.)e(Leschk)o(e,)h Fh(Br)m(oadening)h(of)f (the)h(lowest)g(Landau)f(le)o(vel)i(by)e(a)f(Gaussian)h(r)o(andom)g (potential)j(with)d(an)474 4420 y(arbitr)o(ary)h(corr)n(elation)h (length:)g(an)e(ef)o(\002cient)j(continued\226fr)o(action)i(appr)m(oac) o(h)p FD(,)c(Z.)d(Physik)i(B)g Fg(104)h FD(\(1997\))f(111\226115)330 4503 y([22])34 b(V)-9 b(.)18 b(L.)f(Bonch-Brue)n(vich,)22 b(R.)c(Enderlein,)i(B.)e(Esser)m(,)g(R.)g(K)n(eiper)m(,)j(A.)c(G.)h (Mirono)o(v)l(,)h(I.)f(P)-7 b(.)17 b(Zvyagin,)i(Elektronentheorie)k (ungeord-)474 4586 y(neter)g(Halbleiter)m(,)h(Berlin:)f(VEB)e (Deutscher)i(V)-7 b(erlag)22 b(der)g(W)m(issenschaften,)h(1984)f([in)g (German.)f(Russian)h(original:)i(Mosco)n(w:)474 4670 y(Nauka,)18 b(1981])330 4753 y([23])34 b(E.)13 b(Br)t(\264)-26 b(ezin,)15 b(D.)e(J.)g(Gross,)g(C.)g(Itzykson,)i Fh(Density)g(of)f (states)h(in)f(the)h(pr)n(esence)g(of)f(a)f(str)m(ong)i(ma)o(gnetic)h (\002eld)f(and)f(r)o(andom)g(impurities)p FD(,)474 4836 y(Nucl.)j(Phys.)f(B)h Fg(235)g FD(\(1984\))h(24\22644)330 4919 y([24])34 b(K.)16 b(Broderix,)i(N.)f(Heldt,)g(H.)f(Leschk)o(e,)i Fh(P)-5 b(artition)18 b(function)i(and)d(the)h(density)g(of)g(states)f (for)g(an)g(electr)m(on)j(in)d(the)h(plane)g(subjected)474 5002 y(to)f(a)g(r)o(andom)g(potential)j(and)d(a)g(ma)o(gnetic)j (\002eld)p FD(,)d(Phys.)f(Re)n(v)l(.)h(B)g Fg(40)g FD(\(1989\))h (7479\2267486)330 5085 y([25])34 b(K.)22 b(Broderix,)j(N.)d(Heldt,)i (H.)e(Leschk)o(e,)i Fh(Exact)g(r)n(esults)f(on)g(Landau-le)o(vel)j(br)m (oadening)p FD(,)f(J.)d(Phys.)g(A:)h(Math.)h(Gen.)f Fg(24)g FD(\(1991\))474 5168 y(L825\226L831)330 5251 y([26])34 b(K.)19 b(Broderix,)i(D.)e(Hundertmark,)i(W)-6 b(.)19 b(Kirsch,)h(H.)f(Leschk)o(e,)i Fh(The)f(fate)h(of)f(Lifshits)h(tails)g (in)f(ma)o(gnetic)i(\002elds)p FD(,)e(J.)f(Stat.)h(Phys.)f Fg(80)474 5334 y FD(\(1995\))f(1\22622)p eop %%Page: 21 21 21 20 bop 1025 234 a FI(RANDOM)18 b(SCHR)1465 221 y(\250)1454 234 y(ODINGER)g(OPERA)-6 b(T)o(ORS)18 b(FOR)h(AMORPHOUS)g(SOLIDS)689 b(21)330 425 y FD([27])34 b(K.)14 b(Broderix,)i(D.)e(Hundertmark,)i(H.) d(Leschk)o(e,)j Fh(Self-aver)o(a)o(ging)o(,)i(decomposition)g(and)d (asymptotic)i(pr)m(operties)f(of)f(the)h(density)g(of)474 508 y(states)j(for)f(r)o(andom)h(Sc)o(hr)5 b(\250)-27 b(oding)o(er)20 b(oper)o(ator)o(s)g(with)f(constant)h(ma)o(gnetic)h (\002eld)p FD(,)e(in)f(P)o(ath)h(inte)o(grals)i(from)d(meV)g(to)g(MeV) -5 b(:)19 b(T)m(utzing)474 591 y('92,)e(H.)f(Grabert,)i(A.)e(Inomata,)i (L.)e(S.)g(Schulman,)i(U.)e(W)-5 b(eiss)17 b(\(eds.\))f(Singapore:)k(W) -5 b(orld)17 b(Scienti\002c,)i(1993,)e(pp.)f(98\226107)330 674 y([28])34 b(K.)21 b(Broderix,)i(D.)e(Hundertmark,)i(H.)d(Leschk)o (e,)j Fh(Continuity)i(pr)m(operties)e(of)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)24 b(semigr)m(oups)e(with)h(ma)o(gnetic)h (\002elds)p FD(,)474 757 y(Re)n(v)l(.)17 b(Math.)h(Phys.)e Fg(12)h FD(\(2000\))g(181\226225)330 840 y([29])34 b(K.)20 b(Broderix,)i(H.)e(Leschk)o(e,)h(P)-7 b(.)20 b(M)6 b(\250)-28 b(uller)m(,)22 b Fh(Continuous)h(inte)m(gr)o(al)g(k)o(ernels)f(for)f (unbounded)i(Sc)o(hr)5 b(\250)-27 b(oding)o(er)23 b(semigr)m(oups)e (and)g(their)474 923 y(spectr)o(al)e(pr)m(ojections)p FD(,)g(preprint)g(math-ph/0209020)330 1006 y([30])34 b(R.)16 b(Carmona,)i(J.)e(Lacroix,)i(Spectral)h(theory)g(of)d(random)i (Schr)6 b(\250)-28 b(odinger)19 b(operators,)g(Boston:)f(Birkh)t(\250) -26 b(auser)m(,)19 b(1990)330 1089 y([31])34 b(G.)18 b(Chistyak)o(o)o(v)l(,)j(Y)-7 b(u.)18 b(L)l(yubarskii,)i(L.)d(P)o (astur)m(,)j Fh(On)e(completeness)k(of)d(r)o(andom)g(e)o(xponentials)j (in)d(the)h(Bar)n(gmann-F)-7 b(oc)o(k)19 b(space)p FD(,)g(J.)474 1172 y(Math.)e(Phys.)f Fg(42)h FD(\(2001\))h(3754\2263768)330 1255 y([32])34 b(J.-M.)23 b(Combes,)i(P)-7 b(.)24 b(D.)f(Hislop,)i Fh(Localization)j(for)d(some)g(continuous,)h(r)o(andom)f(Hamiltonians)i (in)e Fj(d)p Fh(-dimensions)p FD(,)h(J.)e(Funct.)474 1338 y(Anal.)17 b Fg(124)h FD(\(1994\))f(149\226180)330 1421 y([33])34 b(J.)20 b(M.)g(Combes,)h(P)-7 b(.)20 b(D.)g(Hislop,)i Fh(Landau)f(Hamiltonians)j(with)e(r)o(andom)f(potentials:)j (Localization)g(and)d(the)h(density)g(of)g(states)p FD(,)474 1504 y(Commun.)16 b(Math.)i(Phys.)e Fg(177)h FD(\(1996\))h(603\226629) 330 1588 y([34])34 b(H.)16 b(L.)g(Cycon,)i(R.)e(G.)g(Froese,)h(W)-6 b(.)16 b(Kirsch,)i(B.)e(Simon,)h(Schr)6 b(\250)-28 b(odinger)19 b(operators,)f(Berlin:)h(Springer)m(,)g(1987)330 1671 y([35])34 b(D.)13 b(Damanik,)i(P)-7 b(.)13 b(Stollmann,)j Fh(Multi-scale)h(analysis)f(implies)g(str)m(ong)e(dynamical)j (localization)p FD(,)g(Geom.)d(Funct.)h(Anal.)f Fg(11)h FD(\(2001\))474 1754 y(11\22629)330 1837 y([36])34 b(S.)18 b(Doi,)h(A.)f(Iw)o(atsuka,)i(T)-5 b(.)18 b(Mine,)h Fh(The)g(uniqueness) i(of)e(the)h(inte)m(gr)o(ated)i(density)e(of)f(states)h(for)f(the)h(Sc) o(hr)5 b(\250)-27 b(oding)o(er)21 b(oper)o(ator)o(s)f(with)474 1920 y(ma)o(gnetic)f(\002elds)p FD(,)f(Math.)f(Z.)f Fg(237)h FD(\(2001\))h(335\226371)330 2003 y([37])34 b(M.)18 b(D.)f(Donsk)o(er)m (,)i(S.)f(R.)g(S.)f(V)-7 b(aradhan,)20 b Fh(Asymptotics)g(of)e(the)h(W) l(iener)h(sausa)o(g)o(e)p FD(,)g(Commun.)e(Pure)g(Appl.)h(Math.)f Fg(28)h FD(\(1975\))g(525\226)474 2086 y(565;)f(Errata:)g Fh(ibid)p FD(.)g(677\226678)330 2169 y([38])34 b(T)-5 b(.)19 b(C.)g(Dorlas,)h(N.)f(Macris,)i(J.)e(V)-9 b(.)19 b(Pul)t(\264)-26 b(e,)20 b Fh(Localisation)j(in)d(a)g(single-band)i (appr)m(oximation)h(to)d(r)o(andom)g(Sc)o(hr)5 b(\250)-27 b(oding)o(er)22 