Content-Type: multipart/mixed; boundary="-------------0311250338853" This is a multi-part message in MIME format. ---------------0311250338853 Content-Type: text/plain; name="03-510.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-510.keywords" Schrodinger operator with magnetic field in a domain with corners, spectral theory, corners effects, superconductivity semi-classical analysis, Agmon estimates ---------------0311250338853 Content-Type: application/postscript; name="article2theo.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="article2theo.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: article2theo.dvi %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips article2theo.dvi -o article2theo.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.11.06:1327 %%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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Fz(dep)r(ends)i(only)f(on)g(the)h(magnetic)f(\014eld)g(and)g(not)h (on)f(the)h(c)n(hoice)e(of)i(the)f(corresp)r(onding)440 2600 y(magnetic)d(p)r(oten)n(tial.)37 b(Then,)26 b(w)n(e)f(denote)h(b)n (y)g Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))26 b(the)g(b)r(ottom)g(of)g(the)g(sp)r(ectrum)440 2700 y(of)i Fu(P)588 2712 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)771 2700 y Fz(.)37 b(W)-7 b(e)28 b(omit)g Fu(h)f Fz(in)h(the)g(notations)f(when) h Fu(h)23 b Fz(=)f(1.)565 2899 y(Berno\013-Stern)n(b)r(erg)34 b([3)o(],)39 b(Hel\013er-Mohamed)c([10)o(],)k(Hel\013er-Morame)34 b([11)o(],)39 b(Lu-P)n(an)440 2999 y([15)o(,)c(16)o(])f(ha)n(v)n(e)f (already)g(analyzed)h(the)g(case)g(of)g(regular)f(domains)g(and)h(our)g (aim)g(is)g(to)440 3098 y(giv)n(e)c(the)h(pro)r(of)f(of)h(the)g (results)g(whic)n(h)g(w)n(ere)e(announced)i(in)g([5)o(])g(in)h(order)d (to)i(establish)440 3198 y(similar)j(results)f(for)h(domains)g(with)h (corners.)55 b(In)35 b(the)f(previous)g(analysis,)g(the)h(mo)r(del)440 3298 y(op)r(erator)24 b Fx(\000r)907 3267 y Fv(2)907 3320 y Ft(A)960 3328 y Fr(0)1022 3298 y Fz(on)h Fw(R)f Fx(\002)18 b Fw(R)1350 3267 y Fv(+)1437 3298 y Fz(with)26 b(a)f(constan)n(t)g(magnetic)f(\014eld)i Fx(A)2620 3310 y Fv(0)2683 3298 y Fz(of)g(in)n(tensit)n(y)f(equal)440 3397 y(to)i(1)h(de\014ned)g(b)n(y)f(:)1502 3542 y Fx(A)1568 3554 y Fv(0)1614 3542 y Fz(:)51 b(=)1786 3486 y(1)p 1786 3523 42 4 v 1786 3599 a(2)1851 3425 y Fp(\022)1986 3491 y Fu(x)2033 3503 y Fv(2)1954 3591 y Fx(\000)p Fu(x)2066 3603 y Fv(1)2145 3425 y Fp(\023)2220 3542 y Fu(;)891 b Fz(\(1.2\))440 3731 y(w)n(as)33 b(pla)n(ying)h(an)g(imp)r(ortan)n(t)g (role.)56 b(Our)34 b(new)h(mo)r(del)f(to)g(analyze)g(the)h(case)e(of)i (a)f(non)440 3830 y(smo)r(oth)c(domain)h(is)f(the)h(op)r(erator)e Fx(\000r)1739 3800 y Fv(2)1739 3853 y Ft(A)1792 3861 y Fr(0)1859 3830 y Fz(in)i(an)f(angular)f(sector.)45 b(W)-7 b(e)31 b(denote)g(b)n(y)f(\012)3257 3842 y Fs(\013)440 3930 y Fz(a)f(sector)f(with)h(an)g(angle)f(equal)h(to)g Fu(\013)g Fz(and)g(b)n(y)g Fu(\026)p Fz(\()p Fu(\013)p Fz(\))h(the)g(b)r(ottom)f(of)g(the)h(sp)r(ectrum)f(for)440 4030 y Fx(\000r)574 3999 y Fv(2)574 4052 y Ft(A)627 4060 y Fr(0)691 4030 y Fz(in)f(\012)848 4042 y Fs(\013)895 4030 y Fz(.)37 b(If)28 b Fu(\013)c Fz(=)f Fu(\031)s Fz(,)28 b(w)n(e)f(write)g(:)1642 4202 y(\002)1707 4214 y Fv(0)1753 4202 y Fz(:)51 b(=)22 b Fu(\026)p Fz(\()p Fu(\031)s Fz(\))p Fu(:)1033 b Fz(\(1.3\))440 4375 y(This)30 b(real)g(\002)862 4387 y Fv(0)929 4375 y Fz(pla)n(ys)g(an)g(imp)r(ortan)n(t)g(role)g(in)h (the)f(study)h(of)g(regular)d(cases.)45 b(It)31 b(app)r(ears)440 4475 y(for)24 b(the)i(study)f(of)g(the)g(b)r(ottom)h(of)f(the)g(sp)r (ectrum)g(of)g(the)h(op)r(erator)d Fx(\000r)2745 4445 y Fv(2)2745 4498 y Ft(A)2798 4506 y Fr(0)2860 4475 y Fz(on)h(a)h(disk)g(\(cf)440 4575 y([2],)k(p.)h(24)e(and)h([19)o(]\))h (and)f(for)f(the)i(estimate)f(of)g Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))30 b(giv)n(en)e(b)n(y)h(Hel\013er-Morame) 440 4674 y(\(cf)24 b([11)o(],)h(p.617-621\).)33 b(W)-7 b(e)24 b(will)g(pro)n(v)n(e)e(that)i Fu(\026)p Fz(\()p Fu(\013)p Fz(\))g(is)g(b)r(ounded)g(from)f(ab)r(o)n(v)n(e)f(b)n(y)i (\002)3066 4686 y Fv(0)3103 4674 y Fz(.)35 b(The)440 4774 y(main)f(result)f(of)h(this)g(pap)r(er)f(is)h(the)g(construction)f (of)g(an)h(asymptotics)f(for)g Fu(\026)p Fz(\()p Fu(\013)p Fz(\))i(as)e Fu(\013)440 4873 y Fz(tends)28 b(to)f(0)h(:)440 5032 y FA(Theorem)h(1.1.)39 b Fo(Ther)l(e)30 b(exists)f(a)g(r)l(e)l(al) g(se)l(quenc)l(e)f Fz(\()p Fu(m)2167 5044 y Fs(j)2203 5032 y Fz(\))2235 5044 y Fs(j)s Ft(2)p Fn(N)2385 5032 y Fo(with)i Fu(m)2638 5044 y Fv(0)2698 5032 y Fz(=)2823 4999 y Fv(1)p 2796 5013 88 4 v 2796 5022 a Ft(p)p 2850 5022 34 3 v 2850 5070 a Fv(3)2922 5032 y Fo(such)f(that)g(:)1244 5295 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))84 b Fx(\030)f Fu(\013)1767 5192 y Ft(1)1740 5217 y Fp(X)1742 5393 y Fs(j)s Fv(=0)1874 5295 y Fu(m)1947 5307 y Fs(j)1981 5295 y Fu(\013)2034 5261 y Fv(2)p Fs(j)2149 5295 y Fz(as)29 b Fu(\013)23 b Fx(!)h Fz(0)p Fu(:)633 b Fz(\(1.4\))1851 5662 y(2)p eop %%Page: 3 3 3 2 bop 440 531 a Fz(T)-7 b(o)24 b(establish)g(this)h(result,)g(w)n(e)f (\014rst)g(determine)h(the)g(b)r(ottom)f(of)h(the)g(essen)n(tial)e(sp)r (ectrum)440 631 y(and)28 b(then)i(construct)e(a)g(regular)f(function)i (whose)f(Ra)n(yleigh)f(quotien)n(t)i(is)f(less)g(than)h(the)440 731 y(b)r(ottom)f(of)f(the)h(essen)n(tial)f(sp)r(ectrum.)440 830 y(The)g(analysis)e(of)h(the)h(mo)r(del)f Fx(\000r)1535 800 y Fv(2)1535 853 y Ft(A)1588 861 y Fr(0)1652 830 y Fz(in)g(\012)1807 842 y Fs(\013)1881 830 y Fz(is)g(useful)h(to)f(appro) n(ximate)f(the)i(case)f(of)g(the)440 930 y(op)r(erator)j Fu(P)831 942 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)1046 930 y Fz(where)h Fu(h)h Fz(is)g(a)g(small)f(parameter,)h Fu(B)13 b Fz(:)58 b(=)28 b(curl)13 b Fx(A)32 b Fz(is)f(a)f(non)h (constan)n(t)440 1029 y(magnetic)f(p)r(oten)n(tial)h(and)f(\012)g(is)h (a)f(domain)g(with)h(corners,)f(as)f(w)n(e)i(see)f(in)h(the)f(follo)n (wing)440 1129 y(theorem)d(:)440 1289 y FA(Theorem)44 b(1.2.)j Fo(L)l(et)40 b Fz(\012)g Fo(b)l(e)h(a)g(b)l(ounde)l(d)g(op)l (en)f(subset)g(of)i Fw(R)2458 1259 y Fv(2)2541 1289 y Fo(whose)g(b)l(oundary)f(is)g(a)440 1388 y(curviline)l(ar)36 b(p)l(olygon)f(with)g(vertic)l(es)g Fu(S)1705 1400 y Fv(1)1742 1388 y Fu(;)14 b(:)g(:)g(:)g(;)g(S)1978 1400 y Fs(N)2041 1388 y Fo(.)52 b(L)l(et)34 b Fz(\012)e Fx(3)g Fu(x)g Fx(7!)f Fu(B)t Fz(\()p Fu(x)p Fz(\))36 b Fo(b)l(e)e(a)h(p)l (ositive)440 1488 y(magnetic)30 b(\014eld)h(and)f(let)f(us)h(de\014ne)g (:)1210 1662 y Fu(b)23 b Fz(=)37 b(inf)1357 1730 y Fs(x)p Ft(2)p 1440 1682 48 3 v Fv(\012)1500 1662 y Fu(B)t Fz(\()p Fu(x)p Fz(\))85 b(and)d Fu(b)2015 1628 y Ft(0)2061 1662 y Fz(=)57 b(inf)2149 1717 y Fs(x)p Ft(2)p Fs(@)t Fv(\012)2332 1662 y Fu(B)t Fz(\()p Fu(x)p Fz(\))p Fu(:)601 b Fz(\(1.5\))440 1884 y Fo(We)38 b(denote)g(by)h Fu(\013)1042 1896 y Fv(1)1079 1884 y Fu(;)14 b(:)g(:)g(:)g(;)g(\013)1317 1896 y Fs(N)1417 1884 y Fo(the)38 b(angles)h(for)g(e)l(ach)g(vertex.)62 b(Then,)41 b(for)e Fu(h)f Fo(smal)t(l,)j(the)440 1983 y(smal)t(lest)d(eigenvalue)g Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))38 b Fo(for)g(the)g(Neumann)-8 b('s)36 b(r)l(e)l(alization)i(of)h Fx(\000)p Fz(\()p Fu(h)p Fx(r)23 b(\000)h Fu(i)p Fx(A)p Fz(\))3267 1953 y Fv(2)440 2083 y Fo(admits)30 b(the)g(fol)t(lowing)j(asymptotics)d(:)692 2303 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))24 b(=)e Fu(h)44 b Fz(inf)1373 2186 y Fp(\022)1434 2303 y Fu(b;)14 b Fz(\002)1572 2315 y Fv(0)1608 2303 y Fu(b)1644 2268 y Ft(0)1667 2303 y Fu(;)100 b Fz(inf)1704 2356 y Fs(j)s Fv(=1)p Fs(;:::;N)1990 2303 y Fu(\026)p Fz(\()p Fu(\013)2125 2315 y Fs(j)2160 2303 y Fz(\))p Fu(B)t Fz(\()p Fu(S)2342 2315 y Fs(j)2378 2303 y Fz(\))2410 2186 y Fp(\023)2490 2303 y Fz(+)18 b Fx(O)r Fz(\()p Fu(h)2721 2268 y Fv(5)p Fs(=)p Fv(4)2826 2303 y Fz(\))p Fu(:)253 b Fz(\(1.6\))565 2523 y(W)-7 b(e)44 b(notice)f(that)h(for)g(regular)e(domains,)47 b(Theorem)c(1.2)g(giv)n(es)f(the)j(estimate)e(of)440 2622 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))37 b(obtained)g(b)n(y)f(Hel\013er-Morame)g([11)o(],)j(p.)f(617-621.)61 b(The)37 b(condition)g(\(1.6\))440 