This is a multi-part message in MIME format. ---------------0206061001475 Content-Type: text/plain; name="02-255.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-255.keywords" Random Schroedinger Operators; Edge states ---------------0206061001475 Content-Type: application/postscript; name="edge.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="edge.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: edge.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -N0 -f edge.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.06.06:0907 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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y(theory)k(with)e(conjugate)j(op)s(erator)f FD(y)s FE(.)59 b(Since)36 b FD(i)p FE([)p FD(H)r(;)15 b(y)s FE(])37 b(=)e FD(p)2373 2034 y Fy(y)2439 2020 y Fv(\000)24 b FD(B)5 b(x)35 b FE(=)h FD(v)2846 2034 y Fy(y)2887 2020 y FE(,)i(the)f(p)s(ositivit)m(y)265 2212 y(of)k FD(i)p FE([)p FD(H)r(;)15 b(y)s FE(])42 b(in)e(sp)s(ectral)h (subspaces)f(of)i FD(H)48 b FE(leads)40 b(to)i(the)g(a.c.)74 b(nature)41 b(of)g(the)h(sp)s(ectrum)265 2404 y(therein.)63 b(This)36 b(implies)g(the)i(p)s(ositivit)m(y)e(of)j FD(v)1896 2418 y Fy(y)1937 2404 y FE(,)h(that)f(represen)m(ts)f(the)h(v)m(elo)s (cit)m(y)f(op)s(erator)265 2596 y(in)d(the)i FD(y)s Fv(\000)p FE(direction)d(along)j(the)f(edge.)59 b(Since)36 b(the)g(sp)s(ectrum)g (of)g FD(H)2709 2610 y Fy(L)2785 2596 y FE(+)24 b FD(U)46 b FE(is)36 b(absolutely)265 2788 y(con)m(tin)m(uous,)48 b(this)43 b(result)h(sa)m(ys)h(that,)k(if)43 b(the)i(p)s(erturbation)d (is)i(su\016cien)m(tly)f(small)g(with)265 2980 y(resp)s(ect)24 b(to)h(the)f(strength)g(of)g(the)g(magnetic)g(\014eld,)g(part)g(of)g (the)g(a.c.)40 b(sp)s(ectrum)23 b(is)g(preserv)m(ed)265 3172 y(and)30 b(th)m(us)g(states)h(propagating)g(along)f(the)h(edge)g (surviv)m(e.)401 3364 y(F)-8 b(or)39 b(the)g(case)g(with)e(t)m(w)m(o)i (b)s(oundaries,)f(sa)m(y)h(at)g(distance)f FD(L)p FE(,)i(few)e(results) e(are)j(kno)m(wn)265 3555 y([EJK],)h([CHS].)63 b(A)38 b(\014rst)g(mo)s(del)f(consists)g(in)g(appro)m(ximating)g(the)h(strip)e (of)j(size)e FD(L)h FE(b)m(y)g(a)265 3747 y(parab)s(olic)c(c)m(hannel)g (where)h(the)g(con\014ning)f(p)s(oten)m(tial)h(is)f(giv)m(en)h(b)m(y)g (a)h(parab)s(olic)d(w)m(all.)54 b(In)265 3939 y(this)31 b(case)h(it)g(is)f(sho)m(wn)g([EJK)o(])h(that)h(if)d(the)i(p)s (erturbation)e FD(V)52 b FE(is)31 b(p)s(erio)s(dic)e(or)j(if)e FD(V)52 b FE(is)31 b(small)265 4131 y(enough)43 b(and)g(deca)m(y)i (fast)e(enough)g(in)g(the)g FD(y)s Fv(\000)p FE(direction)f(\(that)i (exclude)f(Anderson-lik)m(e)265 4323 y(random)30 b(p)s(oten)m(tials\),) g(the)g(a.c.)42 b(sp)s(ectrum)29 b(surviv)m(es)h(in)f(certain)h(in)m (terv)-5 b(als.)401 4515 y(If)48 b(w)m(e)g(consider)f(an)g(in\014nite)f (strip)g(with)h(edges)h(separated)g(b)m(y)g(a)g(distance)f FD(L)p FE(,)53 b(b)m(y)265 4707 y(a)f(smart)g(adaptation)g(of)g(the)g (conjugate)h(op)s(erator)f(it)f(can)h(b)s(e)f(pro)m(v)m(en)h([CHS])g (that,)265 4899 y(if)42 b FD(V)66 b FE(=)46 b(0)d(the)g(existence)g(of) g(extended)g(states)h(is)e(related)h(to)g(a.c.)80 b(sp)s(ectrum.)d(F)-8 b(or)43 b(a)265 5091 y(general)j(\(random\))f(p)s(oten)m(tial)g(w)m(e)h (exp)s(ect)g(that)g(these)g(results)e(turn)h(out)g(to)i(b)s(e)e(false,) 265 5283 y(indeed)51 b(the)i(p)s(oten)m(tial)f(should)e(create)k(a)f (tunnelling)c(b)s(et)m(w)m(een)54 b(the)e(t)m(w)m(o)i(b)s(oundaries) 1852 5637 y FC(3)p eop %%Page: 4 4 4 3 bop 265 100 a FE(and)45 b(th)m(us)h(propagating)g(edge)g(states)h (along)f(the)g(t)m(w)m(o)h(b)s(oundaries)d(should)g(not)i(exists)265 292 y(for)d(all)f(times.)79 b(In)42 b([CHS])h(the)h(authors)e(ha)m(v)m (e)j(sho)m(wn)d(that)i(suc)m(h)f(states)i(surviv)m(e,)g(but)265 484 y(only)23 b(for)h(a)g(\014nite)f(time)h(giv)m(en)g(b)m(y)g(the)g (quan)m(tum)g(tunnelling)d(time)i(b)s(et)m(w)m(een)i(the)f(t)m(w)m(o)i (edges.)401 868 y(In)45 b(this)f(w)m(ork)i(w)m(e)f(address)g(a)h (similar)c(question)j(but)f(for)h(macroscopic)h(\014nite)f(sys-)265 1060 y(tems)d(with)d(t)m(w)m(o)k(con\014ning)d(w)m(alls)g(\(at)i (distance)f FD(L)p FE(\))g(along)g(the)h FD(x)p Fv(\000)p FE(direction)d(and)i(with)265 1252 y(the)30 b FD(y)s Fv(\000)p FE(direction)d(made)i FD(L)g FE(p)s(erio)s(dic.)38 b(In)29 b(this)f(case,)j(although)e(the)g(sp)s(ectrum)f(is)h(discrete,) 265 1444 y(extended)42 b(edge)h(states)h(exist)e(but)g(w)m(e)h(cannot)g (deriv)m(e)f(them)g(directly)f(from)h(a)h(Mourre)265 1636 y(estimate)31 b(as)g(in)e(the)h(in\014nite)e(case.)401 1828 y(In)f(Theorem)f(1)i(w)m(e)f(sho)m(w)g(that,)i(with)d(large)h (probabilit)m(y)-8 b(,)26 b(the)h(sp)s(ectrum)f(of)h(the)g(family)265 2020 y(of)k(random)e(Hamiltonians)g(\()p FD(U)1358 2035 y Fy(`)1421 2020 y FE(and)h FD(U)1660 2034 y Fy(r)1729 2020 y FE(b)s(eing)e(the)j(t)m(w)m(o)h(con\014ning)d(w)m(alls)g(at)i (distance)f FD(L)p FE(\))1373 2264 y FD(H)1449 2278 y Fy(!)1524 2264 y FE(=)25 b FD(H)1696 2278 y Fy(L)1768 2264 y FE(+)20 b FD(V)1912 2278 y Fy(!)1983 2264 y FE(+)f FD(U)2135 2279 y Fy(`)2189 2264 y FE(+)h FD(U)2342 2278 y Fy(r)265 2509 y FE(in)g(an)h(energy)g(in)m(terv)-5 b(al)20 b(\001)25 b Fv(\032)1276 2435 y Fx(\000)1328 2473 y Fz(1)p 1328 2488 36 4 v 1328 2540 a(2)1373 2509 y FD(B)g FE(+)20 b Fv(k)p FD(V)1656 2523 y Fy(!)1707 2509 y Fv(k)1752 2523 y Fw(1)1827 2509 y FD(;)1877 2473 y Fz(3)p 1877 2488 V 1877 2540 a(2)1923 2509 y FD(B)k Fv(\000)c(k)p FD(V)2205 2523 y Fy(!)2256 2509 y Fv(k)2301 2523 y Fw(1)2376 2435 y Fx(\001)2439 2509 y FE(consists)h(in)e(the)j (union)d(of)i(t)m(w)m(o)265 2701 y(sets)31 b(\006)509 2716 y Fy(`)571 2701 y FE(and)f(\006)814 2715 y Fy(r)852 2701 y FE(,)g(and)g(these)g(sets)h(are)g(small)d(p)s(erturbations)g(of) j(the)f(sp)s(ectra)g FD(\033)s FE(\()p FD(H)3194 2715 y Fy(L)3267 2701 y FE(+)19 b FD(U)3419 2716 y Fy(`)3452 2701 y FE(\))265 2893 y(and)24 b FD(\033)s FE(\()p FD(H)602 2907 y Fy(L)664 2893 y FE(+)9 b FD(U)806 2907 y Fy(r)844 2893 y FE(\).)39 b(As)25 b(in)f([FM1)q(],)j(the)e(eigen)m(v)-5 b(alues)25 b(are)g(then)g(classi\014ed)e(b)m(y)i(their)f(quan)m(tum)265 3085 y(mec)m(hanical)i(curren)m(t)f(along)h(the)g(p)s(erio)s(dic)d (direction)i(\()p FD(J)2233 3099 y Fy(E)2318 3085 y FE(=)g(\()p FD( )2508 3099 y Fy(E)2569 3085 y FD(;)15 b(v)2653 3099 y Fy(y)2695 3085 y FD( )2754 3099 y Fy(E)2813 3085 y FE(\)\).)40 b(W)-8 b(e)27 b(sho)m(w)f(that)265 3277 y(the)41 b(eigenfunctions)f(corresp)s(onding)f(the)j(eigen)m(v)-5 b(alues)41 b(in)f(\006)2464 3292 y Fy(`)2537 3277 y FE(and)h(\006)2791 3291 y Fy(r)2870 3277 y FE(ha)m(v)m(e)h(a)g(uniform)265 3469 y(curren)m(t)29 b(with)f(resp)s(ect)h(to)h FD(L)p FE(,)f(this)f(leads)h(to)h(the)f(existence)g(of)h(extended)f(states)h (along)f(the)265 3661 y(b)s(oundaries.)401 3853 y(The)45 b(ph)m(ysical)e(imp)s(ortance)h(of)h(these)g(propagating)g(states)h(is) e(related)h(to)g(the)g(dia-)265 4045 y(magnetic)38 b(b)s(oundary)e (curren)m(ts)h(adv)m(o)s(cated)i(in)e([H].)63 b(Indeed)37 b(propagating)h(states)g(curry)265 4236 y(curren)m(ts)i(and)f(th)m(us)h (con)m(tribute)g(to)g(the)h(total)f(Hall)f(curren)m(t)h(in)f(the)h (sample,)i(therefore)265 4428 y(they)27 b(pla)m(y)f(an)h(imp)s(ortan)m (t)e(role)i(for)f(the)h(explanation)f(of)g(the)h(in)m(teger)g(quan)m (tum)f(Hall)g(e\013ect)265 4620 y(in)j(the)i(framew)m(ork)f(of)h(the)f (so-called)g(edge)h(theory)-8 b(.)401 4812 y(Finally)g(,)36 b(w)m(e)g(remark)g(that)g(our)f(result)g(is)f(consisten)m(t)j(with)d (the)i(analysis)e(of)i(Com)m(b)s(es)265 5004 y(et)h(al.)57 b([CHS].)h(Indeed)35 b(in)g(our)g(w)m(ork)h(w)m(e)h(sho)m(w)f(the)g (existence)h(of)f(propagating)g(curren)m(t)265 5196 y(carrying)g (states)h(for)f(a)h(\014nite)e(system)i(\(size)f FD(L)25 b Fv(\002)e FD(L)p FE(\))37 b(while)d(in)h([CHS])i(it)e(is)h(sho)m(wn)g (that,)265 5388 y(in)29 b(an)i(in\014nite)e(strip)g(\(of)i(width)e FD(L)p FE(\),)i(the)g(curren)m(t)g(carrying)f(edge)h(states)h(surviv)m (e)e(only)g(for)1852 5637 y FC(4)p eop %%Page: 5 5 5 4 bop 265 100 a FE(a)38 b(\014nite)e(time,)j(and)e(this)f(time)h(is)g (m)m(uc)m(h)g(larger)g(than)g(the)h(time)f(sp)s(en)m(t)g(for)g(an)g (electron,)265 292 y(starting)f(from)g(the)g(b)s(ottom)h(of)g(the)f (sample,)h(to)g(reac)m(h)g(its)f(top)h(\(this)e(time)h(is)g(indeed)e (of)265 484 y(order)c Fv(O)s FE(\()p FD(L)p FE(\))h(since)e(the)i(v)m (elo)s(cit)m(y)g(is)e(of)i(order)f Fv(O)s FE(\(1\)\).)265 930 y FB(2)161 b(The)53 b(Mo)t(del)i(and)e(Main)i(Result)265 1212 y FE(W)-8 b(e)32 b(study)d(the)i(sp)s(ectral)e(prop)s(erties)g(of) i(the)f(family)f(of)i(random)e(Hamiltonians)1155 1485 y FD(H)1231 1499 y Fy(!)1306 1485 y FE(=)c FD(H)1478 1499 y Fy(L)1550 1485 y FE(+)20 b FD(U)1703 1500 