Content-Type: multipart/mixed; boundary="-------------0201031101345" This is a multi-part message in MIME format. ---------------0201031101345 Content-Type: text/plain; name="02-3.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-3.comments" Procedings ICMP London ---------------0201031101345 Content-Type: text/plain; name="02-3.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-3.keywords" anomalous transport ---------------0201031101345 Content-Type: application/postscript; name="review.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="review.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: review.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips review %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 2000.07.31:0944 %%BeginProcSet: tex.pro 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b(3.)h(Within)29 b(this)g(framew)n(ork,)d(the)j(task)456 3454 y(is)h(then)h(to)f(sho)n(w)g(that)g(one)g(actually)g(has)g (anomalous)f(quan)n(tum)h(di\013usion)h(for)f(particular)456 3554 y(ap)r(erio)r(dic)35 b(Sc)n(hr\177)-42 b(odinger)34 b(op)r(erators.)60 b(In)36 b(Sec.)61 b(5)36 b(w)n(e)f(review)g(recen)n (t)h(impro)n(v)n(emen)n(ts)e(of)i(a)456 3653 y(general)28 b(lo)n(w)n(er)f(b)r(ound)j(on)f(quan)n(tum)g(motion)h(due)f(to)g (Guarneri.)41 b(One-dimensional)29 b(mo)r(d-)456 3753 y(els)c(for)h(whic)n(h)g(also)f(upp)r(er)h(b)r(ounds)g(can)f(b)r(e)i (pro)n(v)n(en)d(are)h(presen)n(ted)g(in)i(Sec.)36 b(6.)g(This)26 b(allo)n(ws)456 3853 y(to)f(construct)g(3)p Fs(D)r Ft(-mo)r(dels)g (with)h(absolutely)f(con)n(tin)n(uous)g(\(ac\))h(sp)r(ectrum)g(and)f (sub)r(di\013usiv)n(e)456 3952 y(quan)n(tum)i(transp)r(ort,)g Fr(cf.)37 b Ft(Sec.)g(7.)1477 4170 y Fu(2.)47 b(Di\013usion)30 b(exp)s(onen)m(ts)605 4319 y Ft(Giv)n(en)23 b(a)g(p)r(ositiv)n(e)g (measurable)f(function)i Fs(f)32 b Ft(on)23 b([)p Fs(b;)14 b Fq(1)p Ft(\),)24 b(w)n(e)f(in)n(tro)r(duce)g(an)g(upp)r(er)h(p)r(o)n (w)n(er)456 4419 y(gro)n(wth)k(exp)r(onen)n(ts)h Fs(\013)1182 4389 y Fp(+)1251 4419 y Ft(=)13 b(lim)h(sup)1582 4439 y Fo(T)9 b Fn(!1)1781 4419 y Ft(log\()p Fs(f)g Ft(\()p Fs(T)j Ft(\)\))p Fs(=)i Ft(log)o(\()p Fs(T)e Ft(\))30 b(and)f(a)h(lo)n(w)n(er)e(exp)r(onen)n(t)i Fs(\013)3311 4389 y Fn(\000)3380 4419 y Ft(=)456 4519 y(lim)14 b(inf)686 4531 y Fo(T)9 b Fn(!1)884 4519 y Ft(log)q(\()p Fs(f)g Ft(\()p Fs(T)j Ft(\)\))p Fs(=)i Ft(log)o(\()p Fs(T)e Ft(\))23 b(as)g(w)n(ell)g(as)g(an)g(in)n(termediate)g(Mellin)g(exp)r (onen)n(t)h Fs(\013)3148 4489 y Fo(m)3235 4519 y Ft(whic)n(h)456 4619 y(is)k(the)g(in\014nim)n(um)h(of)f(all)g Fs(\015)h Fq(2)24 b Fu(R)k Ft(suc)n(h)g(that)1909 4552 y Fm(R)1964 4573 y Fn(1)1948 4649 y Fo(b)2048 4619 y Fs(dT)d(f)9 b Ft(\()p Fs(T)j Ft(\))p Fs(=T)2443 4589 y Fp(1+)p Fo(\015)2596 4619 y Ft(is)28 b(\014nite.)39 b(W)-7 b(e)29 b(then)g(write)456 4741 y Fs(f)9 b Ft(\()p Fs(T)j Ft(\))685 4696 y Fq(\030)653 4741 y Fl(T)7 b Fk("1)22 b Fs(T)866 4711 y Fo(\013)909 4686 y Fk(\003)974 4741 y Ft(where)28 b Fq(\003)f Ft(is)g(+,)h Fq(\000)f Ft(or)f Fs(m)p Ft(.)37 b(Similarly)27 b(one)h(de\014nes)f Fs(f)9 b Ft(\()p Fs(\017)p Ft(\))2768 4696 y Fq(\030)2758 4741 y Fl(\017)p Fk(#)p Fj(0)23 b Fs(\017)2899 4711 y Fo(\013)2942 4686 y Fk(\003)2981 4741 y Ft(.)p 456 4864 499 4 v 605 4958 a Fv(1991)k Fi(Mathematics)f(Subje)l(ct)f(Classi\014c) l(ation.)36 b Fv(Primary)22 b(82C70;)i(Secondary)h(81Q10.)605 5041 y(The)f(author)h(w)n(as)e(supp)r(orted)i(in)e(part)h(b)n(y)g(NSF)g (Gran)n(t)g(DMS-0070755.)2756 5202 y Fz(c)2740 5204 y Fh(\015)o Fz(0000)31 b(\(cop)n(yrigh)n(t)f(holder\))1933 5315 y Fp(1)p eop %%Page: 2 2 2 1 bop 456 226 a Fp(2)995 b(HERMANN)29 b(SCHULZ-BALDES)605 425 y Ft(Let)e(no)n(w)f Fs(H)k Ft(:)23 b Fs(`)1105 395 y Fp(2)1142 425 y Ft(\(\003\))g Fq(!)g Fs(`)1428 395 y Fp(2)1465 425 y Ft(\(\003\))k(b)r(e)g(a)g(b)r(ounded)g(self-adjoin)n (t)f(Hamiltonian)h(on)f(a)g(lattice)456 535 y(\003)g(whic)n(h)h(can)f (b)r(e)h(either)g Fu(Z)1333 504 y Fo(D)1420 535 y Ft(or)f Fu(N)1596 504 y Fo(D)1656 535 y Ft(.)37 b(If)1815 514 y Fs(~)1798 535 y(X)c Ft(denotes)27 b(the)g(p)r(osition)f(op)r(erator)f (on)i Fs(`)3141 504 y Fp(2)3178 535 y Ft(\(\003\),)g(w)n(e)456 644 y(supp)r(ose)h(that)i(the)f(curren)n(t)f(op)r(erator)g Fs({)p Ft([)1789 623 y Fs(~)1772 644 y(X)6 b(;)14 b(H)7 b Ft(])29 b(is)g(also)e(b)r(ounded.)42 b(F)-7 b(or)29 b(a)f(giv)n(en)g Fs(q)h Fq(6)p Ft(=)c(0,)k(the)456 744 y(di\013usion)e(exp)r(onen)n(t)h Fs(\014)1199 713 y Fn(\003)1195 764 y Fo(q)1265 744 y Ft(is)g(de\014ned)g(b)n(y)1230 928 y Fm(Z)1313 948 y Fo(T)1276 1116 y Fn(\000)p Fo(T)1418 985 y Fs(dt)p 1404 1022 102 4 v 1404 1098 a Ft(2)p Fs(T)1539 1041 y Fq(h)p Ft(0)p Fq(j)22 b(j)1699 1020 y Fs(~)1681 1041 y(X)7 b Ft(\()p Fs(t)p Ft(\))19 b Fq(\000)1970 1020 y Fs(~)1953 1041 y(X)6 b Fq(j)2051 1006 y Fo(q)2102 1041 y Fq(j)p Ft(0)p Fq(i)2277 995 y(\030)2245 1041 y Fl(T)g Fk("1)46 b Fs(T)2481 1006 y Fo(q)r(\014)2554 981 y Fk(\003)2551 1023 y Fl(q)2619 1041 y Ft(,)-2186 b(\(2.1\))456 1269 y(where)719 1248 y Fs(~)701 1269 y(X)7 b Ft(\()p Fs(t)p Ft(\))33 b(=)f Fs(e)1040 1239 y Fo({H)t(t)1168 1248 y Fs(~)1151 1269 y(X)6 b(e)1265 1239 y Fn(\000)p Fo({H)t(t)1461 1269 y Ft(and)33 b Fq(j)p Ft(0)p Fq(i)h Ft(is)f(the)h(normalized)e (state)h(lo)r(calized)g(at)g(the)h(origin)456 1369 y(of)40 b(\003.)74 b(Other)40 b(initial)g(states)g(ma)n(y)g(giv)n(e)f (di\013eren)n(t)h(exp)r(onen)n(ts.)74 b(It)41 b(can)f(b)r(e)g(easily)g (v)n(eri-)456 1479 y(\014ed)c([)p Fu(GSB3)p Ft(])h(that)g(the)g(time)g (mean)1717 1412 y Fm(R)1773 1433 y Fo(T)1757 1509 y Fn(\000)p Fo(T)1875 1479 y Fs(dt=T)47 b Ft(can)36 b(b)r(e)h(replaced)f(b)n(y)g(a) h(gaussian)e(mean)456 1525 y Fm(R)495 1621 y Fg(R)569 1592 y Fs(dt=T)24 b Ft(exp\()p Fq(\000)p Fs(t)1011 1561 y Fp(2)1048 1592 y Fs(=T)1151 1561 y Fp(2)1187 1592 y Ft(\))j(without)g(c)n(hanging)f(the)h(exp)r(onen)n(ts.)36 b(Moreo)n(v)n(er,)25 b(not)h(substracting)473 1689 y Fs(~)456 1710 y(X)41 b Ft(will)35 b(not)g(c)n(hange)f(the)h(exp)r(onen) n(ts)g(neither.)59 b(Finally)-7 b(,)36 b(let)g(us)f(note)f(that)i(it)f (is)g(p)r(ossible)456 1809 y(to)j(de\014ne)g(di\013usion)h(exp)r(onen)n (ts)e(asso)r(ciated)g(to)h(a)g(sp)r(ectral)g(in)n(terv)-5 b(al)38 b(\001)g(b)n(y)g(in)n(tro)r(ducing)456 1909 y(sp)r(ectral)27 b(pro)5 b(jections)26 b(in)i(\(2.1\))f([)p Fu(SBB1)p Ft(].)605 2009 y(F)-7 b(urther)43 b(w)n(e)f(consider)g(co)n(v)-5 b(arian)n(t)41 b(families)i(of)f(sto)r(c)n(hastic)g(Hamiltonians)h(\()p Fs(H)3224 2021 y Fo(!)3272 2009 y Ft(\))3304 2021 y Fo(!)r Fn(2)p Fp(\012)456 2108 y Ft(where)32 b(\(\012)p Fs(;)14 b(T)7 b(;)14 b Fu(Z)981 2078 y Fo(D)1041 2108 y Fs(;)g Fu(P)p Ft(\))33 b(is)g(a)g(compact)f(space)g(\012)h(furnished)g(with)g (an)g(action)f Fs(T)44 b Ft(of)33 b Fu(Z)3217 2078 y Fo(D)3311 2108 y Ft(and)456 2208 y(an)26 b(in)n(v)-5 b(arian)n(t)25 b(ergo)r(dic)g(probabilit)n(y)g(measure)g Fu(P)i Ft(suc)n(h)f(that)g(the)h(co)n(v)-5 b(ariance)24 b(relation)h(w.r.t.)456 2313 y(the)32 b(translations)f(holds:)47 b Fs(U)9 b Ft(\()p Fs(a)p Ft(\))p Fs(H)1564 2325 y Fo(!)1612 2313 y Fs(U)g Ft(\()p Fs(A)p Ft(\))1804 2283 y Fn(\003)1874 2313 y Ft(=)31 b Fs(H)2039 2325 y Fo(T)2087 2309 y Fl(a)2123 2325 y Fo(!)2171 2313 y