b(oper)o(ator)o(s)474 2252 y(in)17 b(a)g(ma)o(gnetic)i (\002eld)p FD(,)f(Helv)l(.)f(Phys.)f(Acta)i Fg(68)f FD(\(1995\))h (329\226364)330 2335 y([39])34 b(T)-5 b(.)16 b(C.)g(Dorlas,)i(N.)e (Macris,)h(J.)f(V)-9 b(.)17 b(Pul)t(\264)-26 b(e,)17 b Fh(Localization)j(in)d(single)i(Landau)e(bands)p FD(,)g(J.)f(Math.)i (Phys.)e Fg(37)h FD(\(1996\))g(1574\2261595)330 2418 y([40])34 b(T)-5 b(.)19 b(C.)h(Dorlas,)h(N.)e(Macris,)i(J.)f(V)-9 b(.)20 b(Pul)t(\264)-26 b(e,)20 b Fh(The)h(natur)n(e)g(of)f(the)h (spectrum)h(for)e(a)g(Landau)h(Hamiltonian)i(with)f(delta)f(impurities) p FD(,)h(J.)474 2501 y(Stat.)c(Phys.)e Fg(87)h FD(\(1997\))g (847\226875)330 2584 y([41])34 b(T)-5 b(.)12 b(C.)h(Dorlas,)h(N.)f (Macris,)h(J.)f(V)-9 b(.)13 b(Pul)t(\264)-26 b(e,)14 b Fh(Char)o(acterization)j(of)d(the)g(spectrum)h(of)f(the)g(Landau)g (Hamiltonian)i(with)f(delta)g(impurities)p FD(,)474 2667 y(Commun.)h(Math.)i(Phys.)e Fg(204)h FD(\(1999\))h(367\226396)330 2750 y([42])34 b(H.)20 b(v)o(on)g(Dreifus,)h(A.)e(Klein,)i Fh(A)f(ne)o(w)h(pr)m(oof)g(of)f(localization)k(in)d(the)g(Ander)o(son)g (tight)h(binding)g(model)p FD(,)f(Commun.)e(Math.)i(Phys.)474 2833 y Fg(124)c FD(\(1989\))h(285\226299)330 2916 y([43])34 b(H.)13 b(v)o(on)g(Dreifus,)h(A.)e(Klein,)j Fh(Localization)h(for)d(r)o (andom)h(Sc)o(hr)5 b(\250)-27 b(oding)o(er)16 b(oper)o(ator)o(s)e(with) h(corr)n(elated)g(potentials)p FD(,)h(Commun.)d(Math.)474 2999 y(Phys.)j Fg(140)h FD(\(1991\))h(133\226147)330 3082 y([44])34 b(T)-5 b(.)14 b(P)-7 b(.)14 b(Eggarter)m(,)j Fh(Some)e(e)o(xact)i(r)n(esults)f(on)f(electr)m(on)j(ener)n(gy)e(le)o (vels)h(in)f(certain)h(one-dimensional)i(r)o(andom)c(potentials)p FD(,)j(Phys.)c(Re)n(v)l(.)474 3165 y(B)j Fg(5)g FD(\(1972\))g (3863\2263865)330 3248 y([45])34 b(K.)16 b(Efeto)o(v)l(,)h (Supersymmetry)h(in)g(disorder)g(and)f(chaos,)h(Cambridge:)h(Cambridge) g(UP)-7 b(,)16 b(1997)330 3331 y([46])34 b(P)-7 b(.)24 b(Elbau,)i(G.)f(M.)g(Graf,)g Fh(Equality)i(of)f(b)o(ulk)g(and)g(edg)o (e)h(Hall)f(conductance)j(r)n(e)o(visited)p FD(,)e(Commun.)e(Math.)h (Phys.)f Fg(229)h FD(\(2002\))474 3414 y(415\226432)330 3497 y([47])34 b(L.)18 b(Erd)6 b(\005)-28 b(os,)18 b Fh(Lifsc)o(hitz)j(tail)f(in)f(a)g(ma)o(gnetic)j(\002eld:)e(the)g (non-classical)i(r)n(e)m(gime)p FD(,)d(Probab)m(.)h(Theory)f(Relat.)i (Fields)f Fg(112)f FD(\(1998\))h(321\226)474 3580 y(371)330 3663 y([48])34 b(L.)23 b(Erd)6 b(\005)-28 b(os,)22 b Fh(Lifshitz)j(tail)g(in)f(a)g(ma)o(gnetic)i(\002eld:)f(coe)o(xistence)i (of)d(classical)i(and)e(quantum)h(behavior)g(in)f(the)h(bor)n(derline)g (case)p FD(,)474 3746 y(Probab)m(.)18 b(Theory)f(Relat.)i(Fields)f Fg(121)f FD(\(2001\))g(219\226236)330 3829 y([49])34 b(X.)18 b(M.)g(Fernique,)i Fh(Re)m(gularit)t(\264)-26 b(e)22 b(des)d(tr)o(ajectoir)n(es)i(des)e(fonctions)i(al)t(\264)-26 b(eatoir)n(es)21 b(Gaussiennes)p FD(,)f(in)f(Ecole)h(d'Et)t(\264)-26 b(e)19 b(de)g(Probabilit)t(\264)-26 b(es)22 b(de)474 3912 y(Saint-Flour)g(IV)e(-)g(1974,)g(P)-7 b(.-L.)18 b(Hennequin)k(\(ed.\),)e(Lecture)h(Notes)g(in)f(Mathematics)j Fg(480)p FD(,)d(Berlin:)i(Springer)m(,)f(1975,)f(pp.)g(1\22696)474 3995 y([in)d(French])330 4078 y([50])34 b(A.)16 b(Figotin,)i(A.)e (Klein,)i Fh(Localization)i(of)d(classical)i(waves)f(I:)f(acoustic)i (waves)p FD(,)f(Commun.)e(Math.)h(Phys.)g Fg(180)g FD(\(1996\))g (439\226482)330 4161 y([51])34 b(W)-6 b(.)23 b(Fischer)m(,)i(T)-5 b(.)23 b(Hupfer)m(,)h(H.)f(Leschk)o(e,)i(P)-7 b(.)23 b(M)6 b(\250)-28 b(uller)m(,)25 b Fh(Existence)h(of)f(the)f(density)i (of)e(states)h(for)f(multi-dimensional)k(continuum)474 4244 y(Sc)o(hr)5 b(\250)-27 b(oding)o(er)19 b(oper)o(ator)o(s)g(with)f (Gaussian)f(r)o(andom)g(potentials)p FD(,)j(Commun.)c(Math.)h(Phys.)f Fg(190)i FD(\(1997\))f(133\226141)330 4327 y([52])34 b(W)-6 b(.)14 b(Fischer)m(,)j(H.)e(Leschk)o(e,)h(P)-7 b(.)15 b(M)6 b(\250)-28 b(uller)m(,)17 b Fh(Spectr)o(al)g(localization) i(by)d(Gaussian)g(r)o(andom)g(potentials)i(in)e(multi-dimensional)k (contin-)474 4410 y(uous)d(space)p FD(,)g(J.)g(Stat.)g(Phys.)f Fg(101)h FD(\(2000\))h(935\226985)330 4493 y([53])34 b(V)-9 b(.)16 b(F)o(ock,)h Fh(Bemerkung)h(zur)e(Quantelung)j(des)e (harmonisc)o(hen)h(Oszillator)o(s)g(im)f(Ma)o(gnetfeld)p FD(,)j(Z.)15 b(Physik)i Fg(47)g FD(\(1928\))g(446\226448)h([in)474 4576 y(German])330 4659 y([54])34 b(R.)16 b(Friedber)o(g,)j(J.)d(M.)g (Luttinger)m(,)j Fh(Density)f(of)g(electr)m(onic)i(le)o(vels)f(in)e (disor)n(der)n(ed)g(systems)p FD(,)g(Phys.)g(Re)n(v)l(.)g(B)g Fg(12)g FD(\(1975\))g(4460\2264474)330 4742 y([55])34 b(J.)19 b(Fr)6 b(\250)-28 b(ohlich,)21 b(T)-5 b(.)18 b(Spencer)m(,)j Fh(Absence)g(of)f(dif)o(fusion)i(in)e(the)g(Ander)o (son)g(tight)h(binding)h(model)e(for)g(lar)n(g)o(e)g(disor)n(der)g(or)g (low)g(ener)n(gy)p FD(,)474 4825 y(Commun.)c(Math.)i(Phys.)e Fg(88)h FD(\(1983\))h(151\226184)330 4908 y([56])34 b(F)-5 b(.)13 b(Germinet,)j(S.)e(De)g(Bi)t(\036)-26 b(evre,)16 b Fh(Dynamical)h(localization)g(for)e(discr)n(ete)h(and)f(continuous)h (r)o(andom)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)17 b(oper)o(ator)o(s)p FD(,)e(Com-)474 4991 y(mun.)h(Math.)i(Phys.)e Fg(194)h FD(\(1998\))h(323-341)330 5074 y([57])34 b(F)-5 b(.)17 b(Germinet,)i(A.)e(Klein,)i Fh(Bootstr)o(ap)g(multiscale)i(analysis)e (and)f(localization)k(in)c(r)o(andom)g(media)p FD(,)h(Commun.)e(Math.)i (Phys.)e Fg(222)474 5157 y FD(\(2001\))h(415\226448)330 5240 y([58])34 b(F)-5 b(.)17 b(Germinet,)i(A.)f(Klein,)h