Content-Type: multipart/mixed; boundary="-------------9907011315729" This is a multi-part message in MIME format. ---------------9907011315729 Content-Type: text/plain; name="99-254.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-254.comments" Corrected version of paper mp-arc 98-391 ---------------9907011315729 Content-Type: text/plain; name="99-254.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-254.keywords" Random Schr\"odinger operators, diffusion exponents, absence of diffusion, dynamical localization, correlated random potentials ---------------9907011315729 Content-Type: application/postscript; name="newdyn.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="newdyn.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: newdyn.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments 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b(Un)o(til)c(the)i(b)q(eginning)f(of)h(the)60 873 y(90's,)g(it)f(w)o(as)h(generally)e(b)q(eliev)o(ed)g(that)i(the)f (o)q(ccurrence)g(of)h(pure-p)q(oin)o(t)f(sp)q(ectrum)g(w)o(as)h(a)60 933 y(su\016cien)o(t)e(criterion)g(for)h(the)g(v)m(anishing)h(of)g(the) f(di\013usion)g(exp)q(onen)o(ts)744 1036 y Fm(\033)774 1016 y Fv(\000)772 1050 y Fy(di\013)840 1036 y Fo(=)e Fm(\033)922 1016 y Fy(+)920 1050 y(di\013)988 1036 y Fo(=)g(0)8 b Fm(:)622 b Fo(\(1\))60 1140 y(Ho)o(w)o(ev)o(er,)19 b(w)o(e)h(no)o(w)g(kno)o(w)h([10,)f(5])g(that)h(pure-p)q(oin)o(t)f(sp)q (ectrum)f(and)i(ev)o(en)e(exp)q(onen)o(tial)60 1200 y(lo)q(calization)f (are)g(not)h(su\016cien)o(t)e(conditions)i(for)f(getting)37 b(\(1\))19 b(or)g(the)f(v)m(anishing)h(of)f(the)60 1260 y(di\013usion)f(constan)o(t,)f(whic)o(h)g(follo)o(ws)g(already)g(from)f Fm(\033)1084 1239 y Fy(+)1082 1274 y(di\013)1150 1260 y Fm(<)f Fo(1.)22 b(The)16 b(study)g(of)h(dynamical)60 1320 y(prop)q(erties)i(requires)f(additional)i(in)o(v)o(estigations.)29 b(This)19 b(w)o(as)h(done)f(recen)o(tly)e(b)o(y)i(Aizen-)60 1380 y(mann)c(and)i(Graf)g([1])f(who)h(pro)o(v)o(ed)e(the)h(dynamical)f (lo)q(calization)h(prop)q(ert)o(y)522 1512 y Fk(E)561 1442 y Fj(\022)601 1512 y Fo(sup)602 1553 y Fx(T)5 b(>)p Fy(1)682 1512 y Fl(hhj)p Fm(X)t Fl(j)792 1491 y Fy(2)813 1512 y Fl(ii)851 1523 y Fx(T)s(;\037)907 1529 y Fs(I)924 1523 y Fy(\()p Fx(H)970 1514 y Fr(\()p Fs(!)q Fr(\))1016 1523 y Fy(\))p Fx( )1056 1442 y Fj(\023)1107 1512 y Fm(<)13 b Fl(1)60 1648 y Fo(for)20 b(a)g(large)f(class)h(of)g(discrete)e (random)i(mo)q(dels,)f(i.e.)30 b(de\014ned)19 b(on)h Fl(H)g Fo(=)f Fm(`)1503 1630 y Fy(2)1524 1648 y Fo(\()p Fk(Z)1579 1630 y Fx(d)1596 1648 y Fo(\).)31 b(More)60 1708 y(recen)o(tly)l(,)18 b(Germinet)g(and)j(De)e(Bi)o(\022)-23 b(evre)18 b([15)q(])h(sho)o(w)o(ed)h(sup)1153 1720 y Fx(T)5 b(>)p Fy(0)1226 1708 y Fl(hj)p Fm(X)t Fl(j)1317 1690 y Fy(2)1337 1708 y Fl(i)1356 1720 y Fx(T)s(;\037)1412 1726 y Fs(I)1429 1720 y Fy(\()p Fx(H)1475 1710 y Fr(\()p Fs(!)q 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Fo(\))f(if)g(w)o(e)g (replace)g(the)g(c)o(haracteristic)f(function)60 2130 y Fm(\037)91 2137 y Fx(I)131 2130 y Fo(b)o(y)20 b(an)o(y)g(smo)q(oth)g (function)h Fm(f)695 2137 y Fx(I)735 2130 y Fo(with)f(compact)g(supp)q (ort)h(in)f Fm(I)k Fo(\(Theorem)19 b(2.1\).)34 b(F)l(or)60 2190 y(Sc)o(hr\177)-24 b(odinger)13 b(op)q(erators)h(on)f Fm(`)620 2172 y Fy(2)641 2190 y Fo(\()p Fk(Z)696 2172 y Fx(d)713 2190 y Fo(\))g(the)g(metho)q(d)f(of)h(pro)q(of)h(can)f(b)q (e)g(strengthened)g(to)g(yield)60 2250 y(dynamical)k(lo)q(calization)i (\(Theorem)f(2.2\).)30 b(By)19 b(relaxing)f(some)h(of)g(the)g (assumptions,)h(a)60 2310 y(result)c(on)g(the)h(absence)f(of)g (di\013usion)h(is)f(also)h(established)f(\(Theorem)e(2.3\).)133 2370 y(Similar)20 b(to)i(the)g(strategy)g(of)g(F)l(r\177)-24 b(ohlic)o(h)21 b(and)i(Sp)q(encer)e([14],)i(who)g(related)e(the)h(con-) 60 2430 y(ductivit)o(y)g(at)i(some)f(\014xed)g(energy)h Fm(E)j Fo(to)d(the)f(b)q(eha)o(vior)h(of)g(the)g(Green's)f(function)h (at)60 2491 y(this)f(energy)l(,)h(it)f(is)f(the)h(strategy)h(of)f(the)g (presen)o(t)g(pap)q(er)g(to)h(deriv)o(e)d(\(1\))j(from)e(appro-)60 2551 y(priate)e(deca)o(y)f(estimates)g(of)h(the)g(Green's)g(function.) 33 b(Ho)o(w)o(ev)o(er,)18 b(in)i(order)g(to)h(calculate)60 2611 y Fk(E)99 2571 y Fj(\000)124 2611 y Fl(hhj)p Fm(X)t Fl(j)234 2593 y Fy(2)255 2611 y Fl(ii)293 2622 y Fx(T)s(;f)344 2628 y Fs(I)361 2622 y Fy(\()p Fx(H)407 2613 y Fr(\()p Fs(!)q Fr(\))453 2622 y Fy(\))p Fx( )493 2571 y Fj(\001)529 2611 y Fo(w)o(e)13 b(need)g(a)h(more)e(re\014ned)h(analysis)h(that)g (tak)o(es)f(in)o(to)g(accoun)o(t)g(the)60 2671 y(dep)q(endence)g(of)h (the)f(Green's)g(function)g(on)h(the)g(energy)f Fm(E)j Fo(and)f(on)f(the)f(realization)g Fm(!)j Fl(2)e Fo(\012)60 2731 y(of)22 b(the)f(random)h(p)q(oten)o(tial.)37 b(This)22 b(can)g(b)q(e)f(done)h(with)g(the)f(help)g(of)h(v)o(on)g(Dreifus)f(and) 903 2856 y(3)p eop %%Page: 4 4 4 3 bop 60 203 a Fo(Klein's)20 b(estimates)f([11])i(on)h(the)f(deca)o 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b(Section)h(2)g(w)o(e)f (presen)o(t)g(the)h(assumptions)60 745 y(needed)e(and)i(w)o(e)f(state)g (our)h(main)e(results.)26 b(In)18 b(Section)g(3)g(w)o(e)g(consider)g (some)f(examples)60 805 y(to)i(illustrate)f(the)g(applicabilit)o(y)e (of)j(our)g(theorems.)27 b(P)o(erio)q(dic)18 b(con)o(tin)o(uous)g(Sc)o (hr\177)-24 b(odinger)60 866 y(op)q(erators)19 b(p)q(erturb)q(ed)e(b)o (y)g(an)g(allo)o(y-t)o(yp)q(e)f(random)h(p)q(oten)o(tial)g(serv)o(e)f (as)i(applications)f(for)60 926 y(b)q(oth)k(Theorem)f(2.3)g(on)h(the)g (absence)f(of)h(di\013usion)g(and)g(for)f(Theorem)g(2.1)g(on)h(v)m (anish-)60 986 y(ing)i(di\013usion)g(exp)q(onen)o(ts)f(at)h(energies)f (near)g(band-edges.)42 b(Example)20 b(2)j(is)g(concerned)60 1046 y(with)e(an)h(application)f(of)h(Theorem)e(2.2)h(and)h (establishes)f(dynamical)f(lo)q(calization)h(for)60 1106 y(a)h(m)o(ulti-dim)o(ensi)o(onal)d(Anderson)j(mo)q(del)e(on)i(the)g (lattice.)35 b(Finally)l(,)21 b(Example)f(3)i(deals)60 1166 y(with)c(a)g(Sc)o(hr\177)-24 b(odinger)17 b(op)q(erator)i(with)f (a)g(correlated)f(allo)o(y-t)o(yp)q(e)g(random)g(p)q(oten)o(tial.)25 b(W)l(e)60 1227 y(sho)o(w)19 b(that)f(the)g(di\013usion)h(exp)q(onen)o (ts)f(of)h(this)f(mo)q(del)f(get)h(smaller)e(and)j(smaller)d(for)j(en-) 60 1287 y(ergies)k(approac)o(hing)h(the)e(b)q(ottom)h(of)h(the)f(sp)q (ectrum.)40 b(F)l(or)23 b(this)g(purp)q(ose)h(w)o(e)f(need)g(a)60 1347 y(\\v)m(ariable-energy")15 b(m)o(ulti-scale)d(analysis)j(that)h (giv)o(es)e(algebraic)g(deca)o(y)g(estimates)g(whic)o(h)60 1407 y(are)g(uniform)e(in)i(energy)l(.)20 b(Details)13 b(of)i(these)e(calculations)g(are)h(deferred)f(to)i(the)e(App)q(endix.) 60 1467 y(Section)j(4)g(is)g(dev)o(oted)g(to)h(the)f(pro)q(ofs)h(of)g (the)f(theorems.)60 1634 y Fn(2)81 b(Main)26 b(results)60 1743 y Fo(W)l(e)18 b(\014rst)h(describ)q(e)e(precisely)g(the)h(quan)o (tities)f(of)h(in)o(terest.)27 b(F)l(or)18 b(the)g(ph)o(ysical)f(relev) m(ance)60 1804 y(of)g(these)e(quan)o(tities,)g(one)i(can)f(refer)f(for) 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b(d)p Fo(.)903 2856 y(4)p eop %%Page: 5 5 5 4 bop 98 203 a Fo(iii\))23 b(The)d(comm)o(utators)e([)p Fm(H)q(;)8 b(X)701 210 y Fx(j)720 203 y Fo(])19 b(:=)h Fm(H)t(X)909 210 y Fx(j)942 203 y Fl(\000)13 b Fm(X)1034 210 y Fx(j)1053 203 y Fm(H)t Fo(,)20 b Fm(j)j Fo(=)e(1)p Fm(;)8 b(:)g(:)g(:)16 b(;)8 b(d)p Fo(,)20 b(are)g(w)o(ell-de\014ned)182 264 y(and)g(relativ)o(ely)d(op)q(erator)j(b)q(ounded)g(with)f(resp)q (ect)g(to)h Fm(H)t Fo(,)g(i.e.)29 b(for)19 b Fk(P)m Fo(-a.e.)c Fm(!)22 b Fl(2)d Fo(\012)182 324 y(there)11 b(exists)h Fm(b)p Fo(\()p Fm(!)r Fo(\))i Fm(<)f Fl(1)f Fo(suc)o(h)g(that)h Fl(k)p Fo([)p Fm(H)942 306 y Fy(\()p Fx(!)q Fy(\))994 324 y Fm(;)8 b(X)1056 331 y Fx(j)1074 324 y Fo(]\()p Fm(H)1151 306 y Fy(\()p Fx(!)q Fy(\))1206 324 y Fo(+)s(i\))1280 306 y Fv(\000)p Fy(1)1327 324 y Fl(k)13 b(\024)h Fm(b)p Fo(\()p Fm(!)r Fo(\).)20 b(W)l(e)11 b(further)182 384 y(assume)k(that)i Fk(E)486 344 y Fj(\000)512 384 y Fm(b)533 366 y Fy(2)553 384 y Fo(\()p Fm(!)r Fo(\))623 344 y Fj(\001)659 384 y Fm(<)d Fl(1)p Fo(.)99 479 y(iv\))24 b Fl(D)q Fo(\()p Fm(H)t Fo(\))12 b Fl(\\)f(D)q Fo(\()p Fl(j)p Fm(X)t Fl(j)p Fo(\))17 b(is)f(dense)g(in)g Fl(H)p Fo(.)113 573 y(v\))24 b(F)l(or)13 b Fm(L)h(>)g Fo(1,)g Fm(z)h Fl(2)f Fk(C)g Fl(n)5 b Fk(R)12 b Fo(and)i Fm(q)r(;)8 b(q)783 555 y Fv(0)807 573 y Fl(2)14 b Fk(Z)890 555 y Fx(d)921 573 y Fo(with)e Fl(k)p Fm(q)6 b Fl(\000)f Fm(q)1149 555 y Fv(0)1160 573 y Fl(k)1185 580 y Fv(1)1236 573 y Fm(>)13 b Fo(2)p Fm(L)h Fo(there)e(is)h(an)h(op)q(erator)182 640 y Fm(D)223 615 y Fy(\()p Fx(!)q Fy(\))222 654 y Fx(L;q)273 645 y Fg(0)286 640 y Fo(\()p Fm(z)r Fo(\))j(de\014ned)f(on)g Fl(D)q Fo(\()p Fm(H)703 622 y Fy(\()p Fx(!)q Fy(\))757 640 y Fo(\))g(suc)o(h)g(that)470 740 y Fh(1)498 747 y Fx(q)515 738 y Fg(0)528 740 y Fo(\()p Fm(H)591 720 y Fy(\()p Fx(!)q Fy(\))655 740 y Fl(\000)11 b Fm(z)r Fo(\))749 720 y Fv(\000)p Fy(1)804 740 y Fh(1)832 747 y Fx(q)865 740 y Fo(=)j Fh(1)945 747 y Fx(q)962 738 y Fg(0)983 740 y Fm(D)1024 715 y Fy(\()p Fx(!)q Fy(\))1023 754 y Fx(L;q)1074 745 y Fg(0)1088 740 y Fo(\()p Fm(z)r Fo(\)\()p Fm(H)1214 720 y Fy(\()p Fx(!)q Fy(\))1278 740 y Fl(\000)c Fm(z)r Fo(\))1371 720 y Fv(\000)p Fy(1)1427 740 y Fh(1)1455 747 y Fx(q)1708 740 y Fo(\(2\))182 826 y(holds)22 b(for)g Fk(P)l Fo(-alm)o(ost)d(ev)o(ery)h Fm(!)r Fo(.)38 b(Here)20 b Fh(1)970 833 y Fx(q)1011 826 y Fo(acts)i(as)g(the)f(m)o (ultiplication)e(op)q(erator)182 886 y(corresp)q(onding)e(to)g(the)f (function)615 997 y Fh(1)643 1004 y Fx(q)662 997 y Fo(\()p Fm(x)p Fo(\))e(:=)807 927 y Fj(\032)865 967 y Fo(1)58 b(if)16 b Fl(k)p Fm(x)11 b Fl(\000)f Fm(q)r Fl(k)1154 974 y Fv(1)1205 967 y Fm(<)k Fo(1)8 b Fm(;)865 1027 y Fo(0)58 b(else)24 b Fm(:)1708 997 y Fo(\(3\))60 1129 y Fh(Remark)16 b(2.1)24 b Fo(F)l(or)12 b(conditions)g(on)h(the)e (random)h(op)q(erator)h Fm(H)j Fo(to)d(ful\014ll)d(i\),)i(see)g([18,)f (7)q(,)g(25)q(].)60 1189 y(Among)g(others,)i(the)f(measurabilit)o(y)d (of)k Fm(H)j Fo(assures)d(that)g(functions)f(of)g(the)g(t)o(yp)q(e)f (\()p Fm(E)s(;)d(!)r Fo(\))14 b Fl(7!)60 1249 y(k)p Fo(\()p Fm(H)148 1231 y Fy(\()p Fx(!)q Fy(\))214 1249 y Fl(\000)f Fm(E)j Fl(\000)d Fo(i)p Fm(")p Fo(\))426 1231 y Fv(\000)p Fy(1)473 1249 y Fm(')p Fl(k)p Fo(,)20 b(whic)o(h)f(are)g(con)o(tin)o (uous)h(in)f Fm(E)s Fo(,)h(are)f(join)o(tly)g(measurable)f(in)h Fm(E)60 1309 y Fo(and)c Fm(!)r Fo(.)21 b(Assumption)13 b(ii\))g(is)i(true)f(for)g(a)h(large)f(class)h(of)f(op)q(erators)i (with)e Fl(C)j Fo(=)d Fm(C)1548 1291 y Fv(1)1544 1322 y Fy(0)1585 1309 y Fo(\()p Fk(R)1643 1291 y Fx(d)1660 1309 y Fo(\),)g(see)60 1370 y(e.g.)24 b([16].)g(Assumption)16 b(iii\))g(naturally)h(o)q(ccurs)h(in)f(the)g(pro)q(of)h(of)g(Lemma)d (4.1)i(and)h(helps)60 1430 y(to)f(guaran)o(tee)f(the)g(\014niteness,)f (for)i(all)e(strictly)g(p)q(ositiv)o(e)h Fm(T)7 b Fo(,)15 b(of)h(the)g(quan)o(tit)o(y)f Fl(hhj)p Fm(X)t Fl(j)1651 1412 y Fy(2)1672 1430 y Fl(ii)1710 1437 y Fx(T)s(; )60 1490 y Fo(giv)o(en)e(b)q(elo)o(w)h(in)f(De\014nition)h(2.2.)20 b(Assumption)13 b(v\))g(assures)i(the)f(existence)e(of)i(a)g(geometric) 60 1550 y(resolv)o(en)o(t)g(equation,)g(whic)o(h)h(is)f(written)h(do)o (wn)g(in)g(an)g(abstract)h(form)e(in)g(Eq.)h(\(2\))g(in)g(order)60 1610 y(to)22 b(suit)f(b)q(oth)i(the)e(discrete)f(and)i(the)g(con)o(tin) o(uous)f(case.)37 b(Concrete)21 b(examples)f(for)h(the)60 1671 y(op)q(erators)j Fm(D)323 1645 y Fy(\()p Fx(!)q Fy(\))322 1685 y Fx(L;q)373 1675 y Fg(0)386 1671 y Fo(\()p Fm(z)r Fo(\),)f(whic)o(h)f(in)f(the)h(sequel)g(will)f(b)q(e)h(supp)q (osed)h(to)g(exhibit)e(some)g(nice)60 1731 y(deca)o(y)15 b(prop)q(erties,)h(can)h(b)q(e)f(found)h(in)f(Section)g(3)g(\(see)g (also)h([8])f(and)g([11)q(]\).)60 1830 y Fh(De\014nition)i(2.1)24 b Fo(Let)13 b Fm(\032)h Fo(:)f Fk(R)607 1837 y Fy(+)647 1830 y Fl(!)h Fk(R)750 1837 y Fy(+)790 1830 y Fo(b)q(e)f(a)g (non-negativ)o(e)g(function.)20 b(The)13 b(op)q(erator)h Fm(H)1717 1812 y Fy(\()p Fx(!)q Fy(\))60 1890 y Fo(is)i(said)h(to)f(b)q (e)h(\()p Fm(\032;)8 b(E)s(;)g(L;)g(q)r Fo(\)-regular,)15 b(if)564 1982 y(sup)570 2023 y Fx(")p Fv(6)p Fy(=0)645 1982 y Fl(k)8 b Fh(1)707 1989 y Fx(q)734 1982 y Fm(D)775 1957 y Fy(\()p Fx(!)q Fy(\))774 1996 y Fx(L;q)828 1982 y Fo(\()p Fm(E)15 b Fo(+)c(i)p Fm(")p Fo(\))p Fl(k)i(\024)g Fm(\032)p Fo(\()p Fm(L)p Fo(\))1189 1962 y Fy(1)p Fx(=)p Fy(2)1253 1982 y Fm(:)60 2099 y Fh(Multi-Scale)k(Assumption)h(1)51 2179 y Fo(\(M1\))24 b(F)l(or)d Fm(H)26 b Fo(satisfying)21 b(Assumptions)f(2.1.i\))g(and)i(v\),)g(w)o(e)e(supp)q(ose)j(that)e (there)g(exist)182 2239 y Fm(L)215 2246 y Fy(0)257 2239 y Fm(>)h Fo(1,)g Fm(\013)h(>)f Fo(1,)g Fm(\027)k(>)c Fo(0,)g Fm(p)h(>)f(\013)p Fo(\()p Fm(d)15 b Fo(+)f(2\),)23 b(a)e(b)q(ounded)h(in)o(terv)m(al)e Fm(I)26 b Fl(\032)c Fk(R)c Fo(and)k(a)182 2300 y(non-negativ)o(e)16 b(function)g Fm(\032)e Fo(:)g Fk(R)766 2307 y Fy(+)806 2300 y Fl(!)g Fk(R)909 2307 y Fy(+)951 2300 y Fo(suc)o(h)i(that)h(for)g Fm(L)1275 2307 y Fx(k)1310 2300 y Fo(:=)c Fm(L)1414 2282 y Fx(\013)1437 2270 y Fs(k)1408 2312 y Fy(0)1474 2300 y Fo(one)k(has)745 2385 y Fm(\032)p Fo(\()p Fm(L)822 2392 y Fx(k)844 2385 y Fo(\))c Fl(\024)h Fo(exp)p Fl(f\000)p Fo(2)p Fm(L)1131 2364 y Fx(\027)1125 2397 y(k)1152 2385 y Fl(g)i Fm(;)182 2470 y Fo(and)223 2568 y Fk(P)257 2527 y Fj(\010)283 2568 y Fm(!)i Fl(j)e Fm(H)405 2547 y Fy(\()p Fx(!)q Fy(\))475 2568 y Fo(is)g(\()p Fm(\032;)8 b(E)s(;)g(L)684 2575 y Fx(k)705 2568 y Fm(;)g(q)751 2547 y Fv(0)o(0)772 2568 y Fo(\))j Fl(\000)f Fo(regular)17 b(for)f(all)g Fm(E)h Fl(2)d Fm(I)721 2646 y Fo(and)j(for)f(either)f Fm(q)1052 2626 y Fv(0)o(0)1087 2646 y Fo(=)f Fm(q)j Fo(or)g Fm(q)1262 2626 y Fv(0)o(0)1296 2646 y Fo(=)d Fm(q)1372 2626 y Fv(0)1383 2606 y Fj(\011)1426 2646 y Fl(\025)g Fo(1)d Fl(\000)g Fm(L)1597 2623 y Fv(\000)p Fx(p)1597 2660 y(k)1644 2646 y Fm(;)50 b Fo(\(4\))182 2731 y(for)16 b(all)g Fm(k)g Fl(2)e Fk(N)j Fo(and)g(all)f Fm(q)r(;)8 b(q)697 2713 y Fv(0)721 2731 y Fl(2)14 b Fk(Z)804 2713 y Fx(d)838 2731 y Fo(with)i Fl(k)p Fm(q)c Fl(\000)f Fm(q)1082 2713 y Fv(0)1093 2731 y Fl(k)1118 2738 y Fv(1)1169 2731 y Fm(>)j Fo(2)p Fm(L)1278 2738 y Fx(k)1300 2731 y Fo(.)903 2856 y(5)p eop %%Page: 6 6 6 5 bop 133 203 a Fo(Losely)15 b(sp)q(eaking,)g(\(M1\))f(assumes)h(a)f (deca)o(y)g(with)h(go)q(o)q(d)h(probabilit)o(y)e(of)h(the)f(lo)q (calized)60 264 y(resolv)o(en)o(ts)j(\(see)i(the)f(examples)e(in)j (Section)f(3\).)28 b(Suc)o(h)18 b(a)h(deca)o(y)f(can)h(b)q(e)f(sho)o (wn)i(b)o(y)e(p)q(er-)60 324 y(forming)e(a)h(\\v)m(ariable-energy")h(m) o(ulti-scale)c(analysis)j(in)g(the)g(spirit)f(of)i(v)o(on)f(Dreifus)f (and)60 384 y(Klein)g(\(see)h(e.g.)25 b([11)q(,)17 b(20,)h(15]\).)25 b(F)l(or)18 b(the)f(w)o(eak)o(er)f(Theorem)h(2.3)h(to)f(hold,)h(w)o(e)f (only)g(need)60 444 y(a)f(w)o(eak)o(er)f(form)g(of)h(\(M1\),)g(in)f (that)i(the)e(set)h(of)g(ev)o(en)o(ts)f(giving)g(rise)h(to)g(deca)o(y)f (need)h(not)g(b)q(e)60 504 y(uniform)f(in)h(energy)l(.)60 589 y Fh(Multi-Scale)h(Assumption)h(2)51 691 y Fo(\(M2\))24 b(F)l(or)f Fm(H)j Fo(satisfying)d(Assumptions)e(2.1.i\))h(and)h(v\),)g (w)o(e)f(assume)g(that)h(there)f(exist)182 751 y Fm(L)215 758 y Fy(0)257 751 y Fm(>)h Fo(1,)g Fm(\013)g(>)f Fo(1,)h Fm(\014)i(>)d Fo(0,)h Fm(m)f(>)h Fo(\(3)15 b(+)f Fm(d)h Fo(+)g Fm(\014)s Fo(\),)21 b Fm(p)i(>)g Fo(3)15 b(+)g Fm(d)f Fo(+)h Fm(\014)24 b Fo(suc)o(h)d(that)h(for)182 812 y Fm(\032)p Fo(\()p Fm(x)p Fo(\))14 b(:=)f Fm(x)380 793 y Fv(\000)p Fx(m)441 812 y Fo(,)j Fm(L)504 819 y Fx(k)539 812 y Fo(:=)e Fm(L)638 819 y Fy(0)658 793 y Fx(\013)681 782 y Fs(k)718 812 y Fo(and)j(for)g(all)f Fm(E)h Fl(2)d Fm(I)20 b Fo(and)d(all)f Fm(q)f Fl(2)f Fk(Z)1382 793 y Fx(d)1399 812 y Fo(,)i Fl(k)p Fm(q)r Fl(k)1503 