Ft(.)52 b(Their)32 b(di\013usion)g(exp)r(onen)n (ts)p 3211 2245 52 4 v 32 w Fs(\014)3263 2259 y Fn(\003)3263 2334 y Fo(q)3333 2313 y Ft(are)456 2415 y(de\014ned)c(b)n(y)f(\(2.1\))g (with)h(a)f(supplemen)n(tary)g(mean)h(o)n(v)n(er)d Fu(P)j Ft(on)g(the)g(l.h.s..)605 2515 y(Using)c(DuHamel's)h(form)n(ula)e(in)i (\(2.1\),)f(it)h(is)f(straigh)n(tforw)n(ard)e(to)i(v)n(erify)f(that)i (0)e Fq(\024)f Fs(\014)3318 2485 y Fn(\003)3314 2536 y Fo(q)3380 2515 y Fq(\024)456 2636 y Ft(1)34 b(for)g(all)g Fs(q)j(>)d Ft(0)g(\(similarly)g(0)g Fq(\024)p 1591 2568 V 34 w Fs(\014)1642 2582 y Fn(\003)1642 2656 y Fo(q)1715 2636 y Fq(\024)g Ft(1\).)57 b(If)35 b(the)g(l.h.s.)58 b(of)34 b(\(2.1\))g(is)g(b)r(ounded)h(in)g(time)456 2738 y(\(hence)22 b Fs(\014)764 2708 y Fn(\003)760 2759 y Fo(q)826 2738 y Ft(=)h(0\),)g(one)f(talks)g(of)h(strong)e(lo)r (calization;)j(this)e(is)h(the)g(case)f(at)g(high)g(disorder)g(and)456 2838 y(the)29 b(band)f(edges)g(for)g(the)h(Anderson)f(mo)r(del.)41 b(The)28 b(quan)n(tum)h(motion)g(is)f(called)g(ballistic)h(if)456 2938 y Fs(\014)507 2907 y Fn(\003)503 2958 y Fo(q)568 2938 y Ft(=)23 b(1)j(whic)n(h)h(is)f(the)h(case)f(for)g(Blo)r(c)n(h)g (electrons.)36 b Fs(\014)2149 2907 y Fn(\003)2145 2958 y Fo(q)2210 2938 y Ft(=)23 b(1)p Fs(=)p Ft(2)i(corresp)r(onds)g(to)h (the)h(di\013usiv)n(e)456 3037 y(regime)21 b(whic)n(h)h(is)g(exp)r (ected)h(in)f(the)h(w)n(eak)e(lo)r(calization)g(regime)g(of)h(the)h(3)p Fs(D)h Ft(Anderson)d(mo)r(del)456 3137 y(\(lo)n(w)28 b(disorder)g(and)h(cen)n(ter)f(of)h(bands\);)h(it)f(can)g(b)r(e)g(pro)n (v)n(en)f(for)g(the)h(Anderson)g(mo)r(del)g(with)456 3236 y(so-called)39 b(free)i(random)f(v)-5 b(ariables,)43 b(a)e(generalized)e(W)-7 b(egner)41 b Fs(n)p Ft(-orbital)e(mo)r(del)j ([)p Fu(SBB1)o Ft(].)456 3336 y(F)-7 b(or)35 b(an)n(y)g(other)g(v)-5 b(alue)36 b(of)f Fs(\014)1379 3306 y Fn(\003)1375 3357 y Fo(q)1418 3336 y Ft(,)j(w)n(e)d(shall)h(sp)r(eak)f(of)h(anomalous)e (quan)n(tum)h(di\013usion.)62 b(F)-7 b(or)456 3436 y(1)p Fs(D)r Ft(-mo)r(dels)31 b Fs(q)j Fq(7!)e Fs(\014)1117 3406 y Fn(\003)1113 3456 y Fo(q)1188 3436 y Ft(is)g(t)n(ypically)g(a)g (strictly)g(increasing)f(curv)n(e,)i(a)f(phenomenon)g(termed)456 3538 y(quan)n(tum)27 b(in)n(termittency)h(in)g([)p Fu(GM)p Ft(].)1323 3771 y Fu(3.)47 b(Anomalous)30 b(Drude)i(F)-8 b(orm)m(ula)605 3920 y Ft(It)25 b(is)g(implicitly)h(con)n(tained)e(in)h (the)g(Einstein)g(relation)f(b)r(et)n(w)n(een)h(the)g(direct)g (conductiv-)456 4020 y(it)n(y)i(and)f(the)i(di\013usion)f(co)r (e\016cien)n(t)g(that)g(the)h(conductivit)n(y)f(is)g(\014nite)g(only)g (if)g(the)h(motion)f(of)456 4120 y(the)k(electrons)f(is)h(di\013usiv)n (e.)47 b(If)31 b(this)h(is)e(not)h(the)h(case)e(for)g(the)i(coheren)n (t)e(quan)n(tum)h(motion)456 4219 y(\()p Fr(i.e.)37 b Fs(\014)678 4231 y Fp(2)738 4219 y Fq(6)p Ft(=)23 b(1)p Fs(=)p Ft(2\),)j(one)g(can)h(sup)r(erp)r(ose)f(a)h(collisional)e(\(P)n (oisson\))h(pro)r(cess)g(similar)g(to)h(that)g(in)456 4319 y(the)c(classical)f(Drude)i(mo)r(del)f(and)g(this)h(forces)e(the)i (electron)e(motion)h(to)h(b)r(e)f(di\013usiv)n(e.)36 b(These)456 4419 y(collisions)26 b(are)h(mo)r(delizing)h(the)g(in)n (teractions)f(with)h(phonons)g(and)f(other)h(electrons)e(as)i(w)n(ell) 456 4518 y(as)e(the)h(static)g(disorder)e(of)i(impurities)g(if)g(it)g (is)g(not)g(already)e(tak)n(en)h(in)n(to)h(accoun)n(t)f(for)g(in)h(the) 456 4618 y(one-particle)35 b(Hamiltonian)h(\(e.g.,)j(if)e Fs(H)44 b Ft(is)36 b(quasip)r(erio)r(dic)g(without)h(added)g(disorder)e (p)r(o-)456 4717 y(ten)n(tial\).)j(Let)29 b Fs(\034)37 b Ft(b)r(e)29 b(the)f(relaxation)f(time,)i(notably)e(the)i(a)n(v)n (erage)c(time)k(b)r(et)n(w)n(een)f(collisions)456 4817 y(renormalized)18 b(b)n(y)h(their)h(e\016ciency)f(to)h(diminish)g (curren)n(t.)33 b(This)20 b(time)g(dep)r(ends)g(phenomeno-)456 4917 y(logically)29 b(on)h(the)h(temp)r(erature,)h(and)e(the)h(phonon)g (and)f(electron)g(collision)g(times)h(div)n(erge)456 5016 y(in)i(the)h(zero)e(temp)r(erature)i(limit.)55 b(Then)33 b(the)h(\(direction)f(a)n(v)n(eraged\))e(direct)j(conductivit)n(y)456 5116 y(at)e(in)n(v)n(erse)e(temp)r(erature)i Fs(\014)37 b Ft(and)31 b(c)n(hemical)h(p)r(oten)n(tial)g Fs(\026)g Ft(of)g(a)g(F)-7 b(ermi)32 b(gas)f(describ)r(ed)h(b)n(y)g(a)456 5216 y(co)n(v)-5 b(arian)n(t)25 b(Hamiltonian)j(\()p Fs(H)1392 5228 y Fo(!)1440 5216 y Ft(\))1472 5228 y Fo(!)r Fn(2)p Fp(\012)1640 5216 y Ft(is)g(giv)n(en)e(b)n(y)i(Kub)r(o's)f(form) n(ula:)p eop %%Page: 3 3 3 2 bop 1125 226 a Fp(ANOMALOUS)28 b(TRANSPOR)-5 b(T)29 b(IN)g(APERIODIC)g(MEDIA)635 b(3)819 588 y Fs(\033)s Ft(\()p Fs(\014)t(;)14 b(\026;)g(\034)9 b Ft(\))48 b(=)d Fq(T)1391 446 y Fm( )1457 588 y Fs({)p Ft([)1524 567 y Fs(~)1507 588 y(X)6 b(;)14 b(f)1660 600 y Fo(\014)s(;\026)1765 588 y Ft(\()p Fs(H)1866 600 y Fo(!)1914 588 y Ft(\)])1997 471 y Fm(\022)2070 532 y Ft(1)p 2068 569 46 4 v 2068 645 a Fs(\034)2142 588 y Fq(\000)k(L)2282 600 y Fo(H)2336 608 y Fl(!)2383 471 y Fm(\023)2445 488 y Fn(\000)p Fp(1)2548 588 y Ft(\()p Fs({)p Ft([)2647 567 y Fs(~)2630 588 y(X)6 b(;)14 b(H)2811 600 y Fo(!)2859 588 y Ft(]\))2914 446 y Fm(!)3021 588 y Ft(.)456 789 y(Here)29 b Fs(f)695 801 y Fo(\014)s(;\026)829 789 y Ft(is)h(the)g(F)-7 b(ermi-Dirac)29 b(function,)i Fq(L)1932 801 y Fo(H)1995 789 y Ft(\()p Fs(A)p Ft(\))d(=)e Fs({)p Ft([)p Fs(H)r(;)14 b(A)p Ft(])30 b(is)g(the)g(an)n(ti-self-adjoin)n(t)f(Li-)456 889 y(ouville)24 b(\(sup)r(er-\))h(op)r(erator)e(\(so)h(that)h(the)g(app)r(earing)f(op)r (erator)f(in)n(v)n(erse)g(exists)h(for)h Fs(\034)32 b(<)23 b Fq(1)p Ft(\),)456 989 y(and)44 b(\014nally)h Fq(T)67 b Ft(is)45 b(the)g(trace)f(p)r(er)h(unit)h(v)n(olume)e(on)h(co)n(v)-5 b(arian)n(t)43 b(op)r(erators)g(de\014ned)j(b)n(y)456 1088 y Fq(T)21 b Ft(\()p Fs(A)616 1100 y Fo(!)665 1088 y Ft(\))k(=)811 1021 y Fm(R)880 1088 y Fs(d)p Fu(P)p Ft(\()p Fs(!)s Ft(\))14 b Fq(h)p Ft(0)p Fq(j)p Fs(A)1280 1100 y Fo(!)1329 1088 y Fq(j)p Ft(0)p Fq(i)p Ft(.)40 b(This)28 b(Kub)r(o)h(form)n(ula)e(tak)n(es)h(fully)h(in)n(to)f(accoun) n(t)g(the)h(quan-)456 1188 y(tum)e(in)n(terferences)e(of)h(the)h (Hamiltonian)f Fs(H)1889 1200 y Fo(!)1964 1188 y Ft(up)h(to)f(times)h (of)f(order)f(1)p Fs(=tau)p Ft(,)g(but)i(describ)r(es)456 1287 y(in)n(teractions)f(with)i(and)g(dissipation)f(to)g(the)h(rest)f (of)h(the)g(solid)f(phenomenologically)-7 b(.)605 1442 y Ff(Theorem)32 b Ft(3.1)e(\(Anomalous)d(Drude)h(F)-7 b(orm)n(ula)27 b([)p Fu(SBB1)o(,)32 b(SBB2)o Ft(]\))p Ff(.)42 b Fe(L)l(et)d(the)h(temp)l(er-)456 1542 y(atur)l(e)32 b(b)l(e)i(\014xe)l(d)f(and)h(\014nite.)48 b(Then)34 b(one)g(has)g(the)g (fol)t(lowing)h(Mel)t(lin)g(exp)l(onent)e(b)l(ehavior)i(in)456 1641 y(the)30 b(zer)l(o)g(dissip)l(ation)h(limit)g Fs(\034)h Fq(!)23 b(1)p Fe(:)1497 1799 y Fs(\033)s Ft(\()p Fs(\014)t(;)14 b(\026;)g(\034)9 b Ft(\))1907 1753 y Fq(\030)1879 1799 y Fl(\034)c Fk("1)46 b Fs(\034)2091 1765 y Fp(2)p 2124 1716 41 3 v Fo(\014)2165 1731 y Fl(m)2165 1781 y Fj(2)2220 1765 y Fn(\000)p Fp(1)2339 1799 y Fe(.)605 1954 y Ft(It)41 b(w)n(ould)g(b)r(e)g(desirable)f(to)h(pro)n(v)n(e)e(that)i(at)g(zero)f (temp)r(erature)g(only)h(the)g(di\013usion)456 2053 y(exp)r(onen)n(t)35 b(at)g(the)g(F)-7 b(ermi)35 