Fh(A)e(c)o(har)o(acterization)22 b(of)c(the)h(Ander)o(son)g (metal-insulator)j(tr)o(ansport)d(tr)o(ansition)p FD(,)g(preprint)h(mp) p 3343 5240 20 4 v 24 w(arc)f(02-)474 5323 y(19,)e(to)g(appear)h(in)g (Duk)o(e)g(Math.)f(J.)p eop %%Page: 22 22 22 21 bop 278 233 a FI(22)802 b(HAJO)19 b(LESCHKE,)e(PETER)g(M)1860 220 y(\250)1849 233 y(ULLER,)f(AND)j(SIMONE)g(W)-7 b(ARZEL)330 425 y FD([59])34 b(F)-5 b(.)16 b(Germinet,)j(A.)d(Klein,)i Fh(Explicit)i(\002nite)f(volume)g(criteria)g(for)e(localization)k(in)d (continuous)h(r)o(andom)f(media)g(and)g(applications)p FD(,)474 508 y(preprint)h(mp)p 790 508 20 4 v 24 w(arc)e(02-375,)h(to)f (appear)i(in)e(Geom.)f(Funct.)i(Anal.)330 591 y([60])34 b(J.)16 b(I.)g(Gersten,)i(F)-5 b(.)16 b(W)-6 b(.)16 b(Smith,)h(The)g (physics)g(and)h(chemistry)g(of)f(materials,)i(Ne)n(w)e(Y)-7 b(ork:)17 b(W)m(ile)o(y)l(,)h(2001)330 674 y([61])34 b(I.)22 b(Y)-7 b(a.)23 b(Goldsheid,)i(S.)d(Molchano)o(v)l(,)i(L.)e(P)o (astur)m(,)i Fh(A)e(pur)n(e)h(point)h(spectrum)g(of)f(the)h(stoc)o (hastic)i(one\226dimensional)g(Sc)o(hr)5 b(\250)-27 b(oding)o(er)474 757 y(oper)o(ator)p FD(,)18 b(Funct.)f(Anal.)g(Appl.)h Fg(11)f FD(\(1977\))g(1\2268)g([Russian)i(original:)g(Funkts.)e(Anal.)h (Prilozh.)g Fg(11)f FD(\(1977\))g(3\22610])330 840 y([62])34 b(I.)19 b(S.)h(Gradshte)o(yn,)h(I.)e(M.)h(Ryzhik,)h(T)-5 b(able)20 b(of)g(inte)o(grals,)i(series,)f(and)f(products,)h(corrected) i(and)e(enlar)o(ged)h(edition,)g(San)e(Die)o(go:)474 923 y(Academic,)e(1980)330 1006 y([63])34 b(S.)14 b(A.)h(Gredeskul,)h (L.)e(A.)h(P)o(astur)m(,)g Fh(Behavior)i(of)e(the)h(density)h(of)f (states)g(in)f(one-dimensional)k(disor)n(der)n(ed)d(systems)g(near)f (the)h(edg)o(es)474 1089 y(of)h(the)h(spectrum)p FD(,)g(Theor)l(.)f (Math.)g(Phys.)f Fg(23)h FD(\(1975\))h(404-409)g([Russian)g(original:)i (T)-5 b(eor)l(.)17 b(Mat.)g(Fiz.)g Fg(23)g FD(\(1975\))h(132\226139]) 330 1172 y([64])34 b(B.)18 b(I.)g(Halperin,)i(M.)e(Lax,)g Fh(Impurity-band)j(tails)e(in)g(the)g(high-density)j(limit.)d(I.)f (Minimum)i(counting)h(methods)p FD(,)e(Phys.)e(Re)n(v)l(.)i Fg(148)474 1255 y FD(\(1966\))f(722\226740)330 1338 y([65])34 b(B.)16 b(Huck)o(estein,)k Fh(Scaling)f(theory)f(of)f(the)h(inte)m(g)o (er)h(quantum)g(Hall)e(ef)o(fect)p FD(,)i(Re)n(v)l(.)e(Mod.)g(Phys.)f Fg(67)h FD(\(1995\))h(357\226396)330 1421 y([66])34 b(D.)17 b(Hundertmark,)j(W)-6 b(.)16 b(Kirsch,)j(S.)e(W)-5 b(arzel,)19 b Fh(Classical)h(ma)o(gnetic)g(Lifshits)g(tails)f(in)f(thr)n(ee)h (space)g(dimensions:)g(impurity)h(poten-)474 1504 y(tials)e(with)g (slow)f(anisotr)m(opic)j(decay)p FD(,)e(preprint)h(math-ph/0212078,)g (to)e(appear)i(in)e(Mark)o(o)o(v)i(Process.)e(Relat.)h(Fields)330 1588 y([67])34 b(D.)16 b(Hundertmark,)i(B.)e(Simon,)h Fh(A)f(diama)o(gnetic)k(inequality)g(for)c(semigr)m(oup)i(dif)o(fer)n (ences)p FD(,)h(preprint)g(mp)p 2933 1588 V 23 w(arc)f(03-78,)f(to)g (appear)h(in)474 1671 y(J.)e(Reine)j(Ange)n(w)l(.)e(Math.)330 1754 y([68])34 b(T)-5 b(.)23 b(Hupfer)m(,)j(H.)e(Leschk)o(e,)h(S.)f(W) -5 b(arzel,)25 b Fh(P)-5 b(oissonian)25 b(obstacles)i(with)e(Gaussian)h (walls)f(discriminate)i(between)g(classical)g(and)474 1837 y(quantum)18 b(Lifshits)g(tailing)h(in)f(ma)o(gnetic)h(\002elds)p FD(,)f(J.)e(Stat.)h(Phys.)f Fg(97)h FD(\(1999\))h(725\226750)330 1920 y([69])34 b(T)-5 b(.)21 b(Hupfer)m(,)h(H.)f(Leschk)o(e,)i(S.)e(W) -5 b(arzel,)22 b Fh(The)g(multiformity)j(of)d(Lifshits)h(tails)g (caused)g(by)f(r)o(andom)g(Landau)g(Hamiltonians)i(with)474 2003 y(r)n(epulsive)19 b(impurity)f(potentials)i(of)d(dif)o(fer)n(ent)i (decay)g(at)e(in\002nity)p FD(,)i(AMS/IP)e(Stud.)g(Adv)l(.)f(Math.)h Fg(16)g FD(\(2000\))h(233\226247)330 2086 y([70])34 b(T)-5 b(.)25 b(Hupfer)m(,)h(H.)f(Leschk)o(e,)i(S.)d(W)-5 b(arzel,)27 b Fh(Upper)f(bounds)g(on)g(the)g(density)i(of)e(states)h(of)f(single)h (Landau)f(le)o(vels)i(br)m(oadened)f(by)474 2169 y(Gaussian)17 b(r)o(andom)h(potentials)p FD(,)h(J.)d(Math.)h(Phys.)f Fg(42)h FD(\(2001\))h(5626\2265641)330 2252 y([71])34 b(T)-5 b(.)17 b(Hupfer)m(,)i(H.)e(Leschk)o(e,)i(P)-7 b(.)17 b(M)6 b(\250)-28 b(uller)m(,)19 b(S.)f(W)-5 b(arzel,)18 b Fh(The)h(absolute)h(continuity)h(of)d(the)h(inte)m(gr)o(ated)i (density)f(of)f(states)g(for)f(ma)o(gnetic)474 2335 y(Sc)o(hr)5 b(\250)-27 b(oding)o(er)19 b(oper)o(ator)o(s)g(with)f(certain)g (unbounded)h(r)o(andom)e(potentials)p FD(,)j(Commun.)c(Math.)h(Phys.)f Fg(221)i FD(\(2001\))f(229\226254)330 2418 y([72])34 b(T)-5 b(.)29 b(Hupfer)m(,)h(H.)g(Leschk)o(e,)g(P)-7 b(.)29 b(M)6 b(\250)-28 b(uller)m(,)32 b(S.)d(W)-5 b(arzel,)30 b Fh(Existence)i(and)f(uniqueness)g(of)g(the)f(inte)m(gr)o(ated)j (density)f(of)e(states)h(for)474 2501 y(Sc)o(hr)5 b(\250)-27 b(oding)o(er)23 b(oper)o(ator)o(s)f(with)g(ma)o(gnetic)i(\002elds)d (and)h(unbounded)g(r)o(andom)f(potentials)p FD(,)j(Re)n(v)l(.)d(Math.)g (Phys.)f Fg(13)h FD(\(2001\))h(1547\226)474 2584 y(1581)330 2667 y([73])34 b(M.)23 b(Jan\337en,)h(O.)f(V)l(iehwe)o(ger)m(,)i(U.)e (F)o(astenrath,)i(J.)e(Hajdu,)h(Introduction)i(to)e(the)g(theory)h(of)f (the)g(inte)o(ger)i(quantum)e(Hall)h(ef)n(fect,)474 2750 y(W)-5 b(einheim:)18 b(VCH,)f(1994)330 2833 y([74])34 b(E.)16 b(O.)g(Kane,)h Fh(Thomas-F)-5 b(ermi)18 b(appr)m(oac)o(h)g(to)f (impur)n(e)g(semiconductor)j(band)e(structur)n(e)p FD(,)f(Phys.)f(Re)n (v)l(.)h Fg(131)h FD(\(1963\))f(79\22688)330 2916 y([75])34 b(W)-6 b(.)14 b(Kirsch,)i Fh(Random)g(Sc)o(hr)5 b(\250)-27 b(oding)o(er)18 b(oper)o(ator)o(s:)f(a)f(cour)o(se)p FD(,)g(in)g(Schr)6 b(\250)-28 b(odinger)18 b(operators,)f(H.)e(Holden,) h(A.)f(Jensen)i(\(eds.