819 y Fv(1)1554 812 y Fm(>)e Fo(2)p Fm(L)1663 819 y Fx(k)1684 812 y Fo(,)i(w)o(e)182 872 y(ha)o(v)o(e)407 982 y Fk(P)448 941 y Fj(\010)475 982 y Fm(!)i Fl(j)e Fm(H)597 961 y Fy(\()p Fx(!)q Fy(\))666 982 y Fo(is)g(\()p Fm(\032;)8 b(E)s(;)g(L)875 989 y Fx(k)897 982 y Fm(;)g(q)r Fo(\))i Fl(\000)h Fo(regular)1171 941 y Fj(\011)1214 982 y Fl(\025)i Fo(1)f Fl(\000)f Fm(L)1385 958 y Fv(\000)p Fx(p)1385 996 y(k)1448 982 y Fm(:)246 b Fo(\(5\))60 1112 y Fh(De\014nition)18 b(2.2)24 b Fo(T)l(ak)o(e)d Fm( )k Fl(2)f(H)f Fo(suc)o(h)e(that)i(for)f(all)f Fm(t)j(>)f Fo(0,)h(e)1287 1094 y Fv(\000)p Fy(i)p Fx(tH)1370 1112 y Fm( )h Fl(2)f(D)q Fo(\()p Fl(j)p Fm(X)t Fl(j)p Fo(\).)39 b(The)60 1173 y(Cesaro)17 b(mean)e(up)i(to)f(time)e Fm(T)21 b(>)13 b Fo(0)k(of)g(the)f(second)g(momen)o(t)d(of)k Fm( )h Fo(is)e(de\014ned)g(as)511 1314 y Fl(hhj)p Fm(X)t Fl(j)621 1293 y Fy(2)641 1314 y Fl(ii)679 1321 y Fx(T)s(; )753 1314 y Fo(:=)829 1280 y(1)p 824 1302 36 2 v 824 1348 a Fm(T)872 1246 y Fj(Z)922 1259 y Fx(T)900 1359 y Fy(0)958 1314 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(e)1085 1293 y Fv(\000)p Fy(i)p Fx(tH)1168 1314 y Fm( )r Fl(k)1227 1293 y Fy(2)1247 1314 y Fm(dt)16 b(;)60 1450 y Fo(where)21 b(i)g(is)g(the)f(imaginary)g (unit.)36 b(By)20 b(analogy)j(with)e([1])f(and)i([28])f(w)o(e)g(sa)o(y) g(that)h Fm( )g Fo(is)60 1510 y(dynamically)14 b(lo)q(calized)h(if)630 1646 y Fk(E)668 1576 y Fj(\022)708 1646 y Fo(sup)709 1687 y Fx(T)5 b(>)p Fy(1)790 1646 y Fl(hhj)p Fm(X)t Fl(j)900 1625 y Fy(2)920 1646 y Fl(ii)958 1653 y Fx(T)s(; )1018 1576 y Fj(\023)1069 1646 y Fm(<)14 b Fl(1)i Fm(;)60 1782 y Fo(where)c Fk(E)20 b Fo(is)12 b(the)g(exp)q(ectation)g(asso)q(ciated) h(with)f Fk(P)m Fo(.)k(The)d(di\013usion)g(exp)q(onen)o(ts)f(are)g (de\014ned)60 1843 y(as)17 b(in)f([28])478 1982 y Fm(\033)508 1962 y Fy(+)506 1996 y(di\013)560 1982 y Fo(\()p Fm( )r Fo(\))41 b(:=)g(lim)8 b(sup)793 2023 y Fx(T)d Fv(!1)929 1948 y Fo(log)k Fk(E)k Fo(\()s Fl(hhj)p Fm(X)t Fl(j)1171 1930 y Fy(2)1192 1948 y Fl(ii)1230 1955 y Fx(T)s(; )1290 1948 y Fo(\))p 929 1971 380 2 v 1066 2016 a(log)c Fm(T)1330 1982 y(;)478 2118 y(\033)508 2097 y Fv(\000)506 2131 y Fy(di\013)560 2118 y Fo(\()p Fm( )r Fo(\))41 b(:=)g(lim)8 b(inf)786 2149 y Fx(T)d Fv(!1)915 2084 y Fo(log)k Fk(E)k Fo(\()s Fl(hhj)p Fm(X)t Fl(j)1157 2066 y Fy(2)1177 2084 y Fl(ii)1215 2091 y Fx(T)s(; )1275 2084 y Fo(\))p 915 2106 V 1051 2152 a(log)d Fm(T)1316 2118 y(:)60 2250 y Fo(When)16 b(the)g(limit)e(exists,)h(the)h(di\013usion)h(constan)o(t)f (is)g(giv)o(en)g(b)o(y)325 2386 y Fm(D)q Fo(\()p Fm( )r Fo(\))f(:=)f(lim)518 2416 y Fx(")p Fv(!)p Fy(0)595 2386 y Fk(E)634 2316 y Fj(\022)674 2386 y Fm(")697 2365 y Fy(2)724 2318 y Fj(Z)752 2431 y Fi(R)787 2386 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(\()p Fm(H)955 2365 y Fy(\()p Fx(!)q Fy(\))1019 2386 y Fl(\000)j Fm(E)j Fl(\000)c Fo(i)p Fm(")p Fo(\))1224 2365 y Fv(\000)p Fy(1)1271 2386 y Fm( )r Fl(k)1330 2365 y Fy(2)1365 2386 y Fm(dE)1429 2316 y Fj(\023)1491 2386 y Fm(:)60 2528 y Fh(Remark)16 b(2.2)24 b Fo(If)17 b(the)f(upp)q(er)h(di\013usion)g(exp)q(onen)o(t)g(ob)q(eys)g Fm(\033)1216 2507 y Fy(+)1214 2542 y(di\013)1268 2528 y Fo(\()p Fm( )r Fo(\))d Fm(<)h Fo(1,)i(then)f(the)h(di\013u-)60 2588 y(sion)g(constan)o(t)f(v)m(anishes,)g Fm(D)q Fo(\()p Fm( )r Fo(\))f(=)e(0.)133 2681 y(The)j(main)f(results)h(of)h(this)f (pap)q(er)h(are)f(summarized)d(in)j(the)g(follo)o(wing)g(theorems.)903 2856 y(6)p eop %%Page: 7 7 7 6 bop 60 203 a Fh(Theorem)17 b(2.1)24 b Ff(\(V)l(anishing)18 b(di\013usion)g(exp)n(onents\).)60 264 y(Consider)i(a)h(r)n(andom)e(op) n(er)n(ator)g Fm(H)24 b Ff(satisfying)d(Assumptions)g(2.1)f(and)g(the)h (Multi-Sc)n(ale)60 324 y(Assumption)d(\(M1\).)j(L)n(et)c Fm(I)583 306 y Fv(0)612 324 y Ff(b)n(e)g(any)h(c)n(omp)n(act)f(subset)h (of)f Fm(I)k Ff(such)d(that)f Fo(dist\()p Fm(@)s(I)t(;)8 b(@)s(I)1637 306 y Fv(0)1646 324 y Fo(\))14 b Fm(>)g Fo(0)p Ff(.)60 384 y(Then)k(for)f(al)r(l)i(c)n(omp)n(actly)e(supp)n (orte)n(d)f Fm(')e Fl(2)g(D)q Fo(\()p Fm(H)t Fo(\))k Ff(and)g(al)r(l)h Fm(f)1197 391 y Fx(I)1215 382 y Fg(0)1242 384 y Fl(2)14 b Fm(C)1328 366 y Fv(1)1324 396 y Fy(0)1365 384 y Fo(\()p Fm(I)1410 366 y Fv(0)1421 384 y Fo(\))j Ff(one)h(has)710 489 y Fm(\033)740 469 y Fv(\006)738 503 y Fy(di\013)793 449 y Fj(\000)816 489 y Fm(f)840 496 y Fx(I)858 487 y Fg(0)871 489 y Fo(\()p Fm(H)t Fo(\))p Fm(')985 449 y Fj(\001)1022 489 y Fo(=)13 b(0)8 b Fm(:)60 595 y Ff(This)17 b(implies)h(in)g(p)n(articular)f(that)g(the)h (di\013usion)g(c)n(onstant)g(is)f(zer)n(o,)g Fm(D)1408 554 y Fj(\000)1432 595 y Fm(f)1456 602 y Fx(I)1474 593 y Fg(0)1487 595 y Fo(\()p Fm(H)t Fo(\))p Fm(')1601 554 y Fj(\001)1638 595 y Fo(=)c(0)p Ff(.)60 703 y Fh(Remark)j(2.3)24 b Fo(If)14 b(the)h(deca)o(y)e(of)i(the)f(function)g Fm(\032)h Fo(is)f(only)g(algebraic,)g(one)h(can)f(still)g(obtain)60 763 y(an)i(estimate)e(for)i(the)f(di\013usion)i(exp)q(onen)o(ts.)k (Namely)13 b(if)i(there)g(exists)g Fm(n)f(>)g(\013)p Fo(\()p Fm(d)d Fo(+)e(2\))16 b(and)60 823 y Fm(c)p Fo(\()p Fm(n)p Fo(\))e Fm(<)g Fl(1)i Fo(suc)o(h)g(that)611 883 y Fl(8)p Fm(k)e Fl(2)g Fk(N)p Fm(;)42 b(\032)p Fo(\()p Fm(L)894 890 y Fx(k)915 883 y Fo(\))14 b Fm(<)g(c)p Fo(\()p Fm(n)p Fo(\))p Fm(L)1121 863 y Fv(\000)p Fy(2)p Fx(n)1121 897 y(k)1206 883 y Fm(;)60 968 y Fo(then)i(w)o(e)g(get)g(according)h (to)f(Remark)f(4.3.ii\))450 1101 y Fm(\033)480 1081 y Fv(\000)478 1115 y Fy(di\013)532 1061 y Fj(\000)555 1101 y Fm(f)579 1108 y Fx(I)597 1099 y Fg(0)610 1101 y Fo(\()p Fm(H)t Fo(\))p Fm(')724 1061 y Fj(\001)761 1101 y Fl(\024)f Fm(\033)844 1081 y Fy(+)842 1115 y(di\013)896 1061 y Fj(\000)919 1101 y Fm(f)943 1108 y Fx(I)961 1099 y Fg(0)974 1101 y Fo(\()p Fm(H)t Fo(\))p Fm(')1088 1061 y Fj(\001)1125 1101 y Fl(\024)1183 1068 y Fm(\013)p Fo(\()p Fm(d)d Fo(+)g(2\))p 1183 1090 180 2 v 1257 1135 a Fm(n)1367 1101 y(:)133 1219 y Fo(F)l(or)17 b(discrete)e(random)g(Sc)o(hr\177)-24 b(odinger)16 b(op)q(erators)i(a)f(stronger)g(result)e(is)h(stated)h(in) 60 1327 y Fh(Theorem)g(2.2)24 b Ff(\(Dynamic)n(al)17 b(lo)n(c)n(alization\).)60 1388 y(Supp)n(ose)22 b(that)g(the)h (assumptions)f(of)f(The)n(or)n(em)g(2.1)h(ar)n(e)f(ful\014l)r(le)n(d)k (and)d(that)g(the)g(Hilb)n(ert)60 1448 y(sp)n(ac)n(e)17 b(is)g Fl(H)e Fo(=)e Fm(`)367 1430 y Fy(2)388 1448 y Fo(\()p Fk(Z)443 1430 y Fx(d)460 1448 y Fo(\))p Ff(.)22 b(Then)c(one)h(has)e(dynamic)n(al)g(lo)n(c)n(alization)581 1582 y Fk(E)620 1511 y Fj(\022)659 1582 y Fo(sup)661 1622 y Fx(T)5 b(>)p Fy(1)741 1582 y Fl(hhj)p Fm(X)t Fl(j)851 1561 y Fy(2)872 1582 y Fl(ii)910 1589 y Fx(T)s(;f)961 1599 y Fs(I)977 1592 y Fg(0)989 1589 y Fy(\()p Fx(H)s Fy(\))p Fx( )1075 1511 y Fj(\023)1125 1582 y Fm(<)14 b Fl(1)8 b Fm(:)459 b Fo(\(6\))133 1716 y(In)21 b(case)f(one)h(can)g (only)g(establish)f(the)h(w)o(eak)o(er)e(Multi-Scale)h(Assumption)f (\(M2\),)j(a)60 1776 y(w)o(eak)o(er)15 b(dynamical)g(prop)q(ert)o(y)h (can)g(still)f(b)q(e)i(sho)o(wn.)60 1884 y Fh(Theorem)g(2.3)24 b Ff(\(A)o(bsenc)n(e)18 b(of)g(di\013usion\).)60 1945 y(L)n(et)24 b Fm(H)k Ff(b)n(e)d(a)f(r)n(andom)g(op)n(er)n(ator)e (satisfying)j(Assumptions)f(2.1)g(and)h(the)g(Multi-Sc)n(ale-)60 2005 y(Assumption)19 b(\(M2\),)f(and)g(let)h Fm(I)661 1987 y Fv(0)691 2005 y Ff(b)n(e)f(as)h(ab)n(ove.)25 b(Then)19 b(for)f(al)r(l)h(c)n(omp)n(actly)f(supp)n(orte)n(d)f Fm(')f Fl(2)60 2065 y(D)q Fo(\()p Fm(H)t Fo(\))i Ff(and)g(al)r(l)h Fm(f)388 2072 y Fx(I)406 2063 y Fg(0)433 2065 y Fl(2)14 b Fm(C)519 2047 y Fv(1)515 2077 y Fy(0)556 2065 y Fo(\()p Fm(I)601 2047 y Fv(0)612 2065 y Fo(\))p Ff(,)j(we)h(have)706 2170 y Fm(\033)736 2150 y Fy(+)734 2184 y(di\013)788 2130 y Fj(\000)811 2170 y Fm(f)835 2177 y Fx(I)853 2168 y Fg(0)866 2170 y Fo(\()p Fm(H)t Fo(\))p Fm(')980 2130 y Fj(\001)1017 2170 y Fm(<)c Fo(1)k Fm(:)60 2276 y Ff(This)f(implies)h (in)g(p)n(articular)f(that)g(the)h(di\013usion)g(c)n(onstant)g (vanishes,)g Fm(D)1447 2235 y Fj(\000)1471 2276 y Fm(f)1495 2283 y Fx(I)1513 2274 y Fg(0)1526 2276 y Fo(\()p Fm(H)t Fo(\))p Fm(')1640 2235 y Fj(\001)1677 2276 y Fo(=)c(0)p Ff(.)60 2441 y Fn(3)81 b(Applications)60 2551 y Fo(W)l(e)18 b(presen)o(t)f(here)h(some)f(examples)f(of)i(random)g(Sc)o(hr\177)-24 b(odinger)17 b(op)q(erators)j(for)e(whic)o(h)f(the)60 2611 y(assumptions)12 b(of)g(one)h(of)f(the)g(ab)q(o)o(v)o(e)g (theorems)e(are)j(ful\014lled.)18 b(The)12 b(\014rst)g(example)e (concerns)60 2671 y(b)q(oth)k(absence)e(of)h(di\013usion)g(and)h(v)m (anishing)f(di\013usion)g(exp)q(onen)o(ts)g(near)f(band)i(edges.)20 b(The)60 2731 y(second)g(example)d(concerns)j(dynamical)d(lo)q (calization)i(and)h(the)g(third)f(one)h(smaller)d(and)903 2856 y(7)p eop %%Page: 8 8 8 7 bop 60 203 a Fo(smaller)12 b(di\013usion)i(exp)q(onen)o(ts)g(for)g (energies)f(approac)o(hing)i(the)e(b)q(ottom)h(of)g(the)g(sp)q(ectrum.) 60 264 y(Our)d(aim)f(is)i(to)g(use,)f(as)i(m)o(uc)o(h)c(as)j(p)q (ossible,)g(the)f(sp)q(ectral)h(results)f(already)g(kno)o(wn)h(for)f (these)60 324 y(mo)q(dels)18 b(in)g(order)h(to)g(pro)o(v)o(e)f(either)g (\(M1\))h(or)g(\(M2\),)g(th)o(us)g(sho)o(wing)h(that)f(our)g(theorems) 60 384 y(can)d(easily)g(b)q(e)g(applied)g(to)h(a)f(v)o(ery)f(large)h (class)h(of)f(random)g(Sc)o(hr\177)-24 b(odinger)16 b(op)q(erators.)133 444 y(W)l(e)i(will)g(adopt)h(the)f(follo)o(wing)g(notations:)27 b(F)l(or)19 b Fm(L)f(>)f Fo(0)i(and)g Fm(x)e Fl(2)h Fk(R)1457 426 y Fx(d)1474 444 y Fo(,)h(\003)1541 451 y Fx(L)1567 444 y Fo(\()p Fm(x)p Fo(\))f(is)g(the)60 504 y(cub)q(e)12 b Fl(f)p Fm(y)j Fl(2)f Fk(R)321 486 y Fx(d)350 504 y Fl(j)e(k)p Fm(x)r Fl(\000)r Fm(y)r Fl(k)523 511 y Fv(1)572 504 y Fl(\024)i Fm(L)p Fl(g)e Fo(and)g Fm(\037)816 512 y Fy(\003)840 518 y Fs(L)863 512 y Fy(\()p Fx(x)p Fy(\))924 504 y Fo(denotes)g(a)g(smo)q(oth)g(c)o(haracteristic)e(function)60 565 y(of)17 b(\003)150 572 y Fx(L)175 565 y Fo(\()p Fm(x)p Fo(\),)f(i.e.)k Fm(\037)386 572 y Fy(\003)410 578 y Fs(L)433 572 y Fy(\()p Fx(x)p Fy(\))496 565 y Fl(2)14 b Fm(C)582 546 y Fy(2)602 565 y Fo(\()p Fk(R)660 546 y Fx(d)677 565 y Fo(\))i(and)h(for)f(some)g(\014xed)f Fm(\016)h(>)d Fo(0,)408 748 y Fm(\037)439 756 y Fy(\003)463 762 y Fs(L)486 756 y Fy(\()p Fx(x)p Fy(\))549 748 y Fo(=)601 631 y Fj(8)601 676 y(>)601 691 y(<)601 780 y(>)601 795 y(:)645 678 y Fo(1)186 b(if)35 b Fl(k)p Fm(x)11 b Fl(\000)g Fm(y)r Fl(k)1084 685 y Fv(1)1134 678 y Fl(\024)j Fm(L)d Fl(\000)g Fm(\016)18 b(;)645 750 y Fo(0)186 b(if)35 b Fl(k)p Fm(x)11 b Fl(\000)g Fm(y)r Fl(k)1084 757 y Fv(1)1134 750 y Fl(\025)j Fm(L)i(;)645 822 y Fl(2)e Fo([0)p Fm(;)8 b Fo(1])65 b(elsewhere)15 b Fm(:)60 925 y Fh(Example)e(1:)23 b(Random)15 b(p)r(erturbations)g(of) h(p)r(erio)r(dic)e(con)n(tin)n(uous)i(Sc)n(hr\177)-28 b(odinger)60 985 y(op)r(erators)133 1045 y Fo(W)l(e)19 b(consider)h(a)g(sp)q(eci\014c)f(case)g(of)h(the)g(random)f(Sc)o (hr\177)-24 b(odinger)19 b(op)q(erators)i(studied)e(in)60 1105 y([4],)509 1166 y Fm(H)553 1145 y Fy(\()p Fx(!)q Fy(\))620 1166 y Fo(=)14 b Fl(\000)p Fo(\001)c(+)h Fm(V)839 1173 y Fy(p)q(er)903 1166 y Fo(+)g Fm(V)991 1145 y Fy(\()p Fx(!)q Fy(\))1076 1166 y Fo(on)17 b Fm(L)1177 1145 y Fy(2)1197 1166 y Fo(\()p Fk(R)1255 1145 y Fx(d)1272 1166 y Fo(\))f Fm(;)60 1249 y Fo(where)c Fm(V)225 1256 y Fy(p)q(er)289 1249 y Fo(is)g(a)g(b)q(ounded)h(p)q(erio)q(dic)e(p)q(oten)o(tial)h(suc) o(h)g(that)g Fl(\000)p Fo(\001)r(+)r Fm(V)1306 1256 y Fy(p)q(er)1370 1249 y Fo(has)h(a)f(gap)h(\()p Fm(B)1633 1256 y Fv(\000)1663 1249 y Fm(;)8 b(B)1722 1256 y Fy(+)1751 1249 y Fo(\))60 1310 y(in)16 b(its)g(sp)q(ectrum)f(and)i Fm(V)532 1292 y Fy(\()p Fx(!)q Fy(\))599 1310 y Fo(=)c Fm(g)684 1272 y Fj(P)736 1324 y Fx(i)p Fv(2)p Fi(Z)797 1315 y Fs(d)822 1310 y Fm(\025)850 1317 y Fx(i)865 1310 y Fo(\()p Fm(!)r Fo(\))p Fm(u)p Fo(\()p Fm(x)e Fl(\000)f Fm(i)p Fo(\))16 b(with)59 1415 y(\(H1\))24 b(The)18 b(p)q(oten)o(tial)g Fm(u)g Fo(is)f(p)q(ositiv)o(e,)h(with)g(compact)f(supp)q(ort,)i(suc)o (h)f(that)g Fm(u)p Fo(\()p Fm(x)p Fo(\))e Fl(\025)h Fo(1)i(on)182 1475 y([)p Fl(\000)p Fo(1)p Fm(=)p Fo(2)p Fm(;)8 b Fo(1)p Fm(=)p Fo(2])415 1457 y Fx(d)436 1475 y Fo(,)16 b Fl(k)p Fm(u)p Fl(k)544 1482 y Fv(1)595 1475 y Fm(<)e Fl(1)p Fo(.)59 1575 y(\(H2\))24 b(\()p Fm(\025)229 1582 y Fx(i)243 1575 y Fo(\))262 1585 y Fx(i)p Fv(2)p Fi(Z)323 1575 y Fs(d)c Fo(is)f(a)h(stationary)g(pro)q(cess)g(of)f(indep)q(enden)o(t)f (and)i(iden)o(tically)d(distributed)182 1635 y(random)f(v)m(ariables.) 22 b(W)l(e)17 b(assume)f(that)h(the)f(probabilit)o(y)g(distribution)g (of)h Fm(\025)1627 1642 y Fx(i)1658 1635 y Fo(has)h(a)182 1695 y(Leb)q(esgue)c(densit)o(y)f Fm(h)h Fl(2)g Fm(C)684 1677 y Fy(0)703 1695 y Fo(\()p Fk(R)p Fo(\),)c(with)k(compact)e(supp)q (ort)j([)p Fl(\000)p Fm(M)r(;)8 b(M)d Fo(],)13 b(and)h(for)g(some)182 1756 y Fm(\027)j(>)d Fo(0,)i Fk(P)l Fl(fj)11 b Fm(\025)h Fl(\006)e Fm(M)20 b Fl(j)p Fm(<)13 b(")p Fl(g)h(\024)f Fm(")770 1738 y Fx(d)p Fy(+2+)p Fx(\027)882 1756 y Fo(.)59 1855 y(\(H3\))24 b(The)31 b(coupling)g(constan)o(t)g Fm(g)j Fo(satis\014es)d Fm(g)41 b(<)e Fo(\()p Fm(B)1166 1862 y Fy(+)1217 1855 y Fl(\000)21 b Fm(B)1314 1862 y Fv(\000)1343 1855 y Fo(\))p Fm(=)p Fo(\(2)p Fm(M)5 b(u)1509 1862 y Fy(max)1578 1855 y Fo(\),)34 b(where)182 1915 y Fm(u)210 1922 y Fy(max)291 1915 y Fo(=)14 b Fl(k)376 1878 y Fj(P)437 1915 y Fm(u)p Fo(\()p Fm(x)d Fl(\000)f Fm(i)p Fo(\))p Fl(k)633 1922 y Fv(1)670 1915 y Fo(.)60 2021 y(It)22 b(is)h(pro)o(v)o(en)f(in)g([4])h(that)g(there)f(exist)g (non-empt)o(y)f(compact)h(subsets)h(of)g Fk(R)p Fo(,)e Fm(I)1628 2028 y Fy(+)1682 2021 y Fl(6)p Fo(=)k Fl(;)60 2081 y Fo(and)f Fm(I)184 2088 y Fv(\000)239 2081 y Fl(6)p 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Fg(0)690 1315 y Fv(62)p Fy(\003)p Fx(;)p Fv(k)p Fx(u)p Fv(\000)p Fx(u)833 1305 y Fg(0)845 1315 y Fv(k)863 1319 y Fg(1)895 1315 y Fy(=1)940 1254 y Fl(h)p Fh(1)987 1261 y Fx(u)1007 1252 y Fg(0)1021 1254 y Fm(;)j(')p Fl(i)p Fo(\()p Fm(H)1165 1229 y Fy(\()p Fx(!)q Fy(\))1161 1270 y(\003)1185 1276 y Fs(L)1210 1270 y Fy(\()p Fx(q)1241 1260 y Fg(0)1252 1270 y Fy(\))1278 1254 y Fl(\000)11 b Fm(z)r Fo(\))1372 1233 y Fv(\000)p Fy(1)1419 1254 y Fh(1)1447 1261 y Fx(u)1470 1254 y Fo(\()p Fm(x)p Fo(\))16 b Fm(:)133 1412 y Fo(Moreo)o(v)o(er,)d(thanks)i(to)g([11,)f(Theorem)f (2.2)i(and)f(Prop)q(osition)i(A.11],)e(there)f(exists)h(0)g Fm(<)60 1472 y(E)96 1479 y Fy(0)130 1472 y Fm(<)f Fl(1)g Fo(suc)o(h)g(that)g(for)g Fm(I)18 b Fo(=)13 b(\()p Fl(\0001)p Fm(;)8 b Fl(\000)p Fm(E)820 1479 y Fy(0)839 1472 y Fo(])t Fl([)t Fo([)p Fm(E)944 1479 y Fy(0)964 1472 y Fm(;)g Fo(+)p Fl(1)p Fo(\))13 b(and)g(for)g(all)g Fm(q)r(;)8 b(q)1403 1454 y Fv(0)1413 1472 y Fo(,)13 b Fl(k)p Fm(q)6 b Fl(\000)t Fm(q)1560 1454 y Fv(0)1571 1472 y Fl(k)1596 1479 y Fv(1)1647 1472 y Fm(>)14 b Fo(2)p Fm(L)p Fo(,)60 1532 y(w)o(e)i(ha)o(v)o(e)110 1635 y Fk(P)144 1579 y Fj(n)174 1635 y Fm(!)r Fl(j)g(8)p Fm(E)g Fl(2)e Fm(I)t(;)h(H)463 1614 y Fy(\()p Fx(!)q Fy(\))516 1635 y Fo(is)h(\()p Fm(\032;)8 b(E)s(;)g(L)725 1642 y Fx(k)747 1635 y Fm(;)g(q)793 1614 y Fv(0)n(0)813 1635 y Fo(\))p Fl(\000)p Fo(regular)16 b(for)h(either)e Fm(q)1273 1614 y Fv(0)o(0)1307 1635 y Fo(=)f Fm(q)k Fo(or)e Fm(q)1482 1614 y Fv(0)o(0)1517 1635 y Fo(=)e Fm(q)1593 1614 y Fv(0)1604 1579 y Fj(o)1389 1727 y Fl(\025)f Fo(1)f Fl(\000)f Fm(L)1560 1704 y Fv(\000)p Fx(p)1560 1741 y(k)1623 1727 y Fm(;)60 1842 y Fo(with)17 b Fm(p)g(>)f Fo(4)p Fm(d)d Fo(+)f(2,)18 b Fm(\032)p Fo(\()p Fm(x)p Fo(\))e(=)g(e)618 1824 y Fv(\000)p Fx(mx=)p Fy(2)751 1842 y Fo(for)i(some)e(\014xed)h Fm(m)f(>)g Fo(0,)i Fm(L)1271 1849 y Fx(k)1309 1842 y Fo(=)e Fm(L)1396 1817 y Fy(\(3)p Fx(=)p Fy(2\))1478 1805 y Fs(k)1396 1854 y Fy(0)1515 1842 y Fo(for)i Fm(L)1624 1849 y Fy(0)1662 1842 y Fo(\014nite)60 1902 y(dep)q(ending)12 b(only)g(on)h Fm(m)f Fo(and)h Fm(p)p Fo(.)20 