2722 y(tak)n(es)27 b(in)h(the)f(case)g(of)h(a)f(constan)n(t)g(magnetic)g(\014eld)h(a)f (simpler)g(form)h(:)440 2881 y FA(Corollary)33 b(1.3.)40 b Fo(F)-6 b(or)30 b Fu(B)k Fo(c)l(onstant,)29 b(we)h(have)h(:)934 3101 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b(=)g(inf)1522 2984 y Fp(\022)1669 3101 y Fz(inf)1584 3155 y Fs(j)s Fv(=1)p Fs(;:::)o(;N)1870 3101 y Fu(\026)p Fz(\()p Fu(\013)2005 3113 y Fs(j)2040 3101 y Fz(\))p Fu(;)14 b Fz(\002)2174 3113 y Fv(0)2211 2984 y Fp(\023)2286 3101 y Fu(B)t(h)k Fz(+)g Fx(O)r Fz(\()p Fu(h)2650 3067 y Fv(5)p Fs(=)p Fv(4)2755 3101 y Fz(\))p Fu(:)565 3321 y Fz(This)j(article)f(is)h(organized)e(as)h(follo)n(ws.)34 b(In)21 b(Section)g(2,)h(w)n(e)f(recall)f(some)g(results)g(ab)r(out)440 3421 y(the)35 b(Neumann)f(realization)f(of)i(the)f(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erator)f(with)j(constan)n(t)f(magnetic)440 3520 y(\014eld)d(in)g(the)h(half)f(plane)f(and)h(sho)n(w)f(that)h(the)h (b)r(ottom)f(of)g(the)g(essen)n(tial)f(sp)r(ectrum)h(for)440 3620 y(an)38 b(angular)f(sector)g(is)h(\002)1287 3632 y Fv(0)1324 3620 y Fz(,)j(the)d(b)r(ottom)h(of)f(the)h(sp)r(ectrum)f (for)g(the)g(half)h(plane.)68 b(In)440 3719 y(Subsection)26 b(2.3,)g(w)n(e)g(use)g(the)h(analysis)e(of)h(the)h(op)r(erator)d Fu(D)2365 3689 y Fv(2)2363 3740 y Fs(t)2419 3719 y Fz(+)15 b(\()p Fu(t)h Fx(\000)g Fu(\020)2694 3731 y Fv(0)2731 3719 y Fz(\))2763 3689 y Fv(2)2827 3719 y Fz(giv)n(en)26 b(in)g(Sub-)440 3819 y(section)d(2.1)g(to)h(construct,)g(inspired)f(b)n (y)g(P)n(an)g([17)o(],)i(a)e(test)h(function)g(and)g(prop)r(ose)e(a)i (\014rst)440 3919 y(upp)r(er)34 b(b)r(ound)g(of)f(the)h(b)r(ottom)g(of) g(the)g(sp)r(ectrum)g(for)f(an)g(angular)f(sector)h(with)h(angle)440 4018 y(close)27 b(to)750 3986 y Fs(\031)p 750 4000 41 4 v 754 4047 a Fv(2)801 4018 y Fz(.)440 4118 y(Then,)g(from)f(Section)h (3)f(to)h(Section)g(10,)f(w)n(e)g(analyze)f(the)i(b)r(ottom)g(of)g(the) g(sp)r(ectrum)g(for)440 4218 y(the)35 b(Neumann)f(realization)f(of)i (the)f(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erator)f(with)j(constan)n(t) f(magnetic)440 4317 y(\014eld)28 b(in)g(an)f(angular)f(sector)h(when)h (the)g(angle)f(tends)h(to)f(0.)37 b(As)28 b(it)g(is)g(more)e(easy)h(to) h(deal)440 4417 y(with)33 b(a)f(non)g(v)-5 b(ariable)32 b(domain,)h(Section)g(3)f(sho)n(ws)f(ho)n(w)h(to)g(reduce)h(the)f (problem)g(to)h(a)440 4516 y(domain)27 b(indep)r(enden)n(t)g(of)g(the)g (angle)f Fu(\013)i Fz(of)f(the)g(sector)f(and)h(to)f(deal)h(with)g(a)g (new)g(op)r(era-)440 4616 y(tor)g Fu(P)627 4628 y Fs(\013)675 4616 y Fz(.)37 b(This)28 b Fu(P)978 4628 y Fs(\013)1054 4616 y Fz(is)f(the)i(sum)f(of)f(t)n(w)n(o)g(op)r(erators)f(with)i (di\013eren)n(t)g(w)n(eigh)n(t)f(according)f(to)440 4716 y Fu(\013)p Fz(.)41 b(Section)29 b(4)g(is)g(dev)n(oted)f(to)h(the)g (analysis)f(of)h(these)g(t)n(w)n(o)f(k)n(ey-op)r(erators)e(and)j(leads) f(in)440 4815 y(Section)g(5)f(to)g(the)h(construction)f(of)h(a)f (formal)g(asymptotics)f(for)h(the)h(eigen)n(v)-5 b(alue.)440 4915 y(Section)21 b(6)g(establishes)f(an)h(upp)r(er)g(b)r(ound)g(of)g (the)h(b)r(ottom)f(of)g(the)g(sp)r(ectrum)h Fu(\026)p Fz(\()p Fu(\013)p Fz(\))g(thanks)440 5015 y(to)27 b(the)g(min-max)g (principle)f(and)h(the)g(computation)g(of)g(the)g(Ra)n(yleigh)f (quotien)n(t)h(for)f(the)440 5114 y(previous)g(formal)h(solution.)440 5214 y(Section)34 b(7)f(uses)h(Agmon's)f(tec)n(hniques)g(to)h(pro)n(v)n (e)e(the)i(deca)n(y)f(of)h(eigenfunction.)55 b(This)440 5313 y(last)29 b(estimate)h(is)f(useful)h(to)f(giv)n(e)g(a)g(w)n(eak)g (lo)n(w)n(er)f(b)r(ound)i(of)f(the)h(\014rst)f(eigen)n(v)-5 b(alue)29 b(as)g(w)n(e)440 5413 y(explain)e(in)h(Section)g(8.)1851 5662 y(3)p eop %%Page: 4 4 4 3 bop 440 531 a Fz(T)-7 b(o)19 b(pro)n(v)n(e)f(that)j(our)e(formal)g (construction)f(giv)n(e)h(further)h(co)r(e\016cien)n(ts)f(of)h(the)g (asymptotics)440 631 y(of)33 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))i(as)d(it)i(is)f(writen)g(in)h(Theorem)e(1.1,)i(w)n(e)f(estimate) g(the)h(splitting)g(b)r(et)n(w)n(een)f(the)440 731 y(t)n(w)n(o)23 b(\014rst)h(eigen)n(v)-5 b(alues)23 b(and)h(b)r(ound)g(from)g(b)r(elo)n (w)f(the)i(second)e(eigen)n(v)-5 b(alue)23 b(in)h(Section)g(10.)440 830 y(In)i(Section)g(11,)g(the)g(analysis)f(of)h(the)h(Neumann)f (realization)f(of)h(the)g(Sc)n(hr\177)-42 b(odinger)24 b(op)r(er-)440 930 y(ator)j(with)h(constan)n(t)g(magnetic)f(\014eld)h (in)g(an)g(angular)e(sector)h(coupled)h(with)h(the)f(results)440 1029 y(ab)r(out)22 b(the)h(plane)f Fw(R)1080 999 y Fv(2)1145 1029 y Fz(and)g(the)h(half)f(plane)g Fw(R)j Fx(\002)18 b Fw(R)2029 999 y Fv(+)2112 1029 y Fz(are)k(useful)g(to)g(estimate)g (the)h(b)r(ottom)440 1129 y(of)k(the)h(sp)r(ectrum)g(of)g Fx(\000)p Fz(\()p Fu(h)p Fx(r)18 b(\000)g Fu(i)p Fx(A)p Fz(\))1577 1099 y Fv(2)1642 1129 y Fz(in)27 b(a)h(non)f(smo)r(oth)g (domain)g(with)h Fu(B)g Fz(=)22 b(curl)14 b Fx(A)28 b Fz(non)440 1229 y(constan)n(t)f(and)g Fu(h)h Fz(tending)g(to)f(0.)440 1499 y Fy(2)135 b(Some)45 b(remarks)h(ab)t(out)f Fm(\002)1994 1517 y Fz(0)2084 1499 y Fy(and)g(applications)440 1681 y Fz(Let)22 b(us)f(recall)g(from)g([8])g(the)h(link)g(b)r(et)n(w)n(een) g(\002)1873 1693 y Fv(0)1909 1681 y Fz(,)h(in)n(tro)r(duced)f(in)f (\(1.3\),)i(and)e(the)h(op)r(erators)440 1780 y Fu(D)511 1750 y Fv(2)509 1801 y Fs(t)567 1780 y Fz(+)c(\()p Fu(t)g Fx(\000)h Fu(\020)6 b Fz(\))888 1750 y Fv(2)953 1780 y Fz(on)27 b Fw(R)1122 1750 y Fv(+)1184 1780 y Fz(.)440 2008 y Fl(2.1)112 b(Link)38 b(with)e Fk(D)1292 1972 y Fj(2)1289 2033 y Fi(t)1353 2008 y FF(+)22 b(\()p Fk(t)g Fh(\000)h Fk(\020)8 b FF(\))1735 1972 y Fj(2)1811 2008 y Fl(on)38 b Fg(R)2033 1972 y Fj(+)440 2161 y FA(Prop)s(osition)h(2.1.) 45 b Fo(L)l(et)37 b Fu(\025)1359 2173 y Fs(H)1422 2161 y Fz(\()p Fu(\020)6 b Fz(\))38 b Fo(b)l(e)f(the)h(b)l(ottom)f(of)g(the) h(sp)l(e)l(ctrum)e(of)i(the)f(Neumann)440 2261 y(r)l(e)l(alization)31 b(in)f Fu(L)1001 2231 y Fv(2)1038 2261 y Fz(\()p Fw(R)1124 2231 y Fv(+)1185 2261 y Fz(\))g Fo(of)h(the)f(op)l(er)l(ator)g Fu(H)7 b Fz(\()p Fu(\020)f Fz(\))31 b Fo(de\014ne)l(d)f(for)g Fu(\020)g Fx(2)23 b Fw(R)36 b Fo(by)30 b(:)1471 2417 y Fu(H)7 b Fz(\()p Fu(\020)f Fz(\))24 b(=)f Fu(D)1836 2383 y Fv(2)1834 2438 y Fs(t)1891 2417 y Fz(+)18 b(\()p Fu(t)h Fx(\000)f Fu(\020)6 b Fz(\))2212 2383 y Fv(2)2250 2417 y Fu(:)440 2574 y Fo(Ther)l(e)43 b(exists)e(a)i(unique)e Fu(\020)1330 2586 y Fv(0)1409 2574 y Fo(such)h(that)g Fu(\025)1840 2586 y Fs(H)1904 2574 y Fz(\()p Fu(\020)1972 2586 y Fv(0)2009 2574 y Fz(\))k(=)e(\002)2261 2586 y Fv(0)2298 2574 y Fo(,)h Fu(\025)2416 2586 y Fs(H)2480 2574 y Fz(\()p Fu(\020)6 b Fz(\))43 b Fo(is)f(de)l(cr)l(e)l(asing)h(fr) l(om)440 2673 y Fz(])13 b Fx(\000)g(1)p Fu(;)h(\020)710 2685 y Fv(0)748 2673 y Fz(])27 b Fo(onto)h Fz(])13 b(+)g Fx(1)p Fu(;)h Fz(\002)1284 2685 y Fv(0)1321 2673 y Fz(])27 b Fo(and)h(incr)l(e)l(asing)g(fr)l(om)g Fz([)p Fu(\020)2172 2685 y Fv(0)2210 2673 y Fu(;)14 b Fz(+)p Fx(1)p Fz([)27 b Fo(onto)g Fz([\002)2719 2685 y Fv(0)2756 2673 y Fu(;)14 b Fz(1[)p Fo(.)37 b(We)28 b(denote)440 2773 y(by)i Fu(\036)g Fo(the)g(normalize)l(d)h(eigenve)l(ctor)g(asso)l(ciate)l(d)g(to)f Fu(\025)2154 2785 y Fs(H)2218 2773 y Fz(\()p Fu(\020)2286 2785 y Fv(0)2323 2773 y Fz(\))p Fo(,)h(then)e(:)973 2864 y Fp(Z)1056 2885 y Ft(1)1019 3053 y Fv(0)1126 2977 y Fz(\()p Fu(t)19 b Fx(\000)f Fu(\020)1326 2989 y Fv(0)1363 2977 y Fz(\))p Fx(j)p Fu(\036)p Fz(\()p Fu(t)p Fz(\))p Fx(j)1584 2943 y Fv(2)1653 2977 y Fu(dt)23 b Fz(=)g(0)82 b Fo(and)i Fu(\036)p Fz(\(0\))2331 2943 y Fv(2)2392 2977 y Fz(=)2489 2921 y Fu(\025)2537 2891 y Ft(00)2537 2944 y Fs(H)2601 2921 y Fz(\()p Fu(\020)2669 2933 y Fv(0)2706 2921 y Fz(\))p 2489 2958 250 4 v 2556 3034 a(2)p Fu(\020)2634 3046 y Fv(0)2749 2977 y Fu(:)362 b Fz(\(2.1\))565 3180 y(W)-7 b(e)31 b(just)g(sho)n(w)f(brie\015y)g(ho)n(w)g(the)h(op)r (erator)e Fu(H)7 b Fz(\()p Fu(\020)f Fz(\))32 b(app)r(ears)d(in)i(the)g (analysis)e(of)i(the)440 3280 y(op)r(erator)23 b Fx(\000r)906 3250 y Fv(2)906 3303 y Ft(A)959 3311 y Fr(0)1021 3280 y Fz(de\014ned)j(on)f Fw(R)f Fx(\002)18 b Fw(R)1633 3250 y Fv(+)1694 3280 y Fz(.)37 b(W)-7 b(e)25 b(notice)g(that)h(after)f(a)f (gauge)g(transform,)h(w)n(e)440 3380 y(ha)n(v)n(e)h(to)i(study)g(the)g (new)f(op)r(erator)f(:)640 3536 y(\()p Fu(D)741 3548 y Fs(x)779 3556 y Fr(1)834 3536 y Fx(\000)18 b Fu(x)964 3548 y Fv(2)1002 3536 y Fz(\))1034 3502 y Fv(2)1089 3536 y Fz(+)h Fu(D)1244 3502 y Fv(2)1242 3556 y Fs(x)1280 3564 y Fr(2)1343 3536 y Fz(on)27 b Fw(R)e Fx(\002)18 b Fw(R)1674 3502 y Fv(+)1735 3536 y Fu(;)42 b Fz(with)28 b(Neumann)g(condition)f(on)g Fu(x)2891 3548 y Fv(2)2952 3536 y Fz(=)c(0)p Fu(:)440 3692 y Fz(Then,)h(w)n(e)f(mak)n(e)f(a)h (partial)f(F)-7 b(ourier)22 b(transform)g(in)h(the)h(\014rst)f(co)r (ordinate.)34 b(The)23 b(op)r(erator)440 3792 y Fx(\000r)574 3762 y Fv(2)574 3815 y Ft(A)627 3823 y Fr(0)691 3792 y Fz(is)28 b(also)e(unitary)h(equiv)-5 b(alen)n(t)28 b(to)f(:)1395 3960 y(\()p Fu(\030)1463 3972 y Fv(1)1520 3960 y Fx(\000)18 b Fu(x)1650 3972 y Fv(2)1687 3960 y Fz(\))1719 3926 y Fv(2)1775 3960 y Fz(+)g Fu(D)1929 3926 y Fv(2)1927 3980 y Fs(x)1965 3988 y Fr(2)2024 3960 y Fz(=)23 b Fu(H)7 b Fz(\()p Fu(\030)2256 3972 y Fv(1)2294 3960 y Fz(\))p Fu(:)440 4116 y Fz(So)26 b(the)g(study)h(of)f(the)g(op)r (erators)f(family)h Fu(H)7 b Fz(\()p Fu(\020)f Fz(\))27 b(is)f(link)n(ed)g(to)g(the)g(Neumann)h(realization)440 4216 y(of)h Fx(\000r)669 4186 y Fv(2)669 4239 y Ft(A)722 4247 y Fr(0)786 4216 y Fz(on)f Fw(R)e Fx(\002)18 b Fw(R)1117 4186 y Fv(+)1178 4216 y Fz(.)440 4315 y(F)-7 b(urthermore,)26 b(the)i(b)r(ottom)g(of)f(the)h(sp)r(ectrum,)g(\002)2068 4327 y Fv(0)2104 4315 y Fz(,)g(is)f(in)h(the)g(essen)n(tial)e(sp)r (ectrum)i(and)440 4415 y(there)f(is)h(no)f(p)r(oin)n(t)h(sp)r(ectrum)g (for)f(the)h(realization)e(on)h Fw(R)e Fx(\002)18 b Fw(R)2438 4385 y Fv(+)2499 4415 y Fz(.)440 4643 y Fl(2.2)112 b(Bottom)36 b(of)i(the)f(essen)m(tial)g(sp)s(ectrum)440 4796 y Fz(Let)29 b(\012)650 4808 y Fs(\013)723 4796 y Fx(\032)d Fw(R)868 4766 y Fv(2)940 4796 y Fz(b)r(e)k(an)f(angular)e(sector)h(with)i(angle) f Fu(\013)p Fz(.)42 b(The)29 b(P)n(ersson)e(Lemma)i(\(cf)h([18)o(]\)) 440 4896 y(ma)n(y)d(b)r(e)h(generalized)e(for)h(un)n(b)r(ounded)h (domains)f(of)h Fw(R)2195 4866 y Fv(2)2265 4896 y Fz(and)g(Neumann)g (realizations)e(:)440 5041 y FA(Lemma)e(2.2)g(\(P)m(ersson\).)37 b Fo(L)l(et)24 b Fz(\012)h Fo(b)l(e)g(an)g(unb)l(ounde)l(d)f(domain)j (of)e Fw(R)2655 5011 y Fv(2)2724 5041 y Fo(with)g(lipschitzian)440 5140 y(b)l(oundary)32 b(and)f Fu(V)50 b Fo(b)l(e)31 b(a)g(semi-b)l (ounde)l(d)g(fr)l(om)h(b)l(elow)g(r)l(e)l(gular)e(function.)43 b(We)30 b(denote)i(by)440 5240 y Fz(inf)21 b Fu(\033)602 5252 y Fv(ess)689 5240 y Fz(\()p Fx(\000)p Fz(\001)855 5252 y Ft(A)931 5240 y Fz(+)d Fu(V)g Fz(\))30 b Fo(the)g(b)l(ottom)g (of)h(the)e(essential)i(sp)l(e)l(ctrum,)e(then)h(:)1106 5396 y Fz(inf)21 b Fu(\033)1268 5408 y Fv(ess)1355 5396 y Fz(\()p Fx(\000)p Fz(\001)1521 5408 y Ft(A)1597 5396 y Fz(+)d Fu(V)g Fz(\))24 b(=)47 b(lim)1889 5446 y Fs(r)r Ft(!1)2068 5396 y Fz(\006\()p Fx(\000)p Fz(\001)2294 5408 y Ft(A)2370 5396 y Fz(+)18 b Fu(V)5 b(;)14 b(r)r Fz(\))p Fu(;)497 b Fz(\(2.2\))1851 5662 y(4)p eop %%Page: 5 5 5 4 bop 440 531 a Fo(with)30 b(:)490 787 y Fz(\006\()p Fx(\000)p Fz(\001)716 799 y Ft(A)792 787 y Fz(+)18 b Fu(V)5 b(;)14 b(r)r Fz(\))9 b(:)52 b(=)269 b(inf)1208 854 y Fs(\036)p Ft(2)p Fs(C)1345 834 y Fq(1)1341 872 y Fr(0)1405 854 y Fv(\()p 1431 806 48 3 v(\012)p Ft(\\)p Ff({)p Fs(S)1598 862 y Fe(r)1631 854 y Fv(\))p Fs(;\036)p Ft(6)p Fv(=0)1815 616 y Fp(8)1815 691 y(<)1815 841 y(:)1899 633 y(R)1938 730 y Fv(\012)2003 608 y Fp(\020)2053 700 y Fx(jr)2145 712 y Ft(A)2202 700 y Fu(\036)p Fz(\()p Fu(x)p Fz(\))p Fx(j)2387 659 y Fv(2)2443 700 y Fz(+)18 b Fu(V)h Fz(\()p Fu(x)p Fz(\))p Fx(j)p Fu(\036)p Fz(\()p Fu(x)p Fz(\))p Fx(j)2910 670 y Fv(2)2949 608 y Fp(\021)3043 700 y Fu(dx)p 1899 768 1235 4 v 2281 778 a Fp(R)2320 874 y Fv(\012)2386 844 y Fx(j)p Fu(\036)p Fz(\()p Fu(x)p Fz(\))p Fx(j)2592 820 y Fv(2)2660 844 y Fu(dx)3143 616 y Fp(9)3143 691 y(=)3143 841 y(;)3231 787 y Fu(:)3134 986 y Fz(\(2.3\))565 1138 y(W)-7 b(e)24 b(use)h(this)f(lemma)g(to)h (determine)f(the)h(b)r(ottom)g(of)f(the)h(essen)n(tial)e(sp)r(ectrum)i (of)f(the)440 1237 y(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erator)g(in)i (an)f(angular)f(sector:)440 1389 y FA(Prop)s(osition)g(2.3.)37 b Fo(The)27 b(b)l(ottom)g(of)g(the)g(essential)g(sp)l(e)l(ctrum)e(for)j (the)e(Neumann)g(r)l(e)l(ali-)440 1489 y(zation)k(of)h Fx(\000r)920 1459 y Fv(2)920 1512 y Ft(A)973 1520 y Fr(0)1039 1489 y Fo(in)f(an)g(angular)g(se)l(ctor)g Fz(\012)1861 1501 y Fs(\013)1938 1489 y Fo(is)g(e)l(qual)g(to)g Fz(\002)2402 1501 y Fv(0)2439 1489 y Fo(.)440 1641 y(Pr)l(o)l(of)i(:)42 b Fz(W)-7 b(e)30 b(estimate)f(\006\()p Fx(\000)p Fz(\001)1436 1653 y Ft(A)1489 1661 y Fr(0)1526 1641 y Fu(;)14 b(r)r Fz(\))30 b(for)f Fu(r)g(>)c Fz(0)k(and)g(sho)n(w)g(that)g(it)h(tends)g (to)f(\002)3049 1653 y Fv(0)3115 1641 y Fz(when)440 1740 y Fu(r)34 b Fz(tends)e(to)f(in\014nit)n(y)-7 b(.)49 b(W)-7 b(e)32 b(use)f(a)g(partition)g(of)h(the)g(unit)n(y)f(whic)n(h)h(shares) e(the)i(sector)e(in)440 1840 y(three)f(sub)r(domains)g(and)g(w)n(e)f (compare)g(to)h(the)h(mo)r(dels)f Fw(R)2314 1810 y Fv(2)2386 1840 y Fz(and)g Fw(R)c Fx(\002)18 b Fw(R)2765 1810 y Fv(+)2855 1840 y Fz(according)27 b(to)440 1939 y(the)h(supp)r(ort)f(of) h(the)g(cut-o\013)g(functions.)440 2039 y(Let)g Fu(r)e(>)c Fz(0)27 b(and)38 b(~)-52 b Fu(\037)27 b Fz(b)r(e)h(a)g(regular)d (function)k(de\014ned)f(from)f Fw(R)33 b Fz(on)n(to)27 b([0)p Fu(;)14 b Fz(1])27 b(suc)n(h)g(that)h(:)1311 2137 y Fp(\032)1424 2203 y Fz(~)-52 b Fu(\037)p Fz(\()p Fu(\032)p Fz(\))84 b(=)e(0)p Fu(;)h Fx(8)p Fu(\032)22 b Fx(\024)h Fz(0)p Fu(;)1424 2303 y Fz(~)-52 b Fu(\037)p Fz(\()p Fu(\032)p Fz(\))84 b(=)e(1)p Fu(;)h Fx(8)p Fu(\032)22 b Fx(\025)h Fz(1)p Fu(:)3134 2254 y Fz(\(2.4\))440 2476 y(Let)38 b(^)-52 b Fu(\037)641 2488 y Fs(j)699 2476 y Fx(2)23 b Fu(C)842 2446 y Ft(1)836 2497 y Fv(0)913 2476 y Fz(\([)p Fx(\000)1043 2443 y Fv(1)p 1043 2457 34 4 v 1043 2505 a(2)1086 2476 y Fu(;)1133 2443 y Fv(1)p 1133 2457 V 1133 2505 a(2)1176 2476 y Fz(])p Fu(;)14 b Fz([0)p Fu(;)g Fz(1]\))27 b(b)r(e)h(suc)n(h)f(that)h(:)926 2590 y Fp(8)926 2665 y(>)926 2690 y(<)926 2839 y(>)926 2864 y(:)1070 2654 y Fz(supp)c(^)-52 b Fu(\037)1307 2666 y Fs(j)1453 2654 y Fx(\032)1601 2587 y Fp(\002)1645 2617 y Fs(j)s Ft(\000)p Fv(3)p 1645 2635 116 4 v 1687 2682 a(4)1771 2654 y Fu(;)1818 2617 y Fs(j)s Ft(\000)p Fv(1)p 1818 2635 V 1859 2682 a(4)1944 2587 y Fp(\003)1992 2654 y Fu(;)83 b Fx(8)p Fu(j)27 b Fz(=)c(1)p Fu(;)14 b Fz(2)p Fu(;)g Fz(3)p Fu(;)1085 2729 y Fv(3)1041 2754 y Fp(X)1044 2931 y Fs(j)s Fv(=1)1185 2833 y Fz(^)-52 b Fu(\037)1227 2799 y Fv(2)1227 2854 y Fs(j)1264 2833 y Fz(\()p Fu(\022)r Fz(\))84 b(=)f(1)p Fu(;)432 b Fx(8)p Fu(\022)25 b Fx(2)2287 2766 y Fp(\002)2322 2833 y Fx(\000)2397 2801 y Fv(1)p 2396 2815 34 4 v 2396 2862 a(2)2439 2833 y Fu(;)2486 2801 y Fv(1)p 2486 2815 V 2486 2862 a(2)2529 2766 y Fp(\003)2578 2833 y Fu(:)3134 2785 y Fz(\(2.5\))440 3082 y(W)-7 b(e)28 b(de\014ne)g(a)f(cut-o\013)h(function)g(in)g(p)r(olar)e(co)r(ordinates) h(:)548 3292 y Fx(8)p Fz(\()p Fu(\032;)14 b(\022)r Fz(\))23 b Fx(2)g Fw(R)935 3258 y Fv(+)1015 3292 y Fx(\002)1098 3175 y Fp(\025)1142 3292 y Fx(\000)1217 3236 y Fz(1)p 1217 3273 42 4 v 1217 3349 a(2)1268 3292 y Fu(;)1315 3236 y Fz(1)p 1315 3273 V 1315 3349 a(2)1366 3175 y Fp(\024)1424 3292 y Fu(;)41 b(\037)1540 3252 y Fs(r)n(;\013;)p Fv(p)r(ol)1540 3315 y Fs(j)1746 3292 y Fz(\()p Fu(\032;)14 b(\022)r Fz(\))24 b(=)33 b(~)-52 b Fu(\037)2109 3200 y Fp(\020)2168 3236 y Fu(\032)p 2168 3273 43 4 v 2170 3349 a(r)2221 3200 y Fp(\021)2294 3292 y Fz(^)g Fu(\037)2336 3304 y Fs(j)2385 3175 y Fp(\022)2462 3236 y Fu(\022)p 2456 3273 54 4 v 2456 3349 a(\013)2520 3175 y Fp(\023)2595 3292 y Fu(;)41 b(j)28 b Fz(=)23 b(1)p Fu(;)14 b Fz(2)p Fu(;)g Fz(3)p Fu(;)102 b Fz(\(2.6\))440 3509 y(and)27 b(the)h(asso)r(ciated)f (functions)h(in)f(cartesian)g(co)r(ordinates)f Fu(\037)2439 3469 y Fs(r)n(;\013)2439 3532 y(j)2535 3509 y Fz(.)37 b(W)-7 b(e)28 b(notice)f(that)h(:)1393 3663 y Fv(3)1350 3688 y Fp(X)1352 3865 y Fs(j)s Fv(=1)1483 3696 y Fp(\014)1483 3746 y(\014)1511 3767 y Fu(\037)1563 3727 y Fs(r)n(;\013)1563 3790 y(j)1659 