y Fy(`)1756 1485 y FE(+)g FD(U)1909 1499 y Fy(r)1967 1485 y FE(+)g FD(V)2111 1499 y Fy(!)2177 1485 y FD(;)106 b(!)28 b Fv(2)d FE(\012)2545 1499 y Fz(\003)3301 1485 y FE(\(2.1\))265 1757 y(acting)39 b(in)e(the)i(Hilb)s(ert)e(space)i FD(L)1460 1724 y Fz(2)1499 1757 y FE(\()p FA(R)c Fv(\002)25 b FE([)p Fv(\000)1828 1722 y Fy(L)p 1828 1737 48 4 v 1834 1789 a Fz(2)1886 1757 y FD(;)1936 1722 y Fy(L)p 1936 1737 V 1942 1789 a Fz(2)1994 1757 y FE(]\))39 b(with)f(p)s(erio)s(dic)d(b)s (oundary)i(conditions)265 1949 y(along)c FD(y)s FE(:)46 b FD( )s FE(\()p FD(x;)15 b Fv(\000)899 1913 y Fy(L)p 899 1928 V 905 1981 a Fz(2)958 1949 y FE(\))30 b(=)f FD( )s FE(\()p FD(x;)1323 1913 y Fy(L)p 1323 1928 V 1329 1981 a Fz(2)1381 1949 y FE(\).)49 b(W)-8 b(e)34 b(c)m(ho)s(ose)g(the)f (Landau)g(gauge)h(in)d(whic)m(h)h(the)h(kinetic)265 2141 y(part)22 b(has)f(the)h(form)f FD(H)1037 2155 y Fy(L)1114 2141 y FE(=)1220 2105 y Fz(1)p 1220 2120 36 4 v 1220 2173 a(2)1265 2141 y FD(p)1311 2108 y Fz(2)1311 2164 y Fy(x)1358 2141 y FE(+)1442 2105 y Fz(1)p 1441 2120 V 1441 2173 a(2)1486 2141 y FE(\()p FD(p)1567 2155 y Fy(y)1612 2141 y Fv(\000)s FD(B)5 b(x)p FE(\))1847 2108 y Fz(2)1907 2141 y FE(and)21 b(the)h(sp)s(ectrum)e(is)h(giv)m(en)h(b)m (y)f(the)h(Landau)265 2333 y(lev)m(els:)52 b FD(\033)s FE(\()p FD(H)721 2347 y Fy(L)773 2333 y FE(\))35 b(=)949 2260 y Fx(\010)1002 2333 y FE(\()p FD(n)20 b FE(+)1213 2297 y Fz(1)p 1213 2312 V 1213 2365 a(2)1258 2333 y FE(\))p FD(B)5 b FE(;)15 b FD(n)25 b Fv(2)g FA(N)1633 2260 y Fx(\011)1692 2333 y FE(.)57 b(The)36 b(p)s(oten)m(tials)f FD(U)2454 2348 y Fy(`)2523 2333 y FE(and)h FD(U)2768 2347 y Fy(r)2842 2333 y FE(represen)m(ting)f(the)265 2525 y(con\014nemen)m(t)h(along)g(the)g FD(x)p Fv(\000)p FE(direction)e(at)i FD(x)e FE(=)g Fv(\006)2083 2489 y Fy(L)p 2083 2504 48 4 v 2089 2556 a Fz(2)2177 2525 y FE(are)i(supp)s(osed)d(strictly)i(monotonic,)265 2717 y(t)m(wice)c(di\013eren)m(tiable)e(and)g(satisfy)833 2990 y FD(c)872 3004 y Fz(1)912 2990 y Fv(j)p 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4799 y Fv(2)27 b FE(\001)f(=)265 4991 y(\()p FD(B)f Fv(\000)20 b FD(\016)n(;)15 b(B)26 b FE(+)20 b FD(\016)s FE(\))26 b Fv(\032)f FE(\001)1025 5005 y Fy(")1092 4991 y FE(the)31 b(distance)f(b)s(et)m(w)m(een)h(t)m(w)m(o)g (consecutiv)m(e)h(eigen)m(v)-5 b(alues)30 b(satis\014es)1190 5197 y Fx(\014)1190 5251 y(\014)1220 5274 y FD(E)1292 5236 y Fy(\013)1287 5296 y(\024)p Fz(+1)1442 5274 y Fv(\000)20 b FD(E)1605 5236 y Fy(\013)1600 5296 y(\024)1655 5197 y Fx(\014)1655 5251 y(\014)1710 5274 y Fv(\025)1816 5212 y FD(C)p 1816 5253 72 4 v 1821 5336 a(L)2261 5274 y(\013)26 b FE(=)f FD(`;)15 b(r)695 b FE(\(2.10\))1852 5637 y FC(6)p eop %%Page: 7 7 7 6 bop 265 100 a FE(where)22 b FD(C)32 b(>)25 b FE(0)e(\(uniformly)e (in)g FD(\024)p FE(\).)39 b(Moreo)m(v)m(er)25 b(for)e(eac)m(h)h FD(E)2266 67 y Fy(\013)2261 123 y(\024)2341 100 y Fv(2)h FE(\001)d(the)h(quan)m(tum)g(mec)m(hanical)265 292 y(curren)m(t)g(asso) s(ciated)h(to)h(the)e(corresp)s(onding)f(eigenfunctions)g(is)g 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b(to)e(the)f(eigenvalues)g FE(\()p Fr(of)h FD(H)2656 702 y Fy(!)2706 688 y FE(\))f Fr(in)g FE(\001)g Fr(ar)-5 b(e)27 b(extende)-5 b(d.)401 954 y FE(The)36 b(main)f(to)s(ols)h(for)f (the)i(pro)s(of)e(of)h(Theorem)g(1)g(are)g(giv)m(en)g(in)f(section)h (3.)58 b(Basically)265 1146 y(they)27 b(consist)f(in)f(a)h(W)-8 b(egner)28 b(estimate)f(for)f(the)h(random)f(Hamiltonians)e FD(H)2867 1160 y Fy(\013)2943 1146 y FE(\()p FD(\013)i FE(=)f FD(`;)15 b(r)s FE(\))26 b(and)265 1338 y(a)32 b(decoupling)d(sc)m(heme)k(that)f(links)d(the)i(resolv)m(en)m(t)h(of)g (the)g(full)d(Hamiltonian)h FD(H)3087 1352 y Fy(!)3168 1338 y FE(to)i(those)265 1530 y(of)c FD(H)442 1545 y Fy(`)474 1530 y FE(,)h FD(H)604 1544 y Fy(r)669 1530 y FE(and)e FD(H)919 1545 y Fy(b)953 1530 y FE(.)39 b(In)27 b(section)h(4)g(w)m(e)g(pro)m(v)m(e)g(t)m(w)m(o)h(prop)s(ositions)d (that)i(giv)m(e)g(part)f FD(a)p FE(\))h(and)f FD(b)p FE(\))265 1722 y(of)f(Theorem)f(1.)40 b(Finally)24 b(in)g(app)s(endix)f 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FD(:)15 b(:)g(:)q Fv(g)3018 1088 y Fz(1)p Fy(=)p Fz(2)3171 1121 y FE(are)44 b(also)265 1313 y(b)s(ounded)28 b(b)m(y)i(a)h(constan)m(t) 1226 1290 y(~)1205 1313 y FD(C)6 b FE(\()p FD(B)f FE(\))31 b(dep)s(ending)d(only)h(on)i FD(B)j FE(and)c(not)h(on)f FD(L)p FE(.)41 b(This)28 b(leads)i(to)816 1562 y Fv(k)p FD(f)10 b(@)969 1524 y Fy(\013)964 1584 y(x)1034 1562 y FE([)p FD(R)1128 1576 y Fz(0)1168 1562 y FE(\()p FD(z)t FE(\)])1310 1520 y Fy(n)1372 1562 y FD(g)s(')p Fv(k)27 b(\024)e(k)p FD(f)10 b Fv(k)1790 1576 y Fw(1)1885 1539 y FE(^)1865 1562 y FD(C)c FE(\()p FD(B)f FE(\)\()2136 1539 y(~)2115 1562 y FD(C)i(A)p FE(\))2290 1524 y Fy(n)2338 1562 y FD(e)2380 1524 y Fw(\000)2460 1497 y Fs(1)p 2445 1509 61 3 v 2445 1550 a(12)2518 1524 y Fz(\026)-37 b Fy(\015)2556 1468 y Fw(p)p 2615 1468 57 3 v 56 x Fy(B)s(D)2735 1562 y Fv(k)p FD(')p Fv(k)27 b FD(:)319 b FE(\(3.32\))265 1811 y(Therefore,)30 b(if)g FD(V)839 1825 y Fz(0)908 1811 y FE(\(i.e.)41 b FD(A)p FE(\))31 b(is)f(small)f(enough)h(the)g 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Fz(\026)-37 b Fy(\015)p 2182 3003 71 4 v 2182 3056 a Fz(12)2282 3024 y Fv(\000)28 b FE(\026)-53 b FD(\026)p FE(.)p 3422 3024 4 62 v 3426 2966 55 4 v 3426 3024 V 3479 3024 4 62 v 401 3279 a(By)35 b(the)f(w)m(a)m(y)h(w)m (e)g(remark)f(that)h(in)e(the)h(pro)s(of)g(ab)s(o)m(v)m(e)h(w)m(e)g(ha) m(v)m(e)g(pro)m(v)m(ed)g(the)f(follo)m(wing)265 3471 y(statemen)m(t)e(\(see)g(\(3.33\))r(\))f(that)g(will)c(b)s(e)j(useful)f (in)g(the)h(next)h(section)1050 3721 y Fv(k)p FE(\(1)22 b Fv(\000)1310 3698 y FE(~)1288 3721 y FD(J)1338 3735 y Fy(\013)1388 3721 y FE(\))1443 3698 y(~)1423 3721 y FD(R)1492 3736 y Fy(b)1526 3721 y FE(\()p FD(z)t FE(\))p FD(g)s Fv(k)28 b(\024)1877 3698 y FE(\026)1857 3721 y FD(C)6 b FE(\()p FD(B)f(;)15 b(V)2130 3735 y Fz(0)2170 3721 y FD(;)g(")p FE(\))p FD(e)2329 3683 y Fw(\000)r Fz(~)-37 b Fy(\015)2425 3627 y Fw(p)p 2484 3627 57 3 v 56 x Fy(B)2541 3627 y Fw(p)p 2600 3627 48 3 v 56 x Fy(L)2677 3721 y FD(:)553 b FE(\(3.36\))265 3970 y(where)21 b FD(g)29 b FE(=)c FD(U)749 3984 y Fy(\013)820 3970 y FE(or)d FD(g)29 b FE(=)c FD(\037)1148 3984 y Fy(B)1230 3970 y FE(\()p FD(B)30 b Fv(\032)25 b FA(R)6 b Fv(\002)s FE([)p Fv(\000)1703 3934 y Fy(L)p 1703 3949 48 4 v 1709 4001 a Fz(2)1767 3970 y FD(;)1817 3934 y Fy(L)p 1817 3949 V 1823 4001 a Fz(2)1875 3970 y FE(]\))22 b(with)f(dist)o(\(supp)13 b FD(g)s(;)i FE(supp\(1)s Fv(\000)2994 3947 y FE(~)2972 3970 y FD(J)3022 3984 y Fy(\013)3072 3970 y FE(\)\))26 b(=)f Fv(O)s FE(\()p FD(D)s FE(\))265 4162 y(and)462 4139 y(~)442 4162 y FD(R)511 4177 y Fy(b)545 4162 y FE(\()p FD(z)t FE(\))31 b(a)g(resolv)m(en)m(t)g(asso)s(ciated)g(to)g(a)g (generic)f(bulk)f(Hamiltonian)g(\()p FD(H)2917 4176 y Fy(L)2989 4162 y FE(+)20 b FD(V)3133 4176 y Fy(!)3183 4162 y Fv(j)3215 4175 y Fz(~)3208 4192 y(\003)3262 4162 y FE(\).)265 4604 y FB(4)161 b(Pro)9 b(jector)55 b(estimates)h(and)g (the)g(pro)t(of)g(of)g(The-)265 4914 y(orem)d(1)265 5196 y FE(In)32 b(this)f(section)h(w)m(e)h(pro)m(v)m(e)g(t)m(w)m(o)h(prop)s (ositions)c(that)j(lead)f(to)h(Theorem)f(1.)47 b(Let)33 b Fv(D)3185 5163 y Fw(0)3237 5196 y FE(=)28 b Fv(f)p FD(\024)h FE(:)265 5388 y FD(E)337 5355 y Fy(\013)332 5411 y(\024)412 5388 y Fv(2)c FE(\001)p FD(;)15 b(\013)26 b FE(=)f FD(`;)15 b(r)s Fv(g)p FE(,)31 b(card\()p Fv(D)1297 5355 y Fw(0)1320 5388 y FE(\))26 b(=)e Fv(O)s FE(\()p FD(L)p FE(\).)42 b(Where)30 b(\001)25 b Fv(\032)g FE(\001)2313 5402 y Fy(")2380 5388 y FE(is)k(giv)m(en)i(in)e(section)h(2.)1828 5637 y FC(14)p eop %%Page: 15 15 15 14 bop 265 100 a FF(Prop)s(osition)46 b(3.)g Fr(F)-7 b(or)41 b FD(L)f Fr(lar)-5 b(ge)41 b(enough,)h(with)f(pr)-5 b(ob)g(ability)42 b(gr)-5 b(e)g(ater)42 b(then)e FE(1)26 b Fv(\000)g FD(L)3218 67 y Fw(\000)p Fy(\027)3316 100 y Fr(,)41 b(we)265 292 y(have)33 b(for)g(al)5 b(l)33 b FD(\024)26 b Fv(2)f(D)986 259 y Fw(0)1347 484 y Fv(k)p FD(P)34 b Fv(\000)20 b FD(P)1633 498 y Fy(\013)1683 484 y FE(\()p FD(E)1790 447 y Fy(\013)1785 507 y(\024)1840 484 y FE(\))p Fv(k)26 b(\024)f FD(e)2084 447 y Fw(\000)p Fy(\015)2179 390 y Fw(p)p 2238 390 57 3 v 57 x Fy(B)2295 390 y Fw(p)p 2353 390 48 3 v 2353 447 a Fy(L)3301 484 y FE(\(4.1\))265 725 y Fr(wher)-5 b(e)39 b FD(P)585 739 y Fy(\013)635 725 y FE(\()p FD(E)742 692 y Fy(\013)737 747 y(\024)792 725 y FE(\))f Fr(is)g(the)g(pr)-5 b(oje)g(ctor)40 b(asso)-5 b(ciate)g(d)41 b(to)d FD(H)2131 739 y Fy(\013)2218 725 y Fr(onto)h FD(E)2502 692 y Fy(\013)2497 747 y(\024)2590 725 y Fr(and)f FD(P)51 b Fr(is)38 b(the)g(pr)-5 