b(This)13 b(giv)o(es)e(exactly)g(the)h(Multi-Scale)f (Assumption)g(\(M1\).)60 1963 y(One)16 b(can)g(th)o(us)h(apply)f (Theorem)f(2.2)h(with)g(an)o(y)h(compact)e(subset)h Fm(I)1352 1944 y Fv(0)1380 1963 y Fo(of)g Fm(I)t Fo(.)60 2045 y Fh(Example)g(3:)25 b(Anderson)19 b(mo)r(del)d(with)j(correlated)e(p)r (oten)n(tials)133 2106 y Fo(W)l(e)f(consider)g(the)g(Hamiltonian)f(on)h Fm(L)872 2088 y Fy(2)892 2106 y Fo(\()p Fk(R)950 2088 y Fx(d)967 2106 y Fo(\))528 2217 y Fm(H)572 2196 y Fy(\()p Fx(!)q Fy(\))639 2217 y Fo(=)d Fl(\000)p Fo(\001)e(+)832 2170 y Fj(X)830 2279 y Fx(i)p Fv(2)p Fi(Z)890 2270 y Fs(d)914 2217 y Fm(\025)942 2224 y Fx(i)956 2217 y Fo(\()p Fm(!)r Fo(\))p Fm(u)p Fo(\()p Fm(x)g Fl(\000)g Fm(i)p Fo(\))d Fm(;)488 b Fo(\(8\))60 2370 y(for)16 b(whic)o(h)f(the)h(random) f(p)q(oten)o(tials)h Fm(\025)788 2377 y Fx(i)802 2370 y Fo(\()p Fm(!)r Fo(\))p Fm(u)p Fo(\()p Fm(x)10 b Fl(\000)g Fm(i)p Fo(\))15 b(at)i(eac)o(h)e(site)g Fm(i)h Fo(are)f(correlated.)21 b(This)60 2430 y(mo)q(del)d(has)j(b)q(een)e(studied)g(e.g.)31 b(b)o(y)19 b([12)q(,)g(9].)31 b(Correlated)19 b(p)q(oten)o(tials)h (means)f(here)g(that)60 2491 y(the)e(coupling)g(constan)o(ts)g Fm(\025)585 2498 y Fx(i)617 2491 y Fo(are)g(not)h(indep)q(enden)o(tly)d (distributed.)23 b(In)17 b(this)g(case,)g(it)f(has)60 2551 y(b)q(een)f(pro)o(v)o(en)g(that)h(the)f(b)q(ottom)h(of)f(the)h(sp) q(ectrum)e(is)h(dense)g(pure)h(p)q(oin)o(t.)21 b(Our)15 b(goal)i(here)60 2611 y(is)g(to)h(pro)o(v)o(e,)e(with)h(the)g(help)g (of)g(a)h(\\v)m(ariable-energy")f(m)o(ulti-scale)e(analysis,)i (dynamical)60 2671 y(lo)q(calization)f(at)g(the)g(b)q(ottom)g(of)h(the) e(sp)q(ectrum.)20 b(Let)d(us)f(\014rst)h(presen)o(t)e(the)h (assumptions)60 2731 y(for)h(this)f(mo)q(del:)891 2856 y(10)p eop %%Page: 11 11 11 10 bop 59 203 a Fo(\(A1\))24 b(The)g(site)g(p)q(oten)o(tial)g Fm(u)p Fo(\()p Fm(x)p Fo(\))f(is)h(non-negativ)o(e,)i(not)f(iden)o (tically)c(zero,)26 b(compactly)182 264 y(supp)q(orted)17 b(and)g Fl(k)p Fm(u)p Fl(k)584 271 y Fv(1)635 264 y Fm(<)d Fl(1)p Fo(.)59 365 y(\(A2\))24 b Fl(f)p Fm(\025)235 372 y Fx(i)249 365 y Fl(g)274 375 y Fx(i)p Fv(2)p Fi(Z)335 365 y Fs(d)14 b Fo(forms)e(a)i(stationary)g(sto)q(c)o(hastic)f(pro)q (cess)h(of)f(iden)o(tically)e(distributed)h(ran-)182 425 y(dom)j(v)m(ariables.)59 527 y(\(A3\))24 b(The)18 b(conditional)h(probabilit)o(y)e(distribution)h(of)h Fm(\025)1144 534 y Fy(0)1164 527 y Fo(,)g(giv)o(en)e Fm(\025)1354 509 y Fv(?)1354 540 y Fy(0)1402 527 y Fo(=)h Fl(f)p Fm(\025)1511 534 y Fx(i)1525 527 y Fm(;)26 b(i)18 b Fl(6)p Fo(=)f(0)p Fl(g)p Fo(,)i(is)182 587 y(absolutely)i(con)o(tin)o (uous)g(with)h(resp)q(ect)f(to)g(Leb)q(esgue)h(measure)f(with)g(densit) o(y)f Fm(h)1750 594 y Fy(0)182 648 y Fo(whic)o(h)15 b(is)i(compactly)d (supp)q(orted,)j Fl(k)p Fm(h)901 655 y Fy(0)920 648 y Fl(k)945 655 y Fv(1)997 648 y Fm(<)c Fl(1)j Fo(and)h Fk(E)t Fo(\()p Fl(j)p Fm(\025)1301 655 y Fy(0)1323 648 y Fl(j)1337 629 y Fy(max)o(\(2)p Fx(;d)p Fy(\))1478 648 y Fo(\))d Fm(<)g Fl(1)p Fo(.)59 749 y(\(A4\))24 b(Let)f Fm(A)f Fo(b)q(e)h(an)o(y)f(giv)o(en)g(ev)o(en)o(t)f(on)i(\003)883 756 y Fx(L)909 749 y Fo(\(0\))g(\(i.e.)40 b(dep)q(ending)22 b(only)h(on)g Fm(\025)1573 756 y Fx(i)1587 749 y Fo(\()p Fm(!)r Fo(\),)h Fm(i)h Fl(2)182 809 y Fo(\003)216 816 y Fx(L)242 809 y Fo(\(0\))s Fl(\\)s Fk(Z)380 791 y Fx(d)397 809 y Fo(\).)20 b(W)l(e)13 b(denote)f(b)o(y)g Fm(A)p Fo(\()p Fm(x)p Fo(\))g(the)g(ev)o(en)o(t)f Fm(A)h Fo(shifted)g(to)g (\003)1361 816 y Fx(L)1387 809 y Fo(\()p Fm(x)p Fo(\).)20 b(F)l(or)13 b(an)o(y)f(giv)o(en)182 870 y Fm(\013)j(>)g Fo(1)i(and)h Fm(\014)f(>)e Fo(1,)i(w)o(e)f(assume)h(that)g(there)f (exist)g Fm(K)1201 877 y Fy(0)1221 870 y Fo(\()p Fm(\013)p Fo(\))i(ev)o(en)d(and)j Fm(C)t Fo(\()p Fm(K)1614 877 y Fy(0)1633 870 y Fo(\))d Fm(<)g Fl(1)182 930 y Fo(suc)o(h)21 b(that)h(for)g(all)f(in)o(teger)g Fm(K)26 b Fl(\025)d Fm(K)898 937 y Fy(0)918 930 y Fo(,)g(one)e(can)h(\014nd)g Fm(\022)q Fo(\()p 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b(of)h(random)f(pro)q(cesses)h(satisfying)g(\(A2\))g({)g(\(A4\))g(are)g (giv)o(en)f(for)h(example)d(in)60 1379 y([12].)k(Under)16 b(these)f(h)o(yp)q(otheses,)h(it)g(is)g(kno)o(wn)g(that)g Fm(H)21 b Fo(is)16 b(an)g(ergo)q(dic)g(family)e(of)j(almost-)60 1439 y(surely)i(essen)o(tially)e(self-adjoin)o(t)i(op)q(erators)i(on)e Fm(C)1024 1421 y Fv(1)1020 1451 y Fy(0)1061 1439 y Fo(\()p Fk(R)1119 1421 y Fx(d)1136 1439 y Fo(\).)31 b(This)19 b(implies)e(i\),)i(ii\))f(and)i(iv\))60 1499 y(of)g(Assumption)f(2.1.) 33 b(F)l(urthermore,)19 b(for)i Fk(P)l Fo(-a.e.)30 b Fm(!)r Fo(,)21 b(the)e(p)q(oten)o(tial)h Fm(V)1438 1481 y Fy(\()p Fx(!)q Fy(\))1511 1499 y Fo(is)g(uniformly)60 1559 y(b)q(ounded)14 b(in)e Fm(!)r Fo(,)h(and)g(th)o(us)g([)p Fm(H)621 1541 y Fy(\()p Fx(!)q Fy(\))674 1559 y Fm(;)8 b(X)736 1566 y Fx(j)754 1559 y Fo(])13 b(is)f(relativ)o(ely)e Fm(H)1079 1541 y Fy(\()p Fx(!)q Fy(\))1132 1559 y Fo(-b)q(ounded,)k (with)f(relativ)o(e)e(b)q(ound)60 1619 y(2\(1)i(+)g Fl(k)p Fm(V)e Fl(k)280 1601 y Fy(2)280 1632 y Fv(1)317 1619 y Fo(\))336 1601 y Fy(1)p Fx(=)p Fy(2)391 1619 y Fo(.)29 b(Assumption)17 b(2.1.v\))i(is)f(simply)e(the)j(geometric)d(resolv)o (en)o(t)i(equation,)60 1680 y(i.e.,)c(for)j(1)d Fl(\024)f Fm(L)h(<)g Fl(1)i Fo(and)h Fm(q)r Fo(,)e Fm(q)654 1661 y Fv(0)665 1680 y Fo(,)h(suc)o(h)g(that)h Fl(k)p Fm(q)12 b Fl(\000)f Fm(q)1044 1661 y Fv(0)1055 1680 y Fl(k)1080 1687 y Fv(1)1131 1680 y Fm(>)j Fo(2)p Fm(L)j Fo(one)f(has,)195 1796 y Fh(1)223 1803 y Fx(q)240 1794 y Fg(0)253 1796 y Fo(\()p Fm(H)316 1776 y Fy(\()p Fx(!)q Fy(\))380 1796 y Fl(\000)11 b Fm(z)r Fo(\))474 1776 y Fv(\000)p Fy(1)521 1796 y Fh(1)549 1803 y Fx(q)582 1796 y Fo(=)j Fh(1)662 1803 y Fx(q)679 1794 y Fg(0)692 1796 y Fo(\()p Fm(H)755 1771 y Fy(\()p Fx(!)q Fy(\))751 1812 y(\003)775 1818 y Fs(L)799 1812 y Fy(\()p Fx(q)830 1803 y Fg(0)841 1812 y Fy(\))868 1796 y Fl(\000)d Fm(z)r Fo(\))962 1776 y Fv(\000)p Fy(1)1009 1796 y Fo([)p Fm(H)1067 1776 y Fy(\()p Fx(!)q Fy(\))1119 1796 y Fm(;)d(\037)1172 1804 y Fy(\003)1196 1810 y Fs(L)1219 1804 y Fy(\()p Fx(q)1250 1795 y Fg(0)1261 1804 y Fy(\))1277 1796 y Fo(]\()p Fm(H)1354 1776 y Fy(\()p Fx(!)q Fy(\))1417 1796 y Fl(\000)j Fm(z)r Fo(\))1511 1776 y Fv(\000)p Fy(1)1558 1796 y Fh(1)1586 1803 y Fx(q)1621 1796 y Fm(;)60 1912 y Fo(where)k Fm(\037)231 1919 y Fy(\003)255 1925 y Fs(L)278 1919 y Fy(\()p Fx(q)309 1910 y Fg(0)320 1919 y Fy(\))351 1912 y Fo(is)g(precisely)e(de\014ned)i(in)g(\(43\))q (.)21 b(Let)15 b Fm(E)1064 1919 y Fv(\000)1108 1912 y Fo(:=)e(inf)s(\(\006\))j(denote)f(the)g(b)q(ottom)g(of)60 1972 y(the)i(almost-sure)f(sp)q(ectrum)f(of)j Fm(H)720 1954 y Fy(\()p Fx(!)q Fy(\))772 1972 y Fo(.)23 b(F)l(ollo)o(wing)17 b([8,)f(12)q(],)g(w)o(e)g(kno)o(w)h(that)h(for)f(all)f Fm(p)f(>)g Fo(0,)60 2032 y Fm(m)e(>)h Fo(0,)h Fm(S)s Fo(,)g Fm(N)5 b Fo(,)15 b(and)h Fm(L)483 2039 y Fy(0)517 2032 y Fm(<)d Fl(1)i Fo(there)g(exists)f Fm(E)s Fo(\()p Fm(L)981 2039 y Fy(0)1001 2032 y Fo(\))g Fm(>)f(E)1121 2039 y Fv(\000)1166 2032 y Fo(suc)o(h)i(that)g(with)g(the)g(notations) 60 2092 y(of)i(the)f(App)q(endix)f(and)i Fm(I)g Fo(=)d([)p Fm(E)659 2099 y Fv(\000)688 2092 y Fm(;)8 b(E)s Fo(\()p Fm(L)801 2099 y Fy(0)821 2092 y Fo(\)],)15 b(w)o(e)h(ha)o(v)o(e)f(the)h (initial)f(deca)o(y:)102 2215 y Fk(P)t Fl(f)m(8)p Fm(E)h Fl(2)e Fm(I)t(;)23 b Fl(9)p Fo(\()p Fm(n)432 2194 y Fv(0)432 2227 y Fx(i)446 2215 y Fo(\))465 2222 y Fx(i)p Fy(=1)p Fx(;:::)o(S)603 2215 y Fo(s)p Fm(:)p Fo(t)p Fm(:)16 b Fe(H)721 2222 y Fx(L)745 2227 y Fr(1)764 2215 y Fo(\()p Fm(E)s(;)8 b(y)868 2222 y Fy(1)887 2215 y Fm(;)g(m;)g Fo(\()p Fm(n)1022 2194 y Fv(0)1022 2227 y Fx(i)1036 2215 y Fo(\)\))32 b(or)17 b Fe(H)1202 2222 y Fx(L)1226 2227 y Fr(1)1245 2215 y Fo(\()p Fm(E)s(;)8 b(y)1349 2222 y Fy(2)1368 2215 y Fm(;)g(m;)g Fo(\()p Fm(n)1503 2194 y Fv(0)1503 2227 y Fx(i)1517 2215 y Fo(\)\))16 b Fl(g)1472 2293 y(\025)e Fo(1)d Fl(\000)g Fm(L)1643 2269 y Fv(\000)p Fx(p)1643 2305 y Fy(0)1707 2293 y Fm(;)60 2403 y Fo(where)24 b Fe(H)245 2410 y Fx(L)269 2415 y Fr(1)312 2403 y Fo(is)g(a)g(deca)o(y) f(prop)q(ert)o(y)h(for)g(resolv)o(en)o(ts)f(precisely)f(de\014ned)i(in) f(\(44\))i(of)f(the)60 2463 y(App)q(endix.)j(Th)o(us,)19 b(according)f(to)h(the)f(estimate)f(\(48\))i(and)g(Lemmas)d(A.1)i(and) 38 b(A.2)17 b(w)o(e)60 2523 y(tak)o(e)i Fm(\013)h Fo(=)f(3)p Fm(=)p Fo(2)i(and)f Fm(K)509 2530 y Fy(0)549 2523 y Fm(>)f Fo(3)p Fm(d)14 b Fo(+)g(5)20 b(suc)o(h)f(that)h Fm(\022)q Fo(\()p Fm(K)1071 2530 y Fy(0)1091 2523 y Fm(;)8 b Fo(3)p Fm(=)p Fo(2\))21 b Fm(>)e Fo(3)p Fm(=d)c Fo(+)e(1)p Fm(=)p Fo(2;)22 b(W)l(e)d(also)h(\014x)60 2584 y Fm(S)d Fo(=)d(4,)h Fm(N)k Fo(=)14 b(4)d(+)f Fm(K)446 2591 y Fy(0)466 2584 y Fo(,)16 b Fm(p)e Fo(=)g(\()p Fm(K)646 2591 y Fy(0)676 2584 y Fo(+)c(1\))p Fm(=)p Fo(2,)17 b Fm(w)e Fo(=)e(2)p Fm(d)e Fo(+)f Fm(K)1096 2591 y Fy(0)1117 2584 y Fm(=)p Fo(2)h(+)f(1)16 b(and)g Fm(m)e(>)g Fo(4)p Fm(w)d Fo(+)f(2\()p Fm(d)h Fl(\000)f Fo(1\).)60 2660 y(Then)19 b(w)o(e)f(obtain,)i(with)f (the)f(notations)i(of)f(the)g(App)q(endix,)f(for)h Fm(L)1345 2667 y Fx(k)1385 2660 y Fo(=)f Fm(L)1474 2630 y Fy(\()1493 2616 y Fr(3)p 1493 2622 16 2 v 1493 2643 a(2)1513 2630 y Fy(\))1527 2618 y Fs(k)1474 2672 y Fy(0)1548 2660 y Fo(,)h(and)g Fm(q)r Fo(,)g Fm(q)1759 2642 y Fv(0)891 2856 y Fo(11)p eop %%Page: 12 12 12 11 bop 60 203 a Fo(suc)o(h)16 b(that)h Fl(k)p Fm(q)12 b Fl(\000)f Fm(q)409 185 y Fv(0)420 203 y Fl(k)445 210 y Fv(1)496 203 y Fm(>)j Fo(2)p Fm(L)605 210 y Fx(k)627 203 y Fo(,)188 323 y Fk(P)222 267 y Fj(n)252 323 y Fl(8)p Fm(E)i Fl(2)e Fm(I)19 b Fo(either)d(for)k(~)-28 b Fm(q)15 b Fo(=)f Fm(q)k Fo(or)i(~)-28 b Fm(q)16 b Fo(=)d Fm(q)935 302 y Fv(0)946 323 y Fm(;)230 429 y Fo(sup)236 468 y Fx(">)p Fy(0)312 429 y Fl(k)p Fo([)p Fl(\000)p Fo(\001)p Fm(;)i(\037)491 406 y Fx(L)515 412 y Fs(k)491 442 y Fx(N)535 429 y Fo(\()t(~)-28 b Fm(q)r Fo(\)]\()p Fm(H)670 436 y Fy(\003)694 443 y Fs(L)715 452 y(k)734 443 y Fr(\()r(~)-17 b Fs(q)q Fr(\))789 429 y Fl(\000)10 b Fm(E)k Fl(\000)d Fo(i)p Fm(")p Fo(\))994 408 y Fv(\000)p Fy(1)1041 429 y Fm(\037)1072 406 y Fx(L)1096 412 y Fs(k)1072 441 y Fy(0)1116 429 y Fo(\()t(~)-28 b Fm(q)r Fo(\))p Fl(k)14 b(\024)f Fm(L)1302 408 y Fv(\000)p Fx(m)1302 443 y(k)1363 374 y Fj(o)1402 429 y Fl(\025)6 b Fo(1)11 b Fl(\000)f Fm(L)1564 405 y Fv(\000)p Fx(p)1564 443 y(k)1628 429 y Fm(:)66 b Fo(\(9\))60 562 y(This)13 b(inequalit)o(y)d(is)j(not)g(the) f(same)g(as)h(the)g(one)f(required)g(in)g(equation)25 b(\(4\))13 b(of)g(Assumption)60 622 y(\(M1\))j(since)e(the)i(deca)o(y)f (of)g(resolv)o(en)o(ts)g(is)g(only)h(algebraic)f(instead)h(of)f(b)q (eing)h(exp)q(onen)o(tial.)60 683 y(Ho)o(w)o(ev)o(er,)11 b Fm(m)g Fo(can)h(b)q(e)h(c)o(hosen)e(larger)h(and)h(larger,)f(if)g (the)f(in)o(terv)m(al)g Fm(I)16 b Fo(of)c(energies)f(gets)h(closer)60 743 y(and)18 b(closer)f(to)h(the)g(b)q(ottom)f(of)h(the)f(sp)q(ectrum.) 24 b(Therefore,)17 b(according)h(to)g(Remark)e(2.3,)60 803 y(the)j(di\013usion)g(exp)q(onen)o(ts)g(con)o(v)o(erge)e(to)j(zero) e(for)h(energy)f(in)o(terv)m(als)g Fm(I)23 b Fo(approac)o(hing)c(the)60 863 y(b)q(ottom)d(of)h(the)f(sp)q(ectrum.)60 1029 y Fn(4)81 b(Pro)r(of)28 b(of)f(the)f(Main)g(results)60 1138 y Fo(Lemma)16 b(4.1)j(guaran)o(tees)f(that)h(the)f(Cesaro)h(mean)e Fl(hhj)p Fm(X)t Fl(j)1159 1120 y Fy(2)1180 1138 y Fl(ii)1218 1145 y Fx(T)s(;')1295 1138 y Fo(of)i(the)f(second)g(momen)o(t)60 1199 y(of)f(p)q(osition)g(is)f(w)o(ell-de\014ned)f(for)h(all)g Fm(T)k(>)14 b Fo(1)j(and)g(all)f Fm(')g Fo(as)h(in)f(Theorem)f(2.1.)22 b(Lemma)14 b(4.2)60 1259 y(exhibits)21 b(the)g(relation)g(b)q(et)o(w)o (een)g(the)h(asymptotic)e(b)q(eha)o(vior)i(in)f Fm(T)28 b Fo(of)22 b Fl(hhj)p Fm(X)t Fl(j)1557 1241 y Fy(2)1578 1259 y Fl(ii)1616 1266 y Fx(T)s(;')1675 1259 y Fo(,)h(for)60 1319 y Fm(')g Fo(lo)q(calized)f(in)h(energy)g(in)g(a)h(compact)e(set)h Fm(I)t Fo(,)h(and)g(the)f(b)q(eha)o(vior)h(of)f(the)g(resolv)o(en)o(ts) 60 1379 y(\()p Fm(H)17 b Fl(\000)c Fm(E)k Fl(\000)c Fo(i)p Fm(")p Fo(\))349 1361 y Fv(\000)p Fy(1)415 1379 y Fo(for)19 b(energies)g Fm(E)e Fo(+)c(i)p Fm(")19 b Fo(approac)o(hing)h(the)f (real)g(axis.)31 b(In)19 b(particular,)g(it)60 1439 y(sho)o(ws)j(that)g (the)f(main)f(con)o(tribution)h(to)g(the)g(second)h(momen)o(t)c(of)k(p) q(osition)f(is)g(due)h(to)60 1500 y(energies)d Fm(E)j Fo(in)d Fm(I)t Fo(.)30 b(The)19 b(pro)q(of)h(of)g(Theorem)e(2.1)i(is)f (then)g(completed)e(with)i(the)g(help)g(of)60 1560 y(Lemma)13 b(4.3.)22 b(T)l(o)16 b(sho)o(w)g(Theorem)e(2.2,)i(w)o(e)f(refer)g(in)g (addtion)h(to)g(Remark)e(4.3.i\).)21 b(On)15 b(the)60 1620 y(other)h(hand,)h(Theorem)e(2.3)h(follo)o(ws)h(from)e(the)h (\014rst)g(t)o(w)o(o)g(lemmas)e(and)j(Lemma)d(4.4.)133 1680 y(The)i(\014rst)f(lemma)e(extends)i(the)g(results)g(of)h(Radin)g (and)g(Simon)e([26)q(])h(to)h(a)f(larger)h(class)60 1740 y(of)h(op)q(erators)g Fm(H)t Fo(,)f(although)i(the)e(set)g Fm(S)j Fo(w)o(e)d(consider)g(is)g(sligh)o(tly)f(di\013eren)o(t.)60 1848 y Fh(Lemma)h(4.1)24 b Ff(L)n(et)e Fm(H)488 1830 y Fy(\()p Fx(!)q Fy(\))564 1848 y Ff(b)n(e)h(a)g(r)n(andom)f(op)n(er)n (ator)g(satisfying)h(Assumptions)g(2.1.i\){iv\))60 1909 y(and)18 b(let)384 2014 y Fm(S)f Fo(:=)c Fl(f)p Fm(')h Fl(2)g(H)p Fm(;)26 b Fl(k)8 b(j)p Fm(X)t Fl(j)p Fm(')p Fl(k)14 b Fm(<)g Fl(1)p Fm(;)25 b Fl(k)p Fm(H)1082 1994 y Fy(\()p Fx(!)q Fy(\))1135 2014 y Fm(')p Fl(k)13 b Fm(<)h Fl(1g)j Fm(:)320 b Fo(\(10\))60 2124 y Ff(Then)18 b(for)e Fk(P)l Ff(-a.e.)k Fm(!)r Ff(,)e Fm(e)499 2106 y Fv(\000)p Fy(i)p Fx(tH)581 2095 y Fr(\()p Fs(!)q Fr(\))645 2124 y Ff(maps)f Fm(S)j Ff(into)d Fm(S)s Ff(,)g(and)g(ther)n(e)g(exists)h(a) f(\014nite)h(c)n(onstant)g Fm(c)1717 2106 y Fy(\()p Fx(!)q Fy(\))60 2185 y Ff(such)g(that)f(for)g(al)r(l)i Fm(t)13 b Fl(2)i Fk(R)p Ff(,)f Fm(d)g Fo(=)g(1)p Fm(;)8 b(:)g(:)g(:)g(;)16 b(j)s Ff(,)h Fm(')d Fl(2)g Fm(S)20 b Ff(and)e Fm( )d Fl(2)f(H)p Ff(,)k(one)g(has)173 2319 y Fl(h)p Fm( )r(;)e Fo([e)292 2298 y Fv(\000)p Fy(i)p Fx(tH)374 2287 y Fr(\()p Fs(!)q Fr(\))421 2319 y Fm(;)8 b(X)483 2326 y Fx(j)502 2319 y Fo(])p Fm(')p Fl(i)13 b Fo(=)h Fl(\000)p Fo(i)693 2251 y Fj(Z)743 2264 y Fx(t)720 2364 y Fy(0)757 2319 y Fl(h)p Fm( )r(;)i Fo(e)862 2298 y Fv(\000)p Fy(i\()p Fx(t)p Fv(\000)p Fx(s)p Fy(\))p Fx(H)1015 2287 y Fr(\()p Fs(!)q Fr(\))1063 2319 y Fo([)p Fm(H)1121 2298 y Fy(\()p Fx(!)q Fy(\))1174 2319 y Fm(;)8 b(X)1236 2326 y Fx(j)1254 2319 y Fo(]e)1290 2298 y Fv(\000)p Fy(i)p Fx(sH)1375 2287 y Fr(\()p Fs(!)q Fr(\))1423 2319 y Fm(')p Fl(i)g Fm(ds)19 b(;)120 b Fo(\(11\))60 2448 y Ff(and)352 2553 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(e)479 2533 y Fv(\000)p Fy(i)p Fx(tH)561 2521 y Fr(\()p Fs(!)q Fr(\))609 2553 y Fm(')p Fl(k)13 b(\024)h Fm(c)753 2533 y Fy(\()p Fx(!)q Fy(\))805 2553 y Fl(j)p Fm(t)p Fl(j)859 2513 y Fj(\000)882 2553 y Fl(k)p Fm(')p Fl(k)d Fo(+)g Fl(k)p Fm(H)1093 2533 y Fy(\()p Fx(!)q Fy(\))1146 2553 y Fm(')p Fl(k)1203 2513 y Fj(\001)1236 2553 y Fo(+)g Fl(k)d(j)p Fm(X)t Fl(j)p Fm(')p Fl(k)18 b Fm(:)204 b Fo(\(12\))60 2659 y Fh(Remark)16 b(4.1)24 b Fo(Strictly)14 b(sp)q(eaking,)i(the)f(set)g Fm(S)j Fo(dep)q(ends)e(on)g Fm(!)r Fo(,)f(but)g(due)g(to)h(Assumption) 60 2719 y(2.1.i\),)f Fm(S)k Fo(is)d Fk(P)m Fo(-alm)o(ost)e(surely)h (constan)o(t.)891 2856 y(12)p eop %%Page: 13 13 13 12 bop 158 203 a Fd(Pro)q(of.)64 b Fo(Since)14 b(there)f(is)i(no)f (p)q(ossible)h(confusion,)g(w)o(e)f(drop)h(the)f(sup)q(erscript)g Fm(!)r Fo(.)21 b(Fix)60 264 y Fm(j)c Fl(2)d(f)p Fo(1)p Fm(;)8 b(:)g(:)g(:)g(;)16 b(d)p Fl(g)g Fo(and)h(let)580 382 y Fm(F)612 389 y Fx(")644 382 y Fo(=)d(\()p Fm(H)h Fo(+)c(i\))922 349 y(1)p 857 371 156 2 v 857 417 a(i)f(+)h Fm("X)993 424 y Fx(j)1017 382 y Fo(\()p Fm(H)k Fo(+)c(i\))1173 362 y Fv(\000)p Fy(1)1236 382 y Fm(:)60 508 y