b(energy)f(en)n(ters)h(this)g(form)n(ula.) 59 b(Note)35 b(\014rst)g(that)g(for)g(a)g(ballistic)456 2156 y(quan)n(tum)26 b(transp)r(ort)f(\()p 1203 2089 52 4 v Fs(\014)1255 2103 y Fo(m)1255 2177 y Fp(2)1341 2156 y Ft(=)d(1\),)27 b(one)f(reco)n(v)n(ers)d(the)k(usual)f(Drude)g (form)n(ula.)35 b(In)27 b(particular,)456 2256 y(the)h(conductivit)n(y) f(div)n(erges)g(in)h(the)g(zero)f(dissipation)g(limit.)39 b(The)27 b(same)h(holds)f(true)h(in)g(the)456 2359 y(sup)r (er-di\013usiv)n(e)23 b(regime)f(\()p 1297 2292 V Fs(\014)1349 2306 y Fo(m)1349 2380 y Fp(2)1435 2359 y Fs(>)h Ft(1)p Fs(=)p Ft(2\).)34 b(Only)23 b(in)h(the)g(di\013usiv)n(e)f(regime)p 2756 2292 V 23 w Fs(\014)2808 2306 y Fo(m)2808 2380 y Fp(2)2894 2359 y Ft(=)f(1)p Fs(=)p Ft(2,)h(the)h(con-)456 2459 y(ductivit)n(y)g(at)g(zero)f(dissipation)h(ma)n(y)g(b)r(e)g (\014nite)h(in)g(accordance)d(with)j(the)f(Einstein)g(relation.)456 2559 y(In)31 b(the)g(sub)r(di\013usiv)n(e)h(regime,)f(the)h (conductivit)n(y)f(v)-5 b(anishes)30 b(in)i(this)f(limit.)49 b(Let)31 b(us)g(remark)456 2658 y(that)25 b(the)h(relaxation)d(time)j (appro)n(ximation)d(and)i(hence)h(the)f(ab)r(o)n(v)n(e)f(Kub)r(o)h (form)n(ula)f(are)g(not)456 2758 y(v)-5 b(alid)31 b(in)h(the)g(strong)f (lo)r(calization)f(regime)h(\(impurit)n(y)h(bands)g(in)g(dop)r(ed)f (semiconductors\))456 2857 y(where)c(hopping)g(conduction)g(is)h (predominan)n(t)f(at)g(lo)n(w)g(temp)r(erature.)605 2957 y(Our)33 b(b)r(elief)h(is)f(that)h(the)g(relaxation)e(time)i(appro)n (ximation)d(is)i(v)-5 b(alid)34 b(in)g(quasicrystals)456 3057 y(o)n(v)n(er)25 b(a)h(wide)h(temp)r(erature)f(range.)35 b(If)27 b(one)g(supp)r(oses)f(a)g(temp)r(erature)g(dep)r(endence)h Fs(\034)9 b Ft(\()p Fs(T)j Ft(\))24 b Fq(\030)456 3156 y Fs(T)517 3126 y Fn(\000)p Fo(\024)640 3156 y Ft(for)29 b(some)g(p)r(ositiv)n(e)g Fs(\024)p Ft(,)h(then)g(the)g(anomalous)e (Drude)i(form)n(ula)e(\(\014x)i Fs(\014)k Ft(for)29 b(simplicit)n(y\)) 456 3270 y(giv)n(es)34 b Fs(\033)s Ft(\()p Fs(\014)t(;)14 b(\026;)g(T)985 3240 y Fo(\024)1028 3270 y Ft(\))36 b Fq(\030)f Fs(T)1257 3240 y Fo(\024)p Fp(\(1)p Fn(\000)p Fp(2)p 1440 3191 41 3 v Fo(\014)1479 3206 y Fl(m)1479 3256 y Fj(2)1535 3240 y Fp(\))1600 3270 y Ft(so)f(that)i(the)g (conductivit)n(y)f(increases)e(with)j(increasing)456 3378 y(temp)r(erature)27 b(whenev)n(er)f(the)i(quan)n(tum)g(motion)f (is)h(sub)r(di\013usiv)n(e)g(\()p 2637 3310 52 4 v Fs(\014)2688 3324 y Fo(m)2688 3398 y Fp(2)2775 3378 y Fs(<)22 b Ft(1)p Fs(=)p Ft(2\).)36 b(This)28 b(giv)n(es)456 3477 y(a)f(qualitativ)n(e)g (explanation)f(of)i(the)g(exp)r(erimen)n(tal)f(observ)-5 b(ation)26 b(discussed)h(ab)r(o)n(v)n(e.)1472 3673 y Fu(4.)47 b(Sp)s(ectral)32 b(dimensions)605 3823 y Ft(Here)24 b(w)n(e)g(in)n(tro)r(duce)g(dimensional)g(prop)r(erties)f(of)h(the)h (\(sp)r(ectral\))f(measures)f(whic)n(h)h(will)456 3922 y(b)r(e)36 b(link)n(ed)g(to)g(the)g(di\013usion)h(exp)r(onen)n(ts)e(b)r (elo)n(w.)62 b(Giv)n(en)36 b(a)g(Borel)f(measure)g Fs(\026)h Ft(on)f Fu(R)p Ft(,)j(its)456 4031 y(lo)r(cal)31 b(dimension)i Fs(d)1097 4001 y Fn(\003)1097 4052 y Fo(\026)1141 4031 y Ft(\()p Fs(E)5 b Ft(\))33 b(at)g Fs(E)j Fq(2)31 b Fu(R)h Ft(are)g(de\014ned)g(b)n(y)g Fs(\026)p Ft(\([)p Fs(E)27 b Fq(\000)22 b Fs(\017;)14 b(E)26 b Ft(+)21 b Fs(\017)p Ft(]\))2906 3986 y Fq(\030)2896 4031 y Fl(\017)p Fk(#)p Fj(0)31 b Fs(\017)3045 3997 y Fo(d)3080 3972 y Fk(\003)3080 4014 y Fl(\026)3119 3997 y Fp(\()p Fo(E)s Fp(\))3227 4031 y Ft(.)51 b(The)456 4134 y(Hausdor\013)28 b(dimension)h(dim)1369 4146 y Fz(H)1418 4134 y Ft(\()p Fs(\026)p Ft(\))h(of)f Fs(\026)g Ft(and)h(its)f(pac)n(king)f(dimension)h(dim)2857 4146 y Fz(P)2902 4134 y Ft(\()p Fs(\026)p Ft(\))h(are)e(de\014ned)456 4233 y(to)36 b(b)r(e)g(the)h Fs(\026)p Ft(-essen)n(tial)e(suprem)n(um)h (of)g(resp)r(ectiv)n(ely)f(the)i(lo)n(w)n(er)e(dimensions)g Fs(d)3087 4203 y Fn(\000)3087 4254 y Fo(\026)3144 4233 y Ft(\()p Fs(E)5 b Ft(\))37 b(and)456 4339 y(the)c(upp)r(er)g(ones)f Fs(d)1080 4309 y Fp(+)1080 4360 y Fo(\026)1135 4339 y Ft(\()p Fs(E)5 b Ft(\).)54 b(F)-7 b(or)32 b(an)h(ac)f(measure,)h(these) g(dimension)g(are)f(equal)g(to)h(1,)h(for)e(a)456 4442 y(pure-p)r(oin)n(t)e(measure)g(equal)h(to)g(0,)h(and)f(for)g(an)n(y)f (other)h(v)-5 b(alue)31 b(in)g(\(0)p Fs(;)14 b Ft(1\),)32 b Fs(\026)f Ft(has)g(a)g(singular)456 4541 y(con)n(tin)n(uous)e(comp)r (onen)n(t.)44 b(If)31 b(dim)1562 4553 y Fz(H)1611 4541 y Ft(\()p Fs(\026)p Ft(\))d Fs(<)f Ft(dim)1983 4553 y Fz(P)2028 4541 y Ft(\()p Fs(\026)p Ft(\),)32 b(then)e Fs(\026)h Ft(has)e(bad)h(scaling)f(prop)r(erties.)456 4641 y(F)-7 b(or)26 b(equiv)-5 b(alen)n(t)27 b(de\014nitions)g(and)g (further)f(prop)r(erties)g(w)n(e)h(refer)f(to)h([)p Fu(P)m(es,)k(GSB2)p Ft(].)36 b(A)28 b(\014ner)456 4741 y(c)n(haracterization)d(of)i Fs(\026)h Ft(is)f(giv)n(en)g(b)n(y)g(its)h(m)n(ultifractal)f (dimensions)h(\()p Fs(q)e Fq(6)p Ft(=)d(1\):)1097 4814 y Fm(Z)1194 4927 y Fs(d\026)p Ft(\()p Fs(E)5 b Ft(\))14 b Fs(\026)p Ft(\([)p Fs(E)25 b Fq(\000)18 b Fs(\017;)c(E)23 b Ft(+)18 b Fs(\017)p Ft(]\))2032 4893 y Fo(q)r Fn(\000)p Fp(1)2209 4881 y Fq(\030)2200 4927 y Fl(\017)p Fk(#)p Fj(0)46 b Fs(\017)2364 4893 y Fp(\()p Fo(q)r Fn(\000)p Fp(1\))p Fo(D)2589 4868 y Fk(\003)2587 4909 y Fl(\026)2627 4893 y Fp(\()p Fo(q)r Fp(\))2743 4927 y Ft(.)456 5113 y(The)35 b(functions)g Fs(q)k Fq(7!)c Fs(D)1264 5083 y Fn(\003)1262 5134 y Fo(\026)1307 5113 y Ft(\()p Fs(q)s Ft(\))g(are)f(non-increasing)f(and)i(moreo)n(v)n(er)e Fs(q)38 b Fq(7!)e Ft(\()p Fs(q)27 b Fq(\000)c Ft(1\))p Fs(D)3207 5083 y Fn(\003)3205 5134 y Fo(\026)3249 5113 y Ft(\()p Fs(q)s Ft(\))36 b(is)456 5216 y(conca)n(v)n(e.)e(Other)25 b(than)g(dim)1362 5228 y Fz(H)1410 5216 y Ft(\()p Fs(\026)p Ft(\))i(and)e(dim)1848 5228 y Fz(P)1893 5216 y Ft(\()p Fs(\026)p Ft(\),)i(the)f(m)n(ultifractal)f(dimensions)g(dep)r(end)h(on) p eop %%Page: 4 4 4 3 bop 456 226 a Fp(4)995 b(HERMANN)29 b(SCHULZ-BALDES)456 425 y Fs(\026)35 b Ft(in)g(its)g(measure)f(class)g([)p Fu(SBB1)o Ft(].)59 b(Under)35 b(the)h(h)n(yp)r(othesis)e(that)h Fs(D)2749 395 y Fn(\000)2747 446 y Fo(\026)2805 425 y Ft(\()p Fs(q)s Ft(\))h(=)f Fs(D)3116 395 y Fp(+)3114 446 y Fo(\026)3171 425 y Ft(\()p Fs(q)s Ft(\))h(and)456 528 y Fs(D)525 540 y Fo(\026)569 528 y Ft(\()p Fs(q)s Ft(\))j Fs(<)f Fq(1)f Ft(for)g(some)f Fs(q)42 b(<)c Ft(0,)h(one)e(can)f (directly)h(implemen)n(t)g(large)f(deviation)g(theory)456 627 y(in)c(order)g(to)g(study)h(the)g(\015uctuations)g(of)f(the)h(lo)r (cal)f(dimensions)h([)p Fu(GSB3)o Ft(].)52 b(Let)33 b Fs(f)3164 639 y Fo(\026)3208 627 y Ft(\()p Fs(\013)p Ft(\))h(b)r(e)456 727 y(the)39 b(Legendre)f(transform)g(of)i(the)f (function)h Fs(q)46 b Fq(7!)c Ft(\()p Fs(q)30 b Fq(\000)25 b Ft(1\))p Fs(D)2511 697 y Fn(\003)2509 747 y Fo(\026)2553 727 y Ft(\()p Fs(q)s Ft(\).)73 b(Set)39 b Fs(I)2943 739 y Fo(\013)2991 727 y Ft(\()p Fs(\017)p Ft(\))k(=)f Fq(f)p Fs(E)47 b Fq(2)456 829 y Fu(R)14 b Fq(j)g Fs(\026)p Ft(\([)p Fs(E)30 b Fq(\000)c Fs(\017;)14 b(E)30 b Ft(+)c Fs(\017)p Ft(]\))42 b Fs(<)f(\017)1390 799 y Fo(\013)1437 829 y Fq(g)p Ft(.)70 b(Then)39 b(there)g(exist)g(constan)n(ts)e Fs(c;)14 b(C)45 b Ft(suc)n(h)39 b(that)g(for)f(an)n(y)456 929 y Fs(\013)23 b Fq(\025)g Fs(f)661 941 y Fo(\026)705 929 y Ft(\()p Fs(\013)p Ft(\):)1237 1098 y Fs(c)14 b(\017)1321 1063 y Fo(\013)p Fn(\000)p Fo(f)1448 1071 y Fl(\026)1487 1063 y Fp(\()p Fo(\013)p Fp(\))1632 1098 y Fq(\024)46 b Fs(\026)p Ft(\()p Fs(I)1861 1110 y Fo(\013)1909 1098 y Ft(\()p Fs(\017)p Ft(\)\))h Fq(\024)f Fs(C)20 b(\017)2310 1063 y Fo(\013)p Fn(\000)p Fo(f)2437 1071 y Fl(\026)2476 1063 y Fp(\()p Fo(\013)p Fp(\))2603 1098 y Ft(.)