\),)e(Lecture)474 2999 y(Notes)j(in)f(Physics)g Fg(345)p FD(,)g(Berlin:)i(Springer)m(,)f(1989,)f(pp.)g(264\226370)330 3082 y([76])34 b(W)-6 b(.)19 b(Kirsch,)i(F)-5 b(.)19 b(Martinelli,)k Fh(On)d(the)h(er)n(godic)h(pr)m(operties)f(of)g(the)g (spectrum)h(of)e(g)o(ener)o(al)i(r)o(andom)f(oper)o(ator)o(s)p FD(,)g(J.)e(Reine)j(Ange)n(w)l(.)474 3165 y(Math.)17 b Fg(334)h FD(\(1982\))f(141\226156)330 3248 y([77])34 b(W)-6 b(.)14 b(Kirsch,)h(F)-5 b(.)13 b(Martinelli,)18 b Fh(On)d(the)g(density)i(of)e(states)g(of)h(Sc)o(hr)5 b(\250)-27 b(oding)o(er)17 b(oper)o(ator)o(s)f(with)g(a)e(r)o(andom)h (potential)p FD(,)i(J.)d(Phys.)g(A:)h(Math.)474 3331 y(Gen.)i Fg(15)g FD(\(1982\))h(2139\2262156)330 3414 y([78])34 b(W)-6 b(.)23 b(Kirsch,)i(F)-5 b(.)24 b(Martinelli,)k Fh(On)c(the)h(essential)i(selfadjointness)h(of)d(stoc)o(hastic)i(Sc)o (hr)5 b(\250)-27 b(oding)o(er)27 b(oper)o(ator)o(s)p FD(,)f(Duk)o(e)f(Math.)h(J.)d Fg(50)474 3497 y FD(\(1983\))18 b(1255\2261260)330 3580 y([79])34 b(W)-6 b(.)25 b(Kirsch,)i(F)-5 b(.)25 b(Martinelli,)30 b Fh(Lar)n(g)o(e)d(de)o(viations)i(and)d (Lifshitz)i(singularity)h(of)e(the)g(inte)m(gr)o(ated)j(density)e(of)f (states)g(of)g(r)o(andom)474 3663 y(Hamiltonians)p FD(,)19 b(Commun.)d(Math.)i(Phys.)e Fg(89)h FD(\(1983\))g(27\22640)330 3746 y([80])34 b(W)-6 b(.)19 b(Kirsch,)h(P)-7 b(.)18 b(Stollmann,)j(G.)e(Stolz,)h Fh(Ander)o(son)h(localization)i(for)d(r)o (andom)g(Sc)o(hr)5 b(\250)-27 b(oding)o(er)22 b(oper)o(ator)o(s)f(with) f(long)h(r)o(ang)o(e)f(inter)o(-)474 3829 y(actions)p FD(,)e(Commun.)f(Math.)g(Phys.)f Fg(195)h FD(\(1998\))h(495\226507)330 3912 y([81])34 b(A.)18 b(Kisele)n(v)l(,)j(Y)-9 b(.)19 b(Last,)g Fh(Solutions,)h(spectrum,)g(and)g(dynamics)h(for)e(Sc)o(hr)5 b(\250)-27 b(oding)o(er)22 b(oper)o(ator)o(s)e(on)g(in\002nite)h (domains)p FD(,)f(Duk)o(e)g(Math.)474 3995 y(J.)c Fg(102)h FD(\(2000\))h(125\226150)330 4078 y([82])34 b(C.)16 b(Kittel,)j (Introduction)h(to)d(solid-state)j(physics,)d Fk(7)p FD(th)h(edition,)g(Ne)n(w)g(Y)-7 b(ork:)17 b(W)m(ile)o(y)l(,)g(1996)330 4161 y([83])34 b(A.)14 b(Klein,)j(J.)d(F)-5 b(.)14 b(Perez,)i Fh(On)f(the)h(density)h(of)f(states)g(for)g(r)o(andom)f(potentials)j (in)e(the)g(pr)n(esence)g(of)g(a)f(uniform)h(ma)o(gnetic)i(\002eld)p FD(,)e(Nucl.)474 4244 y(Phys.)g(B)h Fg(251)g FD(\(1985\))h(199\226211) 330 4327 y([84])34 b(K.)16 b(v)o(on)h(Klitzing,)i Fh(The)e(quantized)i (Hall)f(ef)o(fect)p FD(,)h(Re)n(v)l(.)e(Mod.)g(Phys.)f Fg(58)h FD(\(1986\))g(519\226531)330 4410 y([85])34 b(S.)18 b(K)n(otani,)h Fh(On)g(asymptotic)i(behaviour)f(of)f(the)h(spectr)o(a)g (of)f(a)f(one-dimensional)k(Hamiltonian)f(with)f(a)f(certain)h(r)o (andom)f(coef)o(\002-)474 4493 y(cient)p FD(,)f(Publ.)f(Res.)g(Inst.)g (Math.)g(Sci.,)g(K)n(yoto)h(Uni)n(v)l(.)f Fg(12)g FD(\(1976\))h (447\226492)330 4576 y([86])34 b(F)-5 b(.)17 b(Klopp,)h(L.)f(P)o(astur) m(,)i Fh(Lifshitz)g(tails)g(for)g(r)o(andom)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)20 b(oper)o(ator)o(s)g(with)f(ne)m(gative)i (singular)e(P)-5 b(oisson)18 b(potential)p FD(,)i(Com-)474 4659 y(mun.)c(Math.)i(Phys.)e Fg(206)h FD(\(1999\))h(57\226103)330 4742 y([87])34 b(N.)16 b(P)-7 b(.)16 b(K)n(o)o(v)n(alenk)o(o,)j(Y)-9 b(.)16 b(P)-7 b(.)16 b(Krasn)o(y)l(,)h(U.)f(Kre)o(y)l(,)h(Physics)g(of) g(amorphous)h(metals,)f(Berlin:)i(W)m(ile)o(y)g(VCH,)d(2001)330 4825 y([88])34 b(B.)16 b(Kramer)m(,)i(A.)e(MacKinnon,)j Fh(Localization:)g(theory)g(and)e(e)o(xperiment)p FD(,)i(Rep.)e(Prog.)g (Phys.)f Fg(56)h FD(\(1993\))h(1469\2261564)330 4908 y([89])34 b(U.)16 b(Krengel,)i(Er)o(godic)g(theorems,)g(Berlin:)h(de)e (Gruyter)m(,)h(1985)330 4991 y([90])34 b(R.)16 b(K)o(ubo,)h (Statistical)k(mechanics,)d(Amsterdam:)g(North)g(Holland,)g(1965)330 5074 y([91])34 b(P)-7 b(.)16 b(K)o(uchment,)i(Floquet)g(theory)h(for)e (partial)i(dif)n(ferential)i(equations,)e(Boston:)f(Birkh)t(\250)-26 b(auser)m(,)19 b(1993)330 5157 y([92])34 b(I.)15 b(V)-9 b(.)15 b(K)o(ukushkin,)i(S.)d(V)-9 b(.)16 b(Meshk)o(o)o(v)l(,)g(V)-9 b(.)15 b(B.)g(T)n(imofee)n(v)l(,)h Fh(T)-5 b(wo-dimensional)18 b(electr)m(on)g(density)g(of)e(states)g(in)g(a)g(tr)o(ansver)o(se)h(ma) o(gnetic)474 5240 y(\002eld)p FD(,)h(So)o(v)l(.)e(Phys.)g(Usp.)g Fg(31)h FD(\(1988\))h(511\226534)g([Russian)g(original:)i(Usp.)c(Fiz.)h (Nauk)g Fg(155)h FD(\(1988\))f(219\226264])330 5323 y([93])34 b(H.)16 b(K)o(unz,)h Fh(The)g(quantum)i(Hall)e(ef)o(fect)i(for)f (electr)m(ons)h(in)e(a)g(r)o(andom)g(potential)p FD(,)i(Commun.)e (Math.)g(Phys.)f Fg(112)i FD(\(1987\))f(121\226145)p eop %%Page: 23 23 23 22 bop 1025 234 a FI(RANDOM)18 b(SCHR)1465 221 y(\250)1454 234 y(ODINGER)g(OPERA)-6 b(T)o(ORS)18 b(FOR)h(AMORPHOUS)g(SOLIDS)689 b(23)330 425 y FD([94])34 b(L.)16 b(Landau,)h Fh(Diama)o(gnetismus)j (der)d(Metalle)p FD(,)i(Z.)d(Physik)h Fg(64)g FD(\(1930\))g(629\226637) h([in)g(German])330 508 y([95])34 b(Y)-9 b(.)16 b(Last,)g Fh(Quantum)h(dynamics)g(and)g(decompositions)i(of)e(singular)g (continuous)h(spectr)o(a)p FD(,)f(J.)e(Funct.)i(Anal.)g Fg(142)f FD(\(1996\))h(406\226445)330 591 y([96])34 b(H.)k(Leschk)o(e,) h(S.)f(W)-5 b(arzel,)39 b Fh(Quantum-classical)j(tr)o(ansitions)e(in)f (Lifshits)h(tails)f(with)h(ma)o(gnetic)h(\002elds)p FD(,)e(preprint)h (cond-)474 674 y(mat/0310389,)19 b(to)e(appear)i(in)e(Phys.)f(Re)n(v)l (.)h(Lett.)330 757 y([97])34 b(I.)16 b(M.)h(Lifshitz,)g