3696 y Fp(\014)1659 3746 y(\014)1687 3717 y Fv(2)1738 3767 y Fu(\036)23 b Fz(=)g Fu(\036)28 b Fz(on)p 2090 3700 60 4 v 27 w(\012)19 b Fx(\\)g Fd({)p Fu(S)2336 3779 y Fs(r)2372 3767 y Fu(:)739 b Fz(\(2.7\))440 4015 y(F)-7 b(urthermore,)27 b(it)h(is)f(easy)g(to)g(pro)n(v)n(e)f(the)i (relation)f(:)795 4255 y Fx(jjr)910 4267 y Ft(A)967 4255 y Fu(\036)p Fx(jj)1062 4220 y Fv(2)1062 4277 y Fs(L)1108 4261 y Fr(2)1141 4277 y Fv(\(\012\))1267 4255 y Fz(=)1398 4151 y Fv(3)1355 4176 y Fp(X)1358 4353 y Fs(j)s Fv(=1)1489 4255 y Fx(jjr)1604 4267 y Ft(A)1662 4255 y Fz(\()p Fu(\037)1746 4215 y Fs(r)n(;\013)1746 4278 y(j)1842 4255 y Fu(\036)p Fz(\))p Fx(jj)1969 4220 y Fv(2)1969 4277 y Fs(L)2015 4261 y Fr(2)2048 4277 y Fv(\(\012\))2170 4255 y Fx(\000)2296 4151 y Fv(3)2253 4176 y Fp(X)2255 4353 y Fs(j)s Fv(=1)2386 4255 y Fx(jj)p Fu(\036)p Fx(r)p Fu(\037)2602 4215 y Fs(r)n(;\013)2602 4278 y(j)2699 4255 y Fx(jj)2745 4220 y Fv(2)2745 4277 y Fs(L)2791 4261 y Fr(2)2823 4277 y Fv(\(\012\))2927 4255 y Fu(:)184 b Fz(\(2.8\))440 4503 y(By)27 b(construction,)g(there)h (exists)f(a)g(constan)n(t)g Fu(C)34 b Fz(indep)r(enden)n(t)28 b(of)g Fu(\013)g Fz(and)f Fu(r)k Fz(suc)n(h)c(that)h(:)816 4742 y Fx(jjr)931 4754 y Ft(A)989 4742 y Fu(\036)p Fx(jj)1084 4708 y Fv(2)1084 4765 y Fs(L)1130 4748 y Fr(2)1163 4765 y Fv(\(\012\))1349 4742 y Fx(\025)1540 4639 y Fv(3)1497 4664 y Fp(X)1499 4840 y Fs(j)s Fv(=1)1630 4742 y Fx(jjr)1745 4754 y Ft(A)1803 4742 y Fz(\()p Fu(\037)1887 4703 y Fs(r)n(;\013)1887 4766 y(j)1984 4742 y Fu(\036)p Fz(\))p Fx(jj)2111 4708 y Fv(2)2111 4765 y Fs(L)2157 4748 y Fr(2)2189 4765 y Fv(\(\012\))2311 4742 y Fx(\000)2455 4686 y Fu(C)p 2404 4723 168 4 v 2404 4799 a(\013)2457 4775 y Fv(2)2495 4799 y Fu(r)2534 4775 y Fv(2)2582 4742 y Fx(jj)p Fu(\036)p Fx(jj)2723 4708 y Fv(2)2723 4765 y Fs(L)2769 4748 y Fr(2)2802 4765 y Fv(\(\012\))2905 4742 y Fu(:)206 b Fz(\(2.9\))440 4991 y(It)33 b(is)g(w)n(ell)f(kno)n(wn)g(that)h(the)g(b)r(ottom)g(of)g (the)g(sp)r(ectrum)g(of)f Fu(P)2464 5003 y Ft(A)2517 5011 y Fr(0)2550 5003 y Fs(;)p Fv(\012)2654 4991 y Fz(is)h(in)n(v)-5 b(arian)n(t)31 b(under)440 5091 y(rotation)24 b(or)h(translation)f(of)h (the)h(domain)f(\012.)36 b(So,)26 b(using)f(the)g(fact)h(that)f(\002) 2815 5103 y Fv(0)2875 5091 y Fx(\024)e Fz(1)i(and)g(the)440 5190 y(de\014nition)j(of)g(\006\()p Fx(\000)p Fz(\001)1130 5202 y Ft(A)1183 5210 y Fr(0)1219 5190 y Fu(;)14 b(r)r Fz(\),)29 b(w)n(e)e(deduce)h(:)1314 5398 y(\006\()p Fx(\000)p Fz(\001)1540 5410 y Ft(A)1593 5418 y Fr(0)1630 5398 y Fu(;)14 b(r)r Fz(\))24 b Fx(\025)f Fz(\002)1915 5410 y Fv(0)1970 5398 y Fx(\000)2114 5341 y Fu(C)p 2063 5379 V 2063 5455 a(\013)2116 5431 y Fv(2)2154 5455 y Fu(r)2193 5431 y Fv(2)2241 5398 y Fu(:)828 b Fz(\(2.10\))1851 5662 y(5)p eop %%Page: 6 6 6 5 bop 440 531 a Fz(This)28 b(implies,)f(taking)g(the)h(limit)h Fu(r)d Fx(!)d Fz(+)p Fx(1)k Fz(and)g(using)h(\(2.2\))f(:)1486 712 y(inf)21 b Fu(\033)1648 724 y Fv(ess)1735 712 y Fz(\()p Fx(\000)p Fz(\001)1901 724 y Ft(A)1954 732 y Fr(0)1990 712 y Fz(\))j Fx(\025)e Fz(\002)2198 724 y Fv(0)2235 712 y Fu(:)565 892 y Fz(No)n(w,)28 b(w)n(e)g(establish)g(the)h(upp)r (er)g(b)r(ound.)40 b(Let)29 b Fu(\017)24 b(>)g Fz(0)k(and)h Fu( )2502 904 y Fv(1)2564 892 y Fx(2)c Fu(C)2709 862 y Ft(1)2703 913 y Fv(0)2779 892 y Fz(\()p 2811 822 277 4 v Fw(R)g Fx(\002)18 b Fw(R)3027 868 y Fv(+)3088 892 y Fz(\))29 b(b)r(e)g(a)440 992 y(function)f(suc)n(h)f(that)h(:)1257 1223 y(\002)1322 1235 y Fv(0)1382 1223 y Fx(\024)1480 1149 y(jjr)1595 1161 y Ft(A)1652 1149 y Fu( )1706 1161 y Fv(1)1744 1149 y Fx(jj)1790 1119 y Fv(2)1790 1176 y Fs(L)1836 1159 y Fr(2)1868 1176 y Fv(\()p Fn(R)p Ft(\002)p Fn(R)2036 1159 y Fr(+)2076 1176 y Fv(\))p 1480 1204 627 4 v 1543 1280 a Fx(jj)p Fu( )1643 1292 y Fv(1)1680 1280 y Fx(jj)1726 1251 y Fv(2)1726 1309 y Fs(L)1772 1292 y Fr(2)1804 1309 y Fv(\()p Fn(R)p Ft(\002)p Fn(R)1972 1292 y Fr(+)2013 1309 y Fv(\))2139 1223 y Fx(\024)23 b Fz(\002)2292 1235 y Fv(0)2347 1223 y Fz(+)18 b Fu(\017:)605 b Fz(\(2.11\))440 1481 y(By)34 b(translation)f(and)h(rotation)f(from)h Fu( )1754 1493 y Fv(1)1791 1481 y Fz(,)i(w)n(e)e(de\014ne)h(a)e (function)i Fu( )i Fx(2)e Fu(C)2879 1451 y Ft(1)2873 1502 y Fv(0)2949 1481 y Fz(\(\012)24 b Fx(\\)f Fd({)p Fu(S)3236 1493 y Fs(r)3272 1481 y Fz(\))440 1581 y(suc)n(h)k(that)h(:) 1253 1657 y Fx(jjr)1368 1669 y Ft(A)1426 1657 y Fu( )1480 1669 y Fv(1)1517 1657 y Fx(jj)1563 1626 y Fv(2)1563 1684 y Fs(L)1609 1667 y Fr(2)1641 1684 y Fv(\()p Fn(R)p Ft(\002)p Fn(R)1809 1667 y Fr(+)1850 1684 y Fv(\))p 1253 1712 V 1317 1788 a Fx(jj)p Fu( )1417 1800 y Fv(1)1454 1788 y Fx(jj)1500 1759 y Fv(2)1500 1817 y Fs(L)1546 1800 y Fr(2)1578 1817 y Fv(\()p Fn(R)p Ft(\002)p Fn(R)1746 1800 y Fr(+)1787 1817 y Fv(\))1913 1731 y Fz(=)2011 1657 y Fx(jjr)2126 1669 y Ft(A)2183 1657 y Fu( )s Fx(jj)2286 1627 y Fv(2)2286 1684 y Fs(L)2332 1667 y Fr(2)2365 1684 y Fv(\(\012\))p 2011 1712 458 4 v 2074 1788 a Fx(jj)p Fu( )s Fx(jj)2223 1759 y Fv(2)2223 1816 y Fs(L)2269 1800 y Fr(2)2301 1816 y Fv(\(\012\))2478 1731 y Fu(:)591 b Fz(\(2.12\))440 1946 y(It)34 b(is)f(enough)g(to)g(tak)n(e)g(the)h(limit)g(as)f Fu(r)j Fz(tends)e(to)f(in\014nit)n(y)h(and)f Fu(\017)g Fz(to)h(0)f(to)g(ac)n(hiev)n(e)f(the)440 2045 y(pro)r(of.)2587 b Fd(\003)565 2145 y Fz(Using)18 b(this)g(prop)r(osition,)i(as)d(so)r (on)h(w)n(e)g(\014nd)h(a)f(function)h(in)f Fu(H)2478 2115 y Fv(1)2471 2168 y Ft(A)2524 2176 y Fr(0)2561 2145 y Fz(\(\012)2653 2157 y Fs(\013)2701 2145 y Fz(\))h(with)f(a)g(Ra)n (yleigh)440 2245 y(quotien)n(t)k(strictly)f(less)h(than)g(\002)1441 2257 y Fv(0)1478 2245 y Fz(,)h(then)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\))h(is)f(an)g(eigen)n(v)-5 b(alue.)34 b(F)-7 b(or)21 b(example,)i(Jadallah)440 2344 y([13)o(,)i(14)o(])h(constructs) e(a)h(test-function)h(for)f(the)g(quarter)f(plane)h(and)g(w)n(e)g(can)g (use)g(it)h(to)f(see)440 2444 y(that)j Fu(\026)p Fz(\()712 2411 y Fs(\031)p 712 2425 41 4 v 716 2473 a Fv(2)763 2444 y Fz(\))g(is)f(an)h(eigen)n(v)-5 b(alue.)440 2608 y FA(Remark)27 b(2.4.)36 b Fo(It)26 b(would)h(b)l(e)g(inter)l(esting)f (to)g(show)h(that)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\))i Fo(is)e(strictly)h(b)l(ounde)l(d)g(fr)l(om)440 2708 y(ab)l(ove)k(by)f Fz(\002)838 2720 y Fv(0)905 2708 y Fo(for)g Fu(\013)24 b Fx(2)p Fz(]0)p Fu(;)14 b(\031)s Fz([)30 b Fo(and)g(e)l(qual)g(to)g Fz(\002)1910 2720 y Fv(0)1976 2708 y Fo(for)h Fu(\013)23 b Fx(2)h Fz([)p Fu(\031)s(;)14 b Fz(2)p Fu(\031)s Fz([)p Fo(.)440 2940 y Fl(2.3)112 b(First)37 b(upp)s(er)h(b)s(ound)440 3093 y Fz(The)24 b(eigenfunction)f Fu(\036)h Fz(in)n(tro)r(duced)g(in)g (Prop)r(osition)d(2.1)i(will)h(b)r(e)g(used)f(for)h(the)f(construc-)440 3193 y(tion)28 b(of)g(a)f(test)i(function)f(giving)f(an)h(upp)r(er)g(b) r(ound)g(for)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\).)39 b(This)28 b(\014rst)g(upp)r(er)g(b)r(ound)440 3292 y(follo)n(ws)34 b(the)i(idea)f(of)g(P)n(an)f([17)o(])i(who)f(constructs)f(a)h(quasi-mo) r(de)f(in)i(order)e(to)h(reco)n(v)n(er)440 3392 y(Jadallah's)26 b(result)h(\(cf)i([13)o(,)e(14)o(]\))h(and)g(sho)n(w)f(that)g(:)1638 3597 y Fu(\026)1702 3505 y Fp(\020)1761 3541 y Fu(\031)p 1761 3578 51 4 v 1765 3654 a Fz(2)1821 3505 y Fp(\021)1894 3597 y Fu(<)c Fz(\002)2047 3609 y Fv(0)2083 3597 y Fu(:)440 3801 y Fz(W)-7 b(e)27 b(use)f(the)h(same)e(notations)h(as)g(in)g(Prop)r (osition)f(2.1.)36 b(Adapting)26 b(the)h(idea)f(of)g(P)n(an,)g(w)n(e) 440 3901 y(construct)h(a)g(trial)g(function,)i Fu(u)p Fz(,)e(in)h(p)r(olar)e(co)r(ordinates)h(:)707 4102 y Fu(u)p Fz(\()p Fu(\032;)14 b(\022)r Fz(\))23 b(=)f Fu(e)1089 4067 y Fs(i\015)t(\032)12 b Fv(sin)f Fs(\022)1355 4102 y Fu(\036)1418 4009 y Fp(\020)1468 4102 y Fu(\034)e(\032)14 b Fz(sin)1686 4009 y Fp(\020)1746 4045 y Fu(\013)p 1746 4083 54 4 v 1752 4159 a Fz(2)1828 4102 y(+)k Fu(\022)1952 4009 y Fp(\021\021)2093 4102 y Fu(\036)2156 4009 y Fp(\020)2206 4102 y Fu(\034)9 b(\032)14 b Fz(sin)2424 4009 y Fp(\020)2483 4045 y Fu(\013)p 2483 4083 V 2489 4159 a Fz(2)2565 4102 y Fx(\000)k Fu(\022)2689 4009 y Fp(\021\021)2802 4102 y Fu(;)267 b Fz(\(2.13\))440 4305 y(with)28 b(:)1212 4532 y Fu(\015)g Fz(=)1483 4475 y