-2170 b(\(4.1\))456 1256 y(A)21 b(less)f(explicit,)j(but)e(for)f(our)g(purp)r (oses)h(su\016cien)n(t)f(large)g(deviation)g(b)r(ound)h(whic)n(h)g (moreo)n(v)n(er)456 1356 y(do)r(es)26 b(not)h(dep)r(end)g(on)f(the)h(h) n(yp)r(othesis)f Fs(D)1811 1326 y Fn(\000)1809 1377 y Fo(\026)1868 1356 y Ft(\()p Fs(q)s Ft(\))d(=)g Fs(D)2154 1326 y Fp(+)2152 1377 y Fo(\026)2209 1356 y Ft(\()p Fs(q)s Ft(\))k(can)g(b)r(e)g(found)g(in)g([)p Fu(BGT)p Ft(]:)37 b(Let)26 b Fs(\026)456 1458 y Ft(ha)n(v)n(e)g(a)h(compact)g(supp)r (ort,)h(then)g(for)f(an)n(y)g Fs(q)f Fq(2)d Ft([0)p Fs(;)14 b Ft(1])27 b(there)g(exists)h Fs(\013)23 b Ft(=)g Fs(\013)p Ft(\()p Fs(\017;)14 b(q)s Ft(\))28 b(suc)n(h)f(that)900 1659 y Fs(\026)p Ft(\()p Fs(I)1018 1671 y Fo(\013)1066 1659 y Ft(\()p Fs(\017)p Ft(\)\))46 b Fq(\025)g Fs(\017)1387 1625 y Fp(\(1)p Fn(\000)p Fo(q)r Fp(\))p Fo(\013)1663 1603 y Fq(\000)p Ft(1)p 1613 1640 206 4 v 1613 1716 a(log)q(\()p Fs(\017)p Ft(\))1857 1546 y Fm(Z)1953 1659 y Fs(d\026)p Ft(\()p Fs(E)5 b Ft(\))14 b Fs(\026)p Ft(\([)p Fs(E)25 b Fq(\000)18 b Fs(\017;)c(E)23 b Ft(+)18 b Fs(\017)p Ft(]\))2791 1625 y Fo(q)r Fn(\000)p Fp(1)2941 1659 y Ft(.)-2508 b(\(4.2\))456 1866 y(In)23 b(all)g(the)h(de\014nitions)g (and)f(b)r(ounds)g(ab)r(o)n(v)n(e,)g(the)h(indicator)f(function)h(on)f ([)p Fs(E)15 b Fq(\000)10 b Fs(\017;)k(E)g Ft(+)c Fs(\017)p Ft(])23 b(ma)n(y)456 1966 y(b)r(e)28 b(replaced)e(b)n(y)i(a)f(gaussian) f(of)h(width)i Fs(\017)e Ft(and)g(cen)n(tered)g(at)h Fs(E)5 b Ft(.)973 2204 y Fu(5.)47 b(General)32 b(lo)m(w)m(er)f(b)s (ounds)g(on)h(quan)m(tum)g(di\013usion)605 2353 y Ff(Theorem)g Ft(5.1)p Ff(.)40 b Fe(L)l(et)33 b Fs(H)40 b Fe(b)l(e)33 b(a)h(b)l(ounde)l(d)g(tight-binding)g(Hamiltonian)g(on)g Ft(\003)29 b(=)g Fu(Z)3273 2323 y Fo(D)3367 2353 y Fe(or)456 2453 y Fu(N)531 2423 y Fo(D)591 2453 y Fe(.)38 b(Then)31 b(for)f(any)g Fs(q)c(>)d Ft(0)p Fe(:)498 2657 y Ft(\(i\))28 b Fs(D)r(\014)735 2622 y Fn(\000)731 2677 y Fo(q)829 2657 y Fq(\025)36 b Ft(dim)1069 2669 y Fd(H)1116 2657 y Ft(\()p Fs(\026)p Ft(\))30 b Fe(,)115 b Ft(\(ii\))29 b Fs(D)r(\014)1661 2622 y Fp(+)1657 2677 y Fo(q)1753 2657 y Fq(\025)36 b Ft(dim)1993 2669 y Fd(P)2036 2657 y Ft(\()p Fs(\026)p Ft(\))31 b Fe(,)115 b Ft(\(iii\))28 b Fs(D)r(\014)2604 2622 y Fn(\006)2600 2677 y Fo(q)2698 2657 y Fq(\025)36 b Fs(D)2870 2622 y Fn(\006)2868 2677 y Fo(\026)2940 2539 y Fm(\022)3082 2600 y Fs(D)p 3011 2637 213 4 v 3011 2713 a(D)21 b Ft(+)d Fs(q)3234 2539 y Fm(\023)3338 2657 y Fe(.)605 2869 y Ft(P)n(art)k(\(i\))j(is)e(the)h (original)e(Guarneri)h(Inequalit)n(y)g([)p Fu(G)p Ft(],)h(in)g(a)g (form)f(giv)n(en)g(b)n(y)g(Last)g(who)g(re-)456 2969 y(mo)n(v)n(ed)e(the)h(h)n(yp)r(othesis)g(that)h(the)f(sp)r(ectral)g (measure)f(b)r(e)i(exactly)e(scaling)h(citeLas.)34 b(Com)n(b)r(es)456 3069 y(sho)n(w)n(ed)h(that)j(it)f(also)f(holds)g(for)h(con)n(tin)n (uous)e(Sc)n(hr\177)-42 b(odinger)35 b(op)r(erators)g([)p Fu(Com)o Ft(].)65 b(The)37 b(in-)456 3168 y(equalit)n(y)27 b(\(ii\))i(\014rst)e(app)r(eared)g(in)h([)p Fu(GSB2)p Ft(])g(and)g(\(iii\))h(in)f([)p Fu(GSB3)p Ft(],)g(but)g(under)g(the)g (stronger)456 3268 y(h)n(yp)r(othesis)36 b(that)i Fs(D)1135 3238 y Fp(+)1133 3289 y Fo(\026)1190 3268 y Ft(\()p Fs(q)s Ft(\))i(=)f Fs(D)1509 3238 y Fn(\000)1507 3289 y Fo(\026)1565 3268 y Ft(\()p Fs(q)s Ft(\).)67 b(The)38 b(ab)r(o)n(v)n(e)e(form)h(of)g (\(iii\))h(is)f(due)h(to)f([)p Fu(BGT)q Ft(].)66 b(It)456 3368 y(is)28 b(particularly)g(in)n(teresting)g(for)g(1)p Fs(D)r Ft(-mo)r(dels)g(for)h(whic)n(h)f(anomalous)g(di\013usion)h(go)r (es)f(along)456 3467 y(with)c(m)n(ultifractal)g(prop)r(erties)f(of)h (the)g(sp)r(ectral)f(measures)g(\(see)h([)p Fu(GM,)j(Man)p Ft(])d(and)g(Sec.)36 b(6\).)456 3567 y(General)23 b(results)h(ab)r(out) g(the)h(v)-5 b(arious)23 b(dimensions)h(of)g Fs(\026)h Ft(sho)n(w)e(that)i(the)f(inequalit)n(y)g(\(iii\))h(im-)456 3666 y(plies)g(b)r(oth)g(\(i\))h(and)e(\(ii\).)37 b(The)25 b(b)r(ounds)g(\(i\))h(and)f(\(ii\))g(transp)r(ose)f(to)h(co)n(v)-5 b(arian)n(t)23 b(Hamiltonians)456 3766 y([)p Fu(L,)31 b(SBB1,)g(GSB2)p Ft(].)605 3866 y(Note)26 b(that)g(in)g(one)g (dimension,)g(ac)f(sp)r(ectrum)h(\(dim)2278 3878 y Fz(H)2326 3866 y Ft(\()p Fs(\026)p Ft(\))e(=)f(1\))i(implies)h(ballistic)g(quan-) 456 3965 y(tum)i(m)n(ution.)38 b(F)-7 b(or)27 b Fs(D)f Ft(=)d(2,)28 b(di\013usiv)n(e)g(motion)f(and)h(ac)g(sp)r(ectrum)g(ma)n (y)f(co)r(exist;)h(for)f Fs(D)f Ft(=)d(3)456 4065 y(the)j(motion)g(ma)n (y)f(ev)n(en)g(b)r(e)i(sub)r(di\013usiv)n(e)f(with)g Fs(\014)i Fq(\031)22 b Ft(1)p Fs(=)p Ft(3)j(in)h(presence)f(of)h(ac)g (sp)r(ectrum.)36 b(F)-7 b(or)456 4165 y(higher)34 b(dimensional)h(mo)r (dels,)i(slo)n(w)d(quan)n(tum)h(transp)r(ort)g(ma)n(y)f(hence)h(b)r(e)h (hea)n(vily)e(in\015u-)456 4264 y(enced)e(b)n(y)h(prop)r(erties)e(of)i (the)g(generalized)e(eigenfunctions.)52 b(Ho)n(w)32 b(suc)n(h)g (infromation)g(can)456 4364 y(b)r(e)c(incorp)r(orated)e(in)i(the)g(lo)n (w)n(er)e(b)r(ound)i(w)n(as)e(sho)n(wn)h(in)h([)p Fu(KPG,)j(KL)q Ft(].)605 4464 y(The)c(main)g(idea)f(of)h(pro)r(of)f(is)g(as)g(follo)n (ws:)36 b(a)26 b(giv)n(en)g(con)n(tin)n(uit)n(y)g(of)h(a)f(c)n(hosen)g (part)g(of)h(the)456 4563 y(sp)r(ectral)h(measure)g(\(c)n(haracterized) f(b)n(y)h(a)h(sp)r(ectral)f(dimension\))h(forces)f(the)h(corresp)r (onding)456 4663 y(p)r(ortion)h(of)h(the)g(w)n(a)n(v)n(e)f(pac)n(k)n (et)g(out)h(of)g(a)f(ball)h(of)g(a)f(certain)h(size,)g(due)h(to)e(a)h (re\014ned)g(RA)n(GE)456 4762 y(theorem)h(and)g(con)n(trol)g(its)h (size)f(b)n(y)h(\(4.1\).)52 b(Maximising)32 b(o)n(v)n(er)f(the)i(free)g (parameters)e(giv)n(es)456 4862 y(the)d(lo)n(w)n(er)e(b)r(ound)i (\(iii\).)37 b(More)27 b(precisely)-7 b(,)27 b(for)g(a)g(giv)n(en)g Fs(R)d Fq(2)f Fu(R)p Ft(:)570 5120 y Fq(h)p Ft(0)p Fq(j)14 b(j)722 5099 y Fs(~)704 5120 y(X)6 b Fq(j)802 5086 y Fo(q)839 5120 y Ft(\()p Fs(t)p Ft(\))p Fq(j)p Ft(0)p Fq(i)47 b(\025)e Fs(R)1251 5086 y Fo(q)1333 5041 y Fm(X)1302 5223 y Fn(j)p Fo(n)p Fn(j)p Fo(>R)1498 5120 y Fq(jh)p Fs(n)p Fq(j)p Fs(e)1665 5086 y Fn(\000)p Fo({H)t(t)1828 5120 y Fq(j)p Ft(0)p Fq(ij)1948 5086 y Fp(2)2031 5120 y Ft(=)h Fs(R)2206 5086 y Fo(q)2256 4953 y Fm(0)2256 5103 y(@)2329 5120 y Ft(1)18 b Fq(\000)2503 5041 y Fm(X)2472 5223 y Fn(j)p Fo(n)p Fn(j\024)p Fo(R)2668 5120 y Fq(jh)p Fs(n)p Fq(j)p Fs(e)2835 5086 y Fn(\000)p Fo({H)t(t)2998 5120 y Fq(j)p Ft(0)p Fq(ij)3118 5086 y Fp(2)3156 4953 y Fm(1)3156 5103 y(A)3270 5120 y Ft(.)p eop %%Page: 5 5 5 4 bop 1125 226 a Fp(ANOMALOUS)28 b(TRANSPOR)-5 b(T)29 b(IN)g(APERIODIC)g(MEDIA)635 b(5)456 425 y Ft(No)n(w)27 b(w)n(e)g(decomp)r(ose)g Fq(j)p Ft(0)p Fq(i)c Ft(=)f Fq(j)p Fs( )s Fq(i)d Ft(+)f Fq(j)p Fs( )1686 395 y Fo(c)1720 425 y Fq(i)p Ft(.)37 b(Then)661 502 y Fm(X)629 684 y Fn(j)p Fo(n)p Fn(j\024)p Fo(R)826 581 y Fq(jh)p