Fo(By)k(Assumption)f (2.1.ii\))g(and)i(since)f Fl(C)j Fo(is)d(a)h(core)f(for)h Fm(H)t Fo(,)f(the)g(domain)g(of)h Fm(F)1479 515 y Fx(")1512 508 y Fo(con)o(tains)f(the)60 569 y(dense)h(subset)h Fl(B)e Fo(:=)e(\()p Fm(H)i Fo(+)c(i\))p Fl(C)s Fo(.)21 b(No)o(w)16 b(on)h Fm(B)s Fo(,)e(one)i(can)f(write)378 687 y Fm(F)410 694 y Fx(")442 687 y Fo(:=)d Fl(\000)617 654 y Fm(")p 551 676 V 551 722 a Fo(i)e(+)g Fm("X)688 729 y Fx(j)711 687 y Fo([)p Fm(H)q(;)d(X)828 694 y Fx(j)847 687 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))1017 667 y Fv(\000)p Fy(1)1064 687 y Fm(F)1096 694 y Fx(")1125 687 y Fo(+)1244 654 y(1)p 1179 676 V 1179 722 a(i)f(+)h Fm("X)1315 729 y Fx(j)1355 687 y Fm(:)314 b Fo(\(13\))60 811 y(Th)o(us)17 b(for)f Fm(")e(<)g(")370 818 y Fy(1)405 811 y Fo(small)h(enough,)h(one) h(has)364 927 y Fm(F)396 934 y Fx(")428 927 y Fo(=)480 871 y Fj(\020)509 927 y Fo(1)12 b(+)665 893 y Fm(")p 599 915 V 599 961 a Fo(i)e(+)h Fm("X)735 968 y Fx(j)759 927 y Fo([)p Fm(H)q(;)d(X)876 934 y Fx(j)895 927 y Fo(]\()p Fm(H)14 b Fo(+)d(i\))1064 906 y Fv(\000)p Fy(1)1111 871 y Fj(\021)1141 883 y Fv(\000)p Fy(1)1258 893 y Fo(1)p 1193 915 V 1193 961 a(i)g(+)g Fm("X)1330 968 y Fx(j)1369 927 y Fm(;)60 1053 y Fo(whic)o(h)16 b(implies)d(that)k Fm(F)503 1060 y Fx(")537 1053 y Fo(is)f(uniformly)e(b)q(ounded)j(in)f Fm(")p Fo(.)21 b(F)l(rom)15 b(\(13\))i(one)g(obtains)486 1149 y(\(i)11 b(+)g Fm("X)642 1156 y Fx(j)661 1149 y Fo(\))p Fm(F)712 1156 y Fx(")743 1149 y Fo(=)j(1)e Fl(\000)e Fm(")p Fo([)p Fm(H)q(;)e(X)1020 1156 y Fx(j)1039 1149 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))1209 1129 y Fv(\000)p Fy(1)1255 1149 y Fm(F)1287 1156 y Fx(")1322 1149 y Fm(;)60 1246 y Fo(whic)o(h)k(is)h(an)g(op)q(erator)h(uniformly)c(b)q(ounded)k (in)e Fm(")h Fo(for)g(all)f Fm(")f(<)g(")1279 1253 y Fy(2)1314 1246 y Fo(small)g(enough.)22 b(Hence,)60 1306 y(the)16 b(op)q(erator)307 1424 y(\()p Fm(H)f Fo(+)c(i\))516 1390 y Fm(X)556 1397 y Fx(j)p 468 1413 V 468 1458 a Fo(i)g(+)g Fm("X)605 1465 y Fx(j)628 1424 y Fo(\()p Fm(H)k Fo(+)c(i\))784 1404 y Fv(\000)p Fy(1)873 1424 y Fo(=)41 b([)p Fm(H)q(;)8 b(X)1069 1431 y Fx(j)1088 1424 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))1258 1404 y Fv(\000)p Fy(1)1304 1424 y Fm(F)1336 1431 y Fx(")1365 1424 y Fo(+)g Fm(X)1454 1431 y Fx(j)1473 1424 y Fm(F)1505 1431 y Fx(")1683 1424 y Fo(\(14\))60 1547 y(is)16 b(b)q(ounded.)22 b(No)o(w)16 b(one)h(has)207 1622 y Fm(X)247 1629 y Fx(j)p 159 1644 V 159 1690 a Fo(i)11 b(+)g Fm("X)296 1697 y Fx(j)319 1656 y Fo(e)341 1635 y Fv(\000)p Fy(i)p Fx(tH)438 1656 y Fo(=)j(e)512 1635 y Fv(\000)p Fy(i)p Fx(tH)595 1600 y Fj(\020)625 1656 y Fo(e)647 1635 y Fy(i)p Fx(tH)756 1622 y Fm(X)796 1629 y Fx(j)p 708 1644 V 708 1690 a Fo(i)c(+)h Fm("X)844 1697 y Fx(j)868 1656 y Fo(e)890 1635 y Fv(\000)p Fy(i)p Fx(tH)984 1656 y Fl(\000)1087 1622 y Fm(X)1127 1629 y Fx(j)p 1039 1644 V 1039 1690 a Fo(i)g(+)g Fm("X)1176 1697 y Fx(j)1199 1600 y Fj(\021)1240 1656 y Fo(+)g(e)1311 1635 y Fv(\000)p Fy(i)p Fx(tH)1448 1622 y Fm(X)1488 1629 y Fx(j)p 1399 1644 V 1399 1690 a Fo(i)g(+)g Fm("X)1536 1697 y Fx(j)1576 1656 y Fm(:)93 b Fo(\(15\))60 1782 y(F)l(rom)16 b(\(14\))q(,)h(since)g Fm(X)468 1789 y Fx(j)487 1782 y Fm(=)p Fo(\(i)12 b(+)g Fm("X)669 1789 y Fx(j)687 1782 y Fo(\))18 b(maps)f Fl(D)q Fo(\()p Fm(H)t Fo(\))h(in)o(to)f Fl(D)q Fo(\()p Fm(H)t Fo(\),)i(one)e(obtains)i(for)e(the)h(term)60 1842 y(in)e(paren)o(thesis)g(in)g(the)g(righ)o(t)g(hand)h(side)e(of)20 b(\(15\))q(,)c(in)g(the)g(w)o(eak)g(sense)281 1970 y(e)303 1949 y Fy(i)p Fx(tH)413 1936 y Fm(X)453 1943 y Fx(j)p 364 1958 V 364 2004 a Fo(i)11 b(+)g Fm("X)501 2011 y Fx(j)524 1970 y Fo(e)546 1949 y Fv(\000)p Fy(i)p Fx(tH)641 1970 y Fl(\000)744 1936 y Fm(X)784 1943 y Fx(j)p 696 1958 V 696 2004 a Fo(i)f(+)h Fm("X)832 2011 y Fx(j)870 1970 y Fo(=)i(i)943 1902 y Fj(Z)993 1915 y Fx(t)971 2015 y Fy(0)1016 1970 y Fo(e)1038 1949 y Fy(i)p Fx(sH)1098 1970 y Fo([)p Fm(H)q(;)1228 1936 y(X)1268 1943 y Fx(j)p 1180 1958 V 1180 2004 a Fo(i)d(+)h Fm("X)1316 2011 y Fx(j)1340 1970 y Fo(]e)1376 1949 y Fv(\000)p Fy(i)p Fx(sH)1470 1970 y Fm(ds)17 b(:)134 b Fo(\(16\))60 2093 y(F)l(urthermore,)14 b(b)o(y)i(writing)410 2211 y([)p Fm(H)q(;)540 2177 y(X)580 2184 y Fx(j)p 492 2200 V 492 2245 a Fo(i)10 b(+)h Fm("X)628 2252 y Fx(j)652 2211 y Fo(])i(=)h([)p Fm(H)q(;)8 b(X)848 2218 y Fx(j)867 2211 y Fo(])951 2177 y(1)p 886 2200 V 886 2245 a(i)i(+)h Fm("X)1022 2252 y Fx(j)1057 2211 y Fo(+)g Fm(X)1146 2218 y Fx(j)1164 2211 y Fo([)p Fm(H)q(;)1312 2177 y Fo(1)p 1246 2200 V 1246 2245 a(i)g(+)g Fm("X)1383 2252 y Fx(j)1407 2211 y Fo(])60 2334 y(and)182 2441 y Fm(X)222 2448 y Fx(j)241 2441 y Fo([)p Fm(H)q(;)388 2407 y Fo(1)p 323 2429 V 323 2475 a(i)f(+)h Fm("X)459 2482 y Fx(j)483 2441 y Fo(])224 2578 y(=)41 b Fm(X)343 2585 y Fx(j)370 2507 y Fj(\022)477 2544 y Fo(1)p 412 2566 V 412 2612 a(i)10 b(+)h Fm("X)548 2619 y Fx(j)572 2578 y Fo(\(i)f(+)h Fm("X)727 2585 y Fx(j)746 2578 y Fo(\))p Fm(H)880 2544 y Fo(1)p 814 2566 V 814 2612 a(i)g(+)g Fm("X)951 2619 y Fx(j)985 2578 y Fl(\000)1106 2544 y Fo(1)p 1040 2566 V 1040 2612 a(i)g(+)g Fm("X)1177 2619 y Fx(j)1200 2578 y Fm(H)t Fo(\(i)g(+)g Fm("X)1400 2585 y Fx(j)1419 2578 y Fo(\))1508 2544 y(1)p 1443 2566 V 1443 2612 a(i)f(+)h Fm("X)1579 2619 y Fx(j)1603 2507 y Fj(\023)224 2708 y Fo(=)41 b Fm(X)343 2715 y Fx(j)432 2675 y Fo(1)p 367 2697 V 367 2742 a(i)10 b(+)h Fm("X)503 2749 y Fx(j)527 2708 y Fo([)p Fm("X)604 2715 y Fx(j)622 2708 y Fm(;)d(H)t Fo(])772 2675 y(1)p 707 2697 V 707 2742 a(i)i(+)h Fm("X)843 2749 y Fx(j)883 2708 y Fm(;)891 2856 y Fo(13)p eop %%Page: 14 14 14 13 bop 60 203 a Fo(one)16 b(obtains,)h(together)f(with)g(\(15\))h (and)g(\(16\))153 283 y Fm(X)193 290 y Fx(j)p 104 306 156 2 v 104 351 a Fo(i)11 b(+)g Fm("X)241 358 y Fx(j)265 317 y Fo(e)287 296 y Fv(\000)p Fy(i)p Fx(tH)386 317 y Fo(=)17 b(i)8 b(e)485 296 y Fv(\000)p Fy(i)p Fx(tH)576 249 y Fj(Z)626 262 y Fx(t)604 362 y Fy(0)633 317 y Fo(e)655 296 y Fy(i)p Fx(sH)714 317 y Fo([)p Fm(H)q(;)g(X)831 324 y Fx(j)850 317 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))1020 296 y Fv(\000)p Fy(1)1066 317 y Fo(\()p Fm(H)16 b Fo(+)11 b(i\))1293 283 y(1)p 1228 306 V 1228 351 a(i)f(+)h Fm("X)1364 358 y Fx(j)1387 317 y Fo(e)1409 296 y Fv(\000)p Fy(i)p Fx(sH)1505 317 y Fm(ds)416 457 y Fo(+i)476 389 y Fj(Z)526 402 y Fx(t)504 502 y Fy(0)532 457 y Fm(e)555 436 y Fy(i)p Fx(sH)615 457 y Fm(X)655 464 y Fx(j)744 423 y Fm(")p 678 445 V 678 491 a Fo(i)g(+)g Fm("X)815 498 y Fx(j)838 457 y Fo([)p Fm(X)892 464 y Fx(j)911 457 y Fm(;)d(H)t Fo(]\()p Fm(H)15 b Fo(+)c(i\))1147 436 y Fv(\000)p Fy(1)1193 457 y Fo(\()p Fm(H)16 b Fo(+)11 b(i\))1420 423 y(1)p 1355 445 V 1355 491 a(i)f(+)h Fm("X)1491 498 y Fx(j)1514 457 y Fm(e)1537 436 y Fv(\000)p Fy(i)p Fx(sH)1624 457 y Fm(ds)416 589 y Fo(+)d(e)484 569 y Fv(\000)p Fy(i)p Fx(tH)621 555 y Fm(X)661 562 y Fx(j)p 573 578 V 573 623 a Fo(i)i(+)h Fm("X)709 630 y Fx(j)749 589 y Fm(:)60 701 y Fo(Then)16 b(for)h(all)f Fm(')d Fl(2)h(D)q Fo(\()p Fm(H)t Fo(\),)143 807 y Fl(k)221 773 y Fm(X)261 780 y Fx(j)p 173 795 V 173 841 a Fo(i)c(+)h Fm("X)309 848 y Fx(j)333 807 y Fo(e)355 786 y Fv(\000)p Fy(i)p Fx(tH)438 807 y Fm(')p Fl(k)41 b(\024)h(j)p Fm(t)p Fl(j)8 b(k)p Fo([)p Fm(H)q(;)g(X)813 814 y Fx(j)831 807 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))1001 786 y Fv(\000)p Fy(1)1047 807 y Fl(k)d(k)p Fm(F)1137 814 y Fx(")1155 807 y Fl(k)g(k)p Fo(\()p Fm(H)16 b Fo(+)11 b(i\))p Fm(')p Fl(k)617 939 y Fo(+)p Fl(j)p Fm(t)p Fl(j)d(k)775 906 y Fm("X)838 913 y Fx(j)p 739 928 V 739 974 a Fo(i)i(+)h Fm("X)875 981 y Fx(j)898 939 y Fl(k)d(k)p Fo([)p Fm(H)q(;)g(X)1073 946 y Fx(j)1092 939 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))1262 919 y Fv(\000)p Fy(1)1309 939 y Fl(k)d(k)p Fm(F)1399 946 y Fx(")1417 939 y Fl(k)g(k)p Fo(\()p Fm(H)15 b Fo(+)c(i\))p Fm(')p Fl(k)617 1072 y Fo(+)p Fl(k)733 1038 y Fm(X)773 1045 y Fx(j)p 685 1060 V 685 1106 a Fo(i)f(+)h Fm("X)821 1113 y Fx(j)845 1072 y Fm(')p Fl(k)536 1204 y(\024)42 b Fm(c)p Fl(j)p Fm(t)p Fl(j)692 1164 y Fj(\000)714 1204 y Fl(k)p Fm(H)t(')p Fl(k)11 b Fo(+)g Fl(k)p Fm(')p Fl(k)982 1164 y Fj(\001)1016 1204 y Fo(+)g Fl(k)1143 1171 y Fm(X)1183 1178 y Fx(j)p 1095 1193 V 1095 1238 a Fo(i)f(+)h Fm("X)1231 1245 y Fx(j)1255 1204 y Fm(')p Fl(k)16 b Fm(:)60 1313 y Fo(The)e(constan)o(t)f Fm(c)h Fo(is)f(indep)q(enden)o(t)g(of)h Fm(")p Fo(.)20 b(T)l(aking)14 b(the)f(limit)e Fm(")i Fl(!)h Fo(0,)g(one)g(gets)g(for)f(all)g Fm(')h Fl(2)g Fm(S)408 1395 y Fl(k)p Fm(X)473 1402 y Fx(j)492 1395 y Fo(e)514 1374 y Fv(\000)p Fy(i)p Fx(tH)597 1395 y Fm(')p Fl(k)g(\024)f Fm(c)p Fl(j)p Fm(t)p Fl(j)795 1355 y Fj(\000)818 1395 y Fl(k)p Fm(H)t(')p Fl(k)e Fo(+)g Fl(k)p Fm(')p Fl(k)1086 1355 y Fj(\001)1120 1395 y Fo(+)g Fl(k)p Fm(X)1234 1402 y Fx(j)1252 1395 y Fm(')p Fl(k)16 b Fm(:)60 1477 y Fo(This)h(pro)o(v)o(es)g(\(12\).)24 b(No)o(w,)17 b(since)f(w)o(e)g (kno)o(w)i(that)f(e)1023 1459 y Fv(\000)p Fy(i)p Fx(tH)1123 1477 y Fo(maps)g Fm(S)j Fo(in)o(to)d Fm(S)s Fo(,)f(one)h(has)h(for)g (all)60 1537 y Fm(')e Fo(and)h Fm( )h Fo(in)e Fm(S)s Fo(:)216 1648 y Fl(h)p Fo(e)257 1628 y Fy(i)p Fx(tH)314 1648 y Fm(X)354 1655 y Fx(j)372 1648 y Fo(e)394 1628 y Fv(\000)p Fy(i)p Fx(tH)478 1648 y Fm(';)g( )r Fl(i)41 b Fo(=)h Fl(h)p Fm(X)773 1655 y Fx(j)792 1648 y Fm(';)16 b( )r Fl(i)11 b Fo(+)g(i)989 1580 y Fj(Z)1038 1593 y Fx(t)1016 1693 y Fy(0)1053 1648 y Fl(h)p Fo(e)1094 1628 y Fy(i)p Fx(sH)1154 1648 y Fo([)p Fm(H)q(;)d(X)1271 1655 y Fx(j)1290 1648 y Fo(]e)1326 1628 y Fv(\000)p Fy(i)p Fx(sH)1412 1648 y Fm(';)16 b( )r Fl(i)8 b Fm(ds)17 b(:)69 b Fo(\(17\))60 1756 y(Since)13 b(the)h(in)o(tegrand)h(in)f(\(17\))h(is) f(uniformly)e(b)q(ounded)j(in)f Fm(s)g Fl(2)g Fo([0)p Fm(;)8 b(t)p Fo(])13 b(and)i(since)f Fm(S)j Fo(is)d(dense)60 1817 y(in)i Fl(H)p Fo(,)g(one)h(can)f(extend)g(\(17\))h(to)f(all)g Fm( )f Fl(2)f(H)p Fo(,)j(whic)o(h)e(giv)o(es)h(\(11\).)p 1342 1825 20 43 v 133 1906 a(The)22 b(follo)o(wing)g(lemma)e(is)i(an)h (easy)f(generalization)g(of)g(a)h(result)f(of)g(Mon)o(tc)o(ho)g([23])60 1966 y(stated)c(for)f(op)q(erators)i Fm(H)g Fo(=)d Fl(\000)p Fo(\001)11 b(+)h Fm(V)f Fo(,)17 b(where)g Fm(V)28 b Fo(is)17 b(b)q(ounded)h(b)q(elo)o(w.)24 b(See)17 b(also)h([17])f(for)60 2027 y(related)f(results)g(in)f(the)h(case)h(of)f(random)g(p)q(oten)o (tials)g(whic)o(h)g(are)g(piecewise)f(constan)o(t.)60 2125 y Fh(Lemma)h(4.2)24 b Ff(L)n(et)16 b Fm(H)21 b Ff(b)n(e)c(a)f(r)n (andom)g(op)n(er)n(ator)f(satisfying)i(Assumptions)f(2.1.i\){iv\).)22 b(L)n(et)60 2185 y Fm(f)28 b Fl(2)23 b Fm(C)207 2167 y Fv(1)203 2197 y Fy(0)244 2185 y Fo(\()p Fk(R)p Fo(\))c Ff(b)n(e)k(a)f(non-ne)n(gative)j(function)f(and)e Fm(I)k Fl(\033)d Fo(supp)18 b Fm(f)28 b Ff(any)22 b(c)n(omp)n(act)g(interval) 60 2245 y(such)d(that)g Fm(\016)e Fo(:=)f(dist)o(\()p Fm(@)s(I)t(;)8 b Fo(supp)17 b Fm(f)5 b Fo(\))16 b Fm(>)g Fo(0)p Ff(;)k(then)f(for)f(al)r(l)i Fm(')c Fl(2)g(D)q Fo(\()p Fm(H)t Fo(\))d Fl(\\)f(D)q Fo(\()p Fl(j)p Fm(X)t Fl(j)p Fo(\))20 b Ff(ther)n(e)e(exist)60 2305 y(c)n(onstants)24 b Fm(c)301 2312 y Fy(1)321 2305 y Ff(,)p Fm(c)357 2312 y Fy(2)399 2305 y Ff(and)f Fm(c)520 2312 y Fy(3)563 2305 y Ff(dep)n(ending)h(only)g(on)f Fm(')p Ff(,)i Fm(\016)f Ff(and)f Fl(k)p Fm(f)5 b Fl(k)1279 2313 y Fx(L)1303 2303 y Fg(1)1336 2313 y Fy(\()p Fi(R)-6 b Fy(\))1413 2305 y Ff(such)23 b(that)g(for)g(al)r(l)60 2365 y Fm(")14 b Fo(:=)f(1)p Fm(=T)21 b(>)14 b Fo(0)k Ff(we)g(have,)g(for)f Fk(P)11 b Ff(a.e.)22 b Fm(!)r Ff(:)112 2460 y Fo(1)p 106 2482 36 2 v 106 2528 a Fm(T)155 2426 y Fj(Z)205 2439 y Fx(T)183 2539 y Fy(0)241 2494 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(e)368 2473 y Fv(\000)p Fy(i)p Fx(tH)450 2461 y Fr(\()p Fs(!)q Fr(\))498 2494 y Fm(f)d Fo(\()p Fm(H)590 2473 y Fy(\()p Fx(!)q Fy(\))643 2494 y Fo(\))p Fm(')p Fl(k)719 2473 y Fy(2)738 2494 y Fm(dt)465 2623 y Fl(\024)13 b Fm(c)538 2630 y Fy(1)569 2623 y Fo(+)e Fm(c)639 2630 y Fy(2)659 2623 y Fm(b)680 2602 y Fy(2)699 2623 y Fo(\()p Fm(!)r Fo(\))g(+)g Fm(c)850 2630 y Fy(3)870 2623 y Fm(")901 2555 y Fj(Z)929 2668 y Fx(I)957 2623 y Fl(k)d(j)p Fm(X)t Fl(j)p Fo(\()p Fm(H)1125 2602 y Fy(\()p Fx(!)q Fy(\))1189 2623 y Fl(\000)j Fm(E)j Fo(+)d(i)1358 2589 y Fm(")p 1357 2612 25 2 v 1357 2657 a Fo(2)1386 2623 y(\))1405 2602 y Fv(\000)p Fy(1)1452 2623 y Fm(')p Fl(k)1509 2602 y Fy(2)1529 2623 y Fm(dE)38 b(:)j Fo(\(18\))60 2731 y Ff(The)18 b(r)n(andom)e(variable)j Fm(b)p Fo(\()p Fm(!)r Fo(\))e Ff(was)g(de\014ne)n(d)i(in)f(Assumption)f(2.1.iii\).)891 2856 y Fo(14)p eop %%Page: 15 15 15 14 bop 158 203 a Fd(Pro)q(of.)64 b Fo(W)l(e)18 b(will)f(drop)i(the)e (lab)q(el)h Fm(!)i Fo(in)e(the)g(pro)q(of.)28 b(Since)17 b Fm(H)23 b Fo(is)18 b(self-adjoin)o(t)f(and)60 264 y Fl(j)p Fm(X)t Fl(j)f Fo(is)g(closed,)g(w)o(e)g(ha)o(v)o(e)f(for)i Fm(")c Fo(=)h(1)p Fm(=T)23 b Fo([27)q(,)16 b(Lemma)e(1)i(on)h(p.)f (412])199 374 y(1)p 193 396 36 2 v 193 442 a Fm(T)242 340 y Fj(Z)292 353 y Fx(T)269 452 y Fy(0)327 407 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(e)454 387 y Fv(\000)p Fy(i)p Fx(tH)538 407 y Fm(f)d Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)706 387 y Fy(2)726 407 y Fm(dt)42 b Fl(\024)f Fo(e)8 b Fm(")952 340 y Fj(Z)980 452 y Fi(R)1014 407 y Fl(k)g(j)p Fm(X)t Fl(j)p Fo(e)1141 387 y Fv(\000)p Fx(")1191 373 y Fs(t)p 1190 379 16 2 v 1190 400 a Fr(2)1212 407 y Fo(\002\()p Fm(t)p Fo(\)e)1328 387 y Fv(\000)p Fy(i)p Fx(tH)1411 407 y Fm(f)d Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)1579 387 y Fy(2)1599 407 y Fm(dt)811 537 y Fo(=)896 503 y(e)j Fm(")p 896 525 54 2 v 896 571 a Fo(2)p Fm(\031)963 469 y Fj(Z)991 582 y Fi(R)1025 537 y Fl(k)g(j)p Fm(X)t Fl(j)p Fm(R)1167 544 y Fx(")1186 537 y Fo(\()p Fm(E)s Fo(\))p Fm(f)d Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)1431 516 y Fy(2)1451 537 y Fm(dE)20 b(;)137 b Fo(\(19\))60 675 y(where)15 b Fm(R)237 682 y Fx(")255 675 y Fo(\()p Fm(E)s Fo(\))f(=)g(\()p Fm(H)f Fl(\000)c Fm(E)j Fo(+)d(i)p Fm("=)p Fo(2\))717 657 y Fv(\000)p Fy(1)780 675 y Fo(and)15 b(\002\()p Fm(t)p Fo(\))g(is)g(the)g(Hea)o(viside)f(function.)20 b(In)15 b(order)h(to)60 735 y(b)q(ound)j(the)e(righ)o(t-hand)h(side)f(of)22 b(\(19\))q(,)17 b(w)o(e)g(split)g(the)h(range)g(of)g(in)o(tegration)f (o)o(v)o(er)g Fm(E)j Fo(in)o(to)60 796 y(t)o(w)o(o)c(parts)h(and)g (\014x)f Fm(j)h Fl(2)d(f)p Fo(1)p Fm(;)8 b(:)g(:)g(:)g(;)16 b(d)p Fl(g)136 862 y Fj(Z)164 975 y Fi(R)-6 b Fv(n)o Fx(I)234 930 y Fl(k)p Fm(X)299 937 y Fx(j)317 930 y Fm(R)354 937 y Fx(")373 930 y Fo(\()p Fm(E)s Fo(\))p Fm(f)5 b Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)618 909 y Fy(2)638 930 y Fm(dE)464 1072 y Fl(\024)13 b Fo(4)548 1002 y Fj(\022)586 1004 y(Z)613 1117 y Fi(R)-6 b Fv(n)p Fx(I)683 1072 y Fl(k)p Fm(R)745 1079 y Fx(")764 1072 y Fo(\()p Fm(E)s Fo(\))p Fm(f)5 b Fo(\()p Fm(H)t Fo(\))p Fm(X)992 1079 y Fx(j)1011 1072 y Fm(')p Fl(k)1068 1051 y Fy(2)1087 1072 y Fm(dE)535 b Fo(\(20\))558 1210 y(+)607 1142 y Fj(Z)635 1255 y Fi(R)-6 b Fv(n)p Fx(I)705 1210 y Fl(k)p Fo([)p Fm(X)784 1217 y Fx(j)802 1210 y Fm(;)8 b(f)858 1175 y Fr(1)p 858 1181 16 2 v 858 1202 a(2)881 1210 y Fo(\()p Fm(H)t Fo(\)])p Fm(f)1011 1175 y Fr(1)p 1011 1181 V 1011 1202 a(2)1033 1210 y Fo(\()p Fm(H)t Fo(\))p Fm(R)1152 1217 y Fx(")1171 1210 y Fo(\()p Fm(E)s Fo(\))p Fm(')p Fl(k)1305 1189 y Fy(2)1324 1210 y Fm(dE)298 b Fo(\(21\))558 1347 y(+)607 1279 y Fj(Z)635 1392 y Fi(R)-6 b Fv(n)p Fx(I)705 1347 y Fl(k)p Fm(f)764 1313 y Fr(1)p 764 1319 V 764 1340 a(2)786 1347 y Fo(\()p Fm(H)t Fo(\))p Fm(R)905 1354 y Fx(")924 1347 y Fo(\()p Fm(E)s Fo(\)[)p Fm(H)q(;)8 b(X)1118 1354 y Fx(j)1137 1347 y Fo(])p Fm(R)1188 1354 y Fx(")1206 1347 y Fo(\()p Fm(E)s Fo(\))p Fm(f)1317 1313 y Fr(1)p 1317 1319 V 1317 1340 a(2)1339 1347 y Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)1478 1327 y Fy(2)1498 1347 y Fm(dE)124 b Fo(\(22\))558 1489 y(+)612 1422 y Fj(Z)640 1534 y Fi(R)-6 b Fv(n)p Fx(I)710 1489 y Fl(k)p Fm(f)769 1455 y Fr(1)p 769 1461 V 769 1482 a(4)791 1489 y Fo(\()p Fm(H)t Fo(\))p Fm(R)910 1496 y Fx(")929 1489 y Fo(\()p Fm(E)s Fo(\))p Fm(f)1040 1455 y Fr(1)p 1040 1461 V 1040 1482 a(4)1063 1489 y Fo(\()p Fm(H)t Fo(\)[)p Fm(X)1199 1496 y Fx(j)1217 1489 y Fm(;)8 b(f)1273 1455 y Fr(1)p 1273 1461 V 1273 1482 a(2)1296 1489 y Fo(\()p Fm(H)t Fo(\)])p Fm(')p Fl(k)1449 1469 y Fy(2)1468 1489 y Fm(dE)1532 1419 y Fj(\023)1594 1489 y Fm(:)75 b Fo(\(23\))60 1634 y(Since)15 b(dist\()p Fm(@)s(I)t(;)8 b Fo(supp)16 b Fm(f)5 b Fo(\))14 b Fm(>)g Fo(0,)i(one)g(can)h(b)q(ound)g(the)f(term)f(\(20\)) i(from)e(ab)q(o)o(v)o(e)h(b)o(y)366 1706 y Fj(Z)393 1818 y Fi(R)-6 b Fv(n)p Fx(I)463 1774 y Fl(k)p Fm(R)525 1781 y Fx(")544 1774 y Fo(\()p Fm(E)s Fo(\))p Fm(f)5 b Fo(\()p Fm(H)t Fo(\))p Fl(k)757 1753 y Fy(2)777 1774 y Fl(k)p Fm(X)842 1781 y Fx(j)861 1774 y Fm(')p Fl(k)918 1753 y Fy(2)937 1774 y Fm(dE)45 b Fl(\024)1128 1740 y Fo(2)p Fl(k)p Fm(f)5 b Fl(k)1231 1722 y Fy(2)1231 1752 y Fv(1)p 1128 1762 141 2 v 1187 1808 a Fm(\016)1274 1774 y Fl(k)p Fm(X)1339 1781 y Fx(j)1358 1774 y Fm(')p Fl(k)1415 1753 y Fy(2)1451 1774 y Fm(:)218 b Fo(\(24\))60 1918 y(W)l(e)20 b(no)o(w)h(treat)f(the)g(term)f(\(21\))q(.)33 b(Let)c(~)-32 b Fm(')844 1925 y Fx(E)895 1918 y Fl(\021)20 b Fm(R)991 1925 y Fx(")1010 1918 y Fo(\()p Fm(E)s Fo(\))p Fm(')p Fo(;)i(then)e(ob)o(viously)l(,)g(since)g Fm(')g Fo(is)g(in)60 1978 y Fl(D)q Fo(\()p Fm(H)t Fo(\),)g Fl(k)8 b Fo(~)-32 b Fm(')272 1985 y Fx(E)302 1978 y Fl(k)19 b Fo(and)h Fl(k)p Fm(H)12 b Fo(~)-32 b Fm(')545 1985 y Fx(E)575 1978 y Fl(k)19 b Fo(are)g(\014nite)f(and)i(from)e(Lemma)f(4.1)j(one)f (has,)h(b)o(y)f(using)g(the)60 2038 y(equalit)o(y)c Fm(R)283 2045 y Fx(")301 2038 y Fo(\()p Fm(E)s Fo(\))f(=)g Fl(\000)p Fo(i)505 1998 y Fj(R)537 2012 y Fv(1)528 2056 y Fy(0)583 2038 y Fo(e)605 2020 y Fv(\000)p Fx("t=)p Fy(2)699 2038 y Fo(e)721 2020 y Fy(i)p Fx(t)p Fy(\()p Fx(H)s Fv(\000)p Fx(E)r Fy(\))859 2038 y Fm(dt)326 2184 y Fl(k)8 b(j)p Fm(X)t Fl(j)g Fo(~)-32 b Fm(')463 2191 y Fx(E)493 2184 y Fl(k)42 b(\024)640 2117 y Fj(Z)690 2130 y Fy(+)p Fv(1)668 2229 y Fy(0)763 2184 y Fo(e)785 2164 y Fv(\000)817 2150 y Fs("t)p 817 2156 27 2 v 823 2177 a Fr(2)850 2184 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(e)977 2164 y Fy(i)p Fx(t)p Fy(\()p Fx(H)s Fv(\000)p Fx(E)r Fy(\))1116 2184 y Fm(')p Fl(k)p Fm(dt)560 2321 y Fl(\024)41 b Fm(c)669 2253 y Fj(Z)719 2266 y Fy(+)p Fv(1)697 2366 y Fy(0)792 2321 y Fo(e)814 2300 y Fv(\000)846 2287 y Fs("t)p 846 2293 V 852 2313 a Fr(2)880 2265 y Fj(\020)909 2321 y Fm(t)p Fo(\()p Fl(k)p Fm(')p Fl(k)11 b Fo(+)g Fl(k)p Fm(H)t(')p Fl(k)p Fo(\))g(+)g Fl(k)d(j)p Fm(X)t Fl(j)p Fm(')p Fl(k)p Fo(\))1474 2265 y Fj(\021)560 2420 y Fm(<)42 b Fl(1)16 b Fm(:)963 b Fo(\(25\))60 2530 y(Therefore)32 b(~)-32 b Fm(')321 2537 y Fx(E)375 2530 y Fo(is)24 b(in)g(the)h(set)f Fm(S)j Fo(de\014ned)d(in)h(Lemma)d(4.1.)46 b(By)24 b(writing)g Fm(f)1567 2512 y Fy(1)p Fx(=)p Fy(2)1622 2530 y Fo(\()p Fm(H)t Fo(\))k(=)60 2561 y Fj(R)84 2619 y Fi(R)124 2584 y Fj(d)118 2602 y Fm(f)147 2587 y Fy(1)p Fx(=)p Fy(2)202 2602 y Fo(\()p Fm(t)p Fo(\)e)280 2583 y Fy(i)p Fx(tH)336 2602 y Fm(dt)p Fo(,)41 b(one)c(can)g(pro)o(v)o(e)e(with)i(the)f(same)g (argumen)o(ts)g(as)h(ab)q(o)o(v)o(e)g(that)60 2666 y Fm(f)89 2648 y Fy(1)p Fx(=)p Fy(2)144 2666 y Fo(\()p Fm(H)t Fo(\))p Fm(R)263 2673 y Fx(")282 2666 y Fo(\()p Fm(E)s Fo(\))p Fm(')14 b Fl(2)g Fm(S)s Fo(.)21 b(Then)16 b(w)o(e)g(get)g(from)g(Lemma)e(4.1)891 2856 y(15)p eop %%Page: 16 16 16 15 bop 146 304 a Fl(k)p Fo([)p Fm(f)219 270 y Fr(1)p 219 276 16 2 v 219 297 a(2)241 304 y Fo(\()p Fm(H)t Fo(\))p Fm(;)8 b(X)385 311 y Fx(j)404 304 y Fo(])p Fm(f)452 270 y Fr(1)p 452 276 V 452 297 a(2)474 304 y Fm(R)511 311 y Fx(")529 304 y Fo(\()p Fm(E)s Fo(\))p Fm(')p Fl(k)187 410 y(\024)94 b Fo(sup)268 453 y Fx(\030)q Fv(2H)p Fx(;)p Fv(k)p Fx(\030)q Fv(k)p Fy(=1)453 342 y Fj(Z)481 455 y Fi(R)515 342 y Fj(Z)565 355 y Fx(t)543 455 y Fy(0)588 410 y Fl(j)605 383 y Fj(c)602 410 y Fm(f)636 378 y Fr(1)p 636 384 V 636 405 a(2)659 410 y Fo(\()p Fm(t)p Fo(\))p Fl(j)8 b(jh)p Fm(\030)r(;)16 b Fo(e)845 389 y Fv(\000)p Fy(i\()p Fx(s)p Fv(\000)p Fx(t)p Fy(\))p Fx(H)1000 410 y Fo([)p Fm(H)q(;)8 b(X)1117 417 y Fx(j)1136 410 y Fo(]e)1172 389 y Fy(i)p Fx(sH)1231 410 y Fm(f)1265 376 y Fr(1)p 1265 382 V 1265 402 a(2)1287 410 y Fo(\()p Fm(H)t Fo(\))p Fm(R)1406 417 y Fx(")1425 410 y Fo(\()p Fm(E)s Fo(\))p Fm(')p Fl(ij)p Fm(ds)g(dt)187 551 y Fl(\024)268 483 y Fj(Z)295 596 y Fi(R)330 551 y Fm(t)p Fl(j)365 524 y Fj(c)362 551 y Fm(f)395 519 y Fr(1)p 395 525 V 395 545 a(2)418 551 y Fo(\()p Fm(t)p Fo(\))p Fl(j)g(k)p Fo([)p Fm(H)q(;)g(X)638 558 y Fx(j)656 551 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))826 530 y Fv(\000)p Fy(1)872 551 y Fl(k)d(k)p Fo(\()p Fm(H)16 b Fo(+)11 b(i\))p Fm(f)1121 516 y Fr(1)p 1121 522 V 1121 543 a(4)1143 551 y Fo(\()p Fm(H)t Fo(\))p Fl(k)d(k)p Fm(R)1320 558 y Fx(")1339 551 y Fo(\()p Fm(E)s Fo(\))p Fm(f)1450 516 y Fr(1)p 1450 522 V 1450 543 a(4)1472 551 y Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)p Fm(dt)17 b(;)60 678 y Fo(and,)f(using)h(Assumption)e(2.1.iii\),)f(\(21\))j(is)f(b)q (ounded)i(ab)q(o)o(v)o(e)e(b)o(y)110 748 y Fj(Z)137 861 y Fi(R)-6 b Fv(n)p Fx(I)207 816 y Fl(k)p Fo([)p Fm(f)280 781 y Fr(1)p 280 787 V 280 808 a(2)302 816 y Fo(\()p Fm(H)t Fo(\))p Fm(;)8 b(X)446 823 y Fx(j)465 816 y Fo(])p Fm(f)513 781 y Fr(1)p 513 787 V 513 808 a(2)535 816 y Fm(R)572 823 y Fx(")591 816 y Fo(\()p Fm(E)s Fo(\))p Fm(')p Fl(k)725 795 y Fy(2)744 816 y Fm(dE)694 957 y Fl(\024)14 b Fm(b)768 936 y Fy(2)787 957 y Fo(\(max)817 988 y Fx(E)r Fv(2)p Fx(I)905 957 y Fl(j)p Fm(E)s Fl(j)d Fo(+)g(1\))1075 936 y Fy(2)1095 901 y Fj(\020)1133 889 y(Z)1161 1002 y Fi(R)1195 957 y Fm(t)p Fl(j)1230 930 y Fj(c)1227 957 y Fm(f)1261 925 y Fr(1)p 1261 931 V 1261 951 a(2)1283 957 y Fo(\()p Fm(t)p Fo(\))p Fl(j)p Fm(dt)1396 901 y Fj(\021)1425 912 y Fy(2)1466 923 y Fo(2)p Fl(k)p Fm(f)5 b Fl(k)1569 930 y Fv(1)p 1466 945 141 2 v 1525 991 a Fm(\016)1628 957 y(:)41 b Fo(\(26\))60 1084 y(T)l(o)17 b(b)q(ound)g(the)f(term)f(\(22\))i(one)f(writes)328 1141 y Fj(Z)355 1254 y Fi(R)-6 b Fv(n)p Fx(I)425 1209 y Fl(k)p Fm(f)484 1175 y Fr(1)p 484 1181 16 2 v 484 1202 a(2)507 1209 y Fo(\()p Fm(H)t Fo(\))p Fm(R)626 1216 y Fx(")644 1209 y Fo(\()p Fm(E)s Fo(\)[)p Fm(H)q(;)8 b(X)838 1216 y Fx(j)857 1209 y Fo(])p Fm(R)908 1216 y Fx(")926 1209 y Fo(\()p Fm(E)s Fo(\))p Fm(f)1037 1175 y Fr(1)p 1037 1181 V 1037 1202 a(2)1060 1209 y Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)1199 1188 y Fy(2)1218 1209 y Fm(dE)369 1347 y Fl(\024)449 1279 y Fj(Z)477 1392 y Fi(R)-6 b Fv(n)p Fx(I)547 1291 y Fj(\020)577 1347 y Fl(k)p Fm(f)636 1313 y Fr(1)p 636 1319 V 636 1339 a(2)658 1347 y Fo(\()p Fm(H)t Fo(\))p Fm(R)777 1354 y Fx(")796 1347 y Fo(\()p Fm(E)s Fo(\))p Fl(k)898 1326 y Fy(2)917 1347 y Fl(k)p Fo([)p Fm(H)q(;)8 b(X)1059 1354 y Fx(j)1078 1347 y Fo(]\()p Fm(H)15 b Fo(+)c(i\))1248 1326 y Fv(\000)p Fy(1)1295 1347 y Fl(k)1320 1326 y Fy(2)547 1474 y Fl(\002)16 b(k)p Fo(\()p Fm(H)f Fo(+)c(i\))p Fm(f)817 1440 y Fr(1)p 817 1446 V 817 1466 a(4)839 1474 y Fo(\()p Fm(H)t Fo(\))p Fl(k)946 1453 y Fy(2)966 1474 y Fl(k)p Fm(f)1025 1440 y Fr(1)p 1025 1446 V 1025 1466 a(4)1048 1474 y Fo(\()p Fm(H)t Fo(\))p Fm(R)1167 1481 y Fx(")1185 1474 y Fo(\()p Fm(E)s Fo(\))p Fl(k)1287 1453 y Fy(2)1307 1474 y Fl(k)p Fm(')p Fl(k)1389 1453 y Fy(2)1408 1474 y Fm(dE)1472 1419 y Fj(\021)369 1596 y Fl(\024)456 1562 y Fo(2)p Fm(b)501 1544 y Fy(2)p 454 1585 68 2 v 454 1630 a Fo(3)p Fm(\016)502 1616 y Fy(3)527 1596 y Fl(k)p Fm(f)5 b Fl(k)606 1575 y Fy(2)606 1608 y Fv(1)643 1596 y Fo(\(max)673 1627 y Fx(E)r Fv(2)p Fx(I)761 1596 y Fl(j)p Fm(E)s Fl(j)11 b Fo(+)g(1\))931 1575 y Fy(2)951 1596 y Fl(k)p Fm(')p Fl(k)1033 1575 y Fy(2)1069 1596 y Fm(:)600 b Fo(\(27\))60 1716 y(Finally)l(,)14 b(for)j(\(23\))g(one)f(has)154 1774 y Fj(Z)181 1887 y Fi(R)-6 b Fv(n)p Fx(I)251 1842 y Fl(k)p Fm(f)310 1808 y Fr(1)p 310 1814 16 2 v 310 1834 a(4)333 1842 y Fo(\()p Fm(H)t Fo(\))p Fm(R)452 1849 y Fx(")470 1842 y Fo(\()p Fm(E)s Fo(\))p Fm(f)582 1808 y Fr(1)p 582 1814 V 582 1834 a(4)604 1842 y Fo(\()p Fm(H)t Fo(\)[)p Fm(X)740 1849 y Fx(j)758 1842 y Fm(;)8 b(f)815 1808 y Fr(1)p 815 1814 V 815 1834 a(2)837 1842 y Fo(\()p Fm(H)t Fo(\)])p Fm(')p Fl(k)990 1821 y Fy(2)1009 1842 y Fm(dE)195 1979 y Fl(\024)275 1924 y Fj(\020)313 1912 y(Z)341 2024 y Fi(R)-6 b Fv(n)p Fx(I)411 1979 y Fm(f)445 1945 y Fr(1)p 445 1951 V 445 1972 a(4)468 1979 y Fo(\()p Fm(H)t Fo(\))p Fm(R)587 1986 y Fx(")605 1979 y Fo(\()p Fm(E)s Fo(\))p Fl(k)707 1959 y Fy(2)727 1979 y Fm(dE)791 1924 y Fj(\021\020)911 1979 y Fo(sup)859 2022 y Fx(\030)q Fv(2H)p Fx(;)p Fv(k)p Fx(\030)q Fv(k)p Fy(=1)1045 1979 y Fl(jh)p Fm(f)1112 1945 y Fr(1)p 1112 1951 V 1112 1972 a(4)1135 1979 y Fo(\()p Fm(H)t Fo(\))p Fm(\030)r(;)17 b Fo([)p Fm(f)1319 1945 y Fr(1)p 1319 1951 V 1319 1972 a(2)1341 1979 y Fo(\()p Fm(H)t Fo(\))p Fm(;)8 b(X)1485 1986 y Fx(j)1504 1979 y Fo(])p Fl(kij)1576 1924 y Fj(\021)1605 1935 y Fy(2)195 2146 y Fl(\024)280 2112 y Fo(2)p Fl(k)p Fm(f)d Fl(k)388 2068 y Fr(1)p 389 2074 V 389 2095 a(2)383 2117 y Fv(1)p 280 2134 141 2 v 339 2180 a Fm(\016)435 2146 y Fo(sup)436 2187 y Fx(\030)q Fv(2H)516 2090 y Fj(\020)546 2146 y Fl(j)568 2078 y Fj(Z)596 2191 y Fi(R)634 2119 y Fj(c)630 2146 y Fm(f)665 2114 y Fr(1)p 665 2120 16 2 v 665 2140 a(2)687 2146 y Fo(\()p Fm(t)p Fo(\))p Fl(h)p Fm(f)796 2112 y Fr(1)p 796 2118 V 796 2138 a(4)818 2146 y Fo(\()p Fm(H)t Fo(\))p Fm(\030)r(;)17 b Fo([e)990 2125 y Fy(i)p Fx(tH)1046 2146 y Fm(;)8 b(X)1108 2153 y Fx(j)1126 2146 y Fo(])p Fm(')p Fl(i)p Fm(dt)p Fl(j)1248 2090 y Fj(\021)1278 2102 y Fy(2)195 2308 y Fl(\024)280 2275 y Fo(2)p Fl(k)p Fm(f)d Fl(k)388 2231 y Fr(1)p 389 2237 V 389 2257 a(2)383 2280 y Fv(1)p 280 2297 141 2 v 339 2342 a Fm(\016)426 2253 y Fj(\020)464 2308 y Fo(sup)466 2349 y Fx(\030)q Fv(2H)546 2240 y Fj(Z)574 2353 y Fi(R)608 2308 y Fl(j)625 2282 y Fj(c)622 2308 y Fm(f)656 2276 y Fr(1)p 656 2282 16 2 v 656 2303 a(2)679 2308 y Fo(\()p Fm(t)p Fo(\))p Fl(j)757 2240 y Fj(Z)806 2254 y Fx(t)784 2353 y Fy(0)829 2308 y Fl(jh)p Fm(f)897 2274 y Fr(1)p 897 2280 V 897 2301 a(4)919 2308 y Fo(\()p Fm(H)t Fo(\))p Fm(\030)r(;)17 b Fo(e)1077 2288 y Fv(\000)p Fy(i\()p Fx(s)p Fv(\000)p Fx(t)p Fy(\))p Fx(H)1232 2308 y Fo([)p Fm(H)q(;)8 b(X)1349 2315 y Fx(j)1367 2308 y Fo(]e)1403 2288 y Fy(i)p Fx(sH)1462 2308 y Fm(')p Fl(ij)p Fm(ds)g(dt)1626 2253 y Fj(\021)1657 2264 y Fy(2)195 2451 y Fl(\024)280 2417 y Fo(2)p Fm(b)325 2399 y Fy(2)345 2417 y Fl(k)p Fm(f)d Fl(k)424 2424 y Fv(1)p 280 2439 182 2 v 359 2485 a Fm(\016)475 2451 y Fo(max)486 2482 y Fx(E)r Fv(2)p Fx(I)566 2451 y Fo(\()p Fl(j)p Fm(E)s Fl(j)11 b Fo(+)g(1\))755 2430 y Fy(2)775 2451 y Fl(k)p Fm(')p Fl(k)857 2430 y Fy(2)876 2395 y Fj(\020)914 2383 y(Z)942 2496 y Fi(R)976 2451 y Fm(t)p Fl(j)1011 2424 y Fj(c)1008 2451 y Fm(f)1042 2419 y Fr(1)p 1042 2425 16 2 v 1042 2445 a(2)1064 2451 y Fo(\()p Fm(t)p Fo(\))p Fl(j)1134 2395 y Fj(\021)1163 2406 y Fy(2)1199 2451 y Fm(:)470 b Fo(\(28\))60 2578 y(Inequalities)14 b(\(24\))q(-\(28\))j(imply)d(that)j(there)e(exists)h Fm(c)e(<)g Fl(1)i Fo(indep)q(enden)o(t)f(of)i Fm(!)h Fo(suc)o(h)e(that)524 2635 y Fj(Z)552 2748 y Fi(R)-6 b Fv(n)o Fx(I)622 2703 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(e)749 2682 y Fv(\000)p Fy(i)p Fx(tH)832 2703 y Fm(f)d Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)1000 2682 y Fy(2)1020 2703 y Fm(dE)45 b Fl(\024)c Fm(c)8 b(b)1256 2682 y Fy(2)1292 2703 y Fm(:)377 b Fo(\(29\))891 2856 y(16)p eop %%Page: 17 17 17 16 bop 60 203 a Fo(As)16 b(to)h(the)f(remaining)e(con)o(tribution)i (to)h(\(19\),)f(w)o(e)g(write)102 350 y Fm(")133 282 y Fj(Z)160 395 y Fx(I)189 350 y Fl(k)p Fm(X)254 357 y Fx(j)272 350 y Fm(R)309 357 y Fx(")328 350 y Fo(\()p Fm(E)s Fo(\))p Fm(f)5 b Fo(\()p Fm(H)t Fo(\))p Fm(')p Fl(k)573 329 y Fy(2)593 350 y Fm(dE)402 479 y Fl(\024)13 b Fo(2)p Fm(")509 412 y Fj(Z)538 524 y Fx(I)566 479 y Fl(k)p Fo([)p Fm(X)645 486 y Fx(j)663 479 y Fm(;)8 b(f)d Fo(\()p Fm(H)t Fo(\)])p Fm(R)847 486 y Fx(")865 479 y Fo(\()p Fm(E)s Fo(\))p Fm(')p Fl(k)999 459 y 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1158 y Fx(")1343 1151 y Fo(\()p Fm(E)s Fo(\))p Fm(')p Fl(k)1477 1130 y Fy(2)1496 1151 y Fm(dt)g(dE)20 b(:)41 b Fo(\(31\))60 1297 y(According)23 b(to)g(the)g(fact)h(that)f Fm(R)709 1304 y Fx(")728 1297 y Fo(\()p Fm(E)s Fo(\))p Fm(')g Fo(and)h Fm(f)996 1265 y Fr(1)p 996 1271 V 996 1292 a(2)1019 1297 y Fo(\()p Fm(H)t Fo(\))p Fm(R)1138 1304 y Fx(")1156 1297 y Fo(\()p Fm(E)s Fo(\))p Fm(')f Fo(are)h(b)q(oth)g(in)f(the)g(set)g Fm(S)60 1358 y Fo(de\014ned)16 b(in)f(\(10\))q(,)h(one)g(can)g(pro)o(v)o(e)f(as)i(ab)q(o)o(v)o(e)f (with)g(the)g(help)f(of)h(Lemma)e(4.1)j(that)f(eac)o(h)g(of)60 1418 y(the)g(t)o(w)o(o)g(terms)f(in)h(\(31\))h(are)f(b)q(ounded)h(ab)q (o)o(v)o(e)f(b)o(y)102 1509 y Fj(\020)140 1497 y(Z)167 1609 y Fi(R)202 1564 y Fm(t)p Fl(j)237 1538 y Fj(c)234 1564 y Fm(f)267 1533 y Fr(1)p 267 1539 V 267 1559 a(2)290 1564 y Fo(\()p Fm(t)p Fo(\))p Fl(j)p Fm(dt)403 1509 y Fj(\021)432 1520 y Fy(2)451 1564 y Fl(k)p Fo([)p Fm(H)q(;)8 b(X)593 1571 y Fx(j)612 1564 y Fo(]\()p Fm(H)15 b 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Fy(\))1157 1762 y Fo(\()p Fm(E)14 b Fo(+)d(i)p Fm(")p Fo(\))d Fh(1)1367 1769 y Fx(q)1394 1762 y Fl(k)1419 1741 y Fy(2)1447 1762 y Fl(k)g Fh(1)1508 1769 y Fx(q)1535 1762 y Fm(')p Fl(k)1592 1741 y Fy(2)1620 1762 y Fm(;)49 b Fo(\(36\))60 1924 y(where)16 b(\003)235 1931 y Fx(L)259 1936 y Fr(0)278 1924 y Fo(\(0\))h(is)f(de\014ned)g(as)h(in)f(Section)f (3.)22 b(Then,)173 1985 y Fj(Z)201 2098 y Fy(\012)237 1983 y Fj(\022)273 1985 y(Z)301 2098 y Fx(I)329 2053 y Fl(k)8 b Fh(1)390 2060 y Fx(q)407 2051 y Fg(0)429 2053 y Fm(R)466 2033 y Fy(\()p Fx(!)q Fy(\))519 2053 y Fo(\()p Fm(E)14 b Fo(+)d(i)p Fm(")p Fo(\))p Fm(')p Fl(k)750 2033 y Fy(2)769 2053 y Fm(dE)833 1983 y Fj(\023)879 2053 y Fm(d)p Fk(P)l Fo(\()p Fm(!)r Fo(\))215 2189 y Fl(\024)41 b Fo(\(2)p Fm(L)371 2196 y Fy(0)402 2189 y Fo(+)11 b(1\))494 2169 y Fx(d)594 2142 y Fj(X)523 2252 y Fx(q)q Fv(2)p Fy(\003)588 2258 y Fs(L)609 2265 y Fr(0)628 2252 y Fy(\(0\))p Fv(\\)p Fi(Z)722 2242 y Fs(d)746 2122 y Fj(Z)773 2234 y Fy(\012)809 2119 y Fj(\022)846 2122 y(Z)873 2234 y Fx(I)902 2189 y Fl(k)d Fh(1)963 2196 y Fx(q)980 2187 y Fg(0)1001 2189 y Fm(R)1038 2169 y Fy(\()p Fx(!)q Fy(\))1092 2189 y Fo(\()p Fm(E)14 b Fo(+)d(i)p Fm(")p Fo(\))d Fh(1)1301 2196 y Fx(q)1329 2189 y Fm(')p Fl(k)1386 2169 y Fy(2)1405 2189 y Fm(dE)1469 2119 y Fj(\023)1515 2189 y Fm(d)p Fk(P)l Fo(\()p Fm(!)r Fo(\))215 2374 y Fl(\024)41 b Fo(\(2)p Fm(L)371 2381 y Fy(0)402 2374 y Fo(+)11 b(1\))494 2353 y Fx(d)594 2326 y Fj(X)523 2436 y Fx(q)q Fv(2)p Fy(\003)588 2442 y Fs(L)609 2449 y Fr(0)628 2436 y Fy(\(0\))p Fv(\\)p Fi(Z)722 2427 y Fs(d)746 2288 y Fj(\()794 2306 y(Z)822 2419 y Fy(\012)847 2425 y Fs(k)866 2419 y Fy(\()p Fx(q)q(;q)q Fy(`\))958 2303 y Fj(\022)1003 2306 y(Z)1030 2419 y Fx(I)1058 2374 y Fm(\032)p Fo(\()p Fm(L)1135 2381 y Fx(k)1157 2374 y Fo(\))1176 2318 y Fj(\020)1206 2374 y Fl(k)p Fm(R)1268 2353 y Fy(\()p Fx(!)q Fy(\))1321 2374 y Fo(\()p Fm(E)j Fo(+)d(i)p Fm(")p Fo(\))d Fh(1)1531 2381 y Fx(q)1558 2374 y Fl(k)1583 2353 y Fy(2)531 2543 y Fo(+)p 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203 y Fo(\()p Fm(q)r(;)8 b(q)1226 185 y Fv(0)1236 203 y Fo(\).)20 b(F)l(or)13 b(these)f Fm(!)i Fo(w)o(e)e(pro)q(ceed)60 264 y(b)o(y)18 b(b)q(ounding)i(the)e(resolv)o(en)o(ts)g(according)h(to)g Fl(k)p Fm(R)1007 246 y Fy(\()p Fx(!)q Fy(\))1060 