b(oje)g(ctor)265 917 y(asso)g(ciate)g(d)35 b(to)e FD(H)875 931 y Fy(!)958 917 y Fr(onto)h Fv(f)p FD(z)c Fv(2)24 b FA(C)49 b FE(:)26 b Fv(j)p FD(z)f Fv(\000)20 b FD(E)1764 884 y Fy(\013)1759 939 y(\024)1813 917 y Fv(j)26 b(\024)f FD(e)2002 884 y Fw(\000)6 b Fz(\026)-41 b Fy(\026)2099 827 y Fw(p)p 2158 827 57 3 v 57 x Fy(B)2215 827 y Fw(p)p 2274 827 48 3 v 57 x Fy(L)2326 917 y Fv(g)p Fr(.)265 1174 y(Pr)-5 b(o)g(of.)50 b FE(\(1\):)42 b(Let)31 b Fv(E)i FE(=)25 b Fv(f)p FD(k)k FE(:)d FD(E)1319 1141 y Fy(\013)1314 1202 y Fz(0)p Fy(k)1417 1174 y Fv(2)f FE(\001)p Fv(g)30 b FE(\(card)q(\()p Fv(E)8 b FE(\))26 b(=)f Fv(O)s FE(\()p FD(L)p FE(\)\))31 b(and)f(let)893 1424 y(^)883 1447 y(\012)949 1462 y Fy(`)1006 1447 y FE(=)25 b Fv(f)p FD(!)k Fv(2)c FE(\012)1385 1461 y Fz(\003)1434 1473 y Fu(`)1493 1447 y FE(:)g(dist)o(\()p FD(E)1797 1409 y Fy(r)1792 1470 y Fz(0)p Fy(k)1871 1447 y FD(;)15 b(\033)s FE(\()p FD(H)2077 1462 y Fy(`)2110 1447 y FE(\)\))26 b Fv(\025)f FD(L)2364 1409 y Fw(\000)p Fy(\033)2466 1447 y FD(;)15 b Fv(8)p FD(k)28 b Fv(2)d(E)8 b(g)26 b FD(;)431 b FE(\(4.2\))265 1719 y(with)29 b FD(\033)g(>)c FE(10,)31 b(this)e(set)i(has)f (probabilit)m(y)1404 1991 y FA(P)1459 2005 y Fz(\003)1508 2017 y Fu(`)1542 1991 y FE(\()1587 1968 y(^)1577 1991 y(\012)1643 2006 y Fy(`)1676 1991 y FE(\))c Fv(\025)f FE(1)c Fv(\000)f FD(L)2052 1954 y Fw(\000)p Fz(\()p Fy(\033)r Fw(\000)p Fz(8\))2324 1991 y FD(:)952 b FE(\(4.3\))265 2264 y(Indeed)29 b(for)i(a)f(\014xed)g FD(k)f Fv(2)24 b(E)8 b FE(,)31 b(using)e(Prop)s(osition)f(1)j(and)f(\()p FD(H)7 b FE(1\))31 b(one)g(gets)889 2536 y FA(P)944 2550 y Fz(\003)993 2562 y Fu(`)1043 2462 y Fx(\010)1096 2536 y FD(!)d Fv(2)d FE(\012)1333 2550 y Fz(\003)1382 2562 y Fu(`)1441 2536 y FE(:)h(dist)o(\()p FD(E)1746 2498 y Fy(r)1741 2559 y Fz(0)p Fy(k)1819 2536 y FD(;)15 b(\033)s FE(\()p FD(H)2025 2551 y Fy(`)2059 2536 y FE(\)\))26 b Fv(\025)f FD(L)2313 2498 y Fw(\000)p Fy(\033)2414 2536 y FD(;)46 b FE(for)30 b(one)h FD(k)d Fv(2)d(E)3008 2462 y Fx(\011)736 2753 y Fv(\025)82 b FE(1)21 b Fv(\000)f FD(C)1118 2715 y Fw(0)1141 2753 y FE(\()p FD(h;)15 b(V)1321 2767 y Fz(0)1361 2753 y FE(\))p FD(L)1458 2715 y Fw(\000)p Fy(\033)1560 2753 y FD(L)1622 2715 y Fz(4)1677 2652 y Fx(\020)1741 2715 y Fy(d)1777 2724 y Fs(0)p 1741 2732 71 4 v 1753 2784 a Fy(L)1842 2753 y Fv(\000)20 b FD(L)1995 2715 y Fw(\000)p Fy(\033)2097 2652 y Fx(\021)2151 2675 y Fw(\000)p Fz(2)2271 2753 y Fv(\025)k FE(1)d Fv(\000)f FD(C)7 b FE(\()p FD(h;)15 b(V)2775 2767 y Fz(0)2815 2753 y 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y(\012)25 b(=)1354 4629 y(^)1344 4651 y(\012)1410 4666 y Fy(`)1463 4651 y Fv(\002)20 b FE(\012)1620 4666 y Fy(b)1674 4651 y Fv(\002)1775 4629 y FE(^)1765 4651 y(\012)1831 4665 y Fy(r)1899 4651 y FE(\(\012)2000 4666 y Fy(b)2060 4651 y FE(=)25 b(\012)p Fv(j)2247 4670 y Fz(\003)2296 4682 y Fu(b)2327 4670 y Fw(n)p Fz(\(\003)2438 4682 y Fu(`)2469 4670 y Fw([)p Fz(\003)2565 4678 y Fu(r)2599 4670 y Fz(\))2631 4651 y FE(\))30 b(has)h(probabilit)m(y)1535 4924 y FA(P)1590 4938 y Fz(\003)1644 4924 y FE(\()1689 4901 y(^)1679 4924 y(\012\))25 b Fv(\025)g FE(1)c Fv(\000)f FD(L)2120 4886 y Fw(\000)p Fy(\027)3301 4924 y FE(\(4.8\))265 5196 y(with)29 b FD(\027)i FE(=)25 b FD(\033)e Fv(\000)d FE(9)26 b FD(>)f FE(1.)1828 5637 y FC(15)p eop %%Page: 16 16 16 15 bop 401 100 a FE(\(2\):)44 b(W)-8 b(e)32 b(no)m(w)f(w)m(ork)h (with)e(a)h(giv)m(en)h FD(!)d Fv(2)1862 77 y FE(^)1852 100 y(\012.)43 b(T)-8 b(ak)m(e)40 b(\026)-52 b FD(\026)26 b(>)g FE(0)32 b(as)f(in)f(Prop)s(osition)g(2)h(and)g FD(L)265 292 y 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FE(=)83 b FD(H)1796 928 y Fy(L)1868 914 y FE(+)20 b FD(V)2032 877 y Fy(`)2012 937 y(!)2065 914 y Fv(j)2090 928 y Fz(\003)2139 937 y Fs(1)3301 914 y FE(\(4.9\))1368 1131 y FD(H)1444 1145 y Fz(2)1566 1131 y FE(=)83 b FD(H)1796 1145 y Fy(L)1868 1131 y FE(+)20 b FD(V)2032 1094 y Fy(`)2012 1154 y(!)2065 1131 y Fv(j)2090 1145 y Fz(\003)2139 1154 y Fs(2)2198 1131 y FE(+)g FD(U)2351 1146 y Fy(`)3255 1131 y FE(\(4.10\))265 1369 y(where)57 b(\003)618 1383 y Fz(2)727 1369 y FE(=)868 1296 y Fx(\010)921 1369 y FE(\()p FD(n;)15 b(m)p FE(\))26 b Fv(2)e FA(Z)1342 1337 y Fz(2)1378 1369 y FE(;)15 b FD(n)25 b Fv(2)g FE([)p Fv(\000)1690 1334 y Fy(L)p 1690 1349 48 4 v 1696 1401 a Fz(2)1747 1369 y FD(;)15 b Fv(\000)1868 1334 y Fy(L)p 1868 1349 V 1874 1401 a Fz(2)1947 1369 y FE(+)20 b(\()2083 1334 y Fy(D)p 2083 1349 60 4 v 2095 1401 a Fz(4)2173 1369 y Fv(\000)g FE(1\)])p FD(;)15 b(m)26 b Fv(2)f FE([)p Fv(\000)2707 1334 y Fy(L)p 2707 1349 48 4 v 2713 1401 a Fz(2)2765 1369 y FD(;)2815 1334 y Fy(L)p 2815 1349 V 2821 1401 a Fz(2)2873 1369 y FE(])2898 1296 y Fx(\011)2952 1369 y FE(,)64 b(and)56 b(\003)3307 1383 y Fz(1)3417 1369 y FE(=)265 1561 y(\003)328 1576 y Fy(`)361 1561 y Fv(n)p FE(\003)469 1575 y Fz(2)509 1561 y FE(,)31 b(of)f(course)h FD(H)1023 1575 y Fy(\013)1097 1561 y FE(=)25 b FD(H)1269 1575 y Fz(2)1328 1561 y FE(+)20 b FD(V)1492 1528 y Fy(`)1472 1584 y(!)1525 1561 y Fv(j)1550 1575 y Fz(\003)1599 1584 y Fs(1)1638 1561 y FE(.)401 1753 y(F)-8 b(rom)31 b(the)g(decoupling)d (form)m(ula)i(\(3.18\))j(w)m(e)d(ha)m(v)m(e)577 1992 y FD(R)q FE(\()p FD(z)t FE(\))21 b Fv(\000)f FD(R)944 2007 y Fy(`)977 1992 y FE(\()p FD(z)t FE(\))84 b(=)1331 1836 y Fx( )1403 1905 y(X)1411 2102 y Fy(i)p Fw(2I)1549 1992 y FD(J)1599 2006 y Fy(i)1628 1992 y FD(R)1697 2006 y Fy(i)1725 1992 y FE(\()p FD(z)t FE(\))1864 1969 y(~)1841 1992 y FD(J)1891 2006 y Fy(i)1921 1836 y Fx(!)15 b( )2111 1878 y Fw(1)2081 1905 y Fx(X)2080 2100 y Fy(n)p Fz(=1)2228 1992 y Fv(K)q FE(\()p FD(z)t FE(\))2414 1954 y Fy(n)2462 1836 y Fx(!)2555 1992 y Fv(\000)20 b FE(\(1)h Fv(\000)e FD(J)2887 2007 y Fy(`)2921 1992 y FE(\))p FD(R)3025 2007 y Fy(`)3058 1992 y FE(\()p FD(z)t FE(\))1177 2232 y Fv(\000)83 b FD(J)1381 2247 y Fy(`)1414 2232 y FD(R)1483 2247 y Fy(`)1516 2232 y FE(\()p FD(z)t FE(\)\(1)22 b Fv(\000)1847 2209 y FE(~)1825 2232 y FD(J)1875 2247 y Fy(`)1909 2232 y FE(\))e(+)g FD(J)2105 2247 y Fy(b)2140 2232 y FD(R)2209 2247 y Fy(b)2243 2232 y FE(\()p FD(z)t FE(\))2382 2209 y(~)2359 2232 y FD(J)2409 2247 y Fy(b)2465 2232 y FE(+)g FD(J)2606 2246 y Fy(r)2644 2232 y FD(R)2713 2246 y Fy(r)2751 2232 y FE(\()p FD(z)t FE(\))2890 2209 y(~)2867 2232 y FD(J)2917 2246 y Fy(r)2982 2232 y FD(:)248 b FE(\(4.11\))265 2470 y(in)m(tegrating)30 b(o)m(v)m(er)i FD(@)5 b FE(\000)1032 2484 y Fy(\024)1107 2470 y FE(and)30 b(taking)g(the)h(op)s(erator)g (norm)e(w)m(e)i(get)467 2709 y Fv(k)p FD(P)j Fv(\000)20 b FD(P)753 2724 y Fy(`)786 2709 y FE(\()p FD(E)893 2671 y Fy(`)888 2731 y(\024)933 2709 y FE(\))p Fv(k)84 b(\024)f FD(e)1293 2671 y 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2949 y FE(\)\(1)h Fv(\000)2654 2926 y FE(~)2632 2949 y FD(J)2682 2964 y Fy(`)2716 2949 y FE(\))p Fv(k)1097 3166 y FE(=)83 b FD(a)20 b FE(+)g FD(b)g FE(+)g FD(c)26 b(:)1605 b FE(\(4.12\))265 3404 y(F)-8 b(or)29 b(the)g(\014rst)f(term)h (w)m(e)g(note)g(that)g(for)g FD(L)f FE(large)h(enough)f FD(e)2312 3371 y Fw(\000)6 b Fz(\026)-41 b Fy(\026)2410 3315 y Fw(p)p 2469 3315 57 3 v 56 x Fy(B)2525 3315 y Fw(p)p 2584 3315 48 3 v 56 x Fy(L)2651 3404 y FE(sup)2788 3426 y Fy(z)s Fw(2)p Fy(@)t Fz(\000)2956 3434 y Fu(\024)3016 3404 y Fv(k)p FD(R)3130 3418 y Fy(i)3158 3404 y FE(\()p FD(z)t FE(\))p Fv(k)27 b(\024)e FE(1)265 3596 y(\()p FD(i)33 b Fv(2)f(I)7 b FE(\).)52 b(Indeed,)35 b(for)g FD(i)d FE(=)g FD(`)j FE(w)m(e)g(ha)m(v)m(e)g(sup)1823 3618 y Fy(z)s Fw(2)p Fy(@)t Fz(\000)1991 3626 y Fu(\024)2051 3596 y Fv(k)p FD(R)2165 3611 y Fy(`)2198 3596 y FE(\()p FD(z)t FE(\))p Fv(k)f FE(=)e FD(e)2544 3563 y Fz(\026)-41 b Fy(\026)2580 3507 y Fw(p)p 2639 3507 57 3 v 56 x Fy(B)2696 3507 y Fw(p)p 2755 3507 48 3 v 56 x Fy(L)2841 3596 y FE(b)m(y)35 b(construction,)265 3788 y(for)42 b FD(i)k FE(=)f FD(b)d FE(w)m(e)h(ha)m(v)m(e)h(sup)1195 3810 y Fy(z)s Fw(2)p Fy(@)t Fz(\000)1363 3818 y Fu(\024)1422 3788 y Fv(k)p FD(R)1536 3803 y Fy(b)1571 3788 y FE(\()p FD(z)t FE(\))p Fv(k)j FE(=)e FD(")1937 3755 y Fw(\000)p Fz(1)2074 3788 y FE(and)d(for)g FD(i)j FE(=)g FD(r)g FE(sup)2829 3810 y Fy(z)s Fw(2)p Fy(@)t Fz(\000)2997 3818 y Fu(\024)3057 3788 y Fv(k)p FD(R)3171 3802 y Fy(r)3209 3788 y FE(\()p FD(z)t FE(\))p Fv(k)i FE(=)265 3879 y Fx(\020)319 3980 y FD(L)381 3947 y Fw(\000)p Fz(3)p Fy(\033)538 3980 y Fv(\000)20 b FD(e)671 3947 y Fw(\000)6 b Fz(\026)-41 b Fy(\026)769 3891 y Fw(p)p 828 3891 57 3 v 56 x Fy(B)884 3891 y Fw(p)p 943 3891 48 3 v 56 x Fy(L)995 3879 y Fx(\021)1050 3902 y Fw(\000)p Fz(1)1144 3980 y FE(.)41 b(Then,)29 b(applying)f(Prop)s(osition)h(2)i(w)m(e)f(get)1346 4218 y FD(a)25 b Fv(\024)g