Fu(\020)1519 4487 y Fv(0)p 1380 4513 280 4 v 1380 4589 a Fz(2)p Fu(\034)23 b Fz(sin)1607 4556 y Fs(\013)p 1607 4570 44 4 v 1612 4617 a Fv(2)1781 4532 y Fz(and)110 b Fu(\034)33 b Fz(=)2182 4401 y Fp(r)p 2265 4401 245 4 v 2366 4475 a Fz(1)p 2275 4513 225 4 v 2275 4589 a(2)14 b(sin)f Fu(\013)2509 4532 y(:)560 b Fz(\(2.14\))440 4764 y(W)-7 b(e)28 b(obtain)f(the)h(follo)n (wing)f(result)g(:)440 4928 y FA(Prop)s(osition)j(2.5.)40 b Fo(F)-6 b(or)30 b(every)h Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)1764 4896 y Fs(\031)p 1764 4910 41 4 v 1768 4957 a Fv(2)1815 4928 y Fz(])30 b Fo(:)1357 5159 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\024)1680 5103 y Fz(\002)1745 5115 y Fv(0)p 1646 5140 170 4 v 1646 5216 a Fz(sin)14 b Fu(\013)1844 5159 y Fx(\000)1960 5103 y Fz(cos)f Fu(\013)p 1937 5140 225 4 v 1937 5216 a Fz(4)h(sin)f Fu(\013)2171 5159 y(\036)p Fz(\(0\))2326 5125 y Fv(4)2364 5159 y Fu(:)705 b Fz(\(2.15\))440 5401 y Fo(F)-6 b(urthermor)l(e,)30 b(for)h(every)f Fu(\013)24 b Fx(2)1452 5308 y Fp(i)1501 5368 y Fs(\031)p 1501 5382 41 4 v 1505 5429 a Fv(2)1571 5401 y Fx(\000)18 b Fz(2)c(arctan)1954 5308 y Fp(\020)2013 5360 y Fs(\036)p Fv(\(0\))2138 5335 y Fr(4)p 2013 5381 158 4 v 2034 5429 a Fv(4\002)2118 5437 y Fr(0)2180 5308 y Fp(\021)2244 5401 y Fu(;)2291 5368 y Fs(\031)p 2291 5382 41 4 v 2295 5429 a Fv(2)2342 5308 y Fp(h)2381 5401 y Fo(,)30 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))h Fo(is)f(an)g (eigenvalue.)1851 5662 y Fz(6)p eop %%Page: 7 7 7 6 bop 440 531 a Fo(Pr)l(o)l(of)31 b(:)39 b Fz(A)27 b(c)n(hange)g(of)g(co)r(ordinates)g(sends)g(the)h(sector)e(on)n(to)h(a) g(quarter)g(plane)g Fu(Q)g Fz(:)1142 640 y Fp(\022)1250 706 y Fz(~)-47 b Fu(x)1252 806 y Fz(~)g Fu(y)1334 640 y Fp(\023)1418 757 y Fz(=)23 b Fu(\034)1565 640 y Fp(\022)1668 706 y Fz(sin)1793 674 y Fs(\013)p 1793 688 44 4 v 1798 735 a Fv(2)1969 706 y Fz(cos)2104 674 y Fs(\013)p 2104 688 V 2109 735 a Fv(2)1668 806 y Fz(sin)1793 773 y Fs(\013)p 1793 787 V 1798 834 a Fv(2)1930 806 y Fx(\000)14 b Fz(cos)2143 773 y Fs(\013)p 2143 787 V 2148 834 a Fv(2)2238 640 y Fp(\023)g(\022)2415 706 y Fu(x)2417 806 y(y)2504 640 y Fp(\023)2579 757 y Fu(:)490 b Fz(\(2.16\))440 982 y(The)28 b(expression)e(of)h Fu(u)h Fz(in)f(the)h(new)g(co)r(ordinates)e(is)i(:) 621 1208 y(~)-47 b Fu(u)o Fz(\()5 b(~)-47 b Fu(x)q(;)20 b Fz(~)-48 b Fu(y)s Fz(\))23 b(=)g(exp)1107 1091 y Fp(\022)1168 1208 y Fu(i)1328 1152 y(\015)p 1207 1189 290 4 v 1207 1265 a Fz(2)p Fu(\034)g Fz(cos)1443 1232 y Fs(\013)p 1443 1246 44 4 v 1448 1294 a Fv(2)1506 1208 y Fz(\()5 b(~)-47 b Fu(x)19 b Fx(\000)24 b Fz(~)-48 b Fu(y)s Fz(\))1763 1091 y Fp(\023)1838 1208 y Fu(\036)p Fz(\()5 b(~)-47 b Fu(x)q Fz(\))p Fu(\036)p Fz(\()6 b(~)-48 b Fu(y)t Fz(\))23 b(=)g Fu(e)2307 1174 y Fs(i\020)2360 1182 y Fr(0)2393 1174 y Fv(\()t(~)-37 b Fs(x)o Ft(\000)5 b Fv(~)-38 b Fs(y)r Fv(\))2574 1208 y Fu(\036)p Fz(\()5 b(~)-47 b Fu(x)q Fz(\))p Fu(\036)p Fz(\()6 b(~)-48 b Fu(y)t Fz(\))p Fu(:)176 b Fz(\(2.17\))440 1450 y(After)28 b(computing)g Fx(j)p Fz(\()p Fx(r)19 b(\000)f Fu(i)p Fx(A)1389 1462 y Fv(0)1426 1450 y Fz(\))p Fu(u)p Fx(j)1529 1420 y Fv(2)1593 1450 y Fz(with)29 b(the)f(c)n(hoice)e(\(2.14\))h(of)h Fu(\015)k Fz(and)27 b Fu(\034)9 b Fz(,)29 b(w)n(e)e(obtain)g(:)535 1684 y Fu(q)572 1696 y Ft(A)625 1704 y Fr(0)658 1696 y Fs(;)p Fv(\012)725 1704 y Fe(\013)772 1684 y Fz(\()p Fu(u)p Fz(\))83 b(=)1197 1627 y Fu(\034)1242 1597 y Fv(2)p 1124 1665 229 4 v 1124 1741 a Fu(\034)24 b Fz(sin)14 b Fu(\013)1377 1567 y Fp(\022)1438 1571 y(Z)1521 1591 y Ft(1)1484 1759 y Fv(0)1591 1684 y Fz(\(\()5 b(~)-47 b Fu(x)20 b Fx(\000)e Fu(\020)1841 1696 y Fv(0)1878 1684 y Fz(\))1910 1649 y Fv(2)1948 1684 y Fu(\036)p Fz(\()5 b(~)-47 b Fu(x)q Fz(\))2109 1649 y Fv(2)2165 1684 y Fz(+)18 b Fu(\036)2297 1649 y Ft(0)2321 1684 y Fz(\()5 b(~)-47 b Fu(x)q Fz(\))2433 1649 y Fv(2)2470 1684 y Fz(\))28 b Fu(d)5 b Fz(~)-47 b Fu(x)2662 1571 y Fp(Z)2745 1591 y Ft(1)2708 1759 y Fv(0)2829 1684 y Fx(j)p Fu(\036)p Fz(\()6 b(~)-48 b Fu(y)t Fz(\))p Fx(j)3033 1649 y Fv(2)3098 1684 y Fu(dy)1124 1916 y Fz(+)1203 1803 y Fp(Z)1286 1824 y Ft(1)1249 1992 y Fv(0)1356 1916 y Fz(\(\()6 b(~)-48 b Fu(y)22 b Fx(\000)c Fu(\020)1602 1928 y Fv(0)1639 1916 y Fz(\))1671 1882 y Fv(2)1709 1916 y Fu(\036)p Fz(\()6 b(~)-48 b Fu(y)s Fz(\))1866 1882 y Fv(2)1922 1916 y Fz(+)18 b Fu(\036)2054 1882 y Ft(0)2078 1916 y Fz(\()6 b(~)-48 b Fu(y)s Fz(\))2186 1882 y Fv(2)2224 1916 y Fz(\))27 b Fu(d)6 b Fz(~)-48 b Fu(y)2412 1803 y Fp(Z)2495 1824 y Ft(1)2458 1992 y Fv(0)2579 1916 y Fx(j)p Fu(\036)p Fz(\()5 b(~)-47 b Fu(x)q Fz(\))p Fx(j)2786 1882 y Fv(2)2852 1916 y Fu(dx)2942 1799 y Fp(\023)1114 2152 y Fz(+)1189 2096 y(2)p Fu(\034)1276 2066 y Fv(2)1327 2096 y Fz(cos)13 b Fu(\013)p 1189 2133 317 4 v 1233 2209 a(\034)23 b Fz(sin)14 b Fu(\013)1529 2035 y Fp(\022)1590 2039 y(Z)1673 2059 y Ft(1)1637 2228 y Fv(0)1744 2152 y Fz(\()5 b(~)-47 b Fu(x)19 b Fx(\000)f Fu(\020)1961 2164 y Fv(0)1999 2152 y Fz(\))p Fu(\036)p Fz(\()5 b(~)-47 b Fu(x)q Fz(\))2192 2118 y Fv(2)2257 2152 y Fu(d)5 b Fz(~)-47 b Fu(x)2389 2039 y Fp(Z)2472 2059 y Ft(1)2436 2228 y Fv(0)2543 2152 y Fz(\()6 b(~)-48 b Fu(y)21 b Fx(\000)d Fu(\020)2756 2164 y Fv(0)2794 2152 y Fz(\))p Fu(\036)p Fz(\()6 b(~)-48 b Fu(y)t Fz(\))2984 2118 y Fv(2)3049 2152 y Fu(d)6 b Fz(~)-48 b Fu(y)1124 2384 y Fx(\000)1199 2328 y Fz(1)p 1199 2365 42 4 v 1199 2441 a(4)1264 2271 y Fp(Z)1347 2292 y Ft(1)1310 2460 y Fv(0)1417 2384 y Fz(\()p Fu(\036)p Fz(\()5 b(~)-47 b Fu(x)r Fz(\))1611 2350 y Fv(2)1648 2384 y Fz(\))1680 2350 y Ft(0)1732 2384 y Fu(d)5 b Fz(~)-47 b Fu(x)1864 2271 y Fp(Z)1947 2292 y Ft(1)1910 2460 y Fv(0)2017 2384 y Fz(\()p Fu(\036)p Fz(\()6 b(~)-48 b Fu(y)t Fz(\))2207 2350 y Fv(2)2245 2384 y Fz(\))2277 2350 y Ft(0)2328 2384 y Fu(d)6 b Fz(~)-48 b Fu(y)2442 2267 y Fp(\023)2517 2384 y Fu(:)440 2610 y Fz(W)-7 b(e)23 b(use)f(prop)r(erties)g(of)h Fu(\036)g Fz(recalled)f(in)g(Prop)r (osition)f(2.1)h(and)h(deduce)f(from)h(the)g(min-max)440 2710 y(principle)28 b(:)1110 2852 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))c Fx(\024)1407 2796 y Fu(q)1444 2808 y Ft(A)1497 2816 y Fr(0)1529 2808 y Fs(;)p Fv(\012)1596 2816 y Fe(\013)1643 2796 y Fz(\()p Fu(u)p Fz(\))p 1399 2833 364 4 v 1399 2909 a Fx(jj)p Fu(u)p Fx(jj)1539 2881 y Fv(2)1539 2938 y Fs(L)1585 2921 y Fr(2)1617 2938 y Fv(\(\012)1690 2946 y Fe(\013)1732 2938 y Fv(\))1795 2852 y Fz(=)1927 2796 y(\002)1992 2808 y Fv(0)p 1893 2833 170 4 v 1893 2909 a Fz(sin)14 b Fu(\013)2091 2852 y Fx(\000)2207 2796 y Fz(cos)f Fu(\013)p 2184 2833 225 4 v 2184 2909 a Fz(4)h(sin)f Fu(\013)2418 2852 y(\036)p Fz(\(0\))2573 2818 y Fv(4)2611 2852 y Fu(:)458 b Fz(\(2.18\))440 3071 y(According)27 b(to)h(Prop)r(osition)e(2.3,)i Fu(\026)p Fz(\()p Fu(\013)p Fz(\))h(is)f(an)g(eigen)n(v)-5 b(alue)27 b(as)g(so)r(on)g(as)h Fu(\026)p Fz(\()p Fu(\013)p Fz(\))d Fu(<)e Fz(\002)3077 3083 y Fv(0)3114 3071 y Fz(,)28 b(it)g(is)440 3171 y(enough)f(to)g(solv)n(e)g(:)934 3344 y(\002)999 3356 y Fv(0)p 901 3381 170 4 v 901 3457 a Fz(sin)13 b Fu(\013)1098 3400 y Fx(\000)1214 3344 y Fz(cos)g Fu(\013)p 1191 3381 225 4 v 1191 3457 a Fz(4)h(sin)f Fu(\013)1426 3400 y(\036)p Fz(\(0\))1581 3365 y Fv(4)1641 3400 y Fu(<)23 b Fz(\002)1794 3412 y Fv(0)1854 3400 y Fx(\()-14 b(\))23 b Fz(tan)2163 3308 y Fp(\020)2223 3344 y Fu(\031)p 2223 3381 51 4 v 2227 3457 a Fz(4)2301 3400 y Fx(\000)2394 3344 y Fu(\013)p 2394 3381 54 4 v 2400 3457 a Fz(2)2458 3308 y Fp(\021)2530 3400 y Fu(<)2628 3344 y(\036)p Fz(\(0\))2783 3313 y Fv(4)p 2628 3381 193 4 v 2653 3457 a Fz(4\002)2760 3469 y Fv(0)2831 3400 y Fu(:)3240 3614 y Fd(\003)565 3714 y Fz(W)-7 b(e)25 b(kno)n(w)f(the)i(estimates)f(of)g(\002)1580 3726 y Fv(0)1642 3714 y Fz(and)g Fu(\036)p Fz(\(0\))h(according)d(to)i (Sain)n(t)g(James-De)f(Gennes)440 3813 y([20)o(])k(and)f(computations)g (whose)g(details)h(are)e(giv)n(en)h(in)h([6)o(])g(:)1441 3994 y(\002)1506 4006 y Fv(0)1566 3994 y Fx(')23 b Fz(0)p Fu(:)p Fz(59)p Fu(;)40 b(\036)p Fz(\(0\))24 b Fx(')e Fz(0)p Fu(:)p Fz(87)p Fu(:)788 b Fz(\(2.19\))440 4174 y FA(Remark)31 b(2.6.)40 b Fo(Appr)l(oximately,)31 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))g Fo(is)f(an)g(eigenvalue)h(for)g