264 y Fo(\()p Fm(E)d Fo(+)c(i)p Fm(")p Fo(\))p Fl(k)18 b(\024)f Fm(")1359 246 y Fv(\000)p Fy(1)1406 264 y Fo(.)29 b(This)19 b(yields)e(the)60 324 y(term)e(2)p Fm(\032)p Fo(\()p Fm(L)278 331 y Fx(k)300 324 y Fo(\))p Fm(")342 306 y Fv(\000)p Fy(2)389 324 y Fl(j)p Fm(I)t Fl(j)g Fo(in)h(\(35\).)22 b(F)l(or)16 b Fm(!)22 b(=)-30 b Fl(2)14 b Fo(\012)852 331 y Fx(k)874 324 y Fo(\()p Fm(q)r(;)8 b(q)963 306 y Fv(0)973 324 y Fo(\))16 b(w)o(e)g(use)595 379 y Fj(Z)623 492 y Fx(I)659 447 y Fl(k)8 b Fm(R)729 426 y Fy(\()p Fx(!)q Fy(\))783 447 y Fo(\()p Fm(E)14 b Fo(+)d(i)p Fm(")p Fo(\))p Fm( )r Fl(k)1016 426 y Fy(2)1034 447 y Fm(dE)18 b Fl(\024)1170 413 y Fm(\031)p 1170 435 30 2 v 1173 481 a(")1213 447 y(;)456 b Fo(\(38\))60 572 y(whic)o(h)13 b(is)g(v)m(alid)g(for)g(all)g Fm( )i Fl(2)g(H)e Fo(with)h Fl(k)p Fm( )r Fl(k)f Fo(=)h(1,)f(and)h(the)f(Multi-Scale)f(Assumption)g (\(M1\),)60 632 y(viz.)j Fk(P)l Fo(\(\012)8 b Fl(n)j Fo(\012)317 639 y Fx(k)339 632 y Fo(\()p Fm(q)r(;)d(q)428 614 y Fv(0)438 632 y Fo(\)\))14 b Fl(\024)g Fm(L)576 608 y Fv(\000)p Fx(p)576 646 y(k)623 632 y Fo(,)i(giving)g(the)g (second)g(term)f(in)h(\(35\).)133 692 y(Finally)j(w)o(e)g(deriv)o(e)f (\(32\))j(as)g(follo)o(ws.)32 b(Giv)o(en)18 b Fm(")p Fo(,)j(there)e(exists)g Fm(k)j Fo(=)e Fm(k)r Fo(\()p Fm(")p Fo(\))g Fl(2)g Fk(N)h Fo(suc)o(h)60 752 y(that)714 851 y(e)736 830 y Fx(L)765 818 y Fs(\027)760 842 y(k)q Fg(\000)p Fr(1)834 851 y Fl(\024)13 b Fm(")909 830 y Fv(\000)p Fy(1)970 851 y Fm(<)h Fo(e)1044 830 y Fx(L)1073 818 y Fs(\027)1068 842 y(k)1103 851 y Fm(:)566 b Fo(\(39\))60 949 y(Th)o(us)17 b(w)o(e)e(get,)h(for)h(some)e(constan)o(ts)i Fm(c)784 956 y Fy(0)820 949 y Fo(and)g Fm(c)936 956 y Fy(4)157 1089 y Fk(E)187 1004 y Fj(\()238 1021 y(Z)266 1134 y Fx(I)286 1089 y Fm(")p Fl(k)8 b(j)p Fm(X)t Fl(j)p Fm(R)451 1068 y Fy(\()p Fx(!)q Fy(\))504 1089 y Fo(\()p Fm(E)14 b Fo(+)d(i)p Fm(")p Fo(\))p Fm(')p Fl(k)735 1068 y Fy(2)754 1089 y Fm(dE)818 1004 y Fj(\))233 1240 y Fl(\024)j Fm(")p Fk(E)347 1170 y Fj(\032)388 1172 y(Z)415 1285 y Fx(I)444 1240 y Fl(k)p Fm(\037)500 1248 y Fy(\003)524 1254 y Fs(L)545 1261 y Fr(0)562 1254 y(+2)p Fs(L)621 1262 y(k)643 1248 y Fy(\(0\))690 1240 y Fl(j)p Fm(X)t Fl(j)p Fm(R)799 1219 y Fy(\()p Fx(!)q Fy(\))852 1240 y Fo(\()p Fm(E)g Fo(+)d(i)p Fm(")p Fo(\))p Fm(')p Fl(k)1083 1219 y Fy(2)1102 1240 y Fm(dE)1166 1170 y Fj(\033)297 1385 y Fo(+)g Fm(")396 1323 y Fv(1)377 1338 y Fj(X)382 1444 y Fx(j)r Fy(=)p Fx(k)659 1338 y Fj(X)474 1446 y Fx(L)498 1451 y Fr(0)515 1446 y Fy(+2)p Fx(L)584 1451 y Fs(j)600 1446 y Fx(<)p Fv(k)p Fx(q)662 1436 y Fg(0)674 1446 y Fv(k)692 1450 y Fg(1)724 1446 y Fv(\024)p Fx(L)775 1451 y Fr(0)793 1446 y Fy(+2)p Fx(L)862 1451 y Fs(j)q Fr(+1)916 1385 y Fk(E)955 1315 y Fj(\032)995 1318 y(Z)1023 1430 y Fx(I)1051 1385 y Fl(k)d Fh(1)1112 1392 y Fx(q)1129 1383 y Fg(0)1151 1385 y Fl(j)p Fm(X)t Fl(j)p Fm(R)1260 1365 y Fy(\()p Fx(!)q Fy(\))1313 1385 y Fo(\()p Fm(E)14 b Fo(+)d(i)p Fm(")p Fo(\))p Fm(')p Fl(k)1544 1365 y Fy(2)1563 1385 y Fm(dE)1627 1315 y Fj(\033)233 1565 y Fl(\024)j Fm(c)307 1572 y Fy(0)335 1479 y Fj( )375 1565 y Fm(L)408 1544 y Fx(d)p Fy(+2)408 1578 y Fx(k)484 1565 y Fo(+)551 1502 y Fv(1)533 1517 y Fj(X)538 1623 y Fx(j)r Fy(=)p Fx(k)613 1565 y Fm(L)646 1544 y Fx(d)p Fy(+2)646 1577 y Fx(j)r Fy(+1)720 1524 y Fj(\000)743 1565 y Fm(\032)p Fo(\()p Fm(L)820 1572 y Fx(j)839 1565 y Fo(\))p Fm(")881 1544 y Fv(\000)p Fy(1)928 1565 y Fl(j)p Fm(I)t Fl(j)c Fo(+)h Fm(L)1074 1541 y Fv(\000)p Fx(p)1074 1577 y(j)1121 1524 y Fj(\001)1144 1479 y(!)233 1739 y Fl(\024)j Fm(c)307 1746 y Fy(0)335 1653 y Fj( )375 1739 y Fo(\(log)9 b(1)p Fm(=")p Fo(\))555 1718 y Fx(\013)p Fy(\()p Fx(d)p Fy(+2\))p Fx(=\027)719 1739 y Fo(+)786 1676 y Fv(1)768 1691 y Fj(X)773 1797 y Fx(j)r Fy(=)p Fx(k)848 1739 y Fm(L)881 1713 y Fx(\013)p Fy(\()p Fx(d)p Fy(+2\))881 1751 y Fx(j)1005 1698 y Fj(\000)1028 1739 y Fo(e)1050 1718 y Fv(\000)p Fy(2)p Fx(L)1124 1706 y Fs(\027)1119 1728 y(j)1145 1739 y Fo(e)1167 1718 y Fx(L)1196 1706 y Fs(\027)1191 1728 y(j)1218 1739 y Fl(j)p Fm(I)t Fl(j)h Fo(+)h Fm(L)1364 1715 y Fv(\000)p Fx(p)1364 1751 y(j)1411 1698 y Fj(\001)1434 1653 y(!)233 1871 y Fl(\024)j Fm(c)307 1878 y Fy(4)327 1871 y Fo(\(log)9 b(1)p Fm(=")p Fo(\))507 1850 y Fx(\013)p Fy(\()p Fx(d)p Fy(+2\))p Fx(=\027)668 1871 y Fm(:)p 119 1978 20 43 v 60 2075 a Fh(Remark)16 b(4.3)89 b Fo(i\))24 b(If)16 b Fl(H)e Fo(=)g Fm(`)667 2057 y Fy(2)687 2075 y Fo(\()p Fk(Z)742 2057 y Fx(d)759 2075 y Fo(\),)i(one)g(can)g(also)g (use)g(\(38\))h(in)e(the)h(in)o(tegral)f(o)o(v)o(er)g Fm(!)h Fl(2)182 2135 y Fo(\012)217 2142 y Fx(k)238 2135 y Fo(\()p Fm(q)r(;)8 b(q)327 2117 y Fv(0)338 2135 y Fo(\))18 b(in)f(\(37\))i(and)g(deriv)o(e)d(the)i(b)q(ound)h(2)p Fm(\031)r(\032)p Fo(\()p Fm(L)1149 2142 y Fx(k)1171 2135 y Fo(\))p Fm(")1213 2117 y Fv(\000)p Fy(1)1278 2135 y Fo(instead)f(of)g(2)p Fm(\032)p Fo(\()p Fm(L)1606 2142 y Fx(k)1628 2135 y Fo(\))p Fm(")1670 2117 y Fv(\000)p Fy(2)1717 2135 y Fl(j)p Fm(I)t Fl(j)182 2195 y Fo(for)h(the)f(\014rst)h (term)e(in)i(\(35\).)29 b(This)19 b(implies)d(that)j(\(32\))h(is)e(b)q (ounded)i(uniformly)c(in)182 2255 y Fm(")p Fo(.)26 b(Th)o(us,)18 b(w)o(e)f(obtain)h(the)g(dynamical)e(lo)q(calization)h(prop)q(ert)o(y)h (\(6\))q(,)g(as)g(claimed)d(in)182 2315 y(Theorem)g(2.2.)111 2414 y(ii\))24 b(If)17 b(the)g(deca)o(y)g(of)g(the)g(function)h Fm(\032)f Fo(in)g(the)g(Multi-Scale)f(Assumption)h(\(M1\))g(is)g(only) 182 2474 y(algebraic)786 2573 y Fm(\032)p Fo(\()p Fm(x)p Fo(\))d Fl(\024)f Fm(c)p Fo(\()p Fm(n)p Fo(\))8 b Fm(x)1067 2552 y Fv(\000)p Fy(2)p Fx(n)1144 2573 y Fm(;)525 b Fo(\(40\))182 2671 y(for)22 b(some)e Fm(n)i Fl(2)h Fk(N)g Fo(with)e Fm(n)h(>)h(\013)p Fo(\()p Fm(d)15 b Fo(+)f(2\))22 b(and)g(some)e (constan)o(t)i Fm(c)p Fo(\()p Fm(n)p Fo(\))h Fm(<)f Fl(1)p Fo(,)g(then)182 2731 y(c)o(ho)q(osing)15 b Fm(k)r Fo(\()p Fm(")p Fo(\))f Fl(2)g Fk(N)i Fo(according)f(to)g Fm(L)887 2713 y Fx(n)887 2744 y(k)q Fv(\000)p Fy(1)967 2731 y Fm(<)f(")1042 2713 y Fv(\000)p Fy(1)1103 2731 y Fl(\024)f Fm(L)1188 2713 y Fx(n)1188 2744 y(k)1226 2731 y Fo(instead)i(of)j (\(39\))q(,)d(w)o(e)f(b)q(ound)891 2856 y(19)p eop %%Page: 20 20 20 19 bop 182 203 a Fo(\(32\))22 b(b)o(y)e Fm(c)383 210 y Fy(4)403 203 y Fm(")426 185 y Fv(\000)p Fx(\013)p Fy(\()p Fx(d)p Fy(+2\))p Fx(=n)607 203 y Fo(.)36 b(This)21 b(implies)d Fm(\033)973 183 y Fy(+)971 217 y(di\013)1025 163 y Fj(\000)1048 203 y Fm(f)1072 210 y Fx(I)1090 201 y Fg(0)1103 203 y Fo(\()p Fm(H)t Fo(\))p Fm(')1217 163 y Fj(\001)1262 203 y Fl(\024)1327 180 y Fx(\013)p Fy(\()p Fx(d)p Fy(+2\))p 1327 192 114 2 v 1373 221 a Fx(n)1446 203 y Fo(,)j(as)h(claimed)c(in) 182 264 y(Remark)d(2.3.)60 373 y Fh(Lemma)h(4.4)24 b Ff(L)n(et)19 b Fm(H)24 b Ff(satisfy)19 b(Assumptions)40 b(2.1.i\),)19 b(v\))h(and)g(\(M2\),)g(then)g(for)f Fm(')f Fl(2)h(H)p Ff(,)60 434 y(normalize)n(d)e(and)h(c)n(omp)n(actly)f(supp)n (orte)n(d,)g(ther)n(e)g(exist)i Fm(c)1115 441 y Fy(2)1149 434 y Fm(<)14 b Fl(1)j Ff(and)h Fm(\035)e(>)e Fo(0)k Ff(such)g(that)g(for)60 494 y(al)r(l)h Fo(0)14 b Fm(<)g(")f(<)h Fo(1)420 621 y Fk(E)458 551 y Fj(\032)499 621 y Fm(")522 601 y Fy(2)550 553 y Fj(Z)577 666 y Fx(I)606 621 y Fl(k)8 b(j)p Fm(X)t Fl(j)p Fo(\()p Fm(H)774 601 y Fy(\()p Fx(!)q Fy(\))838 621 y Fl(\000)j Fm(E)j Fl(\000)c Fo(i)p Fm(")p Fo(\))1043 601 y Fv(\000)p Fy(1)1090 621 y Fm(')p Fl(k)1147 601 y Fy(2)1167 621 y Fm(dE)1231 551 y Fj(\033)1282 621 y Fm(<)k(")1357 601 y Fx(\035)1397 621 y Fm(:)158 760 y Fd(Pro)q(of.)64 b Fo(W)l(e)31 b(assume)g(here,)k(without)d(loss)g(of) g(generalit)o(y)l(,)i(that)e(supp\()p Fm(')p Fo(\))40 b Fl(\032)60 820 y(f)p Fm(x)16 b Fl(j)g(k)p Fm(x)p Fl(k)237 827 y Fv(1)288 820 y Fl(\024)d Fm(L)373 827 y Fy(0)393 820 y Fl(g)p Fo(;)j(then)g(w)o(e)g(obtain,)g(for)h Fm(E)f Fl(2)f Fm(I)t Fo(,)g Fm(")e(>)h Fo(0)j(and)g Fm(J)1277 827 y Fy(0)1310 820 y Fl(2)d Fk(N)279 929 y Fl(kj)p Fm(X)t Fl(j)p Fo(\()p Fm(H)439 909 y Fy(\()p Fx(!)q Fy(\))503 929 y Fl(\000)d Fm(E)j Fl(\000)d Fo(i)p Fm(")p Fo(\))709 909 y Fv(\000)p Fy(1)756 929 y Fm(')p Fl(k)813 909 y Fy(2)321 1011 y Fl(\024)41 b Fo(\(2)p Fm(L)477 1018 y Fy(0)509 1011 y Fo(+)11 b(1\))601 990 y Fy(2)635 964 y Fj(X)629 1070 y Fx(k)q Fv(\025)p Fx(J)694 1075 y Fr(0)814 964 y Fj(X)737 1073 y Fx(q)q Fv(2)p Fy(supp)n(\()p Fx(')p Fy(\))p Fv(\\)p Fi(Z)949 1064 y Fs(d)1228 964 y Fj(X)989 1073 y Fx(q)1006 1064 y Fg(0)1017 1073 y Fv(2)p Fi(Z)1065 1064 y Fs(d)1081 1073 y Fx(;L)1115 1078 y Fr(0)1132 1073 y Fy(+2)p Fx(L)1201 1079 y Fs(k)1220 1073 y Fx(<)p Fv(k)p Fx(q)1282 1064 y Fg(0)1293 1073 y Fv(k)1311 1077 y Fg(1)1343 1073 y Fv(\024)p Fx(L)1394 1078 y Fr(0)1412 1073 y Fy(+2)p Fx(L)1481 1079 y Fs(k)q Fr(+1)548 1106 y Fj(\032)585 1176 y Fl(k)d(j)p Fm(X)t Fl(j)p Fh(1)718 1183 y Fx(q)735 1174 y Fg(0)748 1176 y Fl(k)773 1156 y Fy(2)773 1189 y Fv(1)811 1176 y Fl(k)p Fh(1)864 1183 y Fx(q)881 1174 y Fg(0)894 1176 y Fo(\()p Fm(H)957 1156 y Fy(\()p Fx(!)q Fy(\))1021 1176 y Fl(\000)j Fm(E)j Fl(\000)c Fo(i)p Fm(")p Fo(\))1226 1156 y Fv(\000)p Fy(1)1273 1176 y Fh(1)1301 1183 y Fx(q)1320 1176 y Fl(k)1345 1156 y Fy(2)1365 1176 y Fl(k)p Fh(1)1418 1183 y Fx(q)1437 1176 y Fm(')p Fl(k)1494 1156 y Fy(2)1513 1106 y Fj(\033)401 1286 y Fo(+)k Fm(d)p Fo(\()p Fm(L)530 1293 y Fx(J)549 1298 y Fr(0)580 1286 y Fo(+)d Fm(L)662 1293 y Fy(0)693 1286 y Fo(+)g(1\))785 1265 y Fy(2)805 1286 y Fl(k)p Fo(\()p Fm(H)893 1265 y Fy(\()p Fx(!)q Fy(\))957 1286 y Fl(\000)g Fm(E)j Fl(\000)d Fo(i)p Fm(")p Fo(\))1163 1265 y Fv(\000)p Fy(1)1209 1286 y Fm(')p Fl(k)1266 1265 y Fy(2)1302 1286 y Fm(:)60 1396 y Fo(F)l(or)16 b(\014xed)g Fm(E)k Fo(and)c Fm(q)i Fo(w)o(e)e(de\014ne) 420 1505 y(\012\()p Fm(E)s(;)8 b(q)r Fo(\))13 b(:=)g Fl(f)p Fm(!)19 b Fl(j)d Fm(H)804 1485 y Fy(\()p Fx(!)q Fy(\))873 1505 y Fo(is)g(\()p Fm(\032;)8 b(E)s(;)g(L)1082 1512 y Fx(k)1103 1505 y Fm(;)g(q)r Fo(\))p Fl(\000)p Fo(regular)o Fl(g)17 b Fm(:)60 1615 y Fo(Th)o(us,)f(according)h(to)f (Assumption)f(\(M2\),)h(w)o(e)g(get)202 1754 y Fm(")225 1733 y Fy(2)253 1686 y Fj(Z)311 1683 y(\022)348 1686 y(Z)375 1798 y Fx(I)404 1754 y Fl(kj)p Fm(X)t Fl(j)p Fo(\()p Fm(H)564 1733 y Fy(\()p Fx(!)q Fy(\))628 1754 y Fl(\000)10 b Fm(E)k Fl(\000)d Fo(i)p Fm(")p Fo(\))833 1733 y Fv(\000)p Fy(1)880 1754 y Fm(')p Fl(k)937 1733 y Fy(2)956 1754 y Fm(dE)1020 1683 y Fj(\023)1066 1754 y Fm(d)p Fk(P)l Fo(\()p Fm(!)r Fo(\))227 1870 y Fl(\024)25 b Fo(\(2)p Fm(L)367 1877 y Fy(0)398 1870 y Fo(+)11 b(1\))490 1849 y Fy(2)524 1822 y Fj(X)518 1928 y Fx(k)q Fv(\025)p Fx(J)583 1933 y Fr(0)704 1822 y Fj(X)626 1932 y Fx(q)q Fv(2)p Fy(supp)n(\()p Fx(')p Fy(\))p Fv(\\)p Fi(Z)838 1923 y Fs(d)1088 1822 y Fj(X)878 1930 y Fx(q)895 1921 y Fg(0)906 1930 y Fx(;)6 b(L)946 1935 y Fr(0)963 1930 y Fy(+2)p Fx(L)1032 1936 y Fs(k)1051 1930 y Fv(\024k)p Fx(q)1113 1921 y Fg(0)1125 1930 y Fv(k)1143 1934 y Fg(1)1175 1930 y Fv(\024)p Fx(L)1226 1935 y Fr(0)1243 1930 y Fy(+2)p Fx(L)1312 1936 y Fs(k)q Fr(+1)257 1964 y Fj(\()306 1981 y(Z)334 2094 y Fx(I)362 1979 y Fj(\022)407 1981 y(Z)434 2094 y Fy(\012\()p Fx(E)r(;q)q Fy(\))524 2016 y Fm(d)p Fo(\()p Fm(L)601 2023 y Fy(0)632 2016 y Fo(+)11 b(2)p Fm(L)738 2023 y Fx(k)q Fy(+1)816 2016 y Fo(+)g(1\))908 1997 y Fy(2)p 524 2038 405 2 v 693 2083 a Fm(L)726 2066 y Fx(m)726 2097 y(k)933 2049 y Fm(")956 2029 y Fy(2)976 2049 y Fl(k)p Fo(\()p Fm(H)1064 2029 y Fy(\()p Fx(!)q Fy(\))1128 2049 y Fl(\000)g Fm(E)j Fl(\000)d Fo(i)p Fm(")p Fo(\))1334 2029 y Fv(\000)p Fy(1)1380 2049 y Fl(k)1405 2029 y Fy(2)1425 2049 y Fm(d)p Fk(P)l Fo(\()p Fm(!)r Fo(\))257 2215 y(+)303 2147 y Fj(Z)331 2260 y Fy(\012)356 2251 y Fs(c)373 2260 y Fy(\()p Fx(E)r(;q)q Fy(\))407 2215 y Fm(d)p Fo(\()p Fm(L)484 2222 y Fy(0)515 2215 y Fo(+)g(2)p Fm(L)621 2222 y Fx(k)q Fy(+1)699 2215 y Fo(+)g(1\))791 2195 y Fy(2)811 2215 y Fm(")834 2195 y Fy(2)854 2215 y Fl(k)p Fh(1)907 2222 y Fx(q)924 2213 y Fg(0)937 2215 y Fo(\()p Fm(H)1000 2195 y Fy(\()p Fx(!)q Fy(\))1064 2215 y Fl(\000)g Fm(E)j Fl(\000)d Fo(i)p Fm(")p Fo(\))1270 2195 y Fv(\000)p Fy(1)1316 2215 y Fl(k)1341 2195 y Fy(2)1361 2215 y Fm(d)p Fk(P)m Fo(\()p Fm(!)r Fo(\))1490 2145 y Fj(\023)1523 2215 y Fm(dE)1587 2130 y Fj(\))1683 2215 y Fo(\(41\))257 2366 y(+)p Fm(d)p Fo(\()p Fm(L)372 2373 y Fy(0)404 2366 y Fo(+)g(2)p Fm(L)510 2373 y Fx(J)529 2378 y Fr(0)560 2366 y Fo(+)g(1\))652 2346 y Fy(2)672 2366 y Fk(E)711 2296 y Fj(\032)751 2366 y Fm(")774 2346 y Fy(2)802 2299 y Fj(Z)829 2411 y Fx(I)858 2366 y Fl(k)p Fo(\()p Fm(H)946 2346 y Fy(\()p Fx(!)q Fy(\))1010 2366 y Fl(\000)f Fm(E)k Fl(\000)d Fo(i)p Fm(")p Fo(\))1215 2346 y Fv(\000)p Fy(1)1262 2366 y Fm(')p Fl(k)1319 2346 y Fy(2)1338 2366 y Fm(dE)1402 2296 y Fj(\033)1457 2366 y Fm(:)212 b Fo(\(42\))60 2510 y(By)15 b(using)h(the)f(inequalit)o(y)f Fl(k)p Fo(\()p Fm(H)660 2492 y Fy(\()p Fx(!)q Fy(\))723 2510 y Fl(\000)9 b Fm(E)k Fl(\000)c Fo(i)p Fm(")p Fo(\))924 2492 y Fv(\000)p Fy(1)971 2510 y Fl(k)14 b(\024)f Fm(")1085 2492 y Fv(\000)p Fy(1)1148 2510 y Fo(in)i(\(41\))h(and)h(inequalit)o(y) d(\(38\))i(in)60 2571 y(\(42\))q(,)g(w)o(e)f(deriv)o(e)g(the)h(upp)q (er)g(b)q(ound)267 2709 y Fm(")290 2688 y Fy(2)309 2709 y Fk(E)348 2639 y Fj(\032)388 2641 y(Z)416 2754 y Fx(I)444 2709 y Fl(kj)p Fm(X)t Fl(j)p Fo(\()p Fm(H)604 2688 y Fy(\()p Fx(!)q Fy(\))668 2709 y Fl(\000)11 b Fm(E)j Fl(\000)d Fo(i)p Fm(")p Fo(\))874 2688 y Fv(\000)p Fy(1)920 2709 y Fm(')p Fl(k)977 2688 y Fy(2)997 2709 y Fm(dE)1061 2639 y Fj(\033)1112 2709 y Fl(\024)j Fm(C)t Fo(\()p Fm("L)1279 2688 y Fy(2)1279 2721 y Fx(J)1298 2726 y Fr(0)1328 2709 y Fo(+)d Fl(j)p Fm(I)t Fl(j)p Fm(L)1464 2685 y Fv(\000)p Fx(\014)1464 2723 y(J)1483 2728 y Fr(0)1514 2709 y Fo(\))17 b Fm(:)891 2856 y Fo(20)p eop %%Page: 21 21 21 20 bop 60 203 a Fo(Here)14 b Fm(C)k Fo(is)d(a)h(constan)o(t)f(dep)q (ending)g(only)g(on)h Fm(L)949 210 y Fy(0)984 203 y Fo(and)f Fm(d)p Fo(.)21 b(This)15 b(giv)o(es)g(the)g(exp)q(ected)e(result)60 264 y(for)k(an)g(appropriate)g(dep)q(endence)e(of)i Fm(J)810 271 y Fy(0)846 264 y Fo(on)g Fm(")p Fo(:)k(for)c Fm(K)h Fl(2)d Fk(N)i Fo(suc)o(h)f(that)h(2)p Fm(\013)1477 246 y Fv(\000)p Fx(K)1554 264 y Fm(<)d Fo(1)j(and)g(if)60 324 y Fm(L)93 331 y Fx(J)134 324 y Fm(<)g(")212 306 y Fv(\000)p Fy(1)275 324 y Fl(\024)g Fm(L)364 331 y Fx(J)s Fy(+1)433 324 y Fo(,)h(w)o(e)f(tak)o(e)h Fm(J)673 331 y Fy(0)709 324 y Fo(=)e Fm(J)h Fl(\000)12 b Fm(K)t Fo(.)26 b(W)l(e)18 b(th)o(us)g(obtain)g Fm(\035)h Fo(=)d(min)n Fl(f)p Fm(\014)s(\013)1558 306 y Fv(\000)p Fx(K)r Fv(\000)p Fy(1)1665 324 y Fm(;)g Fo(1)c Fl(\000)60 384 y Fo(2)p Fm(\013)115 366 y Fv(\000)p Fx(K)177 384 y Fl(g)p Fo(.)p 274 393 20 43 v 60 548 a Fn(A)81 b(App)r(endix)60 657 y Fo(W)l(e)17 b(presen)o(t)f(here)g(the)h(t)o(w)o(o)g(lemm)o(as)e (necessary)i(for)g(the)g(\\v)m(ariable-energy")g(m)o(ulti-scale)60 717 y(analysis)g(needed)f(in)g(Example)f(3.)23 b(F)l(or)17 b(the)f(rest)h(of)g(this)f(section,)g(w)o(e)g(\014x)h Fm(\013)e(>)f Fo(1,)j Fm(`)e(<)g Fl(1)p Fo(,)60 778 y Fm(N)31 b(>)26 b Fo(4)e(and)g Fm(S)i Fo(ev)o(en,)e(2)i Fm(<)g(S)j(<)d(N)c Fl(\000)15 b Fo(1.)43 b(F)l(or)24 b Fm(L)i Fo(=)g Fm(N)5 b(`)1249 759 y Fx(\013)1298 778 y Fo(w)o(e)23 b(de\014ne)g(for)h(all)f Fm(n)j Fl(2)60 838 y(f)p Fo(0)p Fm(;)8 b Fo(1)p Fm(;)g Fl(\001)g(\001)g(\001)17 b Fm(;)8 b(N)d Fl(g)423 1018 y Fm(\037)454 997 y Fx(L)454 1030 y(n)480 1018 y Fo(\()p Fm(x)p Fo(\))13 b(=)611 886 y Fj(8)611 930 y(>)611 945 y(>)611 960 y(<)611 1050 y(>)611 1065 y(>)611 1080 y(:)656 938 y Fm(J)683 946 y Fy(\003)707 953 y Fs(L=)p Fr(\(4)p Fs(N)s Fr(\))812 946 y Fy(\()p Fx(x)p Fy(\))p Fx(;)p Fy(0)1048 938 y Fm(n)h Fo(=)f(0)k Fm(;)656 1010 y(J)683 1037 y Fy(\003)707 1044 y Fs(nL=)n(N)790 1037 y Fy(\()p Fx(x)p Fy(\))p Fx(;)853 1021 y Fr(\()p Fs(L=)n(N)s Fr(\))939 1012 y(1)p Fs(=\013)p 852 1030 140 2 v 901 1050 a Fr(4)p