Fs(n)p Fq(j)p Fs(e)993 547 y Fn(\000)p Fo({H)t(t)1156 581 y Fq(j)p Ft(0)p Fq(ij)1276 547 y Fp(2)1359 581 y Fq(\024)46 b Ft(1)18 b Fq(\000)g(k)p Fs( )s Fq(k)1754 547 y Fp(2)1809 581 y Ft(+)1923 502 y Fm(X)1892 684 y Fn(j)p Fo(n)p Fn(j\024)p Fo(R)2088 581 y Fq(jh)p Fs(n)p Fq(j)p Fs(e)2255 547 y Fn(\000)p Fo({H)t(t)2418 581 y Fq(j)p Fs( )s Fq(ij)2553 547 y Fp(2)2609 581 y Ft(+)g(crossed)26 b(terms)i(.)456 813 y(With)23 b(little)g(e\013ort)f(the)h(crossed)e(terms)h(in)h Fs( )j Ft(and)c Fs( )2103 783 y Fo(c)2159 813 y Ft(can)h(b)r(e)f (treated)h(similarly)e(to)h(the)h(third)456 913 y(one,)k(so)g(that)h(w) n(e)f(will)h(neglect)f(them)h(here.)37 b(Hence)27 b(w)n(e)h(get)761 1147 y Fq(h)p Ft(0)p Fq(j)14 b(j)912 1126 y Fs(~)895 1147 y(X)6 b Fq(j)993 1113 y Fo(q)1030 1147 y Ft(\()p Fs(t)p Ft(\))p Fq(j)p Ft(0)p Fq(i)47 b(\025)e Fs(R)1442 1113 y Fo(q)1492 980 y Fm(0)1492 1130 y(@)1565 1147 y Fq(k)p Fs( )s Fq(k)1706 1113 y Fp(2)1761 1147 y Fq(\000)1875 1068 y Fm(X)1844 1250 y Fn(j)p Fo(n)p Fn(j\024)p Fo(R)2040 1147 y Fq(jh)p Fs(n)p Fq(j)p Fs(e)2207 1113 y Fn(\000)p Fo({H)t(t)2370 1147 y Fq(j)p Fs( )s Fq(ij)2505 1113 y Fp(2)2562 1147 y Ft(+)18 b(crossed)26 b(terms)3135 980 y Fm(1)3135 1130 y(A)3249 1147 y Ft(.)-2816 b(\(5.1\))456 1386 y(No)n(w)26 b(for)g(a)h(giv)n(en)f(time)h Fs(T)38 b Ft(w)n(e)27 b(c)n(ho)r(ose)e Fq(j)p Fs( )s Fq(i)f Ft(=)e Fs(\037)p Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Ft(0)p Fq(i)28 b Ft(where)e Fs(\037)h Ft(is)g(the)g(indicator)f(function)456 1486 y(on)j Fs(I)609 1498 y Fo(\013)657 1486 y Ft(\(1)p Fs(=T)12 b Ft(\))29 b(w.r.t.)43 b(the)31 b(sp)r(ectral)e(measure)g Fs(\026)h Ft(of)g Fq(j)p Ft(0)p Fq(i)p Ft(.)44 b(According)28 b(to)i(the)h(large)d(deviation)456 1589 y(b)r(ound)35 b(\(4.1\),)j Fq(k)p Fs( )s Fq(k)1092 1558 y Fp(2)1164 1589 y Ft(=)e Fs(\026)p Ft(\()p Fs(I)7 b Ft(\()p Fs(T)12 b Ft(\)\))36 b Fq(\025)g Fs(c)14 b(T)1795 1558 y Fo(f)1827 1566 y Fl(\026)1866 1558 y Fp(\()p Fo(\013)p Fp(\))p Fn(\000)p Fo(\013)2060 1589 y Ft(.)61 b(W)-7 b(e)36 b(no)n(w)f(ha)n(v)n (e)f(to)h(c)n(ho)r(ose)f Fs(R)j Ft(=)f Fs(R)q Ft(\()p Fs(T)12 b Ft(\))456 1688 y(su\016cien)n(tly)30 b(small)g(suc)n(h)g (that)h(the)g(second)e(term)i(in)g(the)f(l.h.s.)46 b(of)30 b(\(5.1\))g(is)h(of)f(comparable)456 1788 y(size,)h(but)g(smaller)f (than)g(this.)47 b(T)-7 b(o)30 b(b)r(ound)h(this)g(second)f(term,)h(a)g (v)n(ery)e(con)n(v)n(enian)n(t)g(w)n(a)n(y)h(of)456 1887 y(dealing)20 b(with)i(oscillation)e(of)h Fs(e)1431 1857 y Fn(\000)p Fo({H)t(t)1615 1887 y Ft(is)g(to)g(tak)n(e)g(a)g(gaussion)e (mean)i(of)g(\(5.1\))g(up)h(to)f(time)g Fs(T)33 b Ft([)p Fu(L)p Ft(].)456 1987 y(If)e Fs(P)595 1999 y Fo(n)672 1987 y Ft(are)f(the)h(generalized)f(eigenfunctions)h(\(that)h(is)f Fq(h)p Ft(0)p Fq(j)p Fs(g)s Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Fs(n)p Fq(i)28 b Ft(=)2749 1920 y Fm(R)2818 1987 y Fs(d\026)p Ft(\()p Fs(E)5 b Ft(\))p Fs(P)3094 1999 y Fo(n)3141 1987 y Ft(\()p Fs(E)g Ft(\))p Fs(g)s Ft(\()p Fs(E)g Ft(\))456 2088 y(and)617 2021 y Fm(R)686 2088 y Fs(d\026)p Ft(\()p Fs(E)g Ft(\))p Fq(j)p Fs(P)985 2100 y Fo(n)1032 2088 y Ft(\()p Fs(E)g Ft(\))p Fq(j)1185 2058 y Fp(2)1246 2088 y Ft(=)22 b(1\),)28 b(this)g(giv)n(es)e(b)n(y)h(the)h (Cauc)n(h)n(y-Sc)n(h)n(w)n(arz)d(inequalit)n(y)456 2163 y Fm(Z)562 2219 y Fs(dt)p 562 2256 74 4 v 569 2333 a(T)659 2276 y(e)698 2241 y Fn(\000)p Fo(t)775 2216 y Fj(2)807 2241 y Fo(=T)889 2216 y Fj(2)926 2276 y Fq(jh)p Fs(n)p Fq(j)p Fs(e)1093 2241 y Fn(\000)p Fo({H)t(t)1256 2276 y Fq(j)p Fs( )s Fq(ij)1391 2241 y Fp(2)1466 2276 y Fq(\024)1567 2163 y Fm(Z)1613 2351 y Fo(I)5 b Fp(\()p Fo(T)k Fp(\))1738 2276 y Fs(d\026)p Ft(\()p Fs(E)c Ft(\))14 b Fs(P)2028 2288 y Fo(n)2074 2276 y Ft(\()p Fs(E)5 b Ft(\))2204 2241 y Fp(2)2256 2163 y Fm(Z)2302 2351 y Fo(I)g Fp(\()p Fo(T)k Fp(\))2426 2276 y Fs(d\026)p Ft(\()p Fs(E)2617 2241 y Fn(0)2641 2276 y Ft(\))14 b Fs(e)2726 2241 y Fn(\000)p Fp(\()p Fo(E)s Fn(\000)p Fo(E)2960 2216 y Fk(0)2982 2241 y Fp(\))3008 2216 y Fj(2)3040 2241 y Fo(T)3088 2216 y Fj(2)3162 2276 y Fq(\024)36 b Fs(T)3324 2241 y Fn(\000)p Fo(\013)3422 2276 y Fs(:)456 2488 y Ft(No)n(w)21 b(w)n(e)h(need)h(to)f (c)n(ho)r(ose)f Fs(R)q Ft(\()p Fs(T)12 b Ft(\))22 b(suc)n(h)f(that)i Fs(T)1927 2457 y Fo(f)1959 2465 y Fl(\026)1998 2457 y Fp(\()p Fo(\013)p Fp(\))p Fn(\000)p Fo(\013)2215 2488 y Fq(\031)g Fs(R)q Ft(\()p Fs(T)12 b Ft(\))2492 2457 y Fo(D)2551 2488 y Fs(T)2612 2457 y Fn(\000)p Fo(\013)2711 2488 y Ft(.)35 b(This)22 b(is)g(satis\014ed)g(for)456 2597 y Fs(R)q Ft(\()p Fs(T)12 b Ft(\))22 b Fq(\031)h Fs(T)816 2567 y Fo(f)848 2575 y Fl(\026)886 2567 y Fp(\()p Fo(\013)p Fp(\))p Fo(=D)1076 2597 y Ft(.)35 b(Hence)24 b(the)g(gaussian)e(time)i(mean)g(of)f Fq(h)p Ft(0)p Fq(j)14 b(j)2490 2576 y Fs(~)2472 2597 y(X)6 b Fq(j)2570 2567 y Fo(q)2607 2597 y Ft(\()p Fs(t)p Ft(\))p Fq(j)p Ft(0)p Fq(i)24 b Ft(is)g(b)r(ounded)g(b)r(elo)n(w)456 2700 y(b)n(y)30 b Fs(T)635 2670 y Fp(\(1+)p Fo(q)r(=D)r Fp(\))p Fo(f)925 2678 y Fl(\026)964 2670 y Fp(\()p Fo(\013)p Fp(\))p Fn(\000)p Fo(\013)1158 2700 y Ft(.)46 b(Maximising)30 b(the)h(exp)r(onen)n(t)f(o) n(v)n(er)f Fs(\013)i Ft(and)g(using)f(the)h(fact)f(that)h(the)456 2799 y(Legendre)20 b(transform)h(of)h Fs(f)1312 2811 y Fo(\026)1378 2799 y Ft(is)g(link)n(ed)g(to)f(the)h(m)n(ultifractal)g (dimensions)g(allo)n(ws)e(to)i(conclude)456 2899 y(immediately)-7 b(.)37 b(Alternativ)n(ely)-7 b(,)27 b(one)g(can)g(use)h(the)g(b)r(ound) g(\(4.2\).)826 3083 y Fu(6.)47 b(Mo)s(del)30 b(sp)s(eci\014c)i(upp)s (er)f(b)s(ounds)g(on)h(quan)m(tum)g(di\013usion)605 3232 y Ft(The)k(link)g(b)r(et)n(w)n(een)g(di\013usion)g(exp)r(onen)n(ts)f (and)h(sp)r(ectral)f(dimensions)h(is)f(most)h(lik)n(ely)456 3332 y(to)i(b)r(e)h(tigh)n(t)f(for)g(Jacobi)f(matrices)h(on)g(\003)i(=) h Fu(N)d Ft(\(1)p Fs(D)i Ft(next)f(neigh)n(b)r(or)e(hopping)h(mo)r (dels\))456 3431 y(where)28 b(the)h(generalized)e(eigenfunctions)h(are) g(giv)n(en)f(b)n(y)i(the)g(orthogonal)d(p)r(olynomials)i(and)456 3531 y(hence)38 b(en)n(tirely)g(determined)h(b)n(y)f(the)h(sp)r(ectral) f(measure.)69 b(This)38 b(fact)h(is)g(fully)g(exploited)456 3630 y(in)34 b(sub)r(ordinacy)f(theory)g([)p Fu(JL)q Ft(],)i(but)g(here)e(w)n(e)h(only)g(use)f(the)i(generalized)d(Herb)r (ert-Jones-)456 3730 y(Thouless)26 b(form)n(ula)h([)p Fu(ST)q Ft(]:)826 3912 y(lim)797 3962 y Fo(n)p Fn(!1)999 3856 y Ft(1)p 995 3893 50 4 v 995 3969 a Fs(n)1068 3912 y Ft(log)o(\()p Fq(j)p Fs(P)1282 3924 y Fo(n)1328 3912 y Ft(\()p Fs(z)t Ft(\))p Fq(j)p Ft(\))46 b(=)1647 3799 y Fm(Z)1744 3912 y Fs(d)p Fq(N)12 b Ft(\()p Fs(E)5 b Ft(\))14 b(log\()p Fq(j)p Fs(E)24 b Fq(\000)18 b Fs(z)t Fq(j)p Ft(\))g Fq(\000)g Ft(log)o(\(cap\(supp\()p Fs(\026)p Ft(\)\)\))29 b(.)