FE(2)p FD(C)7 b FE(\()p FD(B)e(;)15 b(V)1834 4232 y Fz(0)1874 4218 y FD(;)g(")p FE(\))p FD(e)2033 4181 y Fw(\000)r Fz(~)-37 b Fy(\015)2130 4125 y Fw(p)p 2189 4125 57 3 v 56 x Fy(B)2245 4125 y Fw(p)p 2304 4125 48 3 v 56 x Fy(L)2381 4218 y FD(:)849 b FE(\(4.13\))265 4457 y(F)-8 b(or)23 b(the)f(second)g(and)g(third)e(term)i(w)m(e)h (\014rst)e(observ)m(e)i(that)f(b)m(y)g(the)g(second)h(resolv)m(en)m(t)f (form)m(ula)489 4633 y FD(P)547 4648 y Fy(`)581 4633 y FE(\()p FD(E)688 4600 y Fy(`)683 4656 y(\024)728 4633 y FE(\))p 456 4674 341 4 v 456 4757 a(\()p FD(z)j Fv(\000)20 b FD(E)721 4731 y Fy(`)716 4780 y(\024)761 4757 y FE(\))832 4695 y(=)25 b(\()p FD(z)f Fv(\000)c FD(H)1196 4709 y Fz(1)1235 4695 y FE(\))1270 4657 y Fw(\000)p Fz(1)1365 4695 y FD(P)1423 4710 y Fy(`)1456 4695 y FE(\()p FD(E)1563 4657 y Fy(`)1558 4717 y(\024)1604 4695 y FE(\))g(+)g(\()p FD(z)25 b Fv(\000)20 b FD(H)2019 4709 y Fz(1)2058 4695 y FE(\))2093 4657 y Fw(\000)p Fz(1)2188 4695 y FE([)p FD(V)2286 4657 y Fy(`)2266 4717 y(!)2319 4695 y Fv(j)2344 4709 y Fz(\003)2393 4718 y Fs(2)2452 4695 y FE(+)g FD(U)2605 4710 y Fy(`)2638 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y(with)24 b(p)s(erio)s(dic)f(b)s(oundary)h(conditions)g(along)i FD(y)s FE(.)39 b(The)25 b(exact)i(form)m(ula)e(for)g FD(R)2945 4433 y Fz(0)2985 4419 y FE(\()p Fq(x)p FD(;)15 b Fq(x)3180 4386 y Fw(0)3204 4419 y FE(;)g FD(z)t FE(\))26 b(can)265 4611 y(b)s(e)k(found)f(in)g([FM1)q(].)41 b(W)-8 b(e)32 b(in)m(tro)s(duce)d(the)i(follo)m(wing)d(notation)521 4872 y(\010)587 4834 y Fy(\013)636 4872 y FE(\()p Fv(j)p Fq(x)21 b Fv(\000)f Fq(x)928 4834 y Fw(0)951 4872 y Fv(j)976 4886 y Fy(?)1016 4872 y FE(\))367 5181 y(=)521 4940 y Fx(8)521 5021 y(>)521 5049 y(>)521 5076 y(<)521 5240 y(>)521 5267 y(>)521 5294 y(:)602 5089 y FE(1)g(+)758 5011 y Fx(\014)758 5066 y(\014)789 5089 y FE(ln)879 5015 y Fx(\000)931 5053 y Fy(B)p 931 5068 57 4 v 942 5120 a Fz(2)998 5089 y Fv(j)p Fq(x)g Fv(\000)g Fq(x)1254 5056 y Fw(0)1277 5089 y Fv(j)1302 5056 y Fz(2)1302 5111 y Fy(?)1342 5015 y Fx(\001)1383 5011 y(\014)1383 5066 y(\014)1520 5089 y FD(;)40 b(\013)26 b FE(=)f(0)602 5319 y(1)20 b(+)758 5218 y Fx(h)816 5242 y(\014)816 5296 y(\014)847 5319 y FE(ln)938 5245 y Fx(\000)989 5283 y Fy(B)p 989 5298 V 1000 5350 a Fz(2)1056 5319 y Fv(j)p Fq(x)g Fv(\000)g Fq(x)1312 5286 y Fw(0)1335 5319 y Fv(j)1360 5286 y Fz(2)1360 5342 y Fy(?)1400 5245 y Fx(\001)1441 5242 y(\014)1441 5296 y(\014)1492 5319 y FE(+)1583 5245 y Fx(\000)1625 5319 y FE(1)g(+)1781 5242 y Fx(\014)1781 5296 y(\014)1811 5319 y FE(ln)1902 5245 y Fx(\000)1954 5283 y Fy(B)p 1954 5298 V 1965 5350 a Fz(2)2020 5319 y Fv(j)p Fq(x)h Fv(\000)f Fq(x)2277 5286 y Fw(0)2300 5319 y Fv(j)2325 5286 y Fz(2)2325 5342 y Fy(?)2365 5245 y Fx(\001)2406 5242 y(\014)2406 5296 y(\014)2437 5245 y(\001)2493 5319 y Fv(j)p Fq(x)h Fv(\000)f Fq(x)2750 5286 y Fw(0)2773 5319 y Fv(j)2798 5286 y Fw(\000)p Fz(1)2798 5342 y Fy(?)2892 5218 y Fx(i)3026 5319 y FD(;)41 b(\013)26 b FE(=)f(1)g FD(:)3278 5181 y FE(\(A.1\))1828 5637 y FC(18)p eop %%Page: 19 19 19 18 bop 265 100 a FF(Lemma)30 b(1.)40 b Fr(If)30 b Fv(jI)7 b FD(m)15 b(z)t Fv(j)25 b(\024)g FE(1)p Fr(,)31 b Fv(R)p FD(e)15 b(z)30 b Fv(2)1627 27 y Fx(\003)1675 65 y Fz(1)p 1675 80 36 4 v 1675 132 a(2)1720 100 y FD(B)5 b(;)1844 65 y Fz(3)p 1844 80 V 1844 132 a(2)1889 100 y FD(B)1963 27 y Fx(\002)2031 100 y Fr(then,)31 b(for)g FD(L)g Fr(lar)-5 b(ge)31 b(enough,)g(ther)-5 b(e)31 b(exists)265 292 y FD(C)7 b FE(\()p FD(z)t(;)15 b(B)5 b FE(\))33 b Fr(p)-5 b(ositive)34 b(c)-5 b(onstant)34 b(indep)-5 b(endent)34 b(of)f FD(L)f Fr(such)h(that)h FE(\()p FD(\013)26 b FE(=)f(0)p FD(;)15 b FE(1\))789 539 y Fv(j)p FD(@)867 502 y Fy(\013)862 562 y(x)917 539 y FD(R)986 553 y Fz(0)1026 539 y FE(\()p Fq(x)p FD(;)g Fq(x)1221 502 y Fw(0)1245 539 y FE(;)g FD(z)t FE(\))p Fv(j)84 b(\024)f FD(C)1701 502 y Fw(0)1723 539 y FE(\()p FD(z)t(;)15 b(B)5 b FE(\))p FD(e)1995 502 y Fw(\000)2060 475 y Fu(B)p 2062 487 49 3 v 2071 528 a Fs(8)2120 502 y Fw(j)p Fo(x)p Fw(\000)p Fo(x)2287 478 y Ft(0)2310 502 y Fw(j)2330 478 y Fs(2)2330 519 y Fu(?)2369 539 y FE(\010)2435 502 y Fy(\013)2485 539 y FE(\()p Fv(j)p Fq(x)20 b Fv(\000)g Fq(x)2776 502 y Fw(0)2799 539 y Fv(j)2824 553 y Fy(?)2864 539 y FE(\))1475 756 y Fv(\024)83 b FD(C)7 b FE(\()p FD(z)t(;)15 b(B)5 b FE(\))p FD(e)1973 719 y Fw(\000)r Fz(\026)-37 b Fy(\015)2069 663 y Fw(p)p 2127 663 57 3 v 2127 719 a Fy(B)t Fw(j)p Fo(x)p Fw(\000)p Fo(x)2351 695 y Ft(0)2374 719 y Fw(j)2394 727 y Fu(?)2433 756 y FE(\010)2499 719 y Fy(\013)2548 756 y FE(\()p Fv(j)p Fq(x)21 b Fv(\000)f Fq(x)2840 719 y Fw(0)2863 756 y Fv(j)2888 770 y Fy(?)2928 756 y FE(\))315 b(\(A.2\))265 1003 y Fr(wher)-5 b(e)41 b FD(C)7 b FE(\()p FD(z)t(;)15 b(B)5 b FE(\))39 b(=)g FD(cB)1093 970 y Fz(2)1147 1003 y FE(dist)o(\()p FD(z)t(;)15 b(\033)s FE(\()p FD(H)1581 1017 y Fy(L)1634 1003 y FE(\)\))1704 970 y Fw(\000)p Fz(1)1839 1003 y Fr(with)41 b FD(c)f Fr(a)h(numeric)-5 b(al)41 b(p)-5 b(ositive)41 b(c)-5 b(onstant)42 b(and)268 1195 y FE(\026)-48 b FD(\015)30 b FE(=)466 1160 y Fz(1)p 448 1175 71 4 v 448 1227 a(16)529 1195 y Fr(.)265 1432 y(Pr)-5 b(o)g(of.)50 b FE(As)32 b(in)e([FM1)q(])i(w)m(e)g(can)g(pro)m(v)m(e)h(that)f(\(for)g FD(L)f FE(large)h(enough)f(the)h(logarithmic)e(div)m(er-)265 1624 y(gences)h(app)s(ear)f(only)g(for)g Fv(j)p FD(m)p Fv(j)25 b(\024)g FE(1)31 b(and)f(the)g(sum)g(o)m(v)m(er)h Fv(j)p FD(m)p Fv(j)26 b FD(>)f FE(1)31 b(con)m(v)m(erge\))438 1871 y Fv(j)p FD(@)516 1833 y Fy(\013)511 1893 y(x)567 1871 y FD(R)636 1885 y Fz(0)675 1871 y FE(\()p Fq(x)p FD(;)15 b Fq(x)871 1833 y Fw(0)894 1871 y FE(;)g FD(z)t FE(\))p Fv(j)26 b(\024)1172 1826 y Fy(C)1227 1803 y Ft(0)1250 1826 y Fz(\()p Fy(z)s(;B)s Fz(\))p 1172 1850 244 4 v 1277 1902 a(3)1426 1871 y FD(e)1468 1833 y Fw(\000)1533 1806 y Fu(B)p 1533 1818 49 3 v 1542 1859 a Fs(8)1592 1833 y Fw(j)p Fo(x)p Fw(\000)p Fo(x)1759 1810 y Ft(0)1782 1833 y Fw(j)1802 1810 y Fs(2)1860 1871 y FE(+)1981 1785 y Fx(X)1951 1986 y Fw(j)p Fy(m)p Fw(j\024)p Fz(1)2158 1871 y Fv(j)p FD(@)2236 1833 y Fy(\013)2231 1893 y(x)2286 1871 y FD(R)2356 1833 y Fw(1)2355 1893 y Fz(0)2431 1871 y FE(\()p FD(x)15 b(y)23 b 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Fs(1)1910 2636 y Fz(\))1942 2602 y FE(\()p Fv(j)p Fq(x)20 b Fv(\000)g Fq(x)2233 2569 y Fw(0)2256 2602 y Fv(j)p FE(\))2331 2525 y Fx(\014)2331 2580 y(\014)2362 2602 y FE(ln)2453 2529 y Fx(\000)2505 2567 y Fy(B)p 2505 2582 57 4 v 2516 2634 a Fz(2)2571 2602 y Fv(j)p Fq(x)h Fv(\000)f Fq(x)2828 2569 y Fw(0)2851 2602 y Fv(j)2876 2569 y Fz(2)2916 2529 y Fx(\001)2957 2525 y(\014)2957 2580 y(\014)2988 2502 y(o)3063 2602 y FD(;)107 b(\013)25 b FE(=)g(0)593 2788 y Fy(C)648 2765 y Ft(0)670 2788 y Fz(\()p Fy(z)s(;B)s Fz(\))p 593 2812 244 4 v 697 2864 a(3)846 2833 y FD(e)888 2797 y Fw(\000)953 2770 y Fu(B)p 954 2782 49 3 v 963 2823 a Fs(8)1012 2797 y Fw(j)p Fo(x)p Fw(\000)p Fo(x)1180 2774 y Ft(0)1202 2797 y Fw(j)1222 2774 y Fs(2)1260 2732 y Fx(n)1321 2833 y FE(1)20 b(+)g FF(1)1530 2867 y Fp(B)6 b Fz(\()p Fe(0)g Fy(;)1662 2807 y Fw(p)p 1720 2807 175 3 v 1720 2867 a Fz(2)p Fy(B)1811 2848 y Ft(\000)p Fs(1)1895 2867 y Fz(\))1926 2833 y FE(\()p Fv(j)p Fq(x)21 b Fv(\000)f Fq(x)2218 2800 y Fw(0)2241 2833 y Fv(j)p FE(\))2301 2732 y Fx(h)2360 2756 y(\014)2360 2810 y(\014)2390 2833 y FE(ln)2481 2759 y Fx(\000)2533 2797 y Fy(B)p 2533 2812 57 4 v 2544 2864 a Fz(2)2599 2833 y Fv(j)p Fq(x)h Fv(\000)f Fq(x)2855 2800 y Fw(0)2879 2833 y Fv(j)2904 2800 y Fz(2)2943 2759 y Fx(\001)2985 2756 y(\014)2985 2810 y(\014)583 3063 y FE(+)669 2989 y Fx(\000)710 3063 y FE(1)h(+)867 2986 y Fx(\014)867 3040 y(\014)897 3063 y FE(ln)988 2989 y Fx(\000)1040 3027 y Fy(B)p 1040 3042 V 1051 3094 a Fz(2)1106 3063 y Fv(j)p Fq(x)f Fv(\000)g Fq(x)1362 3030 y Fw(0)1385 3063 y Fv(j)1410 3030 y Fz(2)1450 2989 y Fx(\001)1492 2986 y(\014)1492 3040 y(\014)1522 2989 y(\001)1579 3063 y Fv(j)p Fq(x)g Fv(\000)g Fq(x)1835 3030 y Fw(0)1858 3063 y Fv(j)1883 3030 y Fw(\000)p Fz(1)1978 2962 y Fx(io)2082 3063 y FD(;)106 b(\013)26 b FE(=)f(1)g FD(:)265 3266 y FE(No)m(w,)35 b(using)e Fv(j)p Fq(x)22 b Fv(\000)g Fq(x)1003 3233 y Fw(0)1026 3266 y Fv(j)1051 3280 y Fy(?)1122 3266 y Fv(\024)30 b(j)p Fq(x)23 b Fv(\000)f Fq(x)1484 3233 y Fw(0)1507 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FD(H)928 4771 y Fy(\013)1014 4757 y FE(\()p FD(\013)g FE(=)f FD(`;)15 b(r)s FE(\))38 b(in)d(a)j(energy)f(in)m(terv)-5 b(al)36 b(\001)g(=)g(\()p FD(B)30 b Fv(\000)24 b FD(\016)n(;)15 b(B)31 b FE(+)24 b FD(\016)s FE(\))38 b Fv(\032)e FE(\001)3293 4771 y Fy(")3366 4757 y FE(are)265 4949 y(extended)29 b(in)e(the)i(sense)g(that)g(its)f(curren)m(t)h(is)e(strictly)h(p)s (ositiv)m(e/negativ)m(e)h(uniformly)d(in)h FD(L)p FE(,)265 5141 y(that)k(is,)f(if)f(w)m(e)i(ha)m(v)m(e)g FD(H)1081 5155 y Fy(\013)1130 5141 y FD( )1192 5108 y Fy(\013)1189 5164 y(\024)1268 5141 y FE(=)25 b FD(E)1436 5108 y Fy(\013)1431 5164 y(\024)1485 5141 y FD( )1547 5108 y Fy(\013)1544 5164 y(\024)1628 5141 y FE(then)1424 5388 y Fv(j)p FE(\()p FD( )1546 5351 y Fy(\013)1543 5411 y(\024)1597 5388 y FD(;)15 b(v)1681 5402 y Fy(y)1723 5388 y FD( )1785 5351 y Fy(\013)1782 5411 y(\024)1835 5388 y FE(\))p Fv(j)25 b(\025)g FD(C)2088 5351 y Fw(0)2136 5388 y FD(>)g FE(0)h FD(:)954 b FE(\(B.1\))1828 5637 y FC(19)p eop %%Page: 20 20 20 19 bop 265 100 a FE(F)-8 b(rom)31 b(the)f(eigen)m(v)-5 b(alue)31 b(equation)f(w)m(e)h(ha)m(v)m(e)1156 384 y Fv(k)p FE(\()p FD(H)1319 346 y Fz(0)1312 406 y Fy(\013)1382 384 y Fv(\000)19 b FD(E)1544 346 y Fy(\013)1539 406 y(\024)1594 384 y FE(\))p FD( )1691 346 y Fy(\013)1688 406 y(\024)1742 384 y Fv(k)1787 346 y Fz(2)1852 384 y FE(=)25 b Fv(k)p FD(V)2066 346 y Fy(\013)2046 406 y(!)2116 384 y FD( )2178 346 y Fy(\013)2175 406 y(\024)2228 384 y Fv(k)2273 346 y Fz(2)2338 384 y Fv(\024)g FD(V)2507 346 y Fz(2)2487 406 y(0)2572 384 y FD(:)685 b FE(\(B.2\))265 667 y(W)-8 b(e)140 b(no)m(w)g(dev)m(elop)f FD( )1328 634 y Fy(\013)1325 689 y(\024)1517 667 y FE(of)g(the)h(eigenfunctions)e(of)h FD(H)2993 634 y Fz(0)2986 689 y Fy(\013)3174 667 y FE(denoted)265 758 y Fx(n)326 859 y FD(\036)380 874 y Fy(nk)465 859 y FE(\()p FD(x;)15 b(y)s FE(\))26 b(=)807 823 y Fy(e)840 800 y Fu(ik)q(y)p 807 838 127 4 v 817 848 a Fw(p)p 876 848 48 3 v 56 x Fy(L)944 859 y FD(')1003 874 y Fy(nk)1089 859 y FE(\()p FD(x)p 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Fp(Z)1975 1334 y FD( )2034 1348 y Fy(n)2081 1334 y FE(\()p FD(k)s FE(\))p FD(\036)2255 1349 y Fy(nk)2342 1334 y FE(\()p FD(x;)15 b(y)s FE(\))26 b FD(;)679 b FE(\(B.3\))265 1656 y(and)30 b(of)g(course)1237 1848 y Fv(k)p FD( )1344 1811 y Fy(\013)1341 1871 y(\024)1394 1848 y Fv(k)1439 1811 y Fz(2)1504 1848 y FE(=)1631 1735 y Fw(1)1601 1762 y Fx(X)1600 1956 y Fy(n)p Fz(=0)1792 1762 y Fx(X)1748 1973 y Fy(k)r Fw(2)1844 1946 y Fs(2)p Fu(\031)p 1844 1958 V 1857 1999 a(L)1922 1973 y Fp(Z)1982 1848 y Fv(j)p FD( )2066 1862 y Fy(n)2113 1848 y FE(\()p FD(k)s FE(\))p Fv(j)2258 1811 y Fz(2)2324 1848 y FE(=)25 b(1)h FD(:)766 b FE(\(B.4\))265 2138 y(F)-8 b(rom)31 b(\(B.3\))h(the)f(equation)f(\(B.2\))i(b)s(ecomes)1186 2308 y Fw(1)1156 2335 y Fx(X)1155 2530 y Fy(n)p Fz(=0)1346 2335 y Fx(X)1303 2546 y Fy(k)r Fw(2)1399 2519 y Fs(2)p Fu(\031)p 1398 2531 V 1411 2572 a(L)1477 2546 y Fp(Z)1536 2422 y Fv(j)p FD( )1620 2436 y Fy(n)1668 2422 y FE(\()p FD(k)s FE(\))p Fv(j)1813 2384 y Fz(2)1868 2422 y FE(\()q FD(E)1976 2384 y Fy(\013)1971 2445 y(nk)2077 2422 y Fv(\000)19 b FD(E)2239 2384 y Fy(\013)2234 2444 y(\024)2289 2422 y FE(\))2325 2380 y Fz(2)2389 2422 y Fv(\024)25 b FD(V)2558 2384 y Fz(2)2538 2444 y(0)3282 2422 y FE(\(B.5\))265 2744 y(th)m(us)30 b(since)g(eac)m(h)h(term)g(in)e(the)h(sum)g(is)f(p)s (ositiv)m(e)h(w)m(e)g(ha)m(v)m(e)1280 2941 y Fx(X)1236 3152 y Fy(k)r Fw(2)1332 3124 y Fs(2)p Fu(\031)p 1332 3136 V 1345 3178 a(L)1410 3152 y Fp(Z)1470 3027 y Fv(j)p FD( )1554 3041 y Fz(0)1594 3027 y FE(\()p FD(k)s FE(\))p Fv(j)1739 2990 y Fz(2)1794 3027 y FE(\()q FD(E)1902 2990 y Fy(\013)1897 3050 y Fz(0)p Fy(k)1995 3027 y Fv(\000)20 b FD(E)2158 2990 y Fy(\013)2153 3050 y(\024)2208 3027 y FE(\))2243 2985 y Fz(2)2308 3027 y Fv(\024)25 b FD(V)2477 2990 y Fz(2)2457 3050 y(0)3282 3027 y FE(\(B.6\))265 3368 y(W)-8 b(e)32 b(remark)e(that)h(for)f FD(n)25 b Fv(\025)g FE(1)30 b(one)h(has)f Fv(j)p FD(E)1746 3335 y Fy(\013)1741 3395 y(nk)1847 3368 y Fv(\000)20 b FD(E)2010 3335 y Fy(\013)2005 3390 y(\024)2060 3368 y Fv(j)25 b(\025)2216 3332 y Fy(B)p 2216 3347 57 4 v 2227 3399 a Fz(2)2303 3368 y Fv(\000)20 b FD(\016)s FE(,)31 b(this)f(leads)f(to)1085 3651 y Fv(k)p FD( )1189 3665 y Fy(?)1229 3651 y Fv(k)1274 3613 y Fz(2)1340 3651 y Fv(\021)1467 3537 y Fw(1)1436 3564 y Fx(X)1436 3759 y Fy(n)p Fz(=1)1627 3564 y Fx(X)1584 3775 y Fy(k)r Fw(2)1680 3748 y Fs(2)p Fu(\031)p 1679 3760 69 3 v 1692 3801 a(L)1758 3775 y Fp(Z)1817 3651 y Fv(j)p FD( )1901 3665 y Fy(n)1948 3651 y FE(\()p FD(k)s FE(\))p Fv(j)2093 3613 y Fz(2)2159 3651 y Fv(\024)2380 3589 y FD(V)2453 3556 y Fz(2)2433 3614 y(0)p 2265 3630 342 4 v 2265 3722 a FE(\()2310 3686 y Fy(B)p 2310 3701 57 4 v 2321 3753 a Fz(2)2397 3722 y Fv(\000)20 b FD(\016)s FE(\))2566 3696 y Fz(2)2642 3651 y FD(:)615 b FE(\(B.7\))265 3978 y(Let)35 b FD(k)482 3945 y Fy(?)556 3978 y FE(suc)m(h)g(that)g Fv(j)p FD(E)1064 3945 y Fy(\013)1059 4006 y Fz(0)p Fy(k)1133 3987 y Fu(?)1196 3978 y Fv(\000)23 b FD(E)1362 3945 y Fy(\013)1357 4001 y(\024)1412 3978 y Fv(j)35 b FE(is)f(minimal,)f(and)h (for)g(a)h(\014xed)f FD(a)h FE(indep)s(enden)m(t)d(of)j FD(L)g FE(let)265 4170 y Fv(A)25 b FE(=)g([)p FD(k)534 4137 y Fy(?)594 4170 y Fv(\000)20 b FD(a;)15 b(k)823 4137 y Fy(?)883 4170 y FE(+)20 b FD(a)p FE(].)41 b(Then)29 b(from)h(\(B.5\))609 4453 y FD(V)683 4416 y Fz(2)662 4476 y(0)805 4453 y Fv(\025)1002 4367 y Fx(X)959 4578 y Fy(k)r Fw(2)1055 4551 y Fs(2)p Fu(\031)p 1054 4563 69 3 v 1067 4604 a(L)1133 4578 y Fp(Z)1192 4453 y Fv(j)p FD( )1276 4467 y Fz(0)1316 4453 y FE(\()p FD(k)s FE(\))p Fv(j)1461 4416 y Fz(2)1517 4453 y FE(\()p FD(E)1624 4416 y Fy(\013)1619 4477 y Fz(0)p Fy(k)1717 4453 y Fv(\000)20 b FD(E)1880 4416 y Fy(\013)1875 4476 y(\024)1930 4453 y FE(\))1965 4412 y Fz(2)2030 4453 y Fv(\025)2147 4367 y Fx(X)2126 4565 y Fy(k)r Fw(2A)2268 4546 y Fu(c)2315 4453 y Fv(j)p FD( )2399 4467 y Fz(0)2439 4453 y FE(\()p FD(k)s FE(\))p Fv(j)2584 4416 y Fz(2)2640 4453 y FE(\()p FD(E)2747 4416 y Fy(\013)2742 4477 y Fz(0)p Fy(k)2841 4453 y Fv(\000)f FD(E)3003 4416 y Fy(\013)2998 4476 y(\024)3053 4453 y FE(\))3089 4412 y Fz(2)805 4742 y Fv(\025)114 b FE(inf)959 4803 y Fy(k)r Fw(2A)1101 4784 y Fu(c)1148 4742 y FE(\()p FD(E)1255 4704 y Fy(\013)1250 4765 y Fz(0)p Fy(k)1349 4742 y Fv(\000)19 b FD(E)1511 4704 y Fy(\013)1506 4764 y(\024)1561 4742 y FE(\))1597 4700 y Fz(2)1673 4655 y Fx(X)1651 4853 y Fy(k)r Fw(2A)1793 4834 y Fu(c)1840 4742 y Fv(j)p FD( )1924 4756 y Fz(0)1964 4742 y FE(\()p FD(k)s FE(\))p Fv(j)2109 4704 y Fz(2)3282 4742 y FE(\(B.8\))265 5025 y(th)m(us)1125 5130 y Fx(X)1104 5328 y Fy(k)r Fw(2A)1246 5309 y Fu(c)1293 5217 y Fv(j)p FD( )1377 5231 y Fz(0)1417 5217 y FE(\()p FD(k)s FE(\))p Fv(j)1562 5179 y Fz(2)1627 5217 y Fv(\024)25 b FD(V)1797 5179 y Fz(2)1776 5239 y(0)1870 5217 y FE(sup)1851 5296 y Fy(k)r Fw(2A)1993 5277 y Fu(c)2040 5217 y FE(\()q FD(E)2148 5179 y Fy(\013)2143 5240 y Fz(0)p Fy(k)2241 5217 y Fv(\000)20 b FD(E)2404 5179 y Fy(\013)2399 5239 y(\024)2454 5217 y FE(\))2489 5175 y Fw(\000)p Fz(2)2624 5217 y FD(:)633 b FE(\(B.9\))1828 5637 y FC(20)p eop %%Page: 21 21 21 20 bop 265 100 a FE(F)-8 b(rom)31 b(\(B.4\))h(and)e(\(B.7\))i(w)m(e) f(get)1224 353 y(1)26 b Fv(\025)1434 267 y Fx(X)1391 477 y Fy(k)r Fw(2)1487 450 y Fs(2)p Fu(\031)p 1486 462 69 3 v 1499 504 a(L)1565 477 y Fp(Z)1624 353 y Fv(j)p FD( )1708 367 y Fz(0)1748 353 y FE(\()p FD(k)s FE(\))p Fv(j)1893 316 y Fz(2)1959 353 y Fv(\025)f FE(1)c Fv(\000)2299 304 y Fy(V)2356 281 y Fs(2)2340 325 y(0)p 2221 332 247 4 v 2221 397 a Fz(\()2258 370 y Fu(B)p 2259 382 49 3 v 2268 423 a Fs(2)2317 397 y Fw(\000)p Fy(\016)r Fz(\))2432 378 y Fs(2)2503 353 y FD(:)708 b FE(\(B.10\))265 650 y(Com)m(bining)28 b(the)j(last)f(equation)g(and)g(\(B.9\))i(w)m(e)f (get)817 816 y Fx(X)812 1014 y Fy(k)r Fw(2A)969 903 y Fv(j)p FD( )1053 917 y Fz(0)1093 903 y FE(\()p FD(k)s FE(\))p Fv(j)1238 865 y Fz(2)1304 903 y Fv(\025)25 b FE(1)c Fv(\000)e FD(V)1630 865 y Fz(2)1609 925 y(0)1684 774 y Fx(\024)1848 867 y Fz(1)p 1742 882 247 4 v 1742 946 a(\()1779 919 y Fu(B)p 1780 931 49 3 v 1789 973 a Fs(2)1838 946 y Fw(\000)p Fy(\016)r Fz(\))1953 928 y Fs(2)2019 903 y FE(+)38 b(sup)2110 981 y Fy(k)r Fw(2A)2252 963 y Fu(c)2284 903 y FE(\()p FD(E)2391 865 y Fy(\013)2386 926 y Fz(0)p Fy(k)2484 903 y Fv(\000)20 b FD(E)2647 865 y Fy(\013)2642 925 y(\024)2697 903 y FE(\))2732 865 y Fw(\000)p Fz(2)2827 774 y Fx(\025)2915 903 y FD(:)296 b FE(\(B.11\))265 1155 y(Decomp)s(ose)32 b(no)m(w)e FD( )993 1122 y Fy(\013)990 1178 y(\024)1073 1155 y FE(as)h FD( )1247 1122 y Fy(\013)1244 1178 y(\024)1322 1155 y FE(=)25 b FD( )1477 1169 y Fz(0)1537 1155 y FE(+)20 b FD( )1687 1169 y Fy(?)1726 1155 y FE(,)31 b(then)780 1408 y Fv(j)p FE(\()p FD( )902 1370 y Fy(\013)899 1430 y(\024)952 1408 y FD(;)15 b(v)1036 1422 y Fy(y)1078 1408 y FD( )1140 1370 y Fy(\013)1137 1430 y(\024)1190 1408 y FE(\))p Fv(j)26 b(\025)f(j)p FE(\()p FD( )1491 1422 y