Fs(N)1048 1010 y Fo(1)d Fl(\024)f Fm(n)h Fl(\024)g Fm(N)i Fl(\000)11 b Fo(1)17 b Fm(;)656 1101 y(J)683 1109 y Fy(\003)707 1115 y Fs(L)730 1109 y Fy(\()p Fx(x)p Fy(\))p Fx(;L=)p Fy(\(4)p Fx(N)t Fy(\))1048 1101 y Fm(n)d Fo(=)f Fm(N)22 b(;)1683 1018 y Fo(\(43\))60 1195 y(where,)15 b(for)h Fm(\016)f(>)f Fo(0,)h Fm(J)457 1202 y Fy(\003)p Fx(;\016)526 1195 y Fo(is)g(a)h(smo)q(oth)g(c)o (haracteristic)e(function)h(suc)o(h)h(that,)f(for)h Fm(@)s Fo(\003)f(b)q(eing)60 1255 y(the)h(b)q(oundary)h(of)g(the)f(b)q(o)o(x)h (\003,)435 1387 y Fm(J)462 1394 y Fy(\003)p Fx(;\016)515 1387 y Fo(\()p Fm(y)r Fo(\))c(=)644 1302 y Fj(\()684 1353 y Fo(1)49 b(if)19 b Fm(y)d Fl(2)e Fo(\003)i(and)h(dist\()p Fm(y)r(;)8 b(@)s Fo(\003\))k Fm(<)i(\016)k(;)684 1425 y Fo(0)49 b(if)19 b Fm(y)d Fl(62)e Fo(\003)i Fm(;)60 1519 y Fo(and)g(for)f Fm(\016)h Fo(=)d(0,)j Fm(J)397 1526 y Fy(\003)p Fx(;\016)465 1519 y Fo(is)f(the)g(c)o(haracteristic)f (function)h(of)h(\003.)k(One)15 b(also)h(de\014nes)f(the)g(frame)60 1580 y Fe(F)90 1587 y Fx(L)117 1580 y Fo(\()p Fm(x;)8 b(n)p Fo(\))15 b(of)h(the)f(b)q(o)o(x)h(\003)5 b Fs(nL)p 519 1586 41 2 v 526 1606 a(N)566 1580 y Fo(\()p 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Fx(w)1274 1923 y Fm(;)60 2042 y Fo(and)k Fe(H)191 2049 y Fx(L)217 2042 y Fo(\()p Fm(E)s(;)8 b(x;)g(m;)g Fo(\()p Fm(n)460 2049 y Fx(i)473 2042 y Fo(\))492 2049 y Fx(i)p Fy(=1)p Fx(;:::)o(;S)623 2042 y Fo(\))17 b(as)f(the)g(prop)q(ert)o(y:)22 b Fl(8)p Fm(i)12 b Fo(=)i(1)p Fm(;)8 b(:)g(:)g(:)f(;)h(S)19 b(;)24 b Fl(8)p Fm(y)15 b Fl(2)f Fe(F)1496 2049 y Fx(L)1522 2042 y Fo(\()p Fm(x;)8 b(n)1620 2049 y Fx(i)1634 2042 y Fo(\),)370 2134 y(sup)376 2173 y Fx(">)p Fy(0)452 2134 y Fl(k)p Fo([)p Fl(\000)p Fo(\001)p Fm(;)g(\037)624 2113 y Fx(`)624 2146 y(N)656 2134 y Fo(\()p Fm(y)r Fo(\)]\()p Fm(H)793 2142 y Fy(\003)817 2148 y Fs(`)832 2142 y Fy(\()p Fx(y)q Fy(\))891 2134 y Fl(\000)j Fm(E)j Fl(\000)d Fo(i)p Fm(")p Fo(\))1097 2113 y Fv(\000)p Fy(1)1143 2134 y Fm(\037)1174 2113 y Fx(`)1174 2146 y Fy(0)1194 2134 y Fo(\()p Fm(y)r Fo(\))p Fl(k)i(\024)h Fm(`)1369 2113 y Fv(\000)p Fx(m)1446 2134 y Fm(:)223 b Fo(\(44\))60 2251 y(Here)14 b Fm(H)214 2258 y Fy(\003)257 2251 y Fo(refers)h(to)h Fm(H)k 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y(discrete)e(mo)q(dels)h(and)h([13])f(for)h (con)o(tin)o(uous)f(mo)q(dels\).)36 b(F)l(or)22 b(self-consistency)l(,) f(w)o(e)g(giv)o(e)60 324 y(here)16 b(the)g(most)f(imp)q(ortan)o(t)h (steps)g(of)h(the)f(pro)q(of)h(of)f(Lemma)f(A.1.)158 421 y Fd(Pro)q(of.)64 b Fo(Let)17 b Fm(W)482 428 y Fx(L;n;x)583 421 y Fo(=)d([)p Fl(\000)p Fo(\001)p Fm(;)8 b(\037)782 403 y Fx(L)782 434 y(n)807 421 y Fo(\()p Fm(x)p Fo(\)],)15 b(\()p Fm(n)g Fl(6)p Fo(=)f(0\))j(and)g Fm(R)1223 428 y Fy(\003)1264 421 y Fo(=)d(\()p Fm(H)1375 428 y Fy(\003)1413 421 y Fl(\000)d Fm(E)k Fl(\000)c Fo(i)p Fm(")p Fo(\))1620 403 y Fv(\000)p Fy(1)1666 421 y Fo(.)22 b(W)l(e)60 482 y(\014rst)c(apply)g Fm(S)j Fo(times)16 b(the)i(geometric)e(resolv)o(en) o(t)g(equation)i(to)h Fm(W)1306 489 y Fx(L;N)r(;x)1401 482 y Fm(R)1438 489 y Fy(\003)1462 495 y Fs(L)1486 489 y Fy(\()p Fx(x)p Fy(\))1535 482 y Fm(\037)1566 463 y Fx(L)1566 494 y Fy(0)1592 482 y Fo(\()p Fm(x)p Fo(\))e(with)60 542 y(the)f(increasing)g(sequence)f(of)i(b)q(o)o(xes)f(\003)798 549 y Fx(n)819 554 y Fs(i)832 549 y Fx(L=)n(N)906 542 y Fo(\()p Fm(x)p Fo(\),)f Fm(i)f Fo(=)f(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(S) s Fo(,)280 652 y Fm(W)326 659 y Fx(L;N)r(;x)421 652 y Fm(R)458 659 y Fy(\003)482 665 y Fs(L)506 659 y Fy(\()p Fx(x)p Fy(\))555 652 y Fm(\037)586 631 y Fx(L)586 664 y Fy(0)612 652 y Fo(\()p Fm(x)p Fo(\))380 724 y(=)41 b Fm(W)505 731 y Fx(L;N)r(;x)600 724 y Fm(R)637 732 y Fy(\003)666 738 y Fs(n)685 746 y(S)708 738 y(L)p 667 749 63 2 v 685 767 a(N)736 732 y Fy(\()p Fx(x)p Fy(\))786 724 y Fm(\037)817 704 y Fx(@)817 737 y(n)838 743 y Fs(S)862 724 y Fo(\()p Fm(x)p Fo(\))p Fm(W)974 731 y Fx(L;n)1029 737 y Fs(S)1051 731 y Fx(;x)1083 724 y Fm(R)1120 732 y Fy(\003)1149 738 y Fs(n)1168 746 y(S)q Fg(\000)p Fr(1)1230 738 y Fs(L)p 1150 752 102 2 v 1187 770 a(N)1258 732 y Fy(\()p Fx(x)p Fy(\))1308 724 y Fm(\037)1339 704 y Fx(\016)1339 737 y(n)1360 743 y Fs(S)q Fg(\000)p Fr(1)459 830 y Fm(:)8 b(:)g(:)g(W)571 837 y 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b(:)g(:)g(:)g(;)g(N)16 b Fl(\000)11 b Fo(1)p Fl(g)p Fo(:)541 2160 y Fl(k)p Fm(W)612 2167 y Fx(L)636 2158 y Fg(0)647 2167 y Fx(;n;z)708 2160 y Fm(R)745 2167 y Fy(\003)769 2177 y Fs(L)790 2170 y Fg(0)806 2160 y Fo(\()p Fm(z)r Fo(\))p Fm(\037)900 2140 y Fx(L)924 2128 y Fg(0)900 2172 y Fy(0)936 2160 y Fl(k)j(\024)h Fo(~)-25 b Fm(c)p Fl(k)p Fm(R)1111 2167 y Fy(\003)1135 2177 y Fs(L)1156 2170 y Fg(0)1172 2160 y Fo(\()p Fm(z)r Fo(\))p Fl(k)16 b Fm(:)p 119 2279 20 43 v 60 2368 a Fo(W)l(e)i(no)o(w)h (state)g(the)f(probabilistic)g(part)h(of)f(this)h(analysis.)28 b(Since)17 b(\(4\))i(requires)f(a)g(result)60 2428 y(whic)o(h)f(is)g (uniform)g(in)g(energy)l(,)g(w)o(e)g(m)o(ust)f(do)i(a)g(\\v)m (ariable-energy")g(m)o(ulti-scale)d(analysis)60 2488 y(lik)o(e)j(in)h([11)q(],)h(adapted)g(to)g(correlated)g(p)q(oten)o (tials.)31 b(One)20 b(of)g(the)g(k)o(ey)e(estimates)h(w)o(e)g(use)60 2548 y(here)d(is)g(a)g(correlated)g(W)l(egner)g(estimate)e(pro)o(v)o (en)i(in)g([9]:)60 2658 y Fh(Prop)r(osition)i(A.1)24 b Ff(\(Comb)n(es-Hislop-Mourr)n(e)17 b(c)n(orr)n(elate)n(d)g(We)n(gner) h(estimate\).)60 2718 y(We)c(assume)h(\(A1\))f({)h(\(A4\))f(for)g(the)h (mo)n(del)k Fo(\(8\))q Ff(.)i(Then)15 b(ther)n(e)f(exist)i Fm(I)h Fo(=)d([)p Fm(\013;)8 b(\014)s Fo(])p Ff(,)13 b(a)h(c)n(omp)n(act)891 2856 y Fo(22)p eop %%Page: 23 23 23 22 bop 60 203 a Ff(interval)20 b(in)e(the)h(almost-sur)n(e)g(sp)n(e) n(ctrum)f(of)g Fm(H)951 185 y Fy(\()p Fx(!)q Fy(\))1004 203 y Ff(,)g Fm(`)1057 210 y Fy(0)1093 203 y Fm(<)e Fl(1)i Ff(and)g Fm(C)1345 210 y Fx(W)1401 203 y Fm(>)e Fo(0)i Ff(such)h(that)g(for)60 264 y Fm(E)26 b Fl(2)d Fm(I)t Ff(,)f(for)g(any)g(b)n(ounde)n(d)h(op)n(en)f(cub)n(es)h Fo(\003)899 271 y Fy(1)919 264 y Ff(,)g Fo(\003)991 271 y Fy(2)1033 264 y Ff(with)f Fo(dist\(\003)1275 271 y Fy(1)1295 264 y Fm(;)8 b Fo(\003)1351 271 y Fy(2)1370 264 y Fo(\))23 b Fl(\025)f Fm(`)1493 271 y Fy(0)1536 264 y Ff(and)g(for)g(al)r(l)60 324 y Fo(0)14 b Fm(<)g(\021)i(<)d Fo(1)220 434 y Fk(E)250 393 y Fj(\000)285 434 y 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Fm(L)1163 875 y Fy(0)1202 868 y Fm(<)e Fl(1)g Ff(and)h Fm(p)f(>)f Fo(0)i Ff(such)g(that)60 928 y Fo(\()p Fm(\013)11 b Fl(\000)g Fo(1\)\()p Fm(d)h Fl(\000)f Fo(1\)\()p Fm(N)16 b Fl(\000)11 b Fm(S)s Fo(\))p Fm(=)p Fo(\()p Fm(\022)q Fo(\()p Fm(N)17 b Fl(\000)11 b Fm(S;)d(\013)p Fo(\))j Fl(\000)g Fm(\013)p Fo(\))28 b Fm(<)f(p)h(<)g(w)17 b Fl(\000)g Fo(2)p Fm(d)p Ff(,)27 b(and)e(for)f(al)r(l)j Fm(x)1647 935 y Fy(1)1666 928 y Ff(,)g Fm(x)1736 935 y Fy(2)1755 928 y Ff(,)60 988 y Fl(k)p Fm(x)113 995 y Fy(1)143 988 y Fl(\000)11 b Fm(x)221 995 y Fy(2)241 988 y Fl(k)266 995 y Fv(1)317 988 y Fm(>)j Fo(2)p Fm(L)426 995 y Fy(0)446 988 y Ff(,)99 1112 y Fk(P)132 1056 y Fj(n)162 1112 y Fl(8)p Fm(E)i Fl(2)e Fm(I)t(;)h Fl(9)p Fo(\()p Fm(n)421 1119 y Fx(i)435 1112 y Fo(\))454 1119 y Fx(i)p Fy(=1)p Fx(;:::S)593 1112 y Fo(s)p Fm(:)p Fo(t)p Fm(:)676 1071 y Fj(\000)699 1112 y Fe(W)751 1119 y Fx(L)775 1124 y Fr(0)794 1112 y Fo(\()p Fm(E)s(;)8 b(x)902 1119 y Fy(1)922 1112 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)1050 1119 y Fx(i)1064 1112 y Fo(\)\))17 b(and)h Fe(H)1251 1119 y Fx(L)1275 1124 y Fr(0)1294 1112 y Fo(\()p Fm(E)s(;)8 b(x)1402 1119 y Fy(1)1422 1112 y Fm(;)g(m;)g Fo(\()p Fm(n)1557 1119 y Fx(i)1570 1112 y Fo(\)\))1626 1071 y Fj(\001)319 1218 y Fo(or)380 1178 y Fj(\000)403 1218 y Fe(W)455 1225 y Fx(L)479 1230 y Fr(0)498 1218 y Fo(\()p Fm(E)s(;)g(x)606 1225 y Fy(2)625 1218 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)753 1225 y Fx(i)767 1218 y Fo(\)\))18 b(and)g Fe(H)955 1225 y Fx(L)979 1230 y Fr(0)998 1218 y Fo(\()p Fm(E)s(;)8 b(x)1106 1225 y Fy(2)1125 1218 y Fm(;)g(m;)g Fo(\()p Fm(n)1260 1225 y Fx(i)1274 1218 y Fo(\)\))1329 1178 y Fj(\001)1352 1163 y(o)1399 1218 y Fl(\025)14 b Fo(1)d Fl(\000)g Fm(L)1570 1194 y Fv(\000)p Fx(p)1570 1230 y Fy(0)1635 1218 y Fm(;)60 1342 y Ff(then)18 b(for)f Fm(L)279 1349 y Fy(1)313 1342 y Fo(=)d Fm(N)5 b(L)442 1324 y Fx(\013)442 1354 y Fy(0)484 1342 y Ff(and)18 b(for)f(al)r(l)i Fm(y)751 1349 y Fy(1)770 1342 y Ff(,)e Fm(y)826 1349 y Fy(2)846 1342 y Ff(,)g Fl(k)p Fm(y)927 1349 y Fy(1)958 1342 y Fl(\000)11 b Fm(y)1032 1349 y Fy(2)1051 1342 y Fl(k)1076 1349 y Fv(1)1127 1342 y Fm(>)j Fo(2)p Fm(L)1236 1349 y Fy(1)1256 1342 y Ff(,)k(we)g(have:)98 1465 y Fk(P)131 1410 y Fj(n)162 1465 y Fl(8)p Fm(E)d Fl(2)f Fm(I)t(;)25 b Fl(9)p Fo(\()p Fm(n)430 1445 y Fv(0)430 1478 y Fx(i)444 1465 y Fo(\))463 1472 y Fx(i)p Fy(=1)p Fx(;:::)o(S)602 1465 y Fo(s)p Fm(:)p Fo(t)p Fm(:)685 1425 y Fj(\000)707 1465 y Fe(W)759 1472 y Fx(L)783 1477 y Fr(1)803 1465 y Fo(\()p Fm(E)s(;)8 b(y)907 1472 y Fy(1)926 1465 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)1054 1445 y Fv(0)1054 1478 y Fx(i)1068 1465 y Fo(\)\))18 b(and)g Fe(H)1256 1472 y Fx(L)1280 1477 y Fr(1)1299 1465 y Fo(\()p Fm(E)s(;)8 b(y)1403 1472 y Fy(1)1422 1465 y Fm(;)g(m;)g Fo(\()p Fm(n)1557 1445 y Fv(0)1557 1478 y Fx(i)1571 1465 y Fo(\)\))1626 1425 y Fj(\001)327 1572 y Fo(or)388 1531 y Fj(\000)411 1572 y Fe(W)463 1579 y Fx(L)487 1584 y Fr(1)507 1572 y Fo(\()p Fm(E)s(;)g(y)611 1579 y Fy(2)630 1572 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)758 1551 y Fv(0)758 1584 y Fx(i)772 1572 y Fo(\)\))17 b(and)h Fe(H)959 1579 y Fx(L)983 1584 y Fr(1)1003 1572 y Fo(\()p Fm(E)s(;)8 b(y)1107 1579 y Fy(2)1126 1572 y Fm(;)g(m;)g Fo(\()p Fm(n)1261 1551 y Fv(0)1261 1584 y Fx(i)1274 1572 y Fo(\)\))1330 1531 y Fj(\001)1353 1516 y(o)1400 1572 y Fl(\025)13 b Fo(1)f Fl(\000)f Fm(L)1571 1548 y Fv(\000)p Fx(p)1571 1584 y Fy(1)1635 1572 y Fm(:)34 b Fo(\(47\))158 1702 y Fd(Pro)q(of.)64 b Fo(Let)20 b Fm(y)463 1709 y Fy(1)502 1702 y Fo(and)h Fm(y)625 1709 y Fy(2)664 1702 y Fo(b)q(e)f(suc)o(h)g(that)g Fl(k)p Fm(y)1006 1709 y Fy(1)1039 1702 y Fl(\000)13 b Fm(y)1115 1709 y Fy(2)1135 1702 y Fl(k)1160 1709 y Fv(1)1217 1702 y Fm(>)20 b Fo(2)p Fm(L)1332 1709 y Fy(1)1352 1702 y Fo(.)32 b(W)l(e)20 b(denote)g(b)o(y)p 1718 1662 53 2 v 19 w Fe(W)60 1763 y Fo(\(resp.)p 201 1722 36 2 v 21 w Fe(H)p Fo(\))c(the)f(compleme)o(n)o(tary)e(ev)o(en)o(ts)h(of)i Fe(W)g Fo(\(resp.)21 b Fe(H)p Fo(\).)g(W)l(e)16 b(start)g(b)o(y)f (estimating)f(the)60 1823 y(probabilit)o(y)h(of)i(the)f(compleme)o(n)o (t)d(of)k(the)f(ev)o(en)o(t)e(that)j(app)q(ears)h(in)d(\(47\))113 1946 y Fk(P)147 1891 y Fj(n)177 1946 y Fl(9)p Fm(E)h Fl(2)e Fm(I)t(;)24 b Fl(8)p Fo(\()p Fm(n)445 1953 y Fx(i)458 1946 y Fo(\))477 1953 y Fx(i)p Fy(=1)p Fx(;:::)o(;S)625 1906 y Fj(\000)p 647 1906 53 2 v 647 1946 a Fe(W)699 1953 y Fx(L)723 1958 y Fr(1)743 1946 y Fo(\()p Fm(E)s(;)8 b(y)847 1953 y Fy(1)866 1946 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)994 1953 y Fx(i)1008 1946 y Fo(\)\))16 b(or)p 1122 1906 36 2 v 17 w Fe(H)1158 1953 y Fx(L)1182 1958 y Fr(1)1201 1946 y Fo(\()p Fm(E)s(;)8 b(y)1305 1953 y Fy(1)1325 1946 y Fm(;)g(m;)g Fo(\()p Fm(n)1460 1953 y Fx(i)1473 1946 y Fo(\)\))1511 1906 y Fj(\001)235 2053 y Fo(and)330 2012 y Fj(\000)p 353 2012 53 2 v 41 x Fe(W)405 2060 y Fx(L)429 2065 y Fr(1)448 2053 y Fo(\()p Fm(E)s(;)g(y)552 2060 y Fy(2)571 2053 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)699 2060 y Fx(i)714 2053 y Fo(\)\))16 b(or)p 827 2012 36 2 v 16 w Fe(H)863 2060 y Fx(L)887 2065 y Fr(1)907 2053 y Fo(\()p Fm(E)s(;)8 b(y)1011 2060 y Fy(2)1030 2053 y Fm(;)g(m;)g Fo(\()p Fm(n)1165 2060 y Fx(i)1178 2053 y Fo(\)\))1216 2012 y Fj(\001)1239 1997 y(o)155 2146 y Fl(\024)41 b Fk(P)l Fl(f9)p Fm(E)14 b Fl(2)g Fm(I)t(;)23 b Fl(8)p Fo(\()p Fm(n)558 2153 y Fx(i)571 2146 y Fo(\))p 606 2106 53 2 v 16 w Fe(W)659 2153 y Fx(L)683 2158 y Fr(1)702 2146 y Fo(\()p Fm(E)s(;)8 b(y)806 2153 y Fy(1)825 2146 y Fm(;)g(m;)g Fo(\()p Fm(n)960 2153 y Fx(i)974 2146 y Fo(\)\))16 b(and)p 1123 2106 V 17 w Fe(W)1175 2153 y Fx(L)1199 2158 y Fr(1)1218 2146 y Fo(\()p Fm(E)s(;)8 b(y)1322 2153 y Fy(2)1342 2146 y Fm(;)g(m;)g Fo(\()p Fm(n)1477 2153 y Fx(i)1490 2146 y Fo(\)\))p Fl(g)235 2219 y Fo(+)p Fk(P)l Fl(f9)p Fm(E)14 b Fl(2)g Fm(I)t(;)23 b Fl(8)p Fo(\()p Fm(n)596 2226 y Fx(i)609 2219 y Fo(\))p 644 2178 36 2 v 16 w Fe(H)680 2226 y Fx(L)704 2231 y Fr(1)724 2219 y Fo(\()p Fm(E)s(;)8 b(y)828 2226 y Fy(1)847 2219 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)975 2226 y Fx(i)989 2219 y Fo(\)\))16 b(and)p 1138 2178 V 17 w Fe(H)1174 2226 y Fx(L)1198 2231 y Fr(1)1217 2219 y Fo(\()p Fm(E)s(;)8 b(y)1321 2226 y Fy(2)1341 2219 y Fm(;)g(m;)g Fo(\()p Fm(n)1476 2226 y Fx(i)1489 2219 y Fo(\)\))p Fl(g)235 2291 y Fo(+)p Fk(P)l Fl(f9)p Fm(E)14 b Fl(2)g Fm(I)t(;)23 b Fl(8)p Fo(\()p Fm(n)596 2298 y Fx(i)609 2291 y Fo(\))p 644 2251 53 2 v 16 w Fe(W)697 2298 y Fx(L)721 2303 y Fr(1)740 2291 y Fo(\()p Fm(E)s(;)8 b(y)844 2298 y Fy(1)863 2291 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)991 2298 y Fx(i)1005 2291 y Fo(\)\))17 b(and)p 1154 2251 36 2 v 16 w Fe(H)1190 2298 y Fx(L)1214 2303 y Fr(1)1234 2291 y Fo(\()p Fm(E)s(;)8 b(y)1338 2298 y Fy(2)1357 2291 y Fm(;)g(m;)g Fo(\()p Fm(n)1492 2298 y Fx(i)1505 2291 y Fo(\)\))p Fl(g)235 2364 y Fo(+)p Fk(P)l Fl(f9)p Fm(E)14 b Fl(2)g Fm(I)t(;)23 b Fl(8)p Fo(\()p Fm(n)596 2371 y Fx(i)609 2364 y Fo(\))p 644 2324 V 16 w Fe(H)680 2371 y Fx(L)704 2376 y Fr(1)724 2364 y Fo(\()p Fm(E)s(;)8 b(y)828 2371 y Fy(1)847 2364 y Fm(;)g(m;)g Fo(\()p Fm(n)982 2371 y Fx(i)996 2364 y Fo(\)\))16 b(and)p 1145 2324 53 2 v 17 w Fe(W)1197 2371 y Fx(L)1221 2376 y Fr(1)1240 2364 y Fo(\()p Fm(E)s(;)8 b(y)1344 2371 y Fy(2)1363 2364 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)1491 2371 y Fx(i)1505 2364 y Fo(\)\))p Fl(g)155 2450 y(\024)213 2403 y Fj(X)193 2511 y Fy(\()p Fx(n)228 2516 y Fs(i)242 2511 y Fy(\))p Fv(2V)287 2450 y Fk(P)320 2395 y Fj(n)351 2450 y Fl(9)p Fm(E)16 b Fl(2)e Fm(I)t(;)24 b Fl(8)p Fm(j;)8 b(k)14 b Fl(2)g(f)p Fo(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(N) 16 b Fl(\000)11 b Fm(S)s Fl(g)p Fm(;)365 2598 y Fo(dist\()p Fm(\033)r Fo(\()p Fm(H)552 2606 y Fy(\003)581 2612 y Fs(n)600 2619 y(j)617 2612 y(L)638 2619 y Fr(1)p 581 2627 74 2 v 605 2645 a Fs(N)662 2606 y Fy(\()p Fx(y)693 2611 y Fr(1)711 2606 y Fy(\))726 2598 y Fo(\))p Fm(;)d(E)s Fo(\))14 b Fm(<)g(L)924 2578 y Fv(\000)p Fx(w)924 2610 y Fy(1)980 2598 y Fo(and)i(dist\()p Fm(\033)r Fo(\()p Fm(H)1261 2606 y Fy(\003)1290 2612 y Fs(n)1309 2620 y(k)1329 2612 y(L)1350 2619 y Fr(1)p 1291 2623 77 2 v 1316 2641 a Fs(N)1374 2606 y Fy(\()p Fx(y)1405 2611 y Fr(2)1423 2606 y Fy(\))1439 2598 y Fo(\))p Fm(;)8 b(E)s Fo(\))13 b Fm(<)h(L)1636 2578 y Fv(\000)p Fx(w)1636 2610 y Fy(1)1684 2543 y Fj(o)251 2706 y Fo(+3)p Fk(P)m Fl(f9)p Fm(E)f Fl(2)h Fm(I)t(;)24 b Fl(8)p Fo(\()p Fm(n)637 2713 y Fx(i)650 2706 y Fo(\))p Fm(;)p 707 2665 36 2 v 24 w Fe(H)743 2713 y Fx(L)767 2718 y Fr(1)786 2706 y Fo(\()p Fm(E)s(;)8 b(y)890 2713 y Fy(1)910 2706 y Fm(;)g(m;)g Fo(\()p Fm(n)1045 2713 y Fx(i)1058 2706 y Fo(\)\))p Fl(g)16 b Fm(;)532 b Fo(\(48\))891 2856 y(23)p eop %%Page: 24 24 24 23 bop 60 203 a Fo(where)17 b Fl(V)k Fo(=)16 b Fl(f)p Fo(\()p Fm(n)381 210 y Fx(i)395 203 y Fo(\))8 b Fl(j)g Fo(1)17 b Fl(\024)g Fm(n)570 210 y Fy(1)606 203 y Fm(<)f(n)689 210 y Fy(2)726 203 y Fm(<)g Fl(\001)8 b(\001)g(\001)17 b Fm(<)f(n)938 210 y Fx(N)t Fv(\000)p Fx(S)1039 203 y Fl(\024)h Fm(N)g Fl(\000)12 b Fo(1)p Fl(g)p Fo(.)26 b(The)18 b(probabilit)o(y)f(in)g(the)60 264 y(\014rst)g(term)d(in)i(\(48\))h(is) f(b)q(ounded)h(from)e(ab)q(o)o(v)o(e)h(b)o(y)421 357 y Fk(P)l Fl(f9)p Fm(E)e Fl(2)g Fm(I)t(;)23 b Fo(T)l(r\()p Fh(E)773 