Fu(\013)23 b Fx(2)h Fz([1)p Fu(:)p Fz(09)p Fu(;)2881 4141 y Fs(\031)p 2881 4155 41 4 v 2885 4203 a Fv(2)2932 4174 y Fz(])p Fo(.)440 4448 y Fy(3)135 b(Reduction)45 b(to)h(a)f(domain)g(indep)t(enden)l(t)g(of)g Fc(\013)440 4630 y Fz(W)-7 b(e)29 b(are)f(in)n(terested)g(in)i(the)f(v)-5 b(ariation)27 b(with)j(resp)r(ect)e(to)h Fu(\013)g Fz(of)g Fu(\026)p Fz(\()p Fu(\013)p Fz(\).)42 b(The)29 b(shap)r(e)f(of)h(the) 440 4730 y(domain)j(\012)800 4742 y Fs(\013)880 4730 y Fz(suggests)g(to)g(use)h(p)r(olar)f(co)r(ordinates)f(\()p Fu(\032;)14 b(\036)p Fz(\))34 b(and)e(so)g(w)n(e)h(denote)f(b)n(y)h (\012)3212 4700 y Fs(pol)3212 4750 y(\013)440 4829 y Fz(the)28 b(domain)f(:)1365 4929 y(\012)1425 4895 y Fs(pol)1425 4950 y(\013)1527 4929 y Fz(:)51 b(=]0)p Fu(;)14 b Fz(+)p Fx(1)p Fz([)p Fx(\002)2018 4837 y Fp(i)2055 4929 y Fx(\000)2130 4873 y Fu(\013)p 2130 4910 54 4 v 2136 4986 a Fz(2)2193 4929 y Fu(;)2240 4873 y(\013)p 2240 4910 V 2246 4986 a Fz(2)2303 4837 y Fp(h)2356 4929 y Fu(:)440 5100 y Fz(This)28 b(c)n(hange)e(of)i(v)-5 b(ariables)26 b(leads)h(to)g(study)h(the)g(new) g(quadratic)e(form)i(:)887 5351 y(^)-48 b Fu(q)s Fz(\()p Fu(u)p Fz(\))23 b(=)1144 5238 y Fp(Z)1190 5426 y Fv(\012)1237 5398 y Fe(pol)1237 5436 y(\013)1339 5209 y Fp( )1405 5351 y Fx(j)p Fu(@)1472 5363 y Fs(\032)1511 5351 y Fu(u)p Fx(j)1582 5316 y Fv(2)1637 5351 y Fz(+)1749 5295 y(1)p 1730 5332 81 4 v 1730 5408 a Fu(\032)1773 5384 y Fv(2)1834 5230 y Fp(\014)1834 5280 y(\014)1834 5330 y(\014)1834 5380 y(\014)1862 5234 y(\022)1923 5351 y Fu(@)1967 5363 y Fs(\036)2029 5351 y Fz(+)18 b Fu(i)2179 5295 y(\032)2222 5264 y Fv(2)p 2179 5332 V 2198 5408 a Fz(2)2269 5234 y Fp(\023)2344 5351 y Fu(u)2392 5230 y Fp(\014)2392 5280 y(\014)2392 5330 y(\014)2392 5380 y(\014)2419 5248 y Fv(2)2456 5209 y Fp(!)2563 5351 y Fu(\032)28 b(d\032)g(d\036:)271 b Fz(\(3.1\))1851 5662 y(7)p eop %%Page: 8 8 8 7 bop 440 531 a Fz(T)-7 b(o)27 b(ha)n(v)n(e)g(a)g Fu(\032)p Fz(-indep)r(enden)n(t)h(Neumann)g(condition,)f(w)n(e)g(mak)n(e)g(a)g (gauge)g(transform)f(:)1011 748 y Fu(u)1059 760 y Fv(1)1096 748 y Fz(\()p Fu(\032;)14 b(\036)p Fz(\))9 b(:)52 b(=)22 b(exp)1601 631 y Fp(\022)1662 748 y Fu(i)1701 692 y(\032)1744 662 y Fv(2)p 1701 729 81 4 v 1720 805 a Fz(2)1791 748 y Fu(\036)1840 631 y Fp(\023)1915 748 y Fu(u)p Fz(\()p Fu(\032;)14 b(\036)p Fz(\))p Fu(;)37 b Fx(8)p Fz(\()p Fu(\032;)14 b(\036)p Fz(\))23 b Fx(2)h Fz(\012)2618 714 y Fs(pol)2618 768 y(\013)2710 748 y Fu(:)440 961 y Fz(So)j(w)n(e)h (deduce)f(a)g(new)h(quadratic)f(form)g(:)819 1174 y(~)-48 b Fu(q)s Fz(\()p Fu(u)933 1186 y Fv(1)970 1174 y Fz(\))9 b(:)51 b(=)1173 1061 y Fp(Z)1219 1250 y Fv(\012)1266 1222 y Fe(pol)1266 1259 y(\013)1368 1057 y Fp(\022)1430 1174 y Fx(j)p Fu(@)1497 1186 y Fs(\032)1535 1174 y Fu(u)1583 1186 y Fv(1)1639 1174 y Fx(\000)18 b Fu(i)27 b(\032\036)h(u)1946 1186 y Fv(1)1983 1174 y Fx(j)2006 1133 y Fv(2)2061 1174 y Fz(+)2174 1118 y(1)p 2154 1155 V 2154 1231 a Fu(\032)2197 1207 y Fv(2)2272 1174 y Fx(j)p Fu(@)2339 1186 y Fs(\036)2383 1174 y Fu(u)2431 1186 y Fv(1)2468 1174 y Fx(j)2491 1140 y Fv(2)2529 1057 y Fp(\023)2631 1174 y Fu(\032)g(d\032)g(d\036:)440 1394 y Fz(W)-7 b(e)34 b(mak)n(e)f(a)h(last)f(c)n(hange)g(of)g(v)-5 b(ariables)33 b(and)h(another)e(gauge)h(transform)g(to)g(obtain)h(a)440 1493 y(quadratic)k(form)h(dep)r(ending)h(on)f Fu(\013)h Fz(but)g(de\014ned)g(on)f(a)g(domain)g(indep)r(enden)n(t)h(of)g Fu(\013)p Fz(,)440 1593 y(whereas)31 b(at)g(the)h(b)r(eginning,)h(w)n (e)e(had)h(a)f(constan)n(t)g(op)r(erator)f(on)i(an)f Fu(\013)p Fz(-dep)r(enden)n(t)i(do-)440 1692 y(main.)k(W)-7 b(e)28 b(use)f(the)h(c)n(hange)f(of)g(v)-5 b(ariables)27 b(\()p Fu(\032;)14 b(\036)p Fz(\))23 b Fx(2)h Fz(\012)2178 1662 y Fs(pol)2178 1713 y(\013)2293 1692 y Fx(!)f Fz(\()p Fu(t;)14 b(\021)s Fz(\))24 b Fx(2)g Fz(\012)2737 1662 y Fv(0)2801 1692 y Fz(with)29 b(:)1080 1914 y(\()p Fu(t;)14 b(\021)s Fz(\))24 b(=)1366 1797 y Fp(\022)1428 1914 y Fu(\013)1491 1858 y(\032)1534 1827 y Fv(2)p 1491 1895 V 1510 1971 a Fz(2)1581 1914 y Fu(;)1630 1858 y(\036)p 1628 1895 54 4 v 1628 1971 a(\013)1691 1797 y Fp(\023)1766 1914 y Fu(;)42 b Fz(\012)1891 1879 y Fv(0)1937 1914 y Fz(:)51 b(=)23 b Fw(R)2153 1879 y Fv(+)2232 1914 y Fx(\002)2315 1797 y Fp(\025)2359 1914 y Fx(\000)2434 1858 y Fz(1)p 2434 1895 42 4 v 2434 1971 a(2)2485 1914 y Fu(;)2532 1858 y Fz(1)p 2532 1895 V 2532 1971 a(2)2583 1797 y Fp(\024)2641 1914 y Fu(:)440 2127 y Fz(Therefore,)j(w)n(e)i(ha)n(v)n(e)e(to)i(study) f(a)h(family)f(of)h(quadratic)e(forms)h Fu(q)2538 2139 y Fs(\013)2613 2127 y Fz(de\014ned)h(b)n(y)g(:)889 2340 y Fu(q)926 2352 y Fs(\013)973 2340 y Fz(\()p Fu(u)p Fz(\))84 b(=)1316 2227 y Fp(Z)1362 2416 y Fv(\012)1409 2399 y Fr(0)1460 2223 y Fp(\022)1521 2340 y Fz(2)p Fu(t)p Fx(j)p Fu(D)1685 2352 y Fs(t)1714 2340 y Fu(u)18 b Fx(\000)g Fu(\021)31 b(u)p Fx(j)2006 2306 y Fv(2)2061 2340 y Fz(+)2214 2284 y(1)p 2154 2321 163 4 v 2154 2397 a(2)p Fu(\013)2249 2373 y Fv(2)2286 2397 y Fu(t)2326 2340 y Fx(j)p Fu(@)2393 2352 y Fs(\021)2433 2340 y Fu(u)p Fx(j)2504 2306 y Fv(2)2541 2223 y Fp(\023)2644 2340 y Fu(dt)d(d\021)s(:)279 b Fz(\(3.2\))440 2566 y(The)28 b(form)f(domain)g(of)h Fu(q)1234 2578 y Fs(\013)1309 2566 y Fz(is)f Fx(V)1450 2535 y Fs(N)1541 2566 y Fz(with)h(:)695 2779 y Fx(V)753 2745 y Fs(N)825 2779 y Fz(:)51 b(=)987 2662 y Fp(\032)1049 2779 y Fu(u)23 b Fx(2)g Fu(L)1255 2745 y Fv(2)1292 2779 y Fz(\(\012)1384 2745 y Fv(0)1421 2779 y Fz(\))1453 2658 y Fp(\014)1453 2708 y(\014)1453 2758 y(\014)1453 2808 y(\014)1548 2723 y Fz(1)p 1519 2760 100 4 v 1519 2776 a Fx(p)p 1588 2776 30 4 v 68 x Fu(t)1628 2779 y(@)1672 2791 y Fs(\021)1712 2779 y Fu(u)g Fx(2)g Fu(L)1918 2745 y Fv(2)1955 2779 y Fz(\(\012)2047 2745 y Fv(0)2085 2779 y Fz(\))p Fu(;)2154 2707 y Fx(p)p 2223 2707 V 72 x Fu(t)p Fz(\()p Fu(@)2329 2791 y Fs(t)2377 2779 y Fx(\000)18 b Fu(i)27 b(\021)s Fz(\))d Fx(2)f Fu(L)2751 2745 y Fv(2)2788 2779 y Fz(\(\012)2880 2745 y Fv(0)2918 2779 y Fz(\))2950 2662 y Fp(\033)3026 2779 y Fu(:)440 3004 y Fz(W)-7 b(e)28 b(de\014ne)g(the)g(sesquilinear)e (form)h Fu(a)1652 3016 y Fs(\013)1727 3004 y Fz(asso)r(ciated)f(to)i Fu(q)2261 3016 y Fs(\013)2336 3004 y Fz(on)f Fx(V)2509 2974 y Fs(N)2600 3004 y Fz(:)635 3218 y Fu(a)679 3230 y Fs(\013)726 3218 y Fz(\()p Fu(u;)14 b(v)s Fz(\))23 b(=)1029 3104 y Fp(Z)1075 3293 y Fv(\012)1122 3277 y Fr(0)1173 3100 y Fp(\022)1234 3218 y Fz(2)p Fu(t)14 b Fz(\()o Fu(D)1420 3230 y Fs(t)1468 3218 y Fx(\000)k Fu(\021)s Fz(\))c Fu(u)p 1716 3145 366 4 v 27 w Fz(\()q Fu(D)1818 3230 y Fs(t)1865 3218 y Fx(\000)k Fu(\021)s Fz(\))c Fu(v)22 b Fz(+)2253 3161 y(1)p 2193 3198 163 4 v 2193 3274 a(2)p Fu(\013)2288 3250 y Fv(2)2325 3274 y Fu(t)2365 3218 y(@)2409 3230 y Fs(\021)2450 3218 y Fu(u)p 2498 3150 128 4 v(@)2542 3230 y Fs(\021)2582 3218 y Fu(v)2625 3100 y Fp(\023)2728 3218 y Fu(dt)27 b(d\021)s(:)196 b Fz(\(3.3\))440 3441 y(So)27 b(the)h(sp)r(ectrum)g(of)g(the)g(op)r(erator)d Fu(P)1686 3453 y Fs(\013)1762 3441 y Fz(asso)r(ciated)h(to)h(the)h (form)g Fu(a)2642 3453 y Fs(\013)2717 3441 y Fz(on)f(\012)2892 3411 y Fv(0)2957 3441 y Fz(satis\014es)g(:)1501 3609 y Fu(\033)s Fz(\()p Fx(\000r)1717 3574 y Fv(2)1717 3629 y Ft(A)1770 3637 y Fr(0)1808 3609 y Fz(\))c(=)g Fu(\013\033)s Fz(\()p Fu(P)2139 3621 y Fs(\013)2188 3609 y Fz(\))p Fu(:)891 b Fz(\(3.4\))440 3777 y(P)n(articularly)-7 b(,)25 b(b)n(y)j(denoting)f Fu(\025)p Fz(\()p Fu(\013)p Fz(\))i(the)f(b)r (ottom)g(of)g Fu(\033)s Fz(\()p Fu(P)2231 3789 y Fs(\013)2279 3777 y Fz(\),)g(w)n(e)f(observ)n(e)f(:)1611 3945 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))e(=)f Fu(\013\025)p Fz(\()p Fu(\013)p Fz(\))p Fu(:)1003 b Fz(\(3.5\))440 4113 y(F)-7 b(urthermore,)29 b Fu( )k Fz(is)d(an)f(eigenfunction)h(asso)r(ciated)f(to)g Fu(\026)p Fz(\()p Fu(\013)p Fz(\))i(\(if)g(it)f(exists\))g(if)g(and)g (only)440 4212 y(if)e Fu( )570 4224 y Fs(\013)645 4212 y Fz(is)g(an)f(eigenfunction)h(asso)r(ciated)e(to)h Fu(\025)p Fz(\()p Fu(\013)p Fz(\),)j(where)d Fu( )j Fz(and)e Fu( )2608 4224 y Fs(\013)2683 4212 y Fz(are)e(link)n(ed)i(b)n(y)f(:)780 4399 y Fx(8)p Fz(\()p Fu(t;)14 b(\021)s Fz(\))23 b Fx(2)h Fz(\012)1164 4369 y Fv(0)1201 4399 y Fu(;)41 