Fh(Structur)n(e)i(of)e(the)h(ener)n(gy)g(spectrum)g(of)f(the)h (impurity)h(bands)e(in)g(disor)n(der)n(ed)h(solid)f(solutions)p FD(,)h(So)o(v)l(.)f(Phys.)f(JETP)474 840 y Fg(17)h FD(\(1963\))h (1159\2261170)g([Russian)g(original:)i(Zh.)c(Eksp.)g(T)-5 b(eor)l(.)17 b(Fiz.)g Fg(44)g FD(\(1963\))h(1723\2261741])330 923 y([98])34 b(I.)14 b(M.)g(Lifshitz,)h Fh(The)g(ener)n(gy)g(spectrum) h(of)f(disor)n(der)n(ed)g(systems)p FD(,)g(Adv)l(.)e(Phys.)h Fg(13)h FD(\(1964\))g(483\226536;)h Fh(Ener)n(gy)f(spectrum)g(structur) n(e)474 1006 y(and)h(quantum)h(states)g(of)f(disor)n(der)n(ed)h (condensed)h(systems)p FD(,)e(So)o(v)l(.)f(Phys.)g(Usp.)h Fg(7)f FD(\(1965\))i(549\226573)g([Russian)g(original:)i(Usp.)c(Fiz.) 474 1089 y(Nauk.)i Fg(83)g FD(\(1964\))h(617\226663])330 1172 y([99])34 b(I.)25 b(M.)f(Lifshitz,)i Fh(Theory)g(of)g (\003uctuating)i(le)o(vels)f(in)f(disor)n(der)n(ed)f(systems)p FD(,)h(So)o(v)l(.)e(Phys.)h(JETP)f Fg(26)h FD(\(1968\))h(462\226479)g ([Russian)474 1255 y(original:)20 b(Zh.)c(Eksp.)g(T)-5 b(eor)l(.)17 b(Fiz.)g Fg(53)g FD(\(1967\))h(743\226758])297 1338 y([100])34 b(I.)22 b(M.)g(Lifshits,)g(S.)g(A.)g(Gredeskul,)h(L.)f (A.)f(P)o(astur)m(,)j(Introduction)h(to)e(the)g(theory)h(of)e (disordered)j(systems,)d(Ne)n(w)h(Y)-7 b(ork:)23 b(W)m(ile)o(y)l(,)474 1421 y(1988)17 b([Russian)h(original:)i(Mosco)n(w:)e(Nauka,)g(1982])297 1504 y([101])34 b(M.)16 b(A.)g(Lifshits,)i(Gaussian)g(random)f (functions,)i(Dordrecht:)g(Kluwer)m(,)f(1995)297 1588 y([102])34 b(J.)14 b(M.)g(Luttinger)m(,)i Fh(Ne)o(w)g(variational)h (method)f(with)g(applications)i(to)d(disor)n(der)n(ed)g(systems)p FD(,)g(Phys.)f(Re)n(v)l(.)h(Lett.)g Fg(37)g FD(\(1976\))g(609\226612) 297 1671 y([103])34 b(N.)20 b(Macris,)i(J.)e(V)-9 b(.)21 b(Pul)t(\264)-26 b(e,)21 b Fh(Density)i(of)e(states)h(of)g(r)o(andom)f (Sc)o(hr)5 b(\250)-27 b(oding)o(er)24 b(oper)o(ator)o(s)e(with)g (uniform)g(ma)o(gnetic)i(\002eld)p FD(,)e(Lett.)f(Math.)474 1754 y(Phys.)16 b Fg(24)h FD(\(1992\))h(307\226321)297 1837 y([104])34 b(F)-5 b(.)13 b(Martinelli,)18 b(H.)13 b(Holden,)i Fh(On)g(absence)h(of)e(dif)o(fusion)j(near)e(the)g(bottom)g (of)g(the)g(spectrum)h(for)e(a)h(r)o(andom)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)17 b(oper)o(ator)474 1920 y(on)g Fj(L)605 1896 y Ff(2)640 1920 y Fk(\()p Fe(R)726 1896 y Fd(\027)764 1920 y Fk(\))p FD(,)g(Commun.)f(Math.)i(Phys.)e Fg(93)h FD(\(1984\))g(197\226217)297 2003 y([105])34 b(F)-5 b(.)24 b(Martinelli,)k(E.)d(Scoppola,)h Fh(Intr)m(oduction)j(to) c(the)h(mathematical)j(theory)e(of)e(Ander)o(son)h(localization)p FD(,)j(Ri)n(vista)e(del)f(Nuo)o(v)o(o)474 2086 y(Cimento)18 b Fg(10)f FD(\(10\))h(\(1987\))f(1\22690)297 2169 y([106])34 b(H.)13 b(Matsumoto,)h Fh(On)f(the)h(inte)m(gr)o(ated)j(density)e(of)f (states)g(for)f(the)i(Sc)o(hr)5 b(\250)-27 b(oding)o(er)15 b(oper)o(ator)o(s)g(with)g(certain)g(r)o(andom)e(electr)m(oma)o(gnetic) 474 2252 y(potentials)p FD(,)19 b(J.)d(Math.)i(Soc.)e(Japan)i Fg(45)f FD(\(1993\))h(197\226214)297 2335 y([107])34 b(D.)16 b(C.)h(Mattis,)h(The)e(theory)j(of)e(magnetism)h(I,)e (corrected)k(2nd)d(printing,)i(Berlin:)g(Springer)m(,)f(1988)297 2418 y([108])34 b(T)-5 b(.)21 b(Mine,)h Fh(The)g(uniqueness)i(of)e(the) h(inte)m(gr)o(ated)i(density)e(of)f(states)h(for)f(the)h(Sc)o(hr)5 b(\250)-27 b(oding)o(er)24 b(oper)o(ator)o(s)f(for)f(the)h(Robin)f (boundary)474 2501 y(conditions)p FD(,)d(Publ.)e(Res.)g(Inst.)g(Math.)g (Sci.,)g(K)n(yoto)h(Uni)n(v)l(.)f Fg(38)g FD(\(2002\))g(355-385)297 2584 y([109])34 b(N.)16 b(F)-5 b(.)16 b(Mott,)h Fh(Electr)m(ons)i(in)e (disor)n(der)n(ed)g(structur)n(es)p FD(,)h(Adv)l(.)e(Phys.)g Fg(16)h FD(\(1967\))h(49\226144)297 2667 y([110])34 b(S.)20 b(Nakamura,)i Fh(A)e(r)n(emark)h(on)g(the)g(Diric)o(hlet-Neumann)k (decoupling)f(and)d(the)h(inte)m(gr)o(ated)h(density)g(of)e(states)p FD(,)h(J.)e(Funct.)h(Anal.)474 2750 y Fg(179)c FD(\(2001\))h (136\226152)297 2833 y([111])34 b(S.)18 b(Nakao,)h Fh(On)f(the)h (spectr)o(al)h(distrib)o(ution)h(of)e(the)g(Sc)o(hr)5 b(\250)-27 b(oding)o(er)21 b(oper)o(ator)f(with)f(r)o(andom)g (potential)p FD(,)i(Japan.)e(J.)e(Math.)i Fg(3)g FD(\(1977\))474 2916 y(111\226139)297 2999 y([112])34 b(L.)17 b(A.)h(P)o(astur)m(,)i Fh(On)e(the)h(Sc)o(hr)5 b(\250)-27 b(oding)o(er)21 b(equation)g(with)f (a)e(r)o(andom)h(potential)p FD(,)i(Theor)l(.)e(Math.)g(Phys.)f Fg(6)g FD(\(1971\))i(299-306)f([Russian)474 3082 y(original:)h(T)-5 b(eor)l(.)17 b(Mat.)g(Fiz.)g Fg(6)g FD(\(1971\))h(415\226424])297 3165 y([113])34 b(L.)22 b(A.)h(P)o(astur)m(,)i Fh(On)e(the)h(distrib)o (ution)i(of)e(the)h(eig)o(en)m(values)i(of)d(the)h(Sc)o(hr)5 b(\250)-27 b(oding)o(er)26 b(equation)g(with)e(a)g(r)o(andom)g (potential)p FD(,)i(Funct.)474 3248 y(Anal.)17 b(Appl.)g Fg(6)g FD(\(1972\))h(163\226165)g([Russian)g(original:)i(Funkts.)d (Anal.)g(Prilozh.)h Fg(6)f FD(\(1972\))h(93\22694])297 3331 y([114])34 b(L.)16 b(A.)h(P)o(astur)m(,)h Fh(Spectr)o(a)h(of)f(r)o (andom)g(self)g(adjoint)h(oper)o(ator)o(s)p FD(,)g(Russ.)e(Math.)h (Surv)o(e)o(ys)g Fg(28)f FD(\(1973\))h(1\22667)g([Russian)h(original:)h (Usp.)474 3414 y(Mat.)d(Nauk)h Fg(28)f FD(\(1973\))g(3\22664])297 3497 y([115])34 b(L.)17 b(A.)h(P)o(astur)m(,)i Fh(Behavior)f(of)g(some) g(W)l(iener)i(inte)m(gr)o(als)f(as)f Fj(t)k Fc(!)g(1)17 b Fh(and)i(the)h(density)g(of)f(states)h(of)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)21 b(equations)f(with)474 3580 y(r)o(andom)d(potential)p FD(,)j(Theor)l(.)d(Math.)g(Phys.)f