-2780 b(\(6.1\))456 4094 y(Here)24 b Fs(z)i Fq(2)e Fu(C)p Fq(n)p Ft(supp)o(\()p Fs(\026)p Ft(\),)i Fq(N)37 b Ft(is)24 b(the)h(densit)n(y)g(of)f(states)g(and)h (cap)o(\()p Fs(J)8 b Ft(\))26 b(denotes)e(the)h(logarithmic)456 4193 y(capacit)n(y)33 b(of)h(a)g(set)h Fs(J)42 b Fq(\032)34 b Fu(C)p Ft(.)58 b(This)34 b(form)n(ula)f(holds)h(\(and)h Fq(N)47 b Ft(exists\))34 b(for)g(a)g(large)f(class)h(of)456 4293 y(Jacobi)26 b(matrices)h(called)g(asymptotically)g(regular)e(in)j ([)p Fu(ST)q Ft(].)605 4393 y(Here)f(w)n(e)h(consider)f(Jacobi)f (matrices)h(asso)r(ciated)g(to)g(fractal)g(measures)g(of)h(the)g(Can)n (tor)456 4492 y(t)n(yp)r(e.)44 b(Let)31 b Fs(I)877 4462 y Fp(1)870 4513 y(0)944 4492 y Ft(and)f Fs(I)1151 4462 y Fp(1)1144 4513 y(1)1219 4492 y Ft(b)r(e)g(t)n(w)n(o)g(disjoin)n(t)g (closed)f(in)n(terv)-5 b(als)30 b(con)n(tained)f(in)i(a)f(closed)f(in)n (terv)-5 b(al)456 4594 y Fs(I)499 4564 y Fp(0)536 4594 y Ft(.)66 b(Let)37 b Fs(S)44 b Ft(:)39 b Fu(R)g Fq(!)g Fu(R)e Ft(b)r(e)h(suc)n(h)e(that)i(its)f(restrictions)f Fs(S)2412 4559 y Fn(\000)p Fp(1)2407 4616 y(0)p Fo(;)p Fp(1)2540 4594 y Ft(:)j Fs(I)2645 4564 y Fp(1)2638 4615 y(0)p Fo(;)p Fp(1)2768 4594 y Fq(!)g Fs(I)2933 4564 y Fp(0)3008 4594 y Ft(are)d(analytic)456 4706 y(bijections)25 b(satisfying)g Fs(S)5 b Ft(\()p Fs(I)1324 4676 y Fp(0)1375 4706 y Fq(n)14 b Ft(\()p Fs(I)1506 4676 y Fp(1)1499 4726 y(0)1557 4706 y Fq([)h Fs(I)1670 4676 y Fp(1)1663 4726 y(1)1707 4706 y Ft(\)\))f Fq(\\)h Fs(I)1898 4676 y Fp(0)1958 4706 y Ft(=)23 b Fq(;)p Ft(.)35 b(The)26 b(in)n(v)n(erse)e(images)g Fs(S)2914 4676 y Fn(\000)p Fo(N)3029 4706 y Ft(\()p Fs(I)3104 4676 y Fp(0)3141 4706 y Ft(\))i(consist)456 4805 y(of)i(2)593 4775 y Fo(N)685 4805 y Ft(in)n(terv)-5 b(als)28 b(of)g(generation)g Fs(N)9 b Ft(.)40 b(A)29 b(co)r(ding)f Fs(\033)h Ft(=)24 b(\()p Fs(\033)2262 4817 y Fo(i)2290 4805 y Ft(\))2322 4817 y Fo(i)p Fn(\025)p Fp(1)2464 4805 y Ft(with)29 b Fs(\033)2701 4817 y Fo(i)2755 4805 y Fq(2)c(f)p Ft(0)p Fs(;)14 b Ft(1)p Fq(g)27 b Ft(p)r(ermits)i(to)456 4907 y(describ)r(e)21 b(them)i(through)e Fs(I)1329 4877 y Fo(N)1322 4927 y(\033)1415 4907 y Ft(=)i Fs(S)1559 4877 y Fn(\000)p Fp(1)1554 4927 y Fo(\033)1592 4935 y Fj(1)1655 4907 y Fq(\016)7 b Fs(:)14 b(:)g(:)7 b Fq(\016)g Fs(S)1913 4877 y Fn(\000)p Fp(1)1908 4927 y Fo(\033)1946 4935 y Fl(N)2004 4907 y Ft(\()p Fs(I)2079 4877 y Fp(0)2116 4907 y Ft(\).)36 b(The)22 b(co)r(des)f(of)h(length)h Fs(N)30 b Ft(are)21 b(denoted)456 5014 y(b)n(y)26 b(\006)630 4984 y Fo(N)693 5014 y Ft(.)36 b(Finally)27 b Fs(S)k Ft(is)26 b(supp)r(osed)g(to)h(b)r(e)g(expansiv)n(e,)e(namely)h(w)n(e)g (supp)r(ose)h(that)f(there)h(exist)456 5113 y(p)r(ositiv)n(e)c(constan) n(ts)g Fs(a)g(<)g Ft(1)g(and)h Fs(b)g Ft(suc)n(h)g(that)g(the)g(lengh)n (ts)g(of)g Fs(I)2475 5083 y Fo(N)2468 5134 y(\033)2562 5113 y Ft(decrease)e(faster)i(than)g Fs(ba)3382 5083 y Fo(N)456 5213 y Ft(uniformly)k(in)i Fs(\033)s Ft(.)41 b(Then)29 b(the)h(set)e Fs(J)34 b Ft(=)1711 5151 y Fm(T)1780 5238 y Fo(N)6 b Fn(\025)p Fp(1)1942 5213 y Fs(S)1998 5183 y Fn(\000)p Fo(N)2112 5213 y Ft(\()p Fs(I)2187 5183 y Fp(0)2225 5213 y Ft(\))29 b(is)g(a)f(self-similar)g(fractal)h(in)n(v) -5 b(arian)n(t)p eop %%Page: 6 6 6 5 bop 456 226 a Fp(6)995 b(HERMANN)29 b(SCHULZ-BALDES)456 425 y Ft(under)k Fs(S)5 b Ft(.)54 b(The)33 b(usual)g(Can)n(tor)f(sets)h (are)g(obtained)g(b)n(y)g(c)n(hossing)f Fs(S)2683 390 y Fn(\000)p Fp(1)2678 447 y(0)p Fo(;)p Fp(1)2805 425 y Ft(to)h(b)r(e)h(linear,)g(here)456 535 y(particular)27 b(fo)r(cus)h(will)g(b)r(e)h(on)f(real)f(Jula)h(sets)g(acco)r(ciated)f (to)h Fs(S)5 b Ft(\()p Fs(E)g Ft(\))25 b(=)f Fs(E)2819 504 y Fp(2)2875 535 y Fq(\000)18 b Fs(\025)p Ft(,)29 b Fs(\025)c(>)f Ft(2.)38 b(F)-7 b(or)456 634 y(the)23 b(follo)n(wing,)f(w)n(e)g(could)h(c)n(ho)r(ose)e Fs(\026)i Ft(to)f(b)r(e)h(an)g(arbitrary)d(Gibbs)j(measure)f(of)g(the)h (dynamical)456 734 y(system)18 b(\()p Fs(J)o(;)c(S)5 b Ft(\),)21 b(but)e(for)f(simplicit)n(y)h(w)n(e)f(only)h(consider)e (the)i(equal)g(w)n(eigh)n(t)f(Bernoulli)g(measure)456 833 y(satisfying)37 b Fs(\026)p Ft(\()p Fs(I)961 803 y Fo(N)954 854 y(\033)1024 833 y Ft(\))k(=)g(2)1245 803 y Fn(\000)p Fo(N)1359 833 y Ft(.)69 b(It)38 b(is)g(kno)n(wn)g([)p Fu(P)m(es)p Ft(])g(that)h Fs(q)k Fq(7!)e Fs(D)2613 803 y Fp(+)2611 854 y Fo(\026)2668 833 y Ft(\()p Fs(q)s Ft(\))h(=)e Fs(D)2990 803 y Fn(\000)2988 854 y Fo(\026)3046 833 y Ft(\()p Fs(q)s Ft(\))f(is)f(then)456 941 y(real)30 b(analytic.)48 b(No)n(w)31 b(the)h(orthogonal)d(p)r(olynomials)h Fs(P)2258 953 y Fo(n)2335 941 y Ft(in)i Fs(L)2493 911 y Fp(2)2530 941 y Ft(\()p Fu(R)p Fs(;)14 b(\026)p Ft(\))31 b(form)g(a)g(three)h (term)456 1041 y(recurrence)37 b(relation,)k(the)e(co)r(e\016cien)n(ts) g(of)g(whic)n(h)g(giv)n(e)f(our)g(Hamiltonian)h Fs(H)45 b Ft(acting)39 b(on)456 1140 y Fs(`)491 1110 y Fp(2)527 1140 y Ft(\()p Fu(N)p Ft(\).)i(It)29 b(can)f(b)r(e)h(v)n(eri\014ed)f (to)h(b)r(e)g(asymptotically)f(regular)e([)p Fu(GSB1)p Ft(].)41 b(If)29 b Fs(J)36 b Ft(is)29 b(a)f(Julia)g(set,)456 1240 y Fs(H)37 b Ft(is)31 b(called)g(a)f(Julia)g(matrix.)47 b(They)30 b(are)g(kno)n(wn)h(to)f(b)r(e)i(limit-p)r(erio)r(dic)f(for)f Fs(\025)h Ft(su\016cien)n(tly)456 1340 y(large)20 b([)p Fu(BBM)o Ft(],)j(but)f(it)g(is)f(an)g(op)r(en)h(problem)f(to)g(sho)n(w) f(that)i(the)g(Jacobi)e(matrices)h(asso)r(ciated)456 1439 y(to)27 b(general)f(self-similar)h(measures)f(of)i(the)g(ab)r(o)n (v)n(e)e(t)n(yp)r(e)i(are)e(almost)h(p)r(erio)r(dic.)605 1591 y Ff(Theorem)32 b Ft(6.1)p Ff(.)40 b Ft([)p Fu(BaSB)p Ft(])30 b Fe(F)-6 b(or)30 b(Julia)g(matric)l(es)g Fs(\014)2242 1561 y Fp(+)2238 1611 y Fo(q)2321 1591 y Fq(\024)22 b Fs(D)2477 1603 y Fo(\026)2522 1591 y Ft(\(1)c Fq(\000)g Fs(q)s Ft(\))30 b Fe(for)h(al)t(l)f Ft(0)23 b Fq(\024)g Fs(q)j Fq(\024)c Ft(2)p Fe(.)605 1745 y Ft(Numerics)40 b(and)g(non-rigorous)e(argumen)n(ts)h([)p Fu(Man)p Ft(])h(sho)n(w)g (that)g(this)h(is)f(actually)g(an)456 1845 y(equalit)n(y)29 b(for)g(all)g Fs(q)h(>)c Ft(0.)43 b(Bounds)30 b(for)f(other)g(mo)r (dels)h(suc)n(h)f(as)g(an)n(y)g(Bernoulli)g(measure)g(on)456 1944 y(linear)23 b(Can)n(tor)f(sets)i(can)f(also)g(b)r(e)h(obtained)g ([)p Fu(GSB1)o Ft(],)h(but)g(they)f(cannot)f(b)r(e)h(expressed)f(only) 456 2044 y(in)34 b(terms)f(of)h(the)g(m)n(ultifractal)f(dimensions)h(b) r(ecause)f(the)h(asymptotic)g(prop)r(erties)e(of)i(the)456 2144 y(generalized)29 b(eigenfunctions)h(en)n(tering)g(via)g(the)h (Thouless)e(form)n(ula)h(do)g(not)h(cancel)f(out)g(as)456 2243 y(for)20 b(Julia)h(matrices.)34 b(Theoretical)20 b(argumen)n(ts)g(and)h(n)n(umerics)g(sho)n(w)f(that)i(a)f(similar)f (form)n(ula)p 456 2275 52 4 v 456 2343 a Fs(\014)507 2363 y Fo(q)577 2343 y Ft(=)33 b Fs(D)744 2355 y Fn(N)811 2343 y Ft(\(1)22 b Fq(\000)g Fs(q)s Ft(\))34 b(holds)f(for)h(phase-a)n (v)n(eraged)c(exp)r(onen)n(ts)j(of)g(1)p Fs(D)j Ft(quasip)r(erio)r(dic) d(mo)r(dels)456 2445 y([)p Fu(Pie)o Ft(].)k(No)28 b(pro)r(of)f(is)g (kno)n(wn)g(at)h(the)g(time)g(b)r(eing.)605 2545 y(Let)h(us)g(dev)n (elop)f(the)h(k)n(ey)f(elemen)n(t)h(of)g(the)g(pro)r(of)g(dev)n(elop)r (ed)f(in)h([)p Fu(GSB1)p Ft(].)41 b(A)n(t)29 b(a)f(giv)n(en)456 2644 y(time,)e(one)f(has)g(to)h(sho)n(w)e(asymptotic)h(fall)h(o\013)g (of)f Fq(jh)p Fs(n)p Fq(j)p Fs(e)2204 2614 y Fn(\000)p Fo({H)t(t)2367 2644 y Fq(j)p Ft(0)p Fq(ij)h Ft(in)g Fs(n)p Ft(.)36 b(Herefore)24 b(w)n(e)h(appro)n(x-)456 2757 y(imate)35 b Fs(e)733 2727 y Fn(\000)p Fo({H)t(t)896 2757 y Fq(j)p Ft(0)p Fq(i)g Ft(up)g(to)g(time)h Fs(T)46 b Ft(fairly)34 b(w)n(ell)h(b)n(y)g Fq(j)p Fs( )s Ft(\()p Fs(t)p Ft(\))p Fq(i)i Ft(=)2422 2695 y Fm(P)2509 2782 y Fo(\033)2568 2757 y Fs(e)2607 2727 y Fn(\000)p Fo({E)2734 2702 