Fz(0)1531 1408 y FD(;)15 b(v)1615 1422 y Fy(y)1657 1408 y FD( )1716 1422 y Fz(0)1756 1408 y FE(\))p Fv(j)21 b(\000)e(j)p FE(\()p FD( )2046 1422 y Fy(?)2087 1408 y FD(;)c(v)2171 1422 y Fy(y)2213 1408 y FD( )2272 1422 y Fy(?)2311 1408 y FE(\))p Fv(j)21 b(\000)f FE(2)p Fv(j)p FE(\()p FD( )2647 1422 y Fy(?)2688 1408 y FD(;)15 b(v)2772 1422 y Fy(y)2814 1408 y FD( )2873 1422 y Fz(0)2912 1408 y FE(\))p Fv(j)264 b FE(\(B.12\))265 1661 y(the)31 b(\014rst)e(term)i(can)f(b)s(e)g (written)f(as)604 1820 y Fx(Z)655 2026 y Fp(R)737 1943 y FE(d)p FD(x)870 1820 y Fx(Z)971 1819 y Fu(L)p 971 1831 43 3 v 977 1872 a Fs(2)921 2026 y Fw(\000)986 1999 y Fu(L)p 985 2011 V 991 2052 a Fs(2)1072 1943 y FE(d)p FD(y)1201 1730 y Fx(8)1201 1811 y(>)1201 1839 y(<)1201 2002 y(>)1201 2030 y(:)1336 1857 y(X)1281 2068 y Fy(k)1320 2049 y Ft(0)1342 2068 y Fw(2)1399 2041 y Fs(2)p Fu(\031)p 1399 2053 69 3 v 1412 2094 a(L)1478 2068 y Fp(Z)1537 1943 y FD( )1599 1906 y Fw(\003)1596 1966 y Fz(0)1639 1943 y FE(\()p FD(k)1724 1906 y Fw(0)1748 1943 y FE(\))1793 1882 y FD(e)1835 1849 y Fw(\000)p Fy(ik)1953 1825 y Ft(0)1975 1849 y Fy(y)p 1793 1922 224 4 v 1836 1941 a Fv(p)p 1912 1941 62 4 v 76 x FD(L)2027 1943 y(')2086 1906 y Fw(\003)2086 1967 y Fz(0)p Fy(k)2160 1948 y Ft(0)2187 1943 y FE(\()p FD(x)p FE(\))2368 1857 y Fx(X)2324 2068 y Fy(k)r Fw(2)2420 2041 y Fs(2)p Fu(\031)p 2420 2053 69 3 v 2433 2094 a(L)2498 2068 y Fp(Z)2558 1943 y FD( )2617 1957 y Fz(0)2656 1943 y FE(\()p FD(k)s FE(\))p FD(v)2820 1957 y Fy(y)2873 1882 y FD(e)2915 1849 y Fy(ik)r(y)p 2873 1922 147 4 v 2877 1941 a Fv(p)p 2953 1941 62 4 v 76 x FD(L)3029 1943 y(')3088 1958 y Fz(0)p Fy(k)3166 1943 y FE(\()p FD(x)p FE(\))3288 1730 y Fx(9)3288 1811 y(>)3288 1839 y(=)3288 2002 y(>)3288 2030 y(;)451 2268 y FE(=)648 2182 y Fx(X)604 2393 y Fy(k)r Fw(2)700 2366 y Fs(2)p Fu(\031)p 700 2378 69 3 v 713 2419 a(L)778 2393 y Fp(Z)838 2268 y Fv(j)p FD( )922 2282 y Fz(0)962 2268 y FE(\()p FD(k)s FE(\))p Fv(j)1107 2231 y Fz(2)1162 2145 y Fx(Z)1213 2351 y Fp(R)1295 2268 y FE(d)p FD(x)15 b FE(\()p FD(k)24 b Fv(\000)c FD(B)5 b(x)p FE(\))15 b Fv(j)p FD(')1870 2283 y Fz(0)p Fy(k)1948 2268 y FE(\()p FD(x)p FE(\))p Fv(j)2095 2231 y Fz(2)451 2566 y FE(=)648 2479 y Fx(X)604 2690 y Fy(k)r Fw(2)700 2663 y Fs(2)p Fu(\031)p 700 2675 V 713 2716 a(L)778 2690 y Fp(Z)838 2566 y Fv(j)p FD( )922 2580 y Fz(0)962 2566 y FE(\()p FD(k)s FE(\))p Fv(j)1107 2528 y Fz(2)1147 2566 y FD(@)1197 2581 y Fz(^)1195 2598 y Fy(k)1238 2566 y FD(E)1310 2528 y Fy(\013)1305 2588 y Fz(0)1360 2566 y FE(\()1397 2542 y(^)1395 2566 y FD(k)t FE(\))1481 2461 y Fx(\014)1481 2516 y(\014)1481 2570 y(\014)1513 2611 y Fz(^)1511 2629 y Fy(k)r Fz(=)p Fy(k)3236 2566 y FE(\(B.13\))265 2899 y(The)30 b(partial)f(deriv)-5 b(ativ)m(e)30 b(of)h FD(E)1338 2866 y Fy(\013)1333 2923 y Fz(0)1418 2899 y FE(is)e(the)i(curren)m(t)f FD(@)2032 2914 y Fz(^)2030 2931 y Fy(k)2073 2899 y FD(E)2145 2866 y Fy(\013)2140 2923 y Fz(0)2195 2899 y FE(\()2232 2875 y(^)2230 2899 y FD(k)s FE(\))2315 2794 y Fx(\014)2315 2849 y(\014)2315 2903 y(\014)2347 2945 y Fz(^)2346 2962 y Fy(k)q Fz(=)p Fy(k)2507 2899 y FE(=)25 b FD(J)2653 2913 y Fy(E)2709 2890 y Fu(\013)2705 2937 y Fs(0)p Fu(k)2778 2899 y FE(,)30 b(th)m(us)602 3190 y Fv(j)p FE(\()p FD( )721 3204 y Fz(0)762 3190 y FD(;)15 b(v)846 3204 y Fy(y)887 3190 y FD( )946 3204 y Fz(0)986 3190 y FE(\))p Fv(j)84 b(\025)1283 3031 y Fx(\014)1283 3086 y(\014)1283 3140 y(\014)1283 3195 y(\014)1283 3249 y(\014)1372 3104 y(X)1329 3315 y Fy(k)r Fw(2)1425 3287 y Fs(2)p Fu(\031)p 1424 3299 V 1438 3341 a(L)1503 3315 y Fp(Z)1562 3190 y Fv(j)p FD( )1646 3204 y Fz(0)1686 3190 y FE(\()p FD(k)s FE(\))p Fv(j)1831 3153 y Fz(2)1872 3190 y FD(J)1922 3204 y Fy(E)1978 3181 y Fu(\013)1974 3228 y Fs(0)p Fu(k)2047 3031 y Fx(\014)2047 3086 y(\014)2047 3140 y(\014)2047 3195 y(\014)2047 3249 y(\014)1130 3515 y Fv(\025)e(j)p FD(J)1358 3529 y Fy(E)1414 3506 y Fu(\013)1410 3560 y Fs(0)1442 3547 y(\026)1440 3560 y Fu(k)1483 3515 y Fv(j)1523 3387 y Fx(\032)1592 3515 y FE(1)21 b Fv(\000)f FD(V)1822 3477 y Fz(2)1802 3537 y(0)1876 3387 y Fx(\024)2040 3479 y Fz(1)p 1934 3494 247 4 v 1934 3559 a(\()1971 3532 y Fu(B)p 1972 3544 49 3 v 1981 3585 a Fs(2)2030 3559 y Fw(\000)p Fy(\016)r Fz(\))2145 3540 y Fs(2)2211 3515 y FE(+)39 b(sup)2302 3594 y Fy(k)r Fw(2A)2444 3575 y Fu(c)2491 3515 y FE(\()q FD(E)2599 3477 y Fy(\013)2594 3538 y Fz(0)p Fy(k)2692 3515 y Fv(\000)20 b FD(E)2855 3477 y Fy(\013)2850 3537 y(\024)2905 3515 y FE(\))2940 3473 y Fw(\000)p Fz(2)3034 3387 y Fx(\025\033)3236 3515 y FE(\(B.14\))265 3767 y(for)33 b(a)i(suitable)831 3743 y(\026)829 3767 y FD(k)f Fv(2)c(A)p FE(,)k(and)f(w)m(e)i(ha)m(v)m(e)f Fv(j)p FD(J)1738 3781 y Fy(E)1794 3759 y Fu(\013)1790 3813 y Fs(0)1822 3800 y(\026)1820 3813 y Fu(k)1864 3767 y Fv(j)d FD(>)f FE(0.)51 b(The)33 b(second)h(term)g(can)g(b)s(e)f(b)s(ounded)265 3959 y(as)e(follo)m(ws)e Fv(j)p FE(\()p FD( )794 3973 y Fy(?)834 3959 y FD(;)15 b(v)918 3973 y Fy(y)960 3959 y FD( )1019 3973 y Fy(?)1059 3959 y FE(\))p Fv(j)26 b(\024)f(k)p FD( )1345 3973 y Fy(?)1385 3959 y Fv(kk)p FD(v)1519 3973 y Fy(y)1561 3959 y FD( )1620 3973 y Fy(?)1660 3959 y Fv(k)h(\024)1877 3922 y Fy(V)1918 3931 y Fs(0)p 1837 3938 158 4 v 1846 3976 a Fu(B)p 1846 3988 49 3 v 1856 4029 a Fs(2)1905 4003 y Fw(\000)p Fy(\016)2004 3959 y Fv(k)p FD(v)2093 3973 y Fy(y)2134 3959 y FD( )2193 3973 y Fy(?)2233 3959 y Fv(k)31 b FE(and)520 4212 y Fv(k)p FD(v)609 4226 y Fy(y)651 4212 y FD( )710 4226 y Fy(?)750 4212 y Fv(k)795 4175 y Fz(2)918 4212 y FE(=)83 b(2)1132 4111 y Fx(\020)1187 4212 y FD( )1246 4226 y Fy(?)1285 4212 y FD(;)1336 4176 y Fz(1)p 1336 4191 36 4 v 1336 4243 a(2)1396 4212 y FE(\()p FD(p)1477 4226 y Fy(y)1539 4212 y Fv(\000)20 b FD(B)5 b(x)p FE(\))1790 4170 y Fz(2)1845 4212 y FD( )1904 4226 y Fy(?)1944 4111 y Fx(\021)918 4429 y Fv(\024)83 b FE(2)1132 4328 y Fx(\020)1187 4429 y FD( )1246 4443 y Fy(?)1285 4429 y FD(;)1325 4328 y Fx(h)1379 4393 y Fz(1)p 1379 4408 V 1379 4460 a(2)1424 4429 y FD(p)1470 4391 y Fz(2)1470 4451 y Fy(x)1534 4429 y FE(+)1634 4393 y Fz(1)p 1634 4408 V 1634 4460 a(2)1695 4429 y FE(\()p FD(p)1776 4443 y Fy(y)1838 4429 y Fv(\000)20 b FD(B)5 b(x)p FE(\))2089 4387 y Fz(2)2149 4429 y FE(+)20 b FD(U)2302 4443 y Fy(\013)2351 4328 y Fx(i)2410 4429 y FD( )2469 4443 y Fy(?)2508 4328 y Fx(\021)918 4646 y FE(+)83 b(2)1132 4545 y Fx(\020)1187 4646 y FD( )1246 4660 y Fz(0)1285 4646 y FD(;)1325 4545 y Fx(h)1379 4610 y Fz(1)p 1379 4625 V 1379 4677 a(2)1424 4646 y FD(p)1470 4608 y Fz(2)1470 4668 y Fy(x)1534 4646 y FE(+)1634 4610 y Fz(1)p 1634 4625 V 1634 4677 a(2)1695 4646 y FE(\()p FD(p)1776 4660 y Fy(y)1838 4646 y Fv(\000)20 b FD(B)5 b(x)p FE(\))2089 4604 y Fz(2)2149 4646 y FE(+)20 b FD(U)2302 4660 y Fy(\013)2351 4545 y Fx(i)2410 4646 y FD( )2469 4660 y Fz(0)2508 4545 y Fx(\021)2588 4646 y FE(=)25 b(2)2744 4572 y Fx(\000)2786 4646 y FD( )2848 4608 y Fy(\013)2845 4668 y(\024)2898 4646 y FD(;)15 b(H)3021 4608 y Fz(0)3014 4668 y Fy(\013)3063 4646 y FD( )3125 4608 y Fy(\013)3122 4668 y(\024)3175 4572 y Fx(\001)918 4863 y FE(=)83 b(2\()p FD( )1214 4825 y Fy(\013)1211 4885 y(\024)1264 4863 y FD(;)15 b(H)1380 4877 y Fy(\013)1430 4863 y FD( )1492 4825 y Fy(\013)1489 4885 y(\024)1542 4863 y FE(\))20 b Fv(\000)g FE(2\()p FD( )1830 4825 y Fy(\013)1827 4885 y(\024)1881 4863 y FD(;)15 b(V)1995 4825 y Fy(\013)1974 4885 y(!)2044 4863 y FD( )2106 4825 y Fy(\013)2103 4885 y(\024)2156 4863 y FE(\))26 b Fv(\024)f FE(2\()p FD(E)2465 4825 y Fy(\013)2460 4885 y(\024)2536 4863 y FE(+)19 b FD(V)2679 4877 y Fz(0)2719 4863 y FE(\))26 b FD(:)431 b FE(\(B.15\))265 5115 y(This)29 b(leads)g(to)i(the)g(b)s(ound)1239 5368 y Fv(j)p FE(\()p FD( )1358 5382 y Fy(?)1398 5368 y FD(;)15 b(v)1482 5382 y Fy(y)1524 5368 y FD( )1583 5382 y Fy(?)1623 5368 y FE(\))p Fv(j)25 b(\024)1855 5331 y Fy(V)1896 5340 y Fs(0)p 1814 5347 158 4 v 1824 5385 a Fu(B)p 1824 5397 49 3 v 1833 5438 a Fs(2)1883 5412 y Fw(\000)p Fy(\016)1981 5286 y Fx(p)p 2072 5286 442 4 v 82 x FE(2\()p FD(E)2224 5342 y Fy(\013)2219 5390 y(\024)2295 5368 y FE(+)20 b FD(V)2439 5382 y Fz(0)2478 5368 y FE(\))723 b(\(B.16\))1828 5637 y FC(21)p eop %%Page: 22 22 22 21 bop 265 100 a FE(A)31 b(similar)c(argumen)m(t)k(giv)m(es)g(the)f (same)h(b)s(ound)d(for)j(the)f(third)f(term.)265 292 y(Finally)581 576 y Fv(j)p FE(\()p FD( )703 538 y Fy(\013)700 598 y(\024)754 576 y FD(;)15 b(v)838 590 y Fy(y)880 576 y FD( )942 538 y Fy(\013)939 598 y(\024)992 576 y FE(\))p Fv(j)83 b(\025)g(j)p FD(J)1364 590 y Fy(E)1420 567 y Fu(\013)1416 621 y Fs(0)1448 608 y(\026)1446 621 y Fu(k)1489 576 y Fv(j)1529 447 y Fx(\032)1598 576 y FE(1)20 b Fv(\000)g FD(V)1828 538 y