Fg(32)h FD(\(1977\))h(615\226620)g([Russian)g(original:)i(T)-5 b(eor)l(.)17 b(Mat.)g(Fiz.)g Fg(32)g FD(\(1977\))h(88\22695])297 3663 y([116])34 b(L.)21 b(A.)h(P)o(astur)m(,)h Fh(Spectr)o(al)h(pr)m (operties)g(of)f(disor)n(der)n(ed)g(systems)g(in)g(the)g(one-body)h (appr)m(oximation)p FD(,)h(Commun.)d(Math.)h(Phys.)e Fg(75)474 3746 y FD(\(1980\))d(179\226196)297 3829 y([117])34 b(L.)15 b(A.)h(P)o(astur)m(,)h Fh(Spectr)o(al)h(pr)m(operties)g(of)f(r) o(andom)g(selfadjoint)i(oper)o(ator)o(s)f(and)e(matrices)i(\(a)f(surve) n(y\))p FD(,)g(T)n(ransl.,)e(Ser)l(.)i(2,)f(Am.)f(Math.)474 3912 y(Soc.)i Fg(188)g FD(\(1999\))h(153\226195)g([Russian)g(original:) i(T)n(r)l(.)15 b(St-Peterbg.)k(Mat.)e(Obshch.)g Fg(4)g FD(\(1996\))h(222\226286])297 3995 y([118])34 b(L.)16 b(P)o(astur)m(,)h(A.)g(Figotin,)h(Spectra)g(of)f(random)h(and)f (almost-periodic)k(operators,)d(Berlin:)h(Springer)m(,)g(1992)297 4078 y([119])34 b(J.)18 b(V)-9 b(.)19 b(Pul)t(\264)-26 b(e,)19 b(M.)f(Scro)n(wston,)i Fh(In\002nite)h(de)m(g)o(ener)o(acy)h (for)e(a)e(Landau)i(Hamiltonian)h(with)f(P)-5 b(oisson)19 b(impurities)p FD(,)i(J.)d(Math.)h(Phys.)f Fg(38)474 4161 y FD(\(1997\))g(6304\2266314)297 4244 y([120])34 b(J.)16 b(V)-9 b(.)16 b(Pul)t(\264)-26 b(e,)17 b(M.)f(Scro)n(wston,)i Fh(The)f(spectrum)h(of)f(a)f(ma)o(gnetic)j(Sc)o(hr)5 b(\250)-27 b(oding)o(er)19 b(oper)o(ator)f(with)g(r)o(andomly)g (located)h(delta)f(impurities)p FD(,)474 4327 y(J.)e(Math.)h(Phys.)f Fg(41)i FD(\(2000\))f(2805\2262825)297 4410 y([121])34 b(R.)16 b(del)i(Rio,)f(S.)g(Jitomirskaya,)h(Y)-9 b(.)17 b(Last,)f(B.)h(Simon,)g Fh(What)g(is)g(localization?)p FD(,)j(Phys.)c(Re)n(v)l(.)i(Lett.)f Fg(75)g FD(\(1995\))h(117\226119) 297 4493 y([122])34 b(R.)19 b(del)g(Rio,)h(S.)e(Jitomirskaya,)j(Y)-9 b(.)18 b(Last,)h(B.)g(Simon,)f Fh(Oper)o(ator)o(s)j(with)f(singular)g (continuous)h(spectrum)f(IV)l(:)e(Hausdorf)o(f)i(dimen-)474 4576 y(sions,)d(r)o(ank)g(one)h(perturbations)h(and)e(localization)p FD(,)j(J.)c(d'Analyse)j(Math.)e Fg(69)g FD(\(1996\))h(153\226200)297 4659 y([123])34 b(D.)16 b(Ruelle,)j(Statistical)h(mechanics:)g (rigorous)e(results,)f(London:)h(Imperial)h(Colle)o(ge)g(Press,)d(1999) 297 4742 y([124])34 b(R.)16 b(Salomon,)i Fh(Density)g(of)g(states)g (for)f(high)g(Landau)h(le)o(vels)h(and)e(r)o(andom)g(potential)p FD(,)j(Z.)15 b(Physik)j(B)e Fg(65)h FD(\(1987\))h(443\226451)297 4825 y([125])34 b(B.)18 b(I.)f(Shklo)o(vskii,)i(A.)f(L.)f(Efros,)g (Electronic)j(properties)h(of)d(doped)h(semiconductors,)h(Berlin:)g (Springer)m(,)f(1984)g([Russian)g(origi-)474 4908 y(nal:)f(Mosco)n(w:)g (Nauka,)g(1979])297 4991 y([126])34 b(B.)16 b(Simon,)h Fh(Sc)o(hr)5 b(\250)-27 b(oding)o(er)19 b(semigr)m(oups)p FD(,)f(Bull.)g(Am.)d(Math.)j(Soc.)f(\(N.S.\))f Fg(7)g FD(\(1982\))i(447\226526;)h(Erratum:)e Fh(ibid)p FD(.)h Fg(11)f FD(\(1984\))h(426)297 5074 y([127])34 b(B.)16 b(Simon,)h Fh(Absence)h(of)f(ballistic)j(motion)p FD(,)e(Commun.)e (Math.)i(Phys.)e Fg(134)h FD(\(1990\))h(209\226212)297 5157 y([128])34 b(A.)e(Sommerfeld,)h Fh(Zur)f(Elektr)m(onentheorie)37 b(der)c(Metalle)i(auf)e(Grund)g(der)g(F)-5 b(ermisc)o(hen)33 b(Statistik.)i(I.)d(T)-6 b(eil:)34 b(Allg)o(emeines,)474 5240 y(Str)5 b(\250)-27 b(omungs=)18 b(und)f(A)o(ustrittsvor)n(g)5 b(\250)-27 b(ang)o(e)p FD(,)19 b(Z.)c(Physik)i Fg(47)g FD(\(1928\))h(1\22642)g([in)f(German])p eop %%Page: 24 24 24 23 bop 278 233 a FI(24)802 b(HAJO)19 b(LESCHKE,)e(PETER)g(M)1860 220 y(\250)1849 233 y(ULLER,)f(AND)j(SIMONE)g(W)-7 b(ARZEL)297 425 y FD([129])34 b(T)-5 b(.)20 b(C.)g(Spencer)m(,)j Fh(The)e(Sc)o(hr)5 b(\250)-27 b(oding)o(er)23 b(equation)g(with)f(a)f (r)o(andom)g(potential:)i(a)e(mathematical)j(r)n(e)o(vie)o(w)p FD(,)e(in)f(Critical)j(phenomena,)474 508 y(random)g(systems,)f(gauge)h (theories,)h(part)f(II,)f(K.)f(Osterw)o(alder)k(and)e(R.)f(Stora,)g (eds.)h(Amsterdam:)g(Noth-Holland,)h(1986,)f(pp.)474 591 y(895\226943)297 674 y([130])34 b(T)-5 b(.)16 b(Spencer)m(,)i Fh(Localization)i(for)d(r)o(andom)g(and)h(quasiperiodic)i(potentials)p FD(,)f(J.)d(Stat.)h(Phys.)f Fg(51)h FD(\(1988\))h(1009\2261019)297 757 y([131])34 b(H.)14 b(L.)g(St)6 b(\250)-28 b(ormer)m(,)16 b(D.)f(C.)f(Tsui,)h(A.)f(C.)h(Gossard,)g Fh(The)g(fr)o(actional)j (quantum)f(Hall)e(ef)o(fect)p FD(,)j(Re)n(v)l(.)d(Mod.)g(Phys.)f Fg(71)h FD(\(1999\))h(S298\226S305)297 840 y([132])34 b(P)-7 b(.)16 b(Stollmann,)i Fh(Lifshitz)g(asymptotics)i(via)d(linear)i (coupling)g(of)e(disor)n(der)p FD(,)g(Math.)g(Phys.)f(Anal.)i(Geom.)e Fg(2)h FD(\(1999\))h(279\226289)297 923 y([133])34 b(P)-7 b(.)16 b(Stollmann,)i(Caught)h(by)e(disorder:)h(bound)g(states)g(in)f (random)h(media,)f(Boston:)i(Birkh)t(\250)-26 b(auser)m(,)19 b(2001)297 1006 y([134])34 b(G.)21 b(Stolz,)h Fh(Localization)i(for)e (r)o(andom)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)24 b(oper)o(ator)o(s)f (with)f(P)-5 b(oisson)22 b(potential)p FD(,)h(Ann.)e(Inst.)h(Henri)g (Poincar)t(\264)-26 b(e,)23 b(Phys.)474 1089 y(Th)t(\264)-26 b(eor)l(.)17 b Fg(63)g FD(\(1995\))h(297\226314)297 1172 y([135])34 b(G.)21 b(Stolz,)h Fh(Str)o(ate)m(gies)i(in)e(localization)j (pr)m(oofs)c(for)h(one-dimensional)j(r)o(andom)d(Sc)o(hr)5 b(\250)-27 b(oding)o(er)23 b(oper)o(ator)o(s)p FD(,)g(Proc.)e(Indian)i (Acad.)474 1255 y(Sci.,)17 b(Math.)g(Sci.)g Fg(112)g FD(\(2002\))h(229\226243)297 1338 y([136])34 b(I.)17 b(M.)g(Suslo)o(v)l(,)h Fh(De)o(velopment)j(of)d(a)g Fk(\(4)f Fc(\000)f Fj(\017)p Fk(\))p Fh(-dimensional)21 b(theory)e(for)f(the)h (density)g(of)g(states)f(of)h(a)e(disor)n(der)n(ed)i(system)f(near)h (the)474 1421 y(Ander)o(son)e(tr)o(ansition)p FD(,)i(Phys.)d(Usp.)g Fg(41)h FD(\(1998\))h(441\226467)g([Russian)g(original:)i(Usp.)c(Fiz.)h (Nauk)h Fg(168)f FD(\(1998\))h(503\226530])297 1504 y([137])34 b(A.