b( )1319 4411 y Fs(\013)1367 4399 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))23 b(=)g Fu(e)1692 4369 y Ft(\000)p Fs(i\021)r(t)1833 4399 y Fu( )1904 4307 y Fp(\020)1953 4303 y(q)p 2036 4303 79 4 v 2046 4366 a Fv(2)p Fs(t)p 2046 4380 59 4 v 2054 4427 a(\013)2128 4399 y Fz(cos)o(\()p Fu(\013\021)s Fz(\))p Fu(;)2437 4303 y Fp(q)p 2521 4303 79 4 v 2531 4366 a Fv(2)p Fs(t)p 2531 4380 59 4 v 2539 4427 a(\013)2613 4399 y Fz(sin\()p Fu(\013\021)s Fz(\))2876 4307 y Fp(\021)2941 4399 y Fu(:)440 4573 y FA(Remark)31 b(3.1.)40 b Fo(F)-6 b(r)l(om)30 b(the)f(expr)l(ession)i(of)f(the)g(form)h Fu(q)2223 4585 y Fs(\013)2270 4573 y Fo(,)f(we)g(imme)l(diately)i(se)l (e)e(that)f(:)910 4736 y Fu(\013)24 b Fx(7!)f Fu(\013\026)p Fz(\()p Fu(\013)p Fz(\))31 b Fo(is)f(incr)l(e)l(asing)h(and)f Fu(\013)23 b Fx(7!)2178 4696 y Fs(\026)p Fv(\()p Fs(\013)p Fv(\))p 2178 4717 136 4 v 2224 4765 a Fs(\013)2353 4736 y Fo(is)31 b(de)l(cr)l(e)l(asing.)440 4891 y(It)e(would)i(b)l(e)f (inter)l(esting)f(to)h(show)g(the)g(monotonicity)h(of)g Fu(\026)f Fo(fr)l(om)g Fz(]0)p Fu(;)14 b(\031)s Fz(])29 b Fo(onto)h Fz(]0)p Fu(;)14 b Fz(\002)3153 4903 y Fv(0)3189 4891 y Fz(])p Fo(.)440 5045 y Fz(As)24 b(w)n(e)g(kno)n(w)f(from)h (Dauge-Hel\013er)g([8)o(],)h(Hel\013er-Morame)e([11)o(],)i(Lu-P)n(an)e ([16)o(])h(and)g(as)g(w)n(e)440 5145 y(recalled)j(in)h(Section)f(2.1,)g (then)h(:)1296 5356 y Fx(8)p Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b(\031)s Fz(])p Fu(;)1719 5300 y(\026)p Fz(\()p Fu(\013)p Fz(\))p 1719 5337 169 4 v 1776 5413 a Fu(\013)1920 5356 y Fx(\025)2018 5300 y Fu(\026)p Fz(\()p Fu(\031)s Fz(\))p 2018 5337 165 4 v 2075 5413 a Fu(\031)2216 5356 y Fz(=)2313 5300 y(\002)2378 5312 y Fv(0)p 2313 5337 102 4 v 2339 5413 a Fu(\031)2425 5356 y(:)686 b Fz(\(3.6\))1851 5662 y(8)p eop %%Page: 9 9 9 8 bop 440 531 a Fy(4)135 b(Analysis)45 b(of)g(the)g(t)l(w)l(o)i(k)l (ey-op)t(erators)440 730 y Fl(4.1)112 b(Presen)m(tation)440 883 y Fz(In)19 b(the)g(expression)e(of)h Fu(a)1189 895 y Fs(\013)1255 883 y Fz(in)h(\(3.3\),)h(t)n(w)n(o)e(forms)g(\(and)g(t)n (w)n(o)g(asso)r(ciated)f(op)r(erators\))g(app)r(ear.)440 983 y(The)29 b(\014rst)g(one)g(is)f Fu(`)h Fz(\(with)h(asso)r(ciated)e Fu(L)p Fz(\))h(whic)n(h)g(will)g(b)r(e)g(de\014ned)h(just)f(b)r(elo)n (w)g(and)g(the)440 1082 y(second)f(is)h(asso)r(ciated)e(to)h(the)h (Neumann)g(realization)e(of)i Fx(\000)p Fu(@)2430 1052 y Fv(2)2425 1103 y Fs(\021)2495 1082 y Fz(in)g(])19 b Fx(\000)2729 1050 y Fv(1)p 2729 1064 34 4 v 2729 1111 a(2)2772 1082 y Fu(;)2819 1050 y Fv(1)p 2819 1064 V 2819 1111 a(2)2862 1082 y Fz([.)40 b(W)-7 b(e)29 b(de\014ne)440 1182 y(the)f(sesquilinear)e(form)h Fu(`)p Fz(,)h(on)f(:)1005 1363 y Fx(V)1063 1329 y Fs(N)1056 1383 y(`)1135 1363 y Fz(:)51 b(=)1297 1271 y Fp(n)1352 1363 y Fu(u)23 b Fx(2)g Fu(L)1558 1329 y Fv(2)1595 1363 y Fz(\(\012)1687 1329 y Fv(0)1724 1363 y Fz(\))1756 1292 y Fp(\014)1756 1342 y(\014)1812 1291 y Fx(p)p 1881 1291 30 4 v 72 x Fu(t)p Fz(\()p Fu(D)2012 1375 y Fs(t)2060 1363 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(u)23 b Fx(2)g Fu(L)2425 1329 y Fv(2)2462 1363 y Fz(\(\012)2554 1329 y Fv(0)2591 1363 y Fz(\))2623 1271 y Fp(o)2716 1363 y Fu(;)440 1544 y Fz(b)n(y)k(:)895 1681 y Fu(`)p Fz(\()p Fu(u;)14 b(v)s Fz(\))23 b(=)1232 1568 y Fp(Z)1278 1757 y Fv(\012)1325 1740 y Fr(0)1376 1681 y Fz(2)p Fu(t)p Fz(\()p Fu(D)1549 1693 y Fs(t)1596 1681 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(u)p 1803 1609 352 4 v Fz(\()p Fu(D)1904 1693 y Fs(t)1952 1681 y Fx(\000)g Fu(\021)s Fz(\))p Fu(v)31 b(dt)d(d\021)s(;)38 b Fx(8)p Fu(u;)14 b(v)24 b Fx(2)g(V)2764 1647 y Fs(N)2757 1702 y(`)2827 1681 y Fu(:)284 b Fz(\(4.1\))440 1878 y(Let)27 b Fx(P)22 b(\012)16 b(S)6 b Fz(\()p 837 1808 116 4 v Fw(R)892 1854 y Fv(+)953 1878 y Fz(\))27 b(b)r(e)g(the)f(space)g(of)h (p)r(olynomial)e(functions)i(in)g Fu(\021)j Fz(whose)25 b(co)r(e\016cien)n(ts)h(are)440 1978 y(in)i Fx(S)6 b Fz(\()p 625 1908 V Fw(R)680 1954 y Fv(+)741 1978 y Fz(\).)37 b(W)-7 b(e)28 b(de\014ne)g(the)g(op)r(erator)e Fu(L)h Fz(on)g Fx(P)e(\012)18 b(S)6 b Fz(\()p 2147 1908 V Fw(R)2202 1954 y Fv(+)2263 1978 y Fz(\))28 b(b)n(y)f(:)1403 2139 y Fu(L)9 b Fz(:)51 b(=)22 b(2\()p Fu(D)1773 2151 y Fs(t)1820 2139 y Fx(\000)c Fu(\021)s Fz(\))p Fu(t)p Fz(\()p Fu(D)2110 2151 y Fs(t)2159 2139 y Fx(\000)g Fu(\021)s Fz(\))p Fu(:)440 2299 y Fz(Then)28 b(w)n(e)f(v)n(erify)g(:)935 2460 y Fx(8)p Fu(u)21 b Fx(2)j(P)h(\012)18 b(S)6 b Fz(\()p 1384 2390 V Fw(R)1439 2436 y Fv(+)1500 2460 y Fz(\))p Fu(;)42 b Fx(8)p Fu(v)25 b Fx(2)e(V)1845 2426 y Fs(N)1838 2480 y(`)1908 2460 y Fu(;)42 b(`)p Fz(\()p Fu(u;)14 b(v)s Fz(\))23 b(=)f Fx(h)p Fu(Lu;)14 b(v)s Fx(i)2559 2485 y Fs(L)2605 2468 y Fr(2)2637 2485 y Fv(\(\012)2710 2468 y Fr(0)2743 2485 y Fv(\))2787 2460 y Fu(:)324 b Fz(\(4.2\))440 2639 y(The)30 b(form)g Fu(q)849 2651 y Fs(\013)927 2639 y Fz(con)n(tains)f(a)h(term)h(in)1659 2606 y Fv(1)p 1638 2620 76 4 v 1638 2667 a Fs(\013)1681 2651 y Fr(2)1724 2639 y Fz(.)45 b(So)30 b(when)g(trying)g(to)g(minimize)h(it,)g(it)g(is) f(quite)440 2738 y(natural)j(to)h(b)r(egin)g(with)g(studying)g(the)h (restriction)e(of)g(the)i(form)e(to)h(functions)g(whic)n(h)440 2838 y(are)27 b(indep)r(enden)n(t)h(of)g Fu(\021)j Fz(to)c(cancel)g (the)h(term)g(in)2036 2805 y Fv(1)p 2014 2819 V 2014 2867 a Fs(\013)2057 2850 y Fr(2)2100 2838 y Fz(.)440 3067 y Fl(4.2)112 b(A)37 b(new)h(k)m(ey)g(op)s(erator)f Fk(L)1774 3031 y Fj(mean)440 3220 y Fz(W)-7 b(e)20 b(de\014ne)g(the)h (form)e Fu(`)1166 3190 y Fv(mean)1344 3220 y Fz(whic)n(h)g(app)r(ears)g (naturally)g(when)h(w)n(e)g(restrict)f Fu(`)g Fz(to)h(functions)440 3320 y(indep)r(enden)n(t)28 b(of)g Fu(\021)s Fz(.)37 b(Then)28 b(the)g(sesquilinear)e(form)h Fu(`)2146 3289 y Fv(mean)2332 3320 y Fz(is)g(de\014ned)h(on)f(:)785 3480 y Fx(V)843 3446 y Fs(N)836 3501 y Fv(mean)1003 3480 y Fz(:)50 b(=)23 b Fx(f)p Fu(f)31 b Fx(2)24 b Fu(L)1414 3446 y Fv(2)1450 3480 y Fz(\()p Fw(R)1537 3446 y Fv(+)1598 3480 y Fz(\))1630 3410 y Fp(\014)1630 3459 y(\014)1685 3409 y Fx(p)p 1755 3409 30 4 v 1755 3480 a Fu(tf)31 b Fx(2)24 b Fu(L)1993 3446 y Fv(2)2029 3480 y Fz(\()p Fw(R)2116 3446 y Fv(+)2177 3480 y Fz(\))p Fu(;)2274 3409 y Fx(p)p 2343 3409 V 71 x Fu(tD)2442 3492 y Fs(t)2471 3480 y Fu(f)31 b Fx(2)24 b Fu(L)2679 3446 y Fv(2)2715 3480 y Fz(\()p Fw(R)2802 3446 y Fv(+)2863 3480 y Fz(\))p Fx(g)p Fu(;)440 3641 y Fz(b)n(y)j(:)669 3847 y Fu(`)704 3812 y Fv(mean)862 3847 y Fz(\()p Fu(u;)14 b(v)s Fz(\))23 b(=)1165 3734 y Fp(Z)1248 3754 y Ft(1)1211 3922 y Fv(0)1332 3847 y Fz(2)1388 3730 y Fp(\022)1448 3847 y Fu(D)1517 3859 y Fs(t)1546 3847 y Fu(u)p 1621 3780 142 4 v 27 w(D)1690 3859 y Fs(t)1719 3847 y Fu(v)f Fz(+)1895 3791 y(1)p 1874 3828 84 4 v 1874 3904 a(12)1967 3847 y Fu(u)p 2015 3801 44 4 v(v)2058 3730 y Fp(\023)2160 3847 y Fu(t)28 b(dt)23 b(;)37 b Fx(8)p Fu(u;)14 b(v)25 b Fx(2)f(V)2708 3812 y Fs(N)2701 3867 y Fv(mean)2881 3847 y Fu(:)230 b Fz(\(4.3\))440 4053 y(The)28 b(asso)r(ciated)e(op)r(erator,)1469 4245 y Fu(L)1526 4211 y Fv(mean)1707 4245 y Fz(=)c(2)p Fu(D)1905 4257 y Fs(t)1948 4245 y Fu(t)14 b(D)2061 4257 y Fs(t)2108 4245 y Fz(+)2206 4189 y Fu(t)p 2201 4226 42 4 v 2201 4302 a Fz(6)2252 4245 y Fu(;)440 4439 y Fz(is)24 b(self-adjoin)n(t)g (and)g(its)g(domain)g(can)f(b)r(e)i(c)n(haracterized)d(\(cf)i([4]\))h (as)e(b)r(eing)h Fx(W)2936 4409 y Fv(1)2929 4460 y(2)2973 4439 y Fz(\()p Fu(R)3069 4409 y Fv(+)3124 4439 y Fz(\))9 b(:)52 b(=)440 4539 y Fx(f)p Fu(u)26 b Fx(2)h Fu(H)714 4509 y Fv(1)751 4539 y Fz(\()p Fw(R)837 4509 y Fv(+)898 4539 y Fz(\))p Fx(j)j Fu(tu)d Fx(2)g Fu(H)1246 4509 y Fv(2)1283 4539 y Fz(\()p Fw(R)1369 4509 y Fv(+)1430 4539 y Fz(\))p Fx(g)p Fz(.)43 b(Its)30 b(sp)r(ectrum)g(is)g(discrete)f(and)h (the)g(eigen)n(v)-5 b(alues)28 b(are)440 4638 y(simple)j(and)g(giv)n (en)f(b)n(y)1215 4606 y Fv(2)p Fs(n)p Fv(+1)p 1215 4620 159 4 v 1250 4628 a Ft(p)p 1305 4628 34 3 v 48 x Fv(3)1383 4638 y Fz(,)i Fu(n)c Fx(\025)g Fz(0.)46 b(Moreo)n(v)n(er)28 b(eac)n(h)j(eigensubspace)e(is)i(included)g(in)440 4768 y Fx(S)6 b Fz(\()p 528 4699 116 4 v Fw(R)583 4744 y Fv(+)644 4768 y Fz(\).)34 b(Let)19 b(us)f(giv)n(e)f(the)i(expression)e(of)h(the) h(normalized)e(eigen)n(v)n(ector)g