y Fl(N)2731 2743 y(\033)2787 2727 y Fo(t)2816 2757 y Fs(\037)2868 2773 y Fo(I)2902 2757 y Fl(N)2897 2790 y(\033)2960 2757 y Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Ft(0)p Fq(i)35 b Ft(where)456 2868 y Fs(E)522 2838 y Fo(N)517 2888 y(\033)608 2868 y Fq(2)23 b Fs(I)729 2838 y Fo(N)722 2888 y(\033)819 2868 y Ft(as)k(long)f(as)g(the)i(length)f(of)g(\(most)g(of)6 b(\))28 b(the)g(in)n(terv)-5 b(als)26 b Fq(j)p Fs(I)2600 2838 y Fo(N)2593 2888 y(\033)2663 2868 y Fq(j)d(\031)g Fs(e)2836 2838 y Fn(\000)p Fo(N)6 b Fp(\003)3023 2868 y Ft(are)26 b(of)h(order)456 2968 y(1)p Fs(=T)38 b Ft(\(here)28 b(\003)g(is)h(the)f(Ly)n(apuno)n(v)f(exp)r(onen)n(t.)39 b(If)29 b(one)f(then)h(moreo)n(v)n(er)d(c)n(ho)r(oses)h(the)h Fs(E)3223 2937 y Fo(N)3218 2988 y(\033)3286 2968 y Ft('s)h(to)456 3067 y(b)r(e)f(part)f(of)g(a)g(regular)f(lattice)i(of)g(spacing)e(1)p Fs(=T)12 b Ft(,)26 b(then)703 3154 y Fm(Z)786 3174 y Fo(T)750 3343 y Fn(\000)p Fo(T)892 3211 y Fs(dt)p 878 3248 102 4 v 878 3324 a Ft(2)p Fs(T)1003 3267 y Fq(jh)p Fs(n)p Fq(j)p Fs(e)1170 3233 y Fn(\000)p Fo({H)t(t)1333 3267 y Fq(j)p Ft(0)p Fq(ij)1453 3233 y Fp(2)1536 3267 y Fq(\024)46 b Ft(correction)26 b(terms)18 b(+)2373 3188 y Fm(X)2340 3371 y Fo(\033)r Fn(2)p Fp(\006)2472 3355 y Fl(N)2540 3267 y Fq(jh)p Fs(n)p Fq(j)p Fs(\037)2720 3283 y Fo(I)2754 3267 y Fl(N)2749 3300 y(\033)2811 3267 y Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Ft(0)p Fq(ij)3071 3233 y Fp(2)3137 3267 y Ft(.)456 3489 y(The)33 b(length)n(y)g(analysis)f(of) h(the)h(correction)d(terms)i(is)g(based)g(on)g(the)h(study)f(of)h(the)f (\015uctu-)456 3588 y(ations)c(of)i(the)f(band)h(lengths)f(of)g (generation)f Fs(N)9 b Ft(,)31 b(ev)n(en)n(tually)e(leading)h(to)g(the) h(m)n(ultifractal)456 3688 y(dimensions)37 b([)p Fu(BaSB)p Ft(].)66 b(Let)37 b(us)h(fo)r(cus)f(on)g(the)g(argumen)n(t)g(for)f(the) i(main)f(term)g([)p Fu(GSB1)p Ft(].)456 3787 y(One)i(has)h Fs(S)860 3757 y Fn(\000)p Fp(1)949 3787 y Ft(\()p Fs(I)1024 3757 y Fo(N)6 b Fn(\000)p Fp(1)1017 3808 y Fo(\033)1172 3787 y Ft(\))44 b(=)f Fs(I)1399 3757 y Fo(N)1392 3808 y(\033)1489 3787 y Fq([)27 b Fs(I)1614 3757 y Fo(N)1607 3810 y(\033)1647 3794 y Fk(0)1717 3787 y Ft(\(for)40 b(suitable)g Fs(\033)2261 3757 y Fn(0)2284 3787 y Ft(\))h(and)e Fs(I)2573 3757 y Fo(N)2566 3808 y(\033)2680 3787 y Fq(\032)k Fs(I)2831 3757 y Fp(1)2824 3808 y(+)2920 3787 y Ft(and)d Fs(I)3137 3757 y Fo(N)3130 3810 y(\033)3170 3794 y Fk(0)3244 3787 y Fq(\032)j Fs(I)3395 3757 y Fp(1)3388 3808 y Fn(\000)456 3891 y Ft(\(or)e(vica)h(v)n(ersa\).)78 b(In)n(tro)r(ducing)42 b Fq(S)6 b(j)p Fs(\036)p Fq(i)48 b Ft(=)f Fq(j)p Fs(\036)29 b Fq(\016)e(S)6 b(i)43 b Ft(on)f Fs(L)2385 3861 y Fp(2)2422 3891 y Ft(\()p Fu(R)p Fs(;)14 b(\026)p Ft(\),)46 b(w)n(e)c(therefore)f (obtain)456 3990 y Fs(\037)508 4007 y Fo(I)542 3990 y Fl(N)537 4023 y(\033)599 3990 y Ft(\()p Fs(H)7 b Ft(\))23 b(=)g Fs(\037)902 4009 y Fo(I)936 3989 y Fj(1)931 4028 y(+)982 3990 y Ft(\()p Fs(H)7 b Ft(\))p Fq(S)f Fs(\037)1230 4007 y Fo(I)1264 3990 y Fl(N)1259 4023 y(\033)1323 3990 y Ft(\()p Fs(H)h Ft(\))28 b(so)e(that)875 4168 y Fq(h)p Fs(n)p Fq(j)p Fs(\037)1032 4185 y Fo(I)1066 4168 y Fl(N)1061 4201 y(\033)1123 4168 y Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Ft(0)p Fq(i)47 b Ft(=)1517 4090 y Fm(X)1539 4268 y Fo(k)q(;l)1637 4168 y Fq(h)p Fs(n)p Fq(j)p Fs(\037)1794 4187 y Fo(I)1828 4167 y Fj(1)1823 4206 y(+)1875 4168 y Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Fs(k)s Fq(i)23 b(h)p Fs(k)s Fq(jS)6 b(j)p Fs(l)r Fq(i)24 b(h)p Fs(l)r Fq(j)p Fs(\037)2536 4195 y Fo(I)2570 4168 y Fl(N)5 b Fk(\000)p Fj(1)2565 4204 y Fl(\033)2700 4168 y Ft(\()p Fs(H)i Ft(\))p Fq(j)p Ft(0)p Fq(i)28 b Ft(.)-2532 b(\(6.2\))456 4397 y(No)n(w)21 b(a)h(Com)n(b)r(es-Thomas)e(argumen)n(t)h(sho)n(ws)g(that)i Fq(h)p Fs(n)p Fq(j)p Fs(\037)2261 4415 y Fo(I)2295 4395 y Fj(1)2290 4434 y(+)2341 4397 y Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Fs(k)s Fq(i)23 b Ft(decreases)d(exp)r(onen)n(tially)456 4512 y(in)27 b Fq(j)p Fs(n)19 b Fq(\000)f Fs(k)s Fq(j)p Ft(.)36 b(Supp)r(osing)28 b(that)g Fs(S)k Ft(is)27 b(analytic,)g(one)g (can)h(dev)n(elop)e Fs(P)2597 4524 y Fo(l)2623 4512 y Ft(\()p Fs(S)5 b Ft(\()p Fs(E)g Ft(\)\))29 b(using)e(Cauc)n(h)n(y's)456 4611 y(form)n(ula)h(with)i(a)f(con)n(tour)g(\000)1381 4623 y Fo(R)1465 4611 y Ft(of)g(radius)g Fs(R)h Ft(bigger)f(than)g(the) h(sp)r(ectral)f(radius)g Fs(R)3127 4623 y Fo(c)3190 4611 y Ft(around)456 4711 y(the)f(cen)n(ter)f Fs(E)907 4723 y Fo(c)968 4711 y Ft(of)h(the)g(sp)r(ectrum.)37 b(Hence)28 b(b)n(y)f(orthogonalit)n(y)e(of)j(the)g Fs(P)2766 4723 y Fo(m)2829 4711 y Ft('s:)737 4892 y Fq(h)p Fs(k)s Fq(jS)6 b(j)p Fs(l)r Fq(i)84 b Ft(=)1221 4813 y Fm(X)1207 4992 y Fo(m)p Fn(\025)p Fo(k)1369 4779 y Fm(Z)1415 4968 y Fp(\000)1456 4976 y Fl(R)1547 4836 y Fs(dz)p 1531 4873 119 4 v 1531 4949 a Ft(2)p Fs(\031)s({)1765 4836 y(P)1818 4848 y Fo(l)1844 4836 y Ft(\()p Fs(S)5 b Ft(\()p Fs(z)t Ft(\)\))p 1692 4873 451 4 v 1692 4949 a(\()p Fs(z)22 b Fq(\000)c Fs(E)1929 4961 y Fo(c)1964 4949 y Ft(\))1996 4925 y Fo(m)p Fp(+1)2190 4779 y Fm(Z)2287 4892 y Fs(d\026)p Ft(\()p Fs(E)5 b Ft(\))24 b Fs(P)2587 4904 y Fo(k)2628 4892 y Ft(\()p Fs(E)5 b Ft(\))14 b(\()p Fs(E)24 b Fq(\000)18 b Fs(E)3033 4904 y Fo(c)3067 4892 y Ft(\))3099 4858 y Fo(m)1060 5171 y Fq(\024)1236 5115 y Ft(\()p Fs(R)1331 5127 y Fo(c)1365 5115 y Fs(=R)q Ft(\))1503 5085 y Fo(k)p 1217 5152 346 4 v 1217 5228 a Ft(1)g Fq(\000)g Fs(R)1423 5240 y Fo(c)1457 5228 y Fs(=R)1615 5171 y Ft(max)1609 5225 y Fo(z)r Fn(2)p Fp(\000)1729 5233 y Fl(R)1790 5171 y Fq(j)p Fs(P)1866 5183 y Fo(l)1892 5171 y Ft(\()p Fs(S)5 b Ft(\()p Fs(z)t Ft(\)\))p Fq(j)28 b Ft(.)p eop %%Page: 7 7 7 6 bop 1125 226 a Fp(ANOMALOUS)28 b(TRANSPOR)-5 b(T)29 b(IN)g(APERIODIC)g(MEDIA)635 b(7)456 425 y Ft(No)n(w)22 b Fs(P)693 437 y Fo(l)719 425 y Ft(\()p Fs(S)5 b Ft(\()p Fs(z)t Ft(\)\))23 b(is)g(b)r(ounded)h(b)n(y)f(the)g(Thouless)g(form)n (ula.)34 b(If)24 b Fs(S)j Ft(is)d(a)e(p)r(olynomial)h(of)g(degree)f(2,) 456 525 y(this)28 b(giv)n(es)e(for)h(large)f Fs(R)j Ft(the)f(b)r(ound)g Fq(h)p Fs(k)s Fq(jS)6 b(j)p Fs(l)r Fq(i)23 b(\024)g Ft(min)p Fq(f)p Fs(c)14 b(R)2286 495 y Fn(\000)p Fo(k)q Fp(+2)p Fo(l)2484 525 y Fs(;)g Ft(1)p Fq(g)26 b Ft(\(as)h Fs(R)i Ft(is)e(arbitrary)-7 b(,)26 b(this)456 624 y(actually)32 b(implies)i(the)g(kno)n(wn)e(renormalization)f(prop)r(ert)n(y)i Fs(P)2482 636 y Fo(l)2507 624 y Ft(\()p Fs(S)5 b Ft(\()p Fs(z)t Ft(\)\))33 b(=)f Fs(P)2917 636 y Fp(2)p Fo(l)2976 624 y Ft(\()p Fs(z)t Ft(\))h([)p Fu(BBM)p Ft(]\).)456 724 y(Replacing)e(this)i(in)n(to)e(\(6.2\),)j(a)d(recurrence)g(argumen) n(t)g(o)n(v)n(er)f Fs(n)i Ft(sho)n(ws)f(that)i Fq(h)p Fs(n)p Fq(j)p Fs(\037)3116 740 y Fo(I)3150 724 y Fl(N)3145 757 y(\033)3207 724 y Ft(\()p Fs(H)7 b Ft(\))p Fq(j)p Ft(0)p Fq(i)456 839 y Ft(decreases)34 b(exp)r(onen)n(tionally)g(in)i Fs(n)g Ft(for)f Fs(n)i Fq(\025)f Ft(2)1988 809 y Fo(N)2087 839 y Fq(\031)g Fs(T)2249 809 y Fp(log\(2\))p Fo(=)p Fp(\003)2539 839 y Ft(=)g Fs(T)2701 809 y Fp(dim)2810 823 y Fz(H)2859 809 y Fp(\()p Fo(\026)p Fp(\))2955 839 y Ft(.)62 b(This)35 b(is)h(the)456 939 y(upp)r(er)27 b(b)r(ound)h(b)n(y)g(the)g(fractal)e(dimension.)796 