-S.)16 b(Sznitman,)i(Bro)n(wnian)h(motion,)e(obstacles)i(and)f (random)f(media,)h(Berlin:)h(Springer)m(,)f(1998)297 1588 y([138])34 b(S.)16 b(Tcheremchantse)n(v)l(,)k Fh(How)d(to)g(pr)m (o)o(ve)h(dynamical)i(localization)p FD(,)f(Commun.)e(Math.)g(Phys.)f Fg(221)i FD(\(2001\))f(27-56)297 1671 y([139])34 b(A.)22 b(T)n(ip,)f Fh(Absolute)j(continuity)i(of)c(the)i(inte)m(gr)o(ated)h (density)f(of)f(states)h(of)e(the)i(quantum)f(Lor)n(entz)g(gas)f(for)h (a)f(class)h(of)g(r)n(epulsive)474 1754 y(potentials)p FD(,)c(J.)d(Phys.)g(A:)h(Math.)h(Gen.)f Fg(27)g FD(\(1994\))g (1057\2261069)297 1837 y([140])34 b(N.)16 b(Ueki,)h Fh(On)g(spectr)o(a) h(of)g(r)o(andom)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)19 b(oper)o(ator)o(s)f(with)g(ma)o(gnetic)i(\002elds)p FD(,)d(Osaka)h(J.)e (Math.)h Fg(31)g FD(\(1994\))h(177\226187)297 1920 y([141])34 b(N.)18 b(Ueki,)i Fh(W)-6 b(e)m(gner)20 b(estimates)h(and)f (localization)i(for)e(Gaussian)f(r)o(andom)h(potentials)p FD(,)h(preprint)g(2002,)f(to)f(appear)i(in)e(Publ.)g(Res.)474 2003 y(Inst.)e(Math.)g(Sci.,)g(K)n(yoto)h(Uni)n(v)l(.)297 2086 y([142])34 b(I.)29 b(V)-7 b(eseli)t(\264)-26 b(c,)30 b Fh(Inte)m(gr)o(ated)i(density)f(of)f(states)g(and)g(W)-6 b(e)m(gner)31 b(estimates)g(for)e(r)o(andom)h(Sc)o(hrding)o(er)h(oper)o (ator)o(s)p FD(,)f(preprint)i(math-)474 2169 y(ph/0307062,)19 b(to)e(appear)h(in)g(Contemp.)f(Math.)297 2252 y([143])34 b(W)-5 b(ei-Min)22 b(W)-5 b(ang,)21 b Fh(Micr)m(olocalization,)26 b(per)n(colation,)d(and)f(Ander)o(son)g(localization)j(for)c(the)i(ma)o (gnetic)g(Sc)o(hr)5 b(\250)-27 b(oding)o(er)24 b(oper)o(ator)474 2335 y(with)18 b(a)f(r)o(andom)g(potential)p FD(,)i(J.)d(Funct.)i (Anal.)f Fg(146)h FD(\(1997\))f(1\22626)297 2418 y([144])34 b(S.)16 b(W)-5 b(arzel,)18 b(On)f(Lifshits)g(tails)h(in)g(magnetic)h (\002elds,)e(Berlin:)i(Logos,)d(2001)i([PhD-Thesis,)f(Uni)n(v)o(ersit)t (\250)-26 b(at)20 b(Erlangen-N)6 b(\250)-28 b(urnber)o(g])297 2501 y([145])34 b(F)-5 b(.)16 b(W)-5 b(e)o(gner)m(,)17 b Fh(Bounds)g(on)g(the)h(density)h(of)e(states)h(in)f(disor)n(der)n(ed) h(systems)p FD(,)f(Z.)e(Physik)j(B)f Fg(44)g FD(\(1981\))g(9\22615)297 2584 y([146])34 b(F)-5 b(.)23 b(W)-5 b(e)o(gner)m(,)24 b Fh(Exact)h(density)g(of)g(states)f(for)g(lowest)i(Landau)e(le)o(vel)i (in)e(white)h(noise)g(potential.)h(Super\002eld)g(r)n(epr)n(esentation) g(for)474 2667 y(inter)o(acting)20 b(systems)p FD(,)d(Z.)f(Physik)h(B)g Fg(51)g FD(\(1983\))h(279\226285)297 2750 y([147])34 b(J.)16 b(W)-5 b(eidmann,)17 b(Lineare)i(Operatoren)g(in)e(Hilbertr)t (\250)-26 b(aumen)20 b(\(T)-5 b(eil)18 b(II:)g(Anwendungen\),)g (Stuttgart:)i(T)-5 b(eubner)m(,)18 b(2003)g([in)f(German])297 2833 y([148])34 b(H.)17 b(W)-5 b(e)o(yl,)17 b Fh(Das)g(asymptotisc)o (he)k(V)-7 b(erteilungsg)o(esetz)21 b(der)c(Eig)o(enwerte)k(linear)n (er)d(partieller)i(Dif)o(fer)n(entialgleic)o(hun)q(g)o(e)q(n)j(\(mit)18 b(einer)474 2916 y(Anwendung)g(auf)g(die)f(Theorie)i(der)e(Hohlr)o (aumstr)o(ahlung\))p FD(,)i(Math.)e(Ann.)g Fg(71)g FD(\(1912\))h (441\226479)g([in)f(German])297 2999 y([149])34 b(D.)16 b(Y)-7 b(oshioka,)18 b(The)e(quantum)j(Hall)f(ef)n(fect,)g(Berlin:)h (Springer)m(,)g(2002)297 3082 y([150])34 b(J.)16 b(Zak,)h Fh(Ma)o(gnetic)i(tr)o(anslation)g(gr)m(oup)p FD(,)e(Phys.)f(Re)n(v)l(.) i Fg(134)f FD(\(1964\))h(A1602\226A1606)297 3165 y([151])34 b(R.)16 b(Zallen,)i(The)f(physics)h(of)f(amorphous)g(solids,)h(Ne)n(w)f (Y)-7 b(ork:)17 b(W)m(ile)o(y)l(,)h(1983)297 3248 y([152])34 b(J.)16 b(Ziman,)h(Models)g(of)g(disorder)m(,)i(Cambridge:)g(Cambridge) g(UP)-7 b(,)15 b(1979)297 3331 y([153])34 b(J.)16 b(Zittartz,)j(J.)d (S.)g(Langer)m(,)i Fh(Theory)f(of)h(bound)f(states)h(in)f(a)g(r)o (andom)g(potential)p FD(,)j(Phys.)c(Re)n(v)l(.)h Fg(148)g FD(\(1966\))h(741\226747)278 3529 y FA(Citation)24 b(Index)278 3713 y Fb([1])690 b(3)278 3804 y([2])653 b(15)278 3896 y([3])578 b(7,)19 b(15)278 3987 y([4])653 b(15)278 4078 y([5])g(15)278 4170 y([6])g(15)278 4261 y([7])g(15)278 4352 y([8])690 b(7)278 4443 y([9])503 b(2,)19 b(7,)g(15)278 4535 y([10])653 b(7)278 4626 y([11])541 b(9,)19 b(11)278 4717 y([12])653 b(7)278 4809 y([13])g(7)278 4900 y([14])g(7)278 4991 y([15])541 b(7,)19 b(15)278 5083 y([16])653 b(7)278 5174 y([17])g(2)278 5265 y([18])541 b(9,)19 b(14)1508 3713 y([19])540 b(9,)19 b(14)1508 3804 y([20])615 b(11)1508 3896 y([21])652 b(9)1508 3987 y([22])577 b(2,)19 b(3)1508 4078 y([23])428 b(9,)19 b(14,)g(18)1508 4170 y([24])615 b(11)1508 4261 y([25])465 b(8,)19 b(9,)g(11)1508 4352 y([26])428 b(9,)19 b(17,)g(18)1508 4443 y([27])540 b(9\22611)1508 4535 y([28])g(5,)19 b(10)1508 4626 y([29])465 b(5,)19 b(6,)g(10)1508 4717 y([30])316 b(2,)19 b(3,)f(5,)h(9,)g(10)1508 4809 y([31])615 b(14)1508 4900 y([32])391 b(12,)19 b(15,)g(19)1508 4991 y([33])652 b(7)1508 5083 y([34])577 b(2,)19 b(7)1508 5174 y([35])540 b(7,)19 b(15)1508 5265 y([36])615 b(10)2737 3713 y([37])g(16)2737 3804 y([38])652 b(9)2737 3896 y([39])g(9)2737 3987 y([40])540 b(9,)19 b(14)2737 4078 y([41])652 b(7)2737 4170 y([42])615 b(15)2737 4261 y([43])g(15)2737 4352 y([44])g(16)2737 4443 y([45])540 b(3,)19 b(13)2737 4535 y([46])652 b(7)2737 