Fz(2)1807 598 y(0)1882 447 y Fx(\024)2046 540 y Fz(1)p 1940 555 247 4 v 1940 620 a(\()1977 592 y Fu(B)p 1978 604 49 3 v 1987 646 a Fs(2)2036 620 y Fw(\000)p Fy(\016)r Fz(\))2151 601 y Fs(2)2217 576 y FE(+)38 b(sup)2308 655 y Fy(k)r Fw(2A)2450 636 y Fu(c)2497 576 y FE(\()p FD(E)2604 538 y Fy(\013)2599 599 y Fz(0)p Fy(k)2697 576 y Fv(\000)20 b FD(E)2860 538 y Fy(\013)2855 598 y(\024)2910 576 y FE(\))2946 534 y Fw(\000)p Fz(2)3040 447 y Fx(\025\033)1135 792 y Fv(\000)83 b FE(3)1385 755 y Fy(V)1426 764 y Fs(0)p 1345 772 158 4 v 1355 809 a Fu(B)p 1355 821 49 3 v 1364 862 a Fs(2)1413 836 y Fw(\000)p Fy(\016)1512 710 y Fx(p)p 1602 710 442 4 v 1602 792 a FE(2\()p FD(E)1754 766 y Fy(\013)1749 815 y(\024)1825 792 y FE(+)20 b FD(V)1969 806 y Fz(0)2009 792 y FE(\))1192 b(\(B.17\))265 1076 y(that)38 b(is)f(strictly)g(p)s(ositiv)m(e)f(for)i(a)g(su\016cien)m (tly)e(small)g FD(V)2220 1090 y Fz(0)2297 1076 y FD(>)h FE(0)h(\(w)m(e)h(can)f(remark)f(that)h(the)265 1268 y(imp)s(ortan)m(t) 30 b(condition)f(is)g FD(V)1238 1282 y Fz(0)1303 1268 y Fv(\034)c FD(B)5 b FE(\).)265 1715 y FB(C)161 b(Ab)t(out)53 b(Hyp)t(othesis)g(1)265 1997 y FE(In)29 b(this)g(section)i(w)m(e)f(giv) m(e)h(some)f(details)f(ab)s(out)h(Hyp)s(othesis)f(1,)i(in)d(particular) h(w)m(e)h(sho)m(ws)g(a)265 2189 y(p)s(ossibilit)m(y)20 b(to)k(realize)g(it.)38 b(This)22 b(is)g(done)i(b)m(y)f(the)h(addition) e(of)i(a)g(magnetic)g(\015ux)f(0)i Fv(\024)g FE(\010)g Fv(\024)g FE(2)p FD(\031)265 2381 y FE(at)38 b(the)f(origin)f(\(i.e.)61 b(along)37 b(the)g(cylinder)e(axes\))k(and)d(taking)h(t)m(w)m(o)i (symmetric)d(con\014ning)265 2573 y(w)m(alls)29 b FD(U)552 2588 y Fy(`)585 2573 y FE(\()p Fv(\000)p FD(x)p FE(\))d(=)f FD(U)962 2587 y Fy(r)1000 2573 y FE(\()p FD(x)p FE(\))h Fv(\021)f FD(U)10 b FE(\()p FD(x)p FE(\).)265 2765 y(In)30 b(this)f(case)i(the)g(pure)e(w)m(alls)g(Hamiltonians)g(are)973 3048 y FD(H)1056 3011 y Fz(0)1049 3071 y Fy(`)1095 3048 y FE([\010])84 b(=)1458 3012 y Fz(1)p 1458 3027 36 4 v 1458 3080 a(2)1504 3048 y FD(p)1550 3011 y Fz(2)1550 3071 y Fy(x)1613 3048 y FE(+)1714 3012 y Fz(1)p 1714 3027 V 1714 3080 a(2)1775 2975 y Fx(\000)1816 3048 y FD(p)1862 3062 y Fy(y)1924 3048 y Fv(\000)20 b FD(B)5 b(x)19 b FE(+)2261 3012 y Fz(\010)p 2261 3027 51 4 v 2262 3080 a Fy(L)2322 2975 y Fx(\001)2364 2993 y Fz(2)2423 3048 y FE(+)h FD(U)10 b FE(\()p Fv(\000)p FD(x)p FE(\))502 b(\(C.1\))973 3265 y FD(H)1056 3228 y Fz(0)1049 3288 y Fy(r)1095 3265 y FE([\010])84 b(=)1458 3229 y Fz(1)p 1458 3244 36 4 v 1458 3296 a(2)1504 3265 y FD(p)1550 3228 y Fz(2)1550 3288 y Fy(x)1613 3265 y FE(+)1714 3229 y Fz(1)p 1714 3244 V 1714 3296 a(2)1775 3191 y Fx(\000)1816 3265 y FD(p)1862 3279 y Fy(y)1924 3265 y Fv(\000)20 b FD(B)5 b(x)19 b FE(+)2261 3229 y Fz(\010)p 2261 3244 51 4 v 2262 3296 a Fy(L)2322 3191 y Fx(\001)2364 3210 y Fz(2)2423 3265 y FE(+)h FD(U)10 b FE(\()p FD(x)p FE(\))26 b FD(:)522 b FE(\(C.2\))265 3548 y(The)30 b(sp)s(ectra)g(of)h(these)g (Hamiltonians)d(are)1083 3832 y FD(\033)s FE(\()p FD(H)1256 3794 y Fz(0)1249 3854 y Fy(\013)1299 3832 y FE([\010]\))e(=)f Fv(f)p FD(E)1689 3794 y Fy(\013)1684 3855 y(nk)1770 3832 y FE(\(\010\))g(:)h FD(n)f Fv(2)f FA(N)7 b FD(;)15 b(k)35 b Fv(2)2425 3796 y Fz(2)p Fy(\031)p 2425 3811 79 4 v 2440 3863 a(L)2513 3832 y FA(Z)p Fv(g)21 b FD(:)612 b FE(\(C.3\))265 4115 y(Here)21 b(w)m(e)h(consider)d(only)h(the)h (\014rst)f(sp)s(ectral)h(branc)m(hes,)h(and)e(w)m(e)i(write)e FD(E)2768 4082 y Fy(\013)2763 4143 y Fz(0)p Fy(k)2841 4115 y FE(\(\010\))25 b Fv(\021)g FD(E)3170 4082 y Fy(\013)3235 4115 y FE(\()q FD(k)s FE(;)15 b(\010\),)265 4307 y(moreo)m(v)m(er)32 b(w)m(e)f(ha)m(v)m(e)g(\()p FD(k)e FE(=)1217 4271 y Fz(2)p Fy(\031)p 1217 4286 V 1232 4338 a(L)1305 4307 y FD(m)p FE(\))1241 4590 y FD(E)1313 4553 y Fy(\013)1378 4516 y Fx(\000)1429 4554 y Fz(2)p Fy(\031)p 1429 4569 V 1444 4621 a(L)1517 4590 y FD(m)p FE(;)15 b(\010)1703 4516 y Fx(\001)1770 4590 y FE(=)25 b FD(E)1938 4553 y Fy(\013)2003 4516 y Fx(\000)2055 4554 y Fz(2)p Fy(\031)p 2055 4569 V 2070 4621 a(L)2143 4590 y FD(m)20 b FE(+)2344 4554 y Fz(\010)p 2344 4569 51 4 v 2345 4621 a Fy(L)2405 4516 y Fx(\001)2487 4590 y FD(:)769 b FE(\(C.4\))265 4873 y(F)-8 b(rom)31 b(the)f(symmetry)g(of)h(the)g(w)m(alls,)e(for)h(\010)25 b(=)g(0)1144 5157 y FD(E)1216 5119 y Fy(`)1265 5083 y Fx(\000)1316 5121 y Fz(2)p Fy(\031)p 1316 5136 79 4 v 1331 5188 a(L)1404 5157 y FD(m)1484 5083 y Fx(\001)1551 5157 y FE(=)g FD(E)1719 5119 y Fy(r)1772 5083 y Fx(\000)1824 5121 y Fz(2)p Fy(\031)p 1824 5136 V 1839 5188 a(L)1912 5157 y FE(\()p Fv(\000)p FD(m)p FE(\))2133 5083 y Fx(\001)2281 5157 y Fv(8)g FD(m)g Fv(2)g FA(Z)668 b FE(\(C.5\))1828 5637 y FC(22)p eop %%Page: 23 23 23 22 bop 265 100 a FE(therefore)1026 376 y FD(E)1098 339 y Fy(`)1146 303 y Fx(\000)1198 341 y Fz(2)p Fy(\031)r(m)p 1198 356 141 4 v 1244 408 a(L)1368 376 y FE(+)1469 341 y Fz(\010)p 1469 356 51 4 v 1470 408 a Fy(L)1530 303 y Fx(\001)1655 376 y FE(=)83 b FD(E)1881 339 y Fy(`)1929 303 y Fx(\000)1981 341 y Fz(2)p Fy(\031)r(m)p 1981 356 141 4 v 2027 408 a(L)2131 303 y Fx(\001)2193 376 y FE(+)20 b FD(@)2334 391 y Fz(^)2332 409 y Fy(k)2375 376 y FD(E)2447 339 y Fy(`)2480 376 y FE(\()2517 352 y(~)2515 376 y FD(k)2562 391 y Fy(`)2596 376 y FE(\))2641 315 y(\010)p 2641 356 66 4 v 2643 439 a FD(L)3281 376 y FE(\(C.6\))886 593 y FD(E)958 556 y Fy(r)1011 492 y Fx(\020)1075 549 y Fz(2)p Fy(\031)r Fz(\()p Fw(\000)p Fy(m)p Fz(\))p 1075 572 251 4 v 1177 625 a Fy(L)1356 593 y FE(+)1457 558 y Fz(\010)p 1457 573 51 4 v 1458 625 a Fy(L)1518 492 y Fx(\021)1655 593 y FE(=)83 b FD(E)1881 556 y Fy(r)1934 492 y Fx(\020)1998 549 y Fz(2)p Fy(\031)r Fz(\()p Fw(\000)p Fy(m)p Fz(\))p 1998 572 251 4 v 2100 625 a Fy(L)2259 492 y Fx(\021)2333 593 y FE(+)20 b FD(@)2474 608 y Fz(^)2472 626 y Fy(k)2515 593 y FD(E)2587 556 y Fy(r)2625 593 y FE(\()2662 569 y(~)2660 593 y FD(k)2707 607 y Fy(r)2746 593 y FE(\))2791 532 y(\010)p 2791 572 66 4 v 2793 656 a FD(L)3281 593 y FE(\(C.7\))265 869 y(for)30 b(a)h(suitable)829 834 y Fz(2)p Fy(\031)p 829 849 79 4 v 844 901 a(L)917 869 y FE(\()p Fv(\000)p FD(m)p FE(\))25 b Fv(\024)1262 845 y FE(~)1259 869 y FD(k)1306 884 y Fy(`)1365 869 y Fv(\024)1471 834 y Fz(2)p Fy(\031)p 1471 849 V 1486 901 a(L)1559 869 y FE(\()p Fv(\000)p FD(m)p FE(\))20 b(+)1901 834 y Fz(\010)p 1901 849 51 4 v 1902 901 a Fy(L)1993 869 y FE(and)2179 834 y Fz(2)p Fy(\031)p 2179 849 79 4 v 2194 901 a(L)2267 869 y FD(m)26 b Fv(\024)2471 845 y FE(~)2468 869 y FD(k)2515 883 y Fy(r)2579 869 y Fv(\024)2685 834 y Fz(2)p Fy(\031)p 2685 849 V 2700 901 a(L)2773 869 y FD(m)20 b FE(+)2974 834 y Fz(\010)p 2974 849 51 4 v 2975 901 a Fy(L)3035 869 y FE(.)41 b(Th)m(us)560 1041 y Fx(\014)560 1095 y(\014)560 1150 y(\014)590 1145 y FD(E)662 1108 y Fy(`)710 1072 y Fx(\000)762 1110 y Fz(2)p Fy(\031)r(m)p 762 1125 141 4 v 808 1177 a(L)933 1145 y FE(+)1034 1110 y Fz(\010)p 1034 1125 51 4 v 1035 1177 a Fy(L)1095 1072 y Fx(\001)1157 1145 y Fv(\000)19 b FD(E)1319 1108 y Fy(r)1373 1045 y Fx(\020)1437 1101 y Fz(2)p Fy(\031)r Fz(\()p Fw(\000)p Fy(m)p Fz(\))p 1437 1125 251 4 v 1538 1177 a Fy(L)1718 1145 y FE(+)1819 1110 y Fz(\010)p 1819 1125 51 4 v 1820 1177 a Fy(L)1879 1045 y Fx(\021)1934 1041 y(\014)1934 1095 y(\014)1934 1150 y(\014)2047 1145 y FE(=)2211 1084 y(\010)p 2211 1125 66 4 v 2213 1208 a FD(L)2302 1041 y Fx(\014)2302 1095 y(\014)2302 1150 y(\014)2332 1145 y FD(@)2382 1160 y Fz(^)2380 1178 y Fy(k)2423 1145 y FD(E)2495 1108 y Fy(r)2533 1145 y FE(\()2570 1121 y(^)2568 1145 y FD(k)2615 1159 y Fy(r)2654 1145 y FE(\))h Fv(\000)g FD(@)2850 1160 y Fz(^)2848 1178 y Fy(k)2891 1145 y FD(E)2963 1108 y Fy(`)2996 1145 y FE(\()3033 1121 y(^)3031 1145 y FD(k)3078 1160 y Fy(`)3112 1145 y FE(\))3147 1041 y Fx(\014)3147 1095 y(\014)3147 1150 y(\014)2047 1362 y Fv(\025)83 b FE(2)2256 1301 y(\010)p 2256 1341 V 2258 1425 a FD(L)2332 1362 y Fv(j)p FD(@)2407 1377 y Fz(^)2405 1395 y Fy(k)2448 1362 y FD(E)2520 1325 y Fy(`)2553 1362 y FE(\()2590 1338 y(^)2588 1362 y FD(k)2635 1377 y Fy(`)2669 1362 y FE(\))p Fv(j)26 b(\025)f FE(2)p Fv(C)2959 1301 y FE(\010)p 2959 1341 V 2961 1425 a FD(L)3281 1362 y FE(\(C.8\))265 1638 y(where)30 b Fv(C)g FD(>)25 b FE(0.)41 b(A)31 b(similar)d(argumen)m(t)i(sho)m(ws)h(that)507 1810 y Fx(\014)507 1864 y(\014)507 1919 y(\014)538 1914 y FD(E)610 1877 y Fy(`)658 1813 y Fx(\020)722 1870 y Fz(2)p Fy(\031)r Fz(\()p Fy(m)p Fw(\000)p Fz(1\))p 722 1893 286 4 v 841 1946 a Fy(L)1038 1914 y FE(+)1139 1879 y Fz(\010)p 1139 1894 51 4 v 1140 1946 a Fy(L)1200 1813 y Fx(\021)1275 1914 y Fv(\000)20 b FD(E)1438 1877 y Fy(r)1491 1813 y Fx(\020)1555 1870 y Fz(2)p Fy(\031)r Fz(\()p Fw(\000)p Fy(m)p