364 y Fy(1)793 357 y Fo(\()p Fm(J)839 364 y Fx(w)867 357 y Fo(\)\)\(T)l(r\()p Fh(E)1030 364 y Fy(2)1050 357 y Fo(\()p Fm(J)1096 364 y Fx(w)1124 357 y Fo(\)\)\))14 b Fl(\025)f Fo(1)p Fl(g)k Fm(;)356 b Fo(\(49\))60 450 y(where)14 b Fm(J)226 457 y Fx(w)268 450 y Fo(=)g(\()p Fm(E)d Fl(\000)d Fm(L)466 430 y Fv(\000)p Fx(w)466 462 y Fy(1)521 450 y Fm(;)g(E)i Fo(+)e Fm(L)668 430 y Fv(\000)p Fx(w)668 462 y Fy(1)724 450 y Fo(\))14 b(and)h Fh(E)887 457 y Fy(1)907 450 y Fo(,)f Fh(E)972 457 y Fy(2)1006 450 y Fo(are,)h(resp)q(ectiv)o(ely)l(,)d(the)i(sp)q(ectral)g(family)60 510 y(of)j Fm(H)156 518 y Fy(\003)180 525 y Fs(n)199 532 y Fr(1)217 525 y Fs(L)238 532 y Fr(1)255 525 y Fs(=)n(N)298 518 y Fy(\()p Fx(y)329 523 y Fr(1)347 518 y Fy(\))379 510 y Fo(and)g Fm(H)514 518 y Fy(\003)538 525 y Fs(n)557 532 y Fr(2)575 525 y Fs(L)596 532 y Fr(1)613 525 y Fs(=)n(N)657 518 y Fy(\()p Fx(y)688 523 y Fr(2)705 518 y Fy(\))721 510 y 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Fy(1)845 840 y Fj(Z)873 952 y Fx(I)901 907 y Fo(T)l(r\()p Fh(E)1007 914 y Fy(1)1027 907 y Fo(\()p Fm(J)1073 914 y Fx(w)1101 907 y Fo(\)\))j(T)l(r\()p Fh(E)1253 914 y Fy(2)1273 907 y Fo(\()p Fm(J)1319 914 y Fx(w)1347 907 y Fo(\)\))487 1037 y Fl(\024)41 b Fo(2)p Fm(L)624 1016 y Fx(w)624 1049 y Fy(1)661 969 y Fj(Z)689 1082 y Fx(I)717 1037 y Fk(E)t Fl(f)p Fo(T)l(r)s(\()p Fh(E)881 1044 y Fy(1)901 1037 y Fo(\()p Fm(J)947 1044 y Fx(w)975 1037 y Fo(\)\))8 b(T)l(r\()p Fh(E)1127 1044 y Fy(2)1147 1037 y Fo(\()p Fm(J)1193 1044 y Fx(w)1221 1037 y Fo(\)\))p Fl(g)p Fm(dE)487 1140 y Fl(\024)41 b Fo(2)p Fm(C)626 1147 y Fx(W)666 1140 y Fl(j)p Fm(I)t Fl(j)p Fm(L)753 1120 y Fx(w)753 1153 y Fy(1)781 1140 y Fm(L)814 1120 y Fv(\000)p Fy(2)p Fx(w)814 1153 y Fy(1)888 1140 y Fl(j)p Fo(\003)5 b Fs(n)960 1147 y Fr(1)977 1140 y Fs(L)998 1147 y Fr(1)p 940 1151 75 2 v 964 1171 a Fs(N)1022 1140 y Fo(\()p Fm(y)1065 1147 y Fy(1)1085 1140 y Fo(\))p Fl(j)j(j)p Fo(\003)d Fs(n)1198 1147 y Fr(2)1215 1140 y Fs(L)1236 1147 y Fr(1)p 1178 1151 V 1202 1171 a Fs(N)1260 1140 y Fo(\()p Fm(y)1303 1147 y Fy(2)1323 1140 y Fo(\))p Fl(j)327 b Fo(\(50\))487 1231 y Fl(\024)41 b Fo(2)591 1211 y Fy(2)p Fx(d)p Fy(+1)674 1231 y Fm(C)709 1238 y Fx(W)749 1231 y Fl(j)p Fm(I)t Fl(j)p Fm(L)836 1211 y Fv(\000)p Fx(w)q Fy(+2)p Fx(d)836 1244 y Fy(1)971 1231 y Fm(:)698 b Fo(\(51\))60 1325 y(In)22 b(inequalit)o(y)e(\(50\))j(w)o (e)e(ha)o(v)o(e)g(used)h(Prop)q(osition)h(A.1.)38 b(The)22 b(second)g(term)e(in)i(\(48\))g(is)60 1385 y(estimated)14 b(as)j(follo)o(ws)82 1478 y Fk(P)l Fl(f9)p Fm(E)d Fl(2)g Fm(I)t(;)23 b Fl(8)p Fo(\()p Fm(n)405 1485 y Fx(i)418 1478 y Fo(\))p Fm(;)p 475 1438 36 2 v 24 w Fe(H)511 1485 y Fx(L)535 1490 y Fr(1)554 1478 y Fo(\()p Fm(E)s(;)8 b(y)658 1485 y Fy(1)678 1478 y Fm(;)g(m;)g Fo(\()p Fm(n)813 1485 y Fx(i)826 1478 y Fo(\)\))p Fl(g)123 1551 y(\024)41 b Fk(P)237 1511 y Fj(\010)263 1551 y Fl(9)p Fm(E)16 b Fl(2)e Fm(I)t(;)8 b Fl(9)p Fo(1)14 b Fl(\024)f Fm(n)585 1558 y Fy(1)619 1551 y Fm(<)h Fl(\001)8 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Fl(2)i(f)p Fo(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(N)16 b Fl(\000)11 b Fm(S)s Fl(g)p Fm(;)23 b Fo(\003)987 1839 y Fx(L)1011 1844 y Fr(0)1031 1832 y Fo(\()p Fm(z)1073 1839 y Fx(i)1086 1832 y Fo(\))17 b(are)f(not)h(\()p Fm(m;)8 b(E)s Fo(\))p Fl(\000)p Fo(go)q(o)q(d)17 b(b)q(o)o(xes)p Fl(g)123 1921 y(\024)41 b Fm(c)p Fo(\()p Fm(N)r(;)8 b(S;)g(d)p Fo(\))p Fm(L)435 1891 y Fy(\(1)p Fv(\000)503 1877 y Fr(1)p 500 1883 21 2 v 500 1904 a Fs(\013)525 1891 y Fy(\)\()p Fx(d)p Fv(\000)p Fy(1\)\()p Fx(N)t Fv(\000)p Fx(S)r Fy(\))435 1933 y(1)741 1921 y Fm(L)774 1895 y Fv(\000)p Fx(p\022)q Fy(\()p Fx(N)t Fv(\000)p Fx(S;\013)p Fy(\))774 1933 y(0)996 1921 y Fm(;)673 b Fo(\(52\))60 2014 y(for)17 b(some)f(uniform)f (constan)o(t)j Fm(c)p Fo(\()p Fm(N)r(;)8 b(S;)g(d)p Fo(\))17 b(dep)q(ending)g(only)g(on)g Fm(N)5 b Fo(,)17 b Fm(S)i Fo(and)f Fm(d)p Fo(.)23 b(In)16 b(the)h(last)60 2074 y(inequalit)o(y)d(w)o(e)i(ha)o(v)o(e)g(used)g(Assumption)f(\(A4\).)21 b(No)o(w,)16 b(\(51\))h(and)g(\(52\))g(giv)o(e)211 2211 y Fk(P)245 2126 y Fj(\()282 2211 y Fl(8)p Fm(E)e Fl(2)f Fm(I)t(;)24 b Fl(9)p Fo(\()p Fm(n)549 2191 y Fv(0)549 2224 y Fx(i)563 2211 y Fo(\))582 2218 y Fx(i)p Fy(=1)p Fx(;:::)o(S)720 2211 y Fo(s)p Fm(:)p Fo(t)p Fm(:)801 2156 y Fj(\020)831 2211 y Fe(W)883 2218 y Fx(L)907 2223 y Fr(1)927 2211 y Fo(\()p Fm(E)s(;)8 b(y)1031 2218 y Fy(1)1050 2211 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)1178 2191 y Fv(0)1178 2224 y Fx(i)1192 2211 y Fo(\)\))16 b(and)h Fe(H)1377 2218 y Fx(L)1401 2223 y Fr(1)1420 2211 y Fo(\()p Fm(E)s(;)8 b(y)1524 2218 y Fy(1)1544 2211 y Fm(;)g(m;)g Fo(\()p Fm(n)1679 2191 y Fv(0)1679 2224 y Fx(i)1692 2211 y Fo(\)\))1746 2156 y Fj(\021)275 2377 y Fo(or)335 2322 y Fj(\020)364 2377 y Fe(W)416 2384 y Fx(L)440 2389 y Fr(1)460 2377 y Fo(\()p Fm(E)s(;)g(y)564 2384 y Fy(2)583 2377 y Fm(;)g(w)q(;)g Fo(\()p Fm(n)711 2357 y Fv(0)711 2390 y Fx(i)725 2377 y Fo(\)\))16 b(and)h Fe(H)910 2384 y Fx(L)934 2389 y Fr(1)953 2377 y Fo(\()p Fm(E)s(;)8 b(y)1057 2384 y Fy(2)1077 2377 y Fm(;)g(m;)g Fo(\()p Fm(n)1212 2357 y Fv(0)1212 2390 y Fx(i)1225 2377 y Fo(\)\))1280 2322 y Fj(\021)1309 2292 y(\))394 2529 y Fl(\025)41 b Fo(1)12 b Fl(\000)e Fo(2)583 2508 y Fy(2)p Fx(d)p Fy(+1)675 2459 y Fj(\022)716 2498 y Fm(N)16 b Fl(\000)11 b Fo(1)712 2558 y Fm(N)16 b Fl(\000)11 b Fm(S)850 2459 y Fj(\023)895 2529 y Fm(C)930 2536 y Fx(W)970 2529 y Fl(j)p Fm(I)t Fl(j)p Fm(L)1057 2508 y Fv(\000)p Fx(w)q Fy(+2)p Fx(d)1057 2541 y Fy(1)474 2659 y Fl(\000)p Fo(3)p Fm(c)p Fo(\()p Fm(N)r(;)d(S;)g(d)p Fo(\))p Fm(N)786 2621 y Fs(p\022)q Fr(\()p Fs(N)s Fg(\000)p Fs(S;\013)p Fr(\))p 786 2631 155 2 v 853 2651 a Fs(\013)948 2659 y Fm(L)981 2629 y Fy(\(1)p Fv(\000)1047 2615 y Fr(1)p 1045 2621 21 2 v 1045 2642 a Fs(\013)1070 2629 y Fy(\)\()p Fx(d)p Fv(\000)p Fy(1\)\()p Fx(N)t Fv(\000)p Fx(S)r Fy(\))p Fv(\000)1317 2612 y Fs(p\022)q Fr(\()p Fs(N)s Fg(\000)p Fs(S;\013)p Fr(\))p 1316 2621 155 2 v 1383 2642 a Fs(\013)981 2671 y Fy(1)394 2731 y Fl(\025)41 b Fo(1)12 b Fl(\000)e Fm(L)592 2708 y Fv(\000)p Fx(p)592 2744 y Fy(1)656 2731 y Fm(:)891 2856 y Fo(24)p eop %%Page: 25 25 25 24 bop 119 212 20 43 v 60 370 a Fn(Ac)n(kno)n(wledgmen)n(ts)60 479 y Fo(The)18 b(authors)g(w)o(an)o(t)g(to)g(thank)g(J.M.)e(Com)o(b)q (es,)h(P)l(.D.)g(Hislop,)g(H.)g(Lesc)o(hk)o(e,)f(R.)h(Mon)o(tc)o(ho)60 540 y(and)j(E.)e(Mourre)h(for)g(helpful)f(and)i(stim)o(ulating)d (discussions)i(ab)q(out)h(v)m(arious)g(asp)q(ects)g(of)60 600 y(this)g(pap)q(er.)32 b(In)20 b(particular,)g(T.)f(Hupfer)h(is)f (ac)o(kno)o(wledged)g(for)h(p)q(oin)o(ting)g(out)h(an)f(error)60 660 y(in)c(a)i(previous)e(v)o(ersion.)22 b(JMB)17 b(also)g(thanks)g(S.) g(De)g(Bievre)d(and)k(F.)e(Germinet)f(for)i(in)o(ter-)60 720 y(esting)i(remarks)f(ab)q(out)i(the)f(m)o(ulti-scale)d(analysis)j (of)h(v)o(on)f(Dreifus)f(and)i(Klein.)28 b(WF)19 b(is)60 780 y(grateful)f(to)h(J.M.)d(Com)o(b)q(es)i(for)g(hospitalit)o(y)f(and) i(supp)q(ort)g(at)g(the)f(Cen)o(tre)f(de)h(Ph)o(ysique)60 840 y(Th)o(\023)-23 b(eorique,)23 b(CNRS,)g(Marseille)e(and)i(at)g(the) g(Univ)o(ersit)o(\023)-23 b(e)20 b(de)j(T)l(oulon)g(et)g(du)g(V)l(ar,)g (and)60 901 y(JMB)f(thanks)i(H.)e(Lesc)o(hk)o(e)g(for)i(hospitalit)o(y) e(at)h(the)g(Univ)o(ersit\177)-24 b(at)22 b(Erlangen-N)q(\177)-26 b(urn)o(b)q(erg)60 961 y(where)19 b(part)g(of)g(this)g(w)o(ork)g(w)o (as)h(done.)29 b(This)19 b(w)o(ork)g(w)o(as)h(\014nancially)e(supp)q (orted)i(b)o(y)e(the)60 1021 y(Europ)q(ean)i(Union)e(through)h(TMR)g (net)o(w)o(ork)e(FMRX-CT)i(96-0001)i(\(JMB\))c(and)i(b)o(y)f(the)60 1081 y(Deutsc)o(he)e(F)l(orsc)o(h)o(ungsgemeinsc)o(haft)e(\(WF\).)60 1248 y Fn(References)84 1357 y Fo([1])24 b(M.)15 b(Aizenman,)e(G.M.)h (Graf:)21 b(Lo)q(calization)16 b(b)q(ounds)h(for)e(an)h(electron)e (gas,)i Ff(J.)h(Phys.)160 1417 y(A)g Fh(31)p Fo(,)f(6783{6806)j (\(1998\).)84 1519 y([2])24 b(M.)16 b(Aizenman,)e(S.)j(Molc)o(hano)o (v:)k(Lo)q(calization)c(at)g(large)g(disorder)f(and)i(at)f(extreme)160 1579 y(energies:)34 b(an)23 b(elemen)o(tary)d(deriv)m(ation,)k Ff(Commun.)g(Math.)f(Phys.)f Fh(157)p Fo(,)i(245{278)160 1639 y(\(1993\).)84 1741 y([3])g(P)l(.W.)16 b(Anderson:)21 b(Absence)15 b(of)i(di\013usion)f(in)g(certain)g(random)g(lattices,)e Ff(Phys.)j(R)n(ev.)160 1801 y Fh(109)p Fo(,)f(1492{1505)k(\(1958\).)84 1903 y([4])k(J.M.)18 b(Barbaroux,)i(J.M.)e(Com)o(b)q(es,)h(P)l(.D.)g (Hislop:)26 b(Lo)q(calization)20 b(near)f(band)h(edges)160 1963 y(for)d(random)f(Sc)o(hr\177)-24 b(odinger)16 b(op)q(erators,)h Ff(Helv.)i(Phys.)e(A)n(cta)f Fh(70)p Fo(,)g(16{43)i(\(1997\).)84 2065 y([5])24 b(J.M.)19 b(Barbaroux,)i(J.M.)e(Com)o(b)q(es,)h(R.)g(Mon) o(tc)o(ho:)28 b(Remarks)19 b(on)h(the)g(relation)g(b)q(e-)160 2125 y(t)o(w)o(een)h(quan)o(tum)g(dynamics)g(and)i(fractal)f(sp)q (ectra,)h Ff(J.)g(Math.)f(A)o(nal.)i(Appl.)e Fh(213)p Fo(,)160 2185 y(698{722)d(\(1997\).)84 2287 y([6])24 b(J.)11 b(Bellissard,)g(A.)g(V)l(an)g(Elst,)h(H.)f(Sc)o(h)o (ulz-Baldes:)17 b(The)11 b(non-comm)o(utativ)o(e)e(geometry)160 2347 y(of)17 b(the)f(quan)o(tum)f(Hall)g(e\013ect,)g Ff(J.)i(Math.)g(Phys.)f Fh(35)p Fo(,)g(5373{5451)k(\(1994\).)84 2449 y([7])k(R.)17 b(Carmona,)g(J.)g(Lacroix:)24 b(Sp)q(ectral)17 b(theory)g(of)h(random)f(Sc)o(hr\177)-24 b(odinger)17 b(op)q(erators,)160 2509 y(Boston,)g(Birkh\177)-24 b(auser,)15 b(1990.)84 2611 y([8])24 b(J.M.)13 b(Com)o(b)q(es,)g(P)l(.D.)g(Hislop:) 19 b(Lo)q(calization)c(for)e(some)g(con)o(tin)o(uous)h(random)f(Hamil-) 160 2671 y(tonians)k(in)f Fm(d)p Fo(-dimensions,)f Ff(J.)i(F)l(unct.)i (A)o(nal.)e Fh(124)p Fo(,)e(149{180)k(\(1994\).)891 2856 y(25)p eop %%Page: 26 26 26 25 bop 84 203 a Fo([9])24 b(J.M.)19 b(Com)o(b)q(es,)h(P)l(.D.)f (Hislop,)h(E.)g(Mourre:)28 b(Correlated)20 b(W)l(egner)g(inequalities)e (for)160 264 y(random)23 b(Sc)o(hr\177)-24 b(odinger)23 b(op)q(erators,)k Ff(Contemp)n(or)n(ary)22 b(Mathematics)h Fh(217)p Fo(,)i(191{203)160 324 y(\(1998\).)60 424 y([10])f(R.)d(Del)g (Rio,)h(S.)f(Jitomirsk)m(a)o(y)o(a,)g(Y.)f(Last,)j(B.)e(Simon:)30 b(Op)q(erators)22 b(with)g(singular)160 484 y(con)o(tin)o(uous)16 b(sp)q(ectrum)f(IV:)f(Hausdor\013)j(dimensions,)d(rank)j(one)f(p)q (erturbations)g(and)160 544 y(lo)q(calization,)g Ff(J.)h(d'A)o(nalyse)h (Math.)d Fh(69)p Fo(,)h(153{200)j(\(1996\).)60 645 y([11])24 b(H.)15 b(v)o(on)h(Dreifus,)g(A.)f(Klein:)20 b(A)15 b(new)i(pro)q(of)g (of)f(lo)q(calization)g(in)f(the)h(Anderson)g(tigh)o(t)160 705 y(binding)g(Mo)q(del,)g Ff(Commun.)h(Math.)g(Phys.)f Fh(124)p Fo(,)g(285{299)j(\(1989\).)60 805 y([12])24 b(H.)d(v)o(on)h(Dreifus,)h(A.)e(Klein:)31 b(Lo)q(calization)22 b(for)h(random)e(Sc)o(hr\177)-24 b(odinger)22 b(op)q(erators)160 866 y(with)16 b(correlated)g(p)q(oten)o(tials,)g Ff(Commun.)h(Math.)g (Phys.)f Fh(140)p Fo(,)g(133{147)j(\(1991\).)60 966 y([13])24 b(W.)14 b(Fisc)o(her,)f(H.)h(Lesc)o(hk)o(e,)f(P)l(.)i(M)q(\177)-26 b(uller:)19 b(Lo)q(calization)c(b)o(y)f(Gaussian)i(random)e(p)q(oten-) 160 1026 y(tials)i(in)g(m)o(ulti-dim)o(ensional)d(con)o(tin)o(uous)j (space.)g Ff(Pr)n(eprint)p Fo(,)f(to)i(b)q(e)f(submitted)f(to)h(J.)160 1086 y(Stat.)g(Ph)o(ys.)g(\(1999\).)60 1187 y([14])24 b(J.)17 b(F)l(r\177)-24 b(ohlic)o(h,)15 b(T.)h(Sp)q(encer:)22 b(Absence)15 b(of)i(di\013usion)g(in)g(the)f(Anderson)h(tigh)o(t)f (binding)160 1247 y(mo)q(del)g(for)i(large)g(disorder)f(or)h(lo)o(w)f (energy)l(,)g Ff(Commun.)i(Math.)f(Phys.)f Fh(88)p Fo(,)g(151{184)160 1307 y(\(1983\).)60 1407 y([15])24 b(F.)17 b(Germinet,)e(S.)j(De)f(Bi)o (\022)-23 b(evre:)22 b(Dynamical)16 b(lo)q(calization)h(for)h(discrete) e(and)i(con)o(tin-)160 1467 y(uous)23 b(random)e(Sc)o(hr\177)-24 b(odinger)21 b(op)q(erators,)j Ff(Commun.)e(Math.)g(Phys.)f Fh(194)p Fo(,)h(323-341)160 1528 y(\(1998\).)60 1628 y([16])i(P)l(.D.)17 b(Hislop,)f(I.M.)g(Sigal:)23 b(In)o(tro)q(duction) 18 b(to)f(Sp)q(ectral)g(Theory)h(with)f(Applications)160 1688 y(to)j(Sc)o(hr\177)-24 b(odinger)20 b(Op)q(erators,)h(Applied)d (Mathematical)g(Sciences)g(113,)k(New)d(Y)l(ork,)160 1748 y(Springer,)d(1996.)60 1849 y([17])24 b(H.)c(Holden,)g(F.)g (Martinelli:)28 b(On)20 b(absence)h(of)f(di\013usion)h(near)g(the)f(b)q (ottom)g(of)h(the)160 1909 y(sp)q(ectrum)f(for)h(a)h(random)e(Sc)o (hr\177)-24 b(odinger)21 b(op)q(erator)h(on)f Fm(L)1264 1891 y Fy(2)1284 1909 y Fo(\()p Fk(R)1342 1891 y Fx(\027)1361 1909 y Fo(\),)h Ff(Commun.)f(Math.)160 1969 y(Phys.)16 b Fh(93)p Fo(,)g(197{217)j(\(1984\).)60 2069 y([18])24 b(W.)16 b(Kirsc)o(h:)k(Random)15 b(Sc)o(hr\177)-24 b(odinger)15 b(Op)q(erators:)22 b(a)16 b(course,)g(in)f(Sc)o(hr\177)-24 b(odinger)16 b(Op)q(er-)160 2130 y(ators,)h(S\034nderb)q(org)g(DK)g (1988,)g(ed.)e(H.)g(Holden)h(and)g(A.)f(Jensen.)h(Lecture)f(Notes)h(in) 160 2190 y(Ph)o(ysics)g(V)l(ol.)f(345,)i(Berlin,)d(Springer,)i(1989.)60 2290 y([19])24 b(W.)30 b(Kirsc)o(h,)i(P)l(.)e(Stollman,)h(G.)f(Stolz:) 49 b(Anderson)30 b(lo)q(calization)f(for)h(Random)160 2350 y(Sc)o(hr\177)-24 b(odinger)18 b(op)q(erators)i(with)e(long)g (range)h(in)o(teractions,)e Ff(Commun.)j(Math.)f(Phys.)160 2410 y Fh(195)p Fo(,)d(495{507)j(\(1998\).)60 2511 y([20])24 b(F.)17 b(Klopp:)23 b(Lo)q(calization)17 b(for)h(some)e(con)o(tin)o (uous)h(random)f(Sc)o(hr\177)-24 b(odinger)17 b(op)q(erators,)160 2571 y Ff(Commun.)h(Math.)f(Phys.)33 b Fh(167)p Fo(,)16 b(553{569)j(\(1995\).)60 2671 y([21])24 b(H.)19 b(Kunz,)h(B.)e (Souillard:)28 b(Sur)19 b(le)g(sp)q(ectre)g(des)h(op)o(\023)-23 b(erateurs)20 b(aux)g(di\013)o(\023)-23 b(erences)18 b(\014nies)160 2731 y(al)o(\023)-23 b(eatoires,)16 b Ff(Commun.)h(Math.)g(Phys.)f Fh(78)p Fo(,)g(201{246)j(\(1980\).)891 2856 y(26)p eop %%Page: 27 27 27 26 bop 60 203 a Fo([22])24 b(I.M.)c(Lifshits,)j(S.A.)d(Gredeskul,)i (L.A.)f(P)o(astur:)32 b(In)o(tro)q(duction)22 b(to)g(the)f(theory)h(of) 160 264 y(disordered)16 b(systems,)f(New)h(Y)l(ork,)f(Wiley)l(,)f (1988.)60 365 y([23])24 b(R.)d(Mon)o(tc)o(ho:)32 b(Propri)o(\023)-23 b(et)o(\023)g(es)21 b(sp)q(ectrales)g(et)g(dynamiques)f(p)q(our)j(des)e (op)o(\023)-23 b(erateurs)22 b(de)160 425 y(Sc)o(hr\177)-24 b(odinger)16 b(\022)-24 b(a)17 b(barri)o(\022)-23 b(eres)16 b(e\016caces)f(dans)i Fm(L)1003 407 y Fy(2)1023 425 y Fo(\()p Fk(R)1081 407 y Fx(d)1098 425 y Fo(\),)f(Thesis,)g(T)l(oulon,)g (1997.)60 527 y([24])24 b(L.A.)15 b(P)o(astur:)21 b(Sp)q(ectra)16 b(of)f(random)g(self-adjoin)o(t)g(op)q(erators,)i Ff(R)o(ussian)f (Math.)h(Surv.)160 587 y Fh(28)p Fo(,)f(1{67)i(\(1973\).)60 689 y([25])24 b(L.A.)17 b(P)o(astur,)g(A.)f(Figotin:)23 b(Sp)q(ectra)18 b(of)f(random)g(and)h(almost-p)q(erio)q(dic)e(op)q (erators,)160 749 y(Berlin,)e(Springer,)i(1992.)60 851 y([26])24 b(C.)16 b(Radin,)g(B.)f(Simon:)k(In)o(v)m(arian)o(t)d (domains)f(for)h(the)g(time-dep)q(enden)o(t)d(Sc)o(hr\177)-24 b(odinger)160 911 y(equation,)16 b Ff(J.)h(Di\013.)g(Equ.)g Fh(29)p Fo(,)f(289{296)j(\(1978\).)60 1013 y([27])24 b(M.)19 b(Reed,)g(B.)g(Simon:)26 b(Metho)q(ds)20 b(of)g(Mo)q(dern)f (Mathematical)f(Ph)o(ysics,)h(I)q(I)q(I:)g(Scat-)160 1073 y(tering)d(Theory)l(,)g(New)g(Y)l(ork,)f(Academic,)e(1981.)60 1175 y([28])24 b(H.)j(Sc)o(h)o(ulz-Baldes,)i(J.)e(Bellissard:)43 b(Anomalous)27 b(transp)q(ort:)46 b(a)28 b(mathematical)160 1235 y(framew)o(ork,)15 b Ff(R)n(ev.)i(Math.)g(Phys.)f Fh(10)g Fo(1{46)i(\(1998\).)60 1337 y([29])24 b(B.I.)16 b(Shklo)o(vskii,)g(A.L.)h(Efros:)25 b(Electronic)17 b(prop)q(erties)g (of)h(dop)q(ed)h(semiconductors,)160 1397 y(Berlin,)14 b(Springer,)i(1984.)60 1499 y([30])24 b(D.J.)11 b(Thouless:)19 b(Electrons)10 b(in)h(disordered)f(systems)g(and)h(the)g(theory)g(of)g (lo)q(calization,)160 1559 y Ff(Phys.)17 b(R)n(ep.)f Fh(13)p Fo(,)g(93{142)i(\(1974\).)891 2856 y(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9907011315729--