4626 y([47])391 b(14,)19 b(17,)g(18)2737 4717 y([48])503 b(17,)19 b(18)2737 4809 y([49])540 b(3,)19 b(14)2737 4900 y([50])615 b(15)2737 4991 y([51])g(12)2737 5083 y([52])428 b(3,)19 b(14,)g(15)2737 5174 y([53])652 b(8)2737 5265 y([54])615 b(16)p eop %%Page: 25 25 25 24 bop 1025 234 a FI(RANDOM)18 b(SCHR)1465 221 y(\250)1454 234 y(ODINGER)g(OPERA)-6 b(T)o(ORS)18 b(FOR)h(AMORPHOUS)g(SOLIDS)689 b(25)278 425 y Fb([55])616 b(15)278 517 y([56])541 b(7,)19 b(15)278 608 y([57])541 b(7,)19 b(15)278 699 y([58])616 b(15)278 790 y([59])541 b(7,)19 b(15)278 882 y([60])653 b(2)278 973 y([61])g(2)278 1064 y([62])392 b(4,)18 b(8,)h(9,)g(16)278 1156 y([63])616 b(16)278 1247 y([64])653 b(6)278 1338 y([65])g(9)278 1430 y([66])616 b(19)278 1521 y([67])504 b(10,)19 b(12)278 1612 y([68])504 b(17,)19 b(18)278 1704 y([69])616 b(17)278 1795 y([70])541 b(9,)19 b(12)278 1886 y([71])541 b(9,)19 b(12)278 1978 y([72])466 b(4,)19 b(6,)g(10)278 2069 y([73])578 b(3,)19 b(7)278 2160 y([74])653 b(5)278 2252 y([75])429 b(2,)19 b(3,)f(5,)h(9)278 2343 y([76])653 b(5)278 2434 y([77])504 b(10,)19 b(11)278 2526 y([78])653 b(3)278 2617 y([79])616 b(16)278 2708 y([80])g(15)278 2800 y([81])653 b(7)278 2891 y([82])578 b(2,)19 b(6)278 2982 y([83])429 b(9,)19 b(14,)g(18)278 3074 y([84])653 b(7)278 3165 y([85])616 b(16)278 3256 y([86])g(19)278 3348 y([87])653 b(2)1508 425 y([88])577 b(3,)19 b(7)1508 517 y([89])540 b(3,)19 b(10)1508 608 y([90])652 b(6)1508 699 y([91])g(6)1508 790 y([92])577 b(7,)19 b(9)1508 882 y([93])652 b(7)1508 973 y([94])g(8)1508 1064 y([95])g(7)1508 1156 y([96])615 b(19)1508 1247 y([97])540 b(6,)19 b(16)1508 1338 y([98])540 b(6,)19 b(16)1508 1430 y([99])652 b(6)1508 1521 y([100])279 b(2,)19 b(3,)f(5,)h(6,)g(19)1508 1612 y([101])615 b(3)1508 1704 y([102])578 b(16)1508 1795 y([103])428 b(7,)19 b(9,)g(14)1508 1886 y([104])578 b(15)1508 1978 y([105])503 b(2,)19 b(19)1508 2069 y([106])391 b(5,)19 b(10,)g(11)1508 2160 y([107])615 b(5)1508 2252 y([108])578 b(10)1508 2343 y([109])615 b(7)1508 2434 y([110])578 b(10)1508 2526 y([111])354 b(10,)19 b(11,)g(16)1508 2617 y([112])503 b(2,)19 b(10)1508 2708 y([113])503 b(2,)19 b(11)1508 2800 y([114])615 b(2)1508 2891 y([115])279 b(2,)19 b(11,)g(16,)g(19)1508 2982 y([116])466 b(2,)18 b(5,)h(6)1508 3074 y([117])615 b(2)1508 3165 y([118])130 b(2,)18 b(5,)h(6,)g(10,)g(16,)g(19)1508 3256 y([119])503 b(9,)19 b(14)1508 3348 y([120])503 b(9,)19 b(14)2737 425 y([121])615 b(7)2737 517 y([122])g(7)2737 608 y([123])578 b(10)2737 699 y([124])503 b(9,)19 b(14)2737 790 y([125])541 b(3,)18 b(6)2737 882 y([126])503 b(5,)19 b(10)2737 973 y([127])615 b(7)2737 1064 y([128])g(6)2737 1156 y([129])g(2)2737 1247 y([130])578 b(15)2737 1338 y([131])615 b(7)2737 1430 y([132])578 b(16)2737 1521 y([133])429 b(2,)18 b(7,)h(15)2737 1612 y([134])578 b(19)2737 1704 y([135])503 b(2,)19 b(19)2737 1795 y([136])578 b(13)2737 1886 y([137])466 b(16,)19 b(17)2737 1978 y([138])615 b(7)2737 2069 y([139])578 b(19)2737 2160 y([140])391 b(5,)19 b(10,)g(11)2737 2252 y([141])578 b(15)2737 2343 y([142])615 b(2)2737 2434 y([143])g(7)2737 2526 y([144])391 b(9,)19 b(18,)g(19)2737 2617 y([145])578 b(12)2737 2708 y([146])503 b(9,)19 b(13)2737 2800 y([147])615 b(7)2737 2891 y([148])g(5)2737 2982 y([149])g(7)2737 3074 y([150])g(5)2737 3165 y([151])g(2)2737 3256 y([152])g(3)2737 3348 y([153])503 b(6,)19 b(13)430 3593 y FD(I)t FE(N)t(S)t(T)t(I)t(T)t(U)t(T)h(F)762 3587 y FD(\250)754 3593 y FE(U)t(R)i FD(T)t FE(H)t(E)t(O)t(R)t(E)t(T)t(I)t (S)t(C)t(H)t(E)c FD(P)t FE(H)t(Y)t(S)t(I)t(K)t FD(,)i(U)t FE(N)t(I)t(V)t(E)t(R)t(S)t(I)t(T)1909 3587 y FD(\250)1901 3593 y FE(A)t(T)f FD(E)t FE(R)t(L)t(A)t(N)t(G)t(E)t(N)t FD(-)t(N)2404 3587 y(\250)2396 3593 y FE(U)t(R)t(N)t(B)s(E)t(R)t(G)s FD(,)g(S)t FE(T)o(A)q(U)t(D)t(T)t(S)t(T)t(R)t(A)t(SS)r(E)h FD(7)t(,)j(D)t(\226)t(9)t(1)t(0)t(5)t(8)280 3676 y(E)t FE(R)t(L)t(A)t(N)t(G)t(E)t(N)t FD(,)17 b(G)t FE(E)t(R)t(M)t(A)t(N)t(Y) 428 3759 y Fh(E-mail)g(addr)n(ess)p FD(:)k Fa (hajo.leschke@physik.uni-erlangen.de)430 3900 y FD(I)t FE(N)t(S)t(T)t(I)t(T)t(U)t(T)26 b(F)768 3894 y FD(\250)760 3900 y FE(U)t(R)j FD(T)t FE(H)t(E)t(O)t(R)t(E)t(T)t(I)t(S)t(C)t(H)t(E) 24 b FD(P)t FE(H)t(Y)t(S)t(I)t(K)t FD(,)k(G)t FE(E)t(O)t(R)t(G)t FD(-)t(A)q FE(U)t(G)t(U)t(S)t(T)n FD(-)s(U)t FE(N)s(I)s(V)t(E)t(R)s(S)s (I)s(T)2432 3894 y FD(\250)2424 3900 y FE(A)t(T)c FD(G)2582 3894 y(\250)2574 3900 y FE(O)t(T)t(T)t(I)t(N)t(G)t(E)t(N)t FD(,)k(T)n FE(A)t(M)t(M)t(A)t(N)t(N)t(S)t(T)t(R)t(A)t(SS)s(E)c FD(1)t(,)280 3983 y(D)t(\226)t(3)t(7)t(0)t(7)t(7)17 b(G)627 3977 y(\250)619 3983 y FE(O)t(T)t(T)t(I)t(N)t(G)t(E)t(N)t FD(,)f(G)t FE(E)t(R)t(M)t(A)t(N)t(Y)428 4066 y Fh(E-mail)h(addr)n(ess)p FD(:)k Fa(peter.mueller@physik.uni-goettingen.de)430 4207 y FD(I)t FE(N)t(S)t(T)t(I)t(T)t(U)t(T)f(F)762 4201 y FD(\250)754 4207 y FE(U)t(R)i FD(T)t FE(H)t(E)t(O)t(R)t(E)t(T)t(I)t (S)t(C)t(H)t(E)c FD(P)t FE(H)t(Y)t(S)t(I)t(K)t FD(,)i(U)t FE(N)t(I)t(V)t(E)t(R)t(S)t(I)t(T)1909 4201 y FD(\250)1901 4207 y FE(A)t(T)f FD(E)t FE(R)t(L)t(A)t(N)t(G)t(E)t(N)t FD(-)t(N)2404 4201 y(\250)2396 4207 y FE(U)t(R)t(N)t(B)s(E)t(R)t(G)s FD(,)g(S)t FE(T)o(A)q(U)t(D)t(T)t(S)t(T)t(R)t(A)t(SS)r(E)h FD(7)t(,)j(D)t(\226)t(9)t(1)t(0)t(5)t(8)280 4290 y(E)t FE(R)t(L)t(A)t(N)t(G)t(E)t(N)t FD(,)17 b(G)t FE(E)t(R)t(M)t(A)t(N)t(Y) 428 4373 y Fh(E-mail)g(addr)n(ess)p FD(:)k Fa (simone.warzel@physik.uni-erlangen.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0312111027136--