1108 y Fu(7.)47 b(Sub)s(di\013usiv)m(e)31 b(transp)s(ort)h(in)f(higher)g (dimensional)e(mo)s(dels)605 1258 y Ft(A)g(simple)g(w)n(a)n(y)f(to)h (construct)f(higher)h(dimensional)f(mo)r(dels)h(with)g(con)n(trollable) f(trans-)456 1363 y(p)r(ort)e(exp)r(onen)n(ts)g(is)h(to)f(tak)n(e)g(a)h (direct)f(sum)h Fs(H)1942 1333 y Fo(\021)2005 1363 y Ft(=)2093 1301 y Fm(P)2180 1322 y Fo(D)2180 1388 y(j)s Fp(=1)2313 1363 y Fs(\021)2354 1375 y Fo(j)2412 1363 y Fu(1)17 b Fq(\012)f Fs(:)e(:)g(:)i Fq(\012)g Fs(H)2821 1375 y Fo(j)2873 1363 y Fq(\012)g Fs(:)e(:)g(:)i Fq(\012)h Fu(1)26 b Ft(acting)456 1478 y(on)j Fs(`)608 1448 y Fp(2)645 1478 y Ft(\()p Fu(N)752 1448 y Fo(D)812 1478 y Ft(\))h(where)f Fs(\021)g Ft(=)d(\()p Fs(\021)1350 1490 y Fp(1)1387 1478 y Fs(;)14 b(:)g(:)g(:)28 b(;)14 b(\021)1627 1490 y Fo(D)1687 1478 y Ft(\))27 b Fq(2)f Fu(R)1899 1448 y Fo(D)1899 1498 y Fp(+)1988 1478 y Ft(and)k(the)f Fs(H)2365 1490 y Fo(j)2400 1478 y Ft('s)h(are)e(Julia)h(matrices.)42 b(In)29 b(fact)456 1580 y(elemen)n(tary)23 b(inequalities)h(imply)h(that)f Fs(\014)1767 1549 y Fp(+)1763 1600 y Fo(q)1823 1580 y Ft(\()p Fs(H)1931 1549 y Fo(\021)1971 1580 y Ft(\))g(=)e(max)2269 1592 y Fo(j)s Fp(=1)p Fo(:::D)2517 1580 y Fs(\014)2568 1549 y Fp(+)2564 1600 y Fo(q)2623 1580 y Ft(\()p Fs(H)2724 1592 y Fo(j)2760 1580 y Ft(\))i(so)g(that)h(the)g(trans-)456 1679 y(p)r(ort)k(is)h(sub)r(di\013usiv)n(e)h(if)f(it)h(is)f(so)f(for)g (eac)n(h)h(of)f(the)i Fs(H)2166 1691 y Fo(j)2201 1679 y Ft('s.)44 b(The)30 b(sp)r(ectral)f(prop)r(erties)g(can)h(b)r(e)456 1779 y(analysed)c(using)h(results)g(from)h([)p Fu(HT)p Ft(]:)605 1927 y Ff(Theorem)k Ft(7.1)p Ff(.)40 b Ft([)p Fu(BeSB)p Ft(])c Fe(L)l(et)g(the)g(sp)l(e)l(ctr)l(al)h(me)l(asur)l(e)f (of)h Fs(H)2604 1897 y Fo(\021)2680 1927 y Fe(w.r.t.)59 b Fq(j)p Ft(0)p Fq(i)23 b(\012)g Fs(:)14 b(:)g(:)23 b Fq(\012)g(j)p Ft(0)p Fq(i)456 2038 y Fe(b)l(e)34 b Fs(\026)612 2008 y Fo(\021)653 2038 y Fe(.)53 b(Then)35 b(for)g(almost)g(al)t(l)h Fs(\021)f Fq(2)d Fu(R)1718 2008 y Fo(D)1718 2058 y Fp(+)1778 2038 y Fe(,)k Ft(dim)1977 2050 y Fz(H)2026 2038 y Ft(\()p Fs(\026)2108 2008 y Fo(\021)2149 2038 y Ft(\))c(=)f(min)q Fq(f)p Ft(1)p Fs(;)2569 1976 y Fm(P)2655 1996 y Fo(D)2655 2063 y(j)s Fp(=1)2788 2038 y Ft(dim)2926 2050 y Fz(H)2975 2038 y Ft(\()p Fs(\026)3057 2050 y Fo(j)3092 2038 y Ft(\))p Fq(g)k Fe(and,)h(if)456 2102 y Fm(P)543 2122 y Fo(D)543 2189 y(j)s Fp(=1)676 2164 y Ft(dim)814 2176 y Fz(H)863 2164 y Ft(\()p Fs(\026)945 2176 y Fo(j)981 2164 y Ft(\))23 b Fs(>)g Ft(1)p Fe(,)29 b(then)h Fs(\026)1455 2134 y Fo(\021)1525 2164 y Fe(is)g(absolutely)h(c)l(ontinuous.)605 2319 y Ft(Hence)26 b(c)n(ho)r(osing)f(dim)1324 2331 y Fz(H)1373 2319 y Ft(\()p Fs(\026)p Ft(\))f(=)e Fs(D)1667 2331 y Fo(\026)1712 2319 y Ft(\(1\))h Fq(\031)f Ft(1)p Fs(=)p Ft(3,)j(the)i(theorem)e(guaran)n(tees)f(the)j(existence)456 2419 y(of)c(a)h(3)p Fs(D)h Ft(mo)r(del)g(with)f(ac)f(sp)r(ectrum)h(for) g(whic)n(h)g(the)g(di\013usion)g(exp)r(onen)n(t)g Fs(\014)2867 2435 y Fp(0)2900 2418 y Fj(+)2974 2419 y Ft(=)f(1)p Fs(=)p Ft(3)f(is)i(only)456 2519 y(sligh)n(tly)j(bigger)f(than)i(imp)r(osed)f (b)n(y)h(the)g(Guarneri)e(Inequalit)n(y)-7 b(.)605 2666 y(I)21 b(w)n(ould)f(lik)n(e)h(to)f(thank)h(Jean-Marie)e(Barbaroux,)g (Jean)h(Bellissard)g(and)g(Italo)g(Guarneri)456 2765 y(for)27 b(the)h(v)n(ery)e(fruitful)j(collab)r(orations)c(leading)i(to) h(man)n(y)f(of)g(the)h(results)f(presen)n(ted)g(here.)1726 2935 y Fu(References)456 3067 y Fv([BaSB])34 b(J.-M.)23 b(Barbaroux,)g(H.)g(Sc)n(h)n(ulz-Baldes,)h(Annales)g(I.H.P)-6 b(.)23 b(Ph)n(ys.)g(th)n(\023)-33 b(eo.)25 b Fc(71)p Fv(,)d(539-559)j(\(1999\).)456 3150 y([BGT])34 b(J.M.)21 b(Barbaroux,)g(F.)g(Germinet,)g(S.)g(Tc)n(heremc)n(han)n(tsev,)i (C.R.A.S.)d Fc(330)p Fv(,)g(s)n(\023)-33 b(erie)21 b(I,)g(409-414)i (\(2000\).)456 3233 y([BBM])34 b(J.)23 b(Bellissard,)f(D.)h(Bessis,)f (P)-6 b(.)24 b(Moussa,)f(Ph)n(ys.)g(Rev.)h(Lett.)g Fc(49)p Fv(,)f(701-704)i(\(1982\).)456 3316 y([BeSB])35 b(J.)23 b(Bellissard,)f(H.)h(Sc)n(h)n(ulz-Baldes,)h(J.)f(Stat.)i(Ph)n(ys.)e Fc(99)p Fv(,)g(587-594)h(\(2000\).)456 3399 y([Com])33 b(J.-M.)19 b(Com)n(b)r(es,)g(in)h(Di\013eren)n(tial)g(Equations)g(with) g(Applications)h(to)f(Mathematical)g(Ph)n(ysics,)h(Ames)583 3482 y(W.F.,)i(Harell)g(E.M.,)e(Hero)r(d)j(J.V.)f(Eds,)g(Academic)h (Press,)e(Boston)j(\(1993\).)456 3566 y([G])34 b(I.)24 b(Guarneri,)f(Europh)n(ys.)h(Lett.)g Fc(10)p Fv(,)f(95-100)h(\(1989\);) h(Europh)n(ys.)f(Lett.)g Fc(21)p Fv(,)f(729-733)i(\(1993\).)456 3649 y([GM])34 b(I.)24 b(Guarneri,)f(G.)g(Man)n(tica,)h(Ph)n(ys.)g (Rev.)f(Lett.)i Fc(73)p Fv(,)d(3379-3383)k(\(1994\).)456 3732 y([GSB1])35 b(I.)23 b(Guarneri,)h(H.)f(Sc)n(h)n(ulz-Baldes,)g (Rev.)h(Math.)g(Ph)n(ys.)f Fc(11)p Fv(,)g(1249-1268)i(\(1999\).)456 3815 y([GSB2])35 b(I.)23 b(Guarneri,)h(H.)f(Sc)n(h)n(ulz-Baldes,)g (Elect.)h(J.)g(Math.)f(Ph)n(ys.)h Fc(5)p Fv(,)f(\(1999\).)456 3898 y([GSB3])35 b(I.)23 b(Guarneri,)h(H.)f(Sc)n(h)n(ulz-Baldes,)g (Lett.)i(Math.)f(Ph)n(ys.)f(49,)h(317-324)h(\(1999\).)456 3981 y([HT])34 b(X.)23 b(Hu,)g(S.)g(J.)h(T)-6 b(a)n(ylor,)23 b(Math.)h(Pro)r(c.)f(Cam)n(b.)f(Phil.)h(So)r(c.)h Fc(115)p Fv(,)e(527-544)j(\(1994\).)456 4064 y([JL])34 b(S.)24 b(Jitomirsk)l(a)n(y)n(a,)e(Y.)h(Last,)h(Acta)h(Math.)e Fc(183)p Fv(,)f(171-189)j(\(1999\).)456 4147 y([KPG])34 b(R.)23 b(Ketzmeric)n(k,)g(G.)h(P)n(esc)n(hel,)g(and)g(T.)f(Geisel,)g (Ph)n(ys.)h(Rev.)f(Lett.,)h Fc(69)p Fv(,)f(695-698,)h(\(1992\).)456 4230 y([L])43 b(Y.)23 b(Last,)h(J.)f(F)-6 b(unct.)25 b(Anal.)e Fc(142)p Fv(,)f(402-445)j(\(1996\).)456 4313 y([KL])34 b(A.)23 b(Kiselev,)g(Y.)g(Last,)h(preprin)n(t,)f(mp-arc)f (99-206.)i(Duk)n(e)g(???)456 4396 y([Ma)n(y])34 b(D.)23 b(Ma)n(y)n(ou,)h(in)f Fb(Lectures)i(on)f(quasicrystals)p Fv(,)g(\(Les)g(Editions)f(de)i(Ph)n(ysique,)e(Les)h(Ulis,)e(1994\).)456 4479 y([Man])34 b(G.)24 b(Man)n(tica,)g(Ph)n(ysica)g Fc(D)j(103)p Fv(,)22 b(576-589,)j(\(1997\);)g(Ph)n(ysica)f Fc(D)j(109)p Fv(,)22 b(113-127)j(\(1997\).)456 4562 y([P)n(es])34 b(Y.)17 b(P)n(esin,)h Fb(Dimension)e(Theory)h(in)g(Dynamical)g(Systems) p Fv(,)g(\(The)i(Univ)n(ersit)n(y)e(of)g(Chicago)h(Press,)f(1997\).)456 4645 y([Pie])34 b(F.)23 b(Pi)n(\023)-33 b(ec)n(hon,)24 b(Ph)n(ys.)g(Rev.)f(Lett.)i Fc(76)p Fv(,)d(4372-4375)k(\(1996\).)456 4728 y([SBB1])34 b(H.)23 b(Sc)n(h)n(ulz-Baldes,)h(J.)g(Bellissard,)e (Rev.)h(Math.)h(Ph)n(ys.)f Fc(10)p Fv(,)g(1-46)h(\(1998\).)456 4811 y([SBB2])34 b(H.)23 b(Sc)n(h)n(ulz-Baldes,)h(J.)g(Bellissard,)e (J.)h(Stat.)i(Ph)n(ys.)e Fc(91)p Fv(,)f(991-1027)j(\(1998\).)456 4894 y([SG])35 b(C.)23 b(Sire)g(and)h(D.)f(Gratias,)h(in)f(NA)-6 b(TO)23 b(ASI)h(Series)g(B:)f(Ph)n(ysics)g(V)-6 b(ol.)23 b(319,)h(\(Plen)n(um,)f(London,)h(1994\).)456 4977 y([ST])34 b(H.)23 b(Stahl,)h(V.)f(T)-6 b(otik,)24 b Fb(General)g(Orthogonal)h(P)n (olynomials)p Fv(,)d(\(Cam)n(bridge)h(Univ)n(ersit)n(y)g(Press,)g (1992\))605 5133 y Fw(Dep)l(ar)l(tment)h(of)h(Ma)l(thema)l(tics,)f (University)h(of)g(Calif)o(ornia)f(a)l(t)h(Ir)-6 b(vine,)25 b(Ir)-6 b(vine,)25 b(CA,)g(92697)605 5216 y Fi(E-mail)h(addr)l(ess)5 b Fv(:)33 b Fa(schuba@math.uci.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0201031101345--