Fz(\))p 1555 1893 251 4 v 1656 1946 a Fy(L)1836 1914 y FE(+)1937 1879 y Fz(\010)p 1937 1894 51 4 v 1938 1946 a Fy(L)1998 1813 y Fx(\021)2052 1810 y(\014)2052 1864 y(\014)2052 1919 y(\014)354 2131 y FE(=)507 1999 y Fx(\014)507 2054 y(\014)507 2109 y(\014)507 2163 y(\014)548 2070 y FE(\010)p 548 2110 66 4 v 550 2194 a FD(L)638 2030 y Fx(h)681 2131 y FD(@)731 2146 y Fz(^)729 2164 y Fy(k)772 2131 y FD(E)844 2094 y Fy(`)878 2131 y FE(\()915 2107 y(^)913 2131 y FD(k)960 2146 y Fy(`)993 2131 y FE(\))h Fv(\000)f FD(@)1190 2146 y Fz(^)1188 2164 y Fy(k)1231 2131 y FD(E)1303 2094 y Fy(r)1341 2131 y FE(\()1378 2107 y(^)1376 2131 y FD(k)1423 2145 y Fy(r)1462 2131 y FE(\))1497 2030 y Fx(i)1560 2131 y Fv(\000)1661 2095 y Fz(2)p Fy(\031)p 1661 2110 79 4 v 1676 2163 a(L)1749 2131 y FD(@)1799 2146 y Fz(^)1797 2164 y Fy(k)1840 2131 y FD(E)1912 2094 y Fy(`)1945 2131 y FE(\()1982 2107 y(^)1980 2131 y FD(k)2027 2146 y Fy(`)2061 2131 y FE(\))2096 1999 y Fx(\014)2096 2054 y(\014)2096 2109 y(\014)2096 2163 y(\014)2152 2131 y Fv(\025)2248 1999 y Fx(\014)2248 2054 y(\014)2248 2109 y(\014)2248 2163 y(\014)2278 2131 y FE(2)2333 2070 y(\010)p 2333 2110 66 4 v 2335 2194 a FD(L)2409 2131 y Fv(j)p FD(@)2484 2146 y Fz(^)2482 2164 y Fy(k)2525 2131 y FD(E)2597 2094 y Fy(`)2631 2131 y FE(\()2668 2107 y(^)2666 2131 y FD(k)2713 2146 y Fy(`)2746 2131 y FE(\))p Fv(j)h(\000)2928 2070 y FE(2)p FD(\031)p 2928 2110 101 4 v 2947 2194 a(L)3039 2131 y Fv(j)p FD(@)3114 2146 y Fz(^)3112 2164 y Fy(k)3155 2131 y FD(E)3227 2094 y Fy(`)3260 2131 y FE(\()3297 2107 y(^)3295 2131 y FD(k)3342 2146 y Fy(`)3376 2131 y FE(\))p Fv(j)3436 1999 y Fx(\014)3436 2054 y(\014)3436 2109 y(\014)3436 2163 y(\014)354 2381 y Fv(\025)82 b FE(2)p Fv(C)616 2319 y(j)p FE(\010)20 b Fv(\000)g FD(\031)s Fv(j)p 616 2360 283 4 v 726 2443 a FD(L)3281 2381 y FE(\(C.9\))265 2657 y(Then,)30 b(for)g(a)h(\014xed)e(0)d FD(<)f FE(\010)1195 2624 y Fy(?)1259 2657 y FD(<)g(\031)34 b FE(or)c FD(\031)e(<)d FE(\010)1794 2624 y Fy(?)1859 2657 y FD(<)g FE(2)p FD(\031)977 2933 y FE(dist)1139 2859 y Fx(\000)1181 2933 y FD(\033)s FE(\()p FD(H)1354 2895 y Fz(0)1347 2956 y Fy(`)1394 2933 y FE([\010)1485 2895 y Fy(?)1524 2933 y FE(]\))c Fv(\\)e FE(\001)1761 2947 y Fy(")1798 2933 y FD(;)c(\033)s FE(\()p FD(H)2011 2895 y Fz(0)2004 2955 y Fy(r)2051 2933 y FE([\010)2142 2895 y Fy(?)2182 2933 y FE(]\))20 b Fv(\\)g FE(\001)2419 2947 y Fy(")2456 2859 y Fx(\001)2523 2933 y Fv(\025)2629 2871 y FD(d)2676 2885 y Fz(0)p 2629 2912 87 4 v 2641 2995 a FD(L)2751 2933 y(:)459 b FE(\(C.10\))265 3379 y FB(Ac)l(kno)l(wledgemen)l(ts)265 3661 y FE(W)-8 b(e)28 b(wish)e(to)h(thank)g(J.M.)h(Com)m(b)s(es,)f(P)-8 b(.)28 b(Exner,)f(J.)g(F)-8 b(r\177)-45 b(ohlic)m(h)26 b(and)h(P)-8 b(.A.)28 b(Martin)f(for)f(helpful)265 3853 y(discussions.)38 b(The)28 b(w)m(ork)i(of)f(C.F.)h(w)m(as)f(supp)s(orted)f(b)m(y)h(a)g (gran)m(t)h(from)f(the)g(F)-8 b(onds)30 b(National)265 4045 y(Suisse)f(de)h(la)g(Rec)m(herc)m(he)i(Scien)m(ti\014que)d(No.)41 b(20)31 b(-)g(55654.98.)265 4491 y FB(References)265 4773 y FE([BCD])50 b(P)-8 b(.)42 b(Briet,)j(J.M.)d(Com)m(b)s(es,)j(P)-8 b(.)42 b(Duclos:)63 b(Sp)s(ectral)41 b(stabilit)m(y)f(under)h(th)m (unneling.)546 4965 y(Comm)m(un.)30 b(Math.)h(Ph)m(ys.)f FF(126)p FE(,)i(133)f(\(1989\))265 5196 y([BG])96 b(F.)32 b(Ben)m(tosela,)i(V.)e(Grecc)m(hi:)44 b(Stark)32 b(W)-8 b(annier)31 b(Ladders.)g(Comm)m(un.)g(Math.)i(Ph)m(ys.)546 5388 y FF(142)p FE(,)e(169)h(\(1991\))1828 5637 y FC(23)p eop %%Page: 24 24 24 23 bop 265 100 a FE([BCH])50 b(J.M.)28 b(Barbaroux,)h(J.M.)f(Com)m (b)s(es,)g(P)-8 b(.D.)30 b(Hislop:)38 b(Lo)s(calization)27 b(near)h(band)f(edges)546 292 y(for)j(random)g(Sc)m(hr\177)-45 b(odinger)29 b(op)s(erators.)i(Helv.)f(Ph)m(ys.)g(Acta)i FF(70)p FE(,)f(16)g(\(1997\))265 511 y([CH])97 b(J.M.)31 b(Com)m(b)s(es,)e(P)-8 b(.D.)32 b(Hislop:)39 b(Landau)29 b(Hamiltonians)f(with)h(random)g(p)s(oten)m(tials:)546 703 y(lo)s(calization)43 b(and)h(the)g(densit)m(y)f(of)h(states.)i (Comm)m(un.)d(Math.)i(Ph)m(ys.)f FF(177)p FE(,)k(603)546 895 y(\(1996\))265 1114 y([CHS])h(J.M.)c(Com)m(b)s(es,)k(P)-8 b(.D.)46 b(Hislop,)i(E.)d(So)s(ccorsi:)69 b(Edge)46 b(states)g(for)f (quan)m(tum)f(Hall)546 1306 y(hamiltonians.)28 b(mp-arc/02-172)265 1525 y([dBP])54 b(S.)23 b(de)h(Bi)m(\022)-43 b(evre,)26 b(J.V.)e(Pul)m(\023)-43 b(e:)37 b(Propagating)24 b(edge)h(states)f(for) g(magnetic)g(Hamiltonian.)546 1717 y(Math.)31 b(Ph)m(ys.)g(Electr.)f (J.)g FF(5)p FE(,)h(no.)g(3)f(\(1999\))265 1936 y([DMP1])50 b(T.C.)28 b(Dorlas,)h(N.)f(Macris,)h(J.V.)f(Pul)m(\023)-43 b(e:)40 b(Lo)s(calisation)27 b(in)g(a)h(single-band)e(appro)m(x-)546 2128 y(imation)31 b(to)h(random)f(Sc)m(hr\177)-45 b(odinger)31 b(op)s(erators)g(in)g(a)h(magnetic)g(\014eld.)f(Helv.)h(Ph)m(ys.)546 2320 y(Acta)g FF(68)p FE(,)f(330)h(\(1995\))265 2539 y([DMP2])50 b(T.C.)30 b(Dorlas,)g(N.)g(Macris,)g(J.V.)g(Pul)m(\023)-43 b(e:)40 b(Lo)s(calization)29 b(in)f(single)h(Landau)f(bands.)546 2731 y(J.)i(Math.)i(Ph)m(ys.)e FF(37)p FE(,)h(1574)h(\(1996\))265 2950 y([DMP3])50 b(T.C.)41 b(Dorlas,)i(N.)e(Macris,)i(J.V.)e(Pul)m (\023)-43 b(e:)61 b(The)40 b(nature)g(of)h(the)f(sp)s(ectrum)g(for)g(a) 546 3142 y(Landau)30 b(Hamiltonian)e(whit)h(delta)i(impurities.)c(J.)j (Stat.)i(Ph)m(ys.)e FF(87)p FE(,)h(847)h(\(1997\))265 3361 y([DMP4])50 b(T.C.)33 b(Dorlas,)h(N.)g(Macris,)f(J.V.)h(Pul)m (\023)-43 b(e:)46 b(Characterization)33 b(of)g(the)g(sp)s(ectrum)f(of) 546 3553 y(the)22 b(Landau)g(Hamiltonian)e(with)h(delta)h(impurities.)c (Comm)m(un.)k(Math.)h(Ph)m(ys.)f FF(204)p FE(,)546 3745 y(367)32 b(\(1999\))265 3963 y([EJK])51 b(P)-8 b(.)31 b(Exner,)f(A.)h(Jo)m(y)m(e,)h(H.)f(Ko)m(v)-5 b(arik:)40 b(Magnetic)32 b(transp)s(ort)e(in)f(a)h(straigh)m(t)h(parab)s(olic)546 4155 y(c)m(hannel.)f(J.)g(Ph)m(ys.)h(A:)f(Math.)i(Gen.)e FF(34)p FE(,)i(9733)g(\(2001\))265 4374 y([F])172 b(C.)69 b(F)-8 b(errari:)118 b(Dynamique)68 b(d'une)h(particule)e(quan)m(tique) i(dans)f(un)g(c)m(hamp)546 4566 y(magn)m(\023)-43 b(etique)31 b(inhomog)m(\022)-43 b(ene.)32 b(Diploma)d(w)m(ork,)i(EPFL)f(\(1999\).) 265 4785 y([F)m(GW])51 b(J.)25 b(F)-8 b(r\177)-45 b(ohlic)m(h,)25 b(G.M.)h(Graf,)h(J.)e(W)-8 b(alc)m(her:)39 b(On)24 b(the)h(extended)g (nature)f(of)i(edge)f(states)546 4977 y(of)31 b(quan)m(tum)e(Hall)h (Hamiltonians.)f(Ann.)h(Henri)f(P)m(oincar)m(\023)-43 b(e)32 b FF(1)p FE(,)e(405)i(\(2000\))265 5196 y([FM1])50 b(C.)38 b(F)-8 b(errari,)39 b(N.)f(Macris:)55 b(In)m(termixture)36 b(of)i(extended)g(edge)g(and)f(lo)s(calized)f(bulk)546 5388 y(energy)31 b(lev)m(els)f(in)f(macroscopic)i(Hall)e(systems.)i Fr(math-ph/0011013)1828 5637 y FC(24)p eop %%Page: 25 25 25 24 bop 265 100 a FE([FM2])50 b(C.)31 b(F)-8 b(errari,)30 b(N.)h(Macris:)42 b(Sp)s(ectral)29 b(prop)s(erties)g(of)i(\014nite)e (quan)m(tum)i(Hall)e(systems.)546 292 y(T)-8 b(o)31 b(app)s(ear)f(in)f (J.)h(Op)s(er.)f(Theor.)i(\(mp-arc/02-121\))265 526 y([H])163 b(B.I.)36 b(Halp)s(erin:)49 b(Quan)m(tized)36 b(Hall)e(conductance,)39 b(curren)m(t-carrying)c(edge)h(states,)546 718 y(and)f(the)g(existence) h(of)g(extended)f(states)i(in)d(a)h(t)m(w)m(o-dimensional)f(disordered) g(p)s(o-)546 910 y(ten)m(tial.)d(Ph)m(ys.)f(Rev.)h(B)g FF(25)p FE(,)g(2185)h(\(1982\))265 1143 y([M])148 b(N.)29 b(Macris:)39 b(Sp)s(ectral)28 b(\015o)m(w)g(and)g(lev)m(el)g(spacing)f (of)i(edge)g(states)h(for)e(quan)m(tum)f(Hall)546 1335 y(Hamiltonians.)i(Preprin)m(t)265 1569 y([MMP])50 b(N.)35 b(Macris,)g(P)-8 b(.A.)36 b(Martin)d(and)h(J.V.)h(Pul)m(\023)-43 b(e:)49 b(On)33 b(Edge)i(States)g(In)f(Semi-In\014nite)546 1761 y(Quan)m(tum)c(Hall)f(Systems.)h(J.)h(Ph)m(ys.)f(A:)h(Math.)g (Gen.)g FF(32)p FE(,)g(1985)h(\(1999\))265 1994 y([PG])98 b(R.E.)25 b(Prange)h(and)e(S.M.)i(Girvin:)36 b Fr(The)28 b(Quantum)g(Hal)5 b(l)28 b(E\013e)-5 b(ct)p FE(.)24 b(New)i(Y)-8 b(ork:)38 b(Grad-)546 2186 y(uate)31 b(T)-8 b(exts)31 b(in)e(Con)m(temp)s(orary)h(Ph)m(ysics,)g(Springer,)e(1987)265 2419 y([vKDP])49 b(K.)42 b(v.)g(Klitzing,)i(G.)e(Dorda,)k(M.)d(P)m(epp) s(er:)63 b(New)42 b(metho)s(d)g(for)f(high-accuracy)546 2611 y(determination)27 b(of)g(the)h(\014ne-structure)f(constan)m(t)i (based)f(on)f(quan)m(tized)h(Hall)e(resis-)546 2803 y(tance.)32 b(Ph)m(ys.)e(Rev.)h(Lett.)g FF(45)p FE(,)g(494)h(\(1980\))265 3037 y([W])138 b(W.M.)32 b(W)-8 b(ang:)44 b(Microlo)s(calization,)30 b(p)s(ercolation)g(and)h(Anderson)f(lo)s(calization)g(for)546 3229 y(the)35 b(magnetic)h(Sc)m(hr\177)-45 b(odinger)33 b(op)s(erator)j(with)d(a)j(random)e(p)s(oten)m(tial.)h(J.)g(of)g(F)-8 b(unct.)546 3421 y(Anal.)30 b FF(146)p FE(,)h(1)g(\(1997\))1828 5637 y FC(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0206061001475--