Fu(u)2761 4738 y Fv(mean)2761 4789 y(1)2937 4768 y Fz(asso)r(ciated)440 4868 y(to)27 b(the)h(\014rst)g(eigen)n(v)-5 b(alue)26 b(of)i Fu(L)1404 4838 y Fv(mean)1562 4868 y Fz(,)f(denoted)h(b)n(y)f Fu(\025)2089 4838 y Fv(mean)2089 4889 y(1)2275 4868 y Fz(and)h(equal)f(to)2794 4835 y Fv(1)p 2767 4849 88 4 v 2767 4858 a Ft(p)p 2822 4858 34 3 v 48 x Fv(3)2892 4868 y Fz(:)1137 5103 y Fu(u)1185 5069 y Fv(mean)1185 5123 y(1)1343 5103 y Fz(\()p Fu(t)p Fz(\))c(=)1610 5047 y(1)p 1558 5084 146 4 v 1558 5161 a(3)1600 5137 y Fv(1)p Fs(=)p Fv(4)1714 5103 y Fz(exp)1854 4986 y Fp(\022)1915 5103 y Fx(\000)2051 5047 y Fu(t)p 1990 5084 153 4 v 1990 5169 a Fz(2)2032 5100 y Fx(p)p 2101 5100 42 4 v 69 x Fz(3)2152 4986 y Fp(\023)2227 5103 y Fu(;)42 b Fx(8)p Fu(t)22 b Fx(2)h Fw(R)2523 5069 y Fv(+)2584 5103 y Fu(:)527 b Fz(\(4.4\))440 5313 y(W)-7 b(e)32 b(can)g(use)f(the)h(function)h Fu(u)1416 5283 y Fv(mean)1416 5334 y(1)1605 5313 y Fz(as)e(trial)g(function)i(to)e(b)r (ound)h(from)g(ab)r(o)n(v)n(e)e Fu(\026)p Fz(\()p Fu(\013)p Fz(\))j(as)440 5413 y(follo)n(ws)27 b(:)1851 5662 y(9)p eop %%Page: 10 10 10 9 bop 440 531 a FA(Prop)s(osition)32 b(4.1.)41 b Fo(F)-6 b(or)31 b Fu(\013)25 b(<)1473 463 y Fx(p)p 1543 463 42 4 v 1543 531 a Fz(3)o(\002)1649 543 y Fv(0)1686 531 y Fo(,)32 b(the)f(b)l(ottom)g(of)g(the)g(sp)l(e)l(ctrum)g(of)h Fu(P)2895 543 y Ft(A)2948 551 y Fr(0)2981 543 y Fs(;)p Fv(\012)3048 551 y Fe(\013)3125 531 y Fo(is)f(an)440 631 y(eigenvalue)g Fu(\026)p Fz(\()p Fu(\013)p Fz(\))g Fo(wich)g(satis\014es)f(:)1656 833 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\024)1973 777 y Fu(\013)p 1945 814 111 4 v 1945 831 a Fx(p)p 2014 831 42 4 v 68 x Fz(3)2065 833 y Fu(:)1046 b Fz(\(4.5\))440 1058 y(Before)25 b(b)r(eing)g(more)g (precise)g(ab)r(out)g(the)h(op)r(erator)e Fu(L)2162 1028 y Fv(mean)2319 1058 y Fz(,)j(let)e(us)h(men)n(tion)f(an)h(impro)n(v-) 440 1157 y(men)n(t)i(of)f(the)h(upp)r(er)g(b)r(ound)g(\(4.5\))f(due)h (to)g(So)r(eren)f(F)-7 b(ournais)26 b(:)440 1339 y FA(Prop)s(osition)i (4.2.)38 b Fo(F)-6 b(or)27 b Fu(\013)d(<)1502 1257 y Ft(p)p 1556 1257 34 3 v 1556 1305 a Fv(3\002)1640 1313 y Fr(0)p 1468 1320 238 4 v 1468 1330 a Fx(p)p 1538 1330 169 4 v 1538 1386 a Fv(1)p Ft(\000)p Fv(\002)1674 1366 y Fr(2)1674 1404 y(0)1716 1339 y Fo(,)k(the)g(b)l(ottom)g(of)g(the)g (sp)l(e)l(ctrum)f(of)h Fu(P)2901 1351 y Ft(A)2954 1359 y Fr(0)2987 1351 y Fs(;)p Fv(\012)3054 1359 y Fe(\013)3128 1339 y Fo(is)g(an)440 1481 y(eigenvalue)j Fu(\026)p Fz(\()p Fu(\013)p Fz(\))g Fo(wich)g(satis\014es)f(:)1560 1683 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\024)1973 1627 y Fu(\013)p 1849 1664 303 4 v 1849 1680 a Fx(p)p 1918 1680 234 4 v 70 x Fz(3)18 b(+)g Fu(\013)2114 1726 y Fv(2)2161 1683 y Fu(:)950 b Fz(\(4.6\))440 1950 y Fo(Pr)l(o)l(of)24 b(:)36 b Fz(The)20 b(idea)g(is)h(to)f(estimate)h(\(3.1\))f(for)g(a)g (function)h(lik)n(e)f Fu(e)2420 1920 y Fs(i)2453 1893 y Fe(\032)2483 1877 y Fr(2)p 2453 1907 63 3 v 2470 1940 a(2)2525 1920 y Fs(\036)p Fv(\(1)p Ft(\000)p Fs(\016)r Fv(\))2739 1950 y Fu(u)p Fz(\()p Fu(\032)p Fz(\))g(and)h(c)n(ho)r(ose) 440 2050 y Fu(u)27 b Fz(and)h(co)r(e\016cien)n(t)f Fu(\016)s Fz(.)37 b(This)28 b(lead)f(to)g(de\014ne)h(the)g(function)g Fu(u)23 b Fx(2)g(V)2545 2020 y Fs(N)2636 2050 y Fz(on)k(\012)2811 2020 y Fv(0)2876 2050 y Fz(b)n(y)g(:)1142 2248 y Fu(u)p Fz(\()p Fu(t;)14 b(\021)s Fz(\))23 b(=)g Fu(e)1515 2213 y Fs(it\021)r(\014)1640 2188 y Fr(2)1677 2248 y Fu(e)1716 2213 y Ft(\000)1779 2186 y Fe(\014)p 1777 2200 38 3 v 1777 2233 a(\013)1838 2191 y(t)p 1835 2200 29 3 v 1835 2233 a Fr(2)1905 2248 y Fz(with)28 b Fu(\014)g Fz(=)2391 2191 y Fu(\013)p 2266 2229 303 4 v 2266 2245 a Fx(p)p 2335 2245 234 4 v 69 x Fz(3)18 b(+)g Fu(\013)2531 2290 y Fv(2)2579 2248 y Fu(:)440 2472 y Fz(Then)28 b(it)g(is)f(easy)g(to)g (compute)h(:)744 2723 y Fu(q)781 2735 y Fs(\013)829 2723 y Fz(\()p Fu(u)p Fz(\))83 b(=)1171 2610 y Fp(Z)1218 2799 y Fv(\012)1265 2782 y Fr(0)1315 2581 y Fp( )1381 2723 y Fz(2)p Fu(t)1467 2603 y Fp(\014)1467 2653 y(\014)1467 2702 y(\014)1467 2752 y(\014)1494 2723 y Fu(\021)s Fz(\()p Fu(\014)1621 2689 y Fv(2)1678 2723 y Fx(\000)18 b Fz(1\))g(+)g Fu(i)1996 2667 y(\014)p 1975 2704 95 4 v 1975 2780 a Fz(2)p Fu(\013)2079 2603 y Fp(\014)2079 2653 y(\014)2079 2702 y(\014)2079 2752 y(\014)2107 2623 y Fv(2)2163 2723 y Fz(+)2263 2667 y Fu(\014)2314 2637 y Fv(4)2351 2667 y Fu(t)p 2256 2704 133 4 v 2256 2780 a Fz(2)p Fu(\013)2351 2756 y Fv(2)2398 2581 y Fp(!)2477 2723 y Fx(j)p Fu(u)p Fz(\()p Fu(t;)c(\021)s Fz(\))p Fx(j)2746 2689 y Fv(2)2812 2723 y Fu(dt)28 b(d\021)1024 2984 y Fz(=)1201 2928 y Fu(\013)p 1181 2965 93 4 v 1181 3041 a Fz(2)p Fu(\014)1298 2867 y Fp(\022)1369 2928 y Fz(\()p Fu(\014)1452 2898 y Fv(2)1508 2928 y Fx(\000)18 b Fz(1\))1665 2898 y Fv(2)p 1369 2965 334 4 v 1515 3041 a Fz(6)1731 2984 y(+)1846 2928 y Fu(\014)1897 2898 y Fv(2)p 1824 2965 133 4 v 1824 3041 a Fz(2)p Fu(\013)1919 3017 y Fv(2)1966 2984 y Fz(\(1)g(+)g Fu(\014)2192 2950 y Fv(2)2230 2984 y Fz(\))2262 2867 y Fp(\023)2337 2871 y(Z)2383 3060 y Fv(\012)2430 3043 y Fr(0)2481 2984 y Fx(j)p Fu(u)p Fz(\()p Fu(t;)c(\021)s Fz(\))p Fx(j)2750 2950 y Fv(2)2815 2984 y Fu(dt)1024 3209 y Fz(=)1312 3153 y(1)p 1181 3190 303 4 v 1181 3207 a Fx(p)p 1251 3207 234 4 v 1251 3276 a Fz(3)k(+)g Fu(\013)1447 3252 y Fv(2)1508 3096 y Fp(Z)1554 3285 y Fv(\012)1601 3268 y Fr(0)1652 3209 y Fx(j)p Fu(u)p Fz(\()p Fu(t;)c(\021)s Fz(\))p Fx(j)1921 3175 y Fv(2)1986 3209 y Fu(dt:)1052 b Fz(\(4.7\))440 3438 y(By)20 b(the)h(min-max)f(principle,)i(w)n(e)e (deduce)h(that)g Fu(\026)p Fz(\()p Fu(\013)p Fz(\))j Fx(\024)2331 3406 y Fs(\013)p 2245 3420 215 4 v 2245 3428 a Ft(p)p 2300 3428 160 3 v 48 x Fv(3+)p Fs(\013)2427 3459 y Fr(2)2469 3438 y Fz(.)35 b(Prop)r(osition)19 b(2.3)g(sho)n(ws) 440 3554 y(that)28 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))h(is)e(an)g (eigen)n(v)-5 b(alue)27 b(for)g(an)n(y)g(angle)g Fu(\013)h Fz(suc)n(h)f(that)2455 3521 y Fs(\013)p 2370 3535 215 4 v 2370 3544 a Ft(p)p 2424 3544 160 3 v 2424 3591 a Fv(3+)p Fs(\013)2551 3575 y Fr(2)2617 3554 y Fu(<)c Fz(\002)2770 3566 y Fv(0)2807 3554 y Fz(.)410 b Fd(\003)440 3735 y FA(Remark)31 b(4.3.)40 b Fo(Appr)l(oximately,)31 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))g Fo(is)f(an)g(eigenvalue)h(for)g Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b Fz(1)p Fu(:)p Fz(26[)p Fo(.)440 3834 y(Combining)28 b(this)g(with)f(R)l(emark)g(2.6,)i(we)f(c) l(onclude)f(with)h(a)f(go)l(o)l(d)h(ac)l(cur)l(acy)g(that)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\))h Fo(is)440 3934 y(an)i(eigenvalue)h (for)f Fu(\013)24 b Fx(2)p Fz(]0)p Fu(;)1336 3901 y Fs(\031)p 1336 3915 41 4 v 1340 3963 a Fv(2)1387 3934 y Fz(])p Fo(.)565 4099 y Fz(F)-7 b(urthermore,)22 b(w)n(e)g(deduce)h(b)n(y)f (standard)f(F)-7 b(redholm)22 b(theory)g(and)g(a)g(regularit)n(y)f (theo-)440 4198 y(rem)27 b(\(cf)i([4)o(]\),)f(the)g(follo)n(wing)f (lemma)g(:)440 4363 y FA(Lemma)42 b(4.4.)k Fo(L)l(et)39 b Fz(\()p Fu(\025)1220 4333 y Fv(mean)1220 4384 y(1)1379 4363 y Fu(;)14 b(u)1464 4333 y Fv(mean)1464 4384 y(1)1621 4363 y Fz(\))40 b Fo(the)g(fundamental)g(state)f(of)h Fu(L)2699 4333 y Fv(mean)2857 4363 y Fo(.)68 b(F)-6 b(or)39 b(every)440 4463 y Fu(f)32 b Fx(2)23 b(S)6 b Fz(\()p 679 4393 116 4 v Fw(R)734 4439 y Fv(+)795 4463 y Fz(\))30 b Fo(ortho)l(gonal)h(to)f Fu(u)1411 4433 y Fv(mean)1411 4483 y(1)1568 4463 y Fo(,)h(ther)l(e)e(exists)h(a)g(unique)f Fu(u)23 b Fx(2)g(S)6 b Fz(\()p 2634 4393 V Fw(R)2689 4439 y Fv(+)2750 4463 y Fz(\))30 b Fo(such)g(that)g(:)1071 4581 y Fp(\032)1255 4643 y Fz(\()p Fu(L)1344 4613 y Fv(mean)1520 4643 y Fx(\000)18 b Fu(\025)1651 4613 y Fv(mean)1651 4663 y(1)1809 4643 y Fz(\))p Fu(u)83 b Fz(=)g Fu(f)38 b Fo(on)30 b Fw(R)2372 4613 y Fv(+)2433 4643 y Fu(;)1175 4685 y Fp(R)1230 4705 y Ft(1)1214 4781 y Fv(0)1314 4752 y Fu(u)p Fz(\()p Fu(t)p Fz(\))p 1486 4679 301 4 v 30 w Fu(u)1534 4723 y Fv(mean)1534 4774 y(1)1692 4752 y Fz(\()p Fu(t)p Fz(\))g Fu(dt)83 b Fz(=)g(0)p Fu(:)3134 4698 y Fz(\(4.8\))440 4941 y Fo(F)-6 b(urthermor)l(e,)30 b(if)h Fu(u)e Fo(is)h(given)g(by)h(\(4.8\),)g(then)f(for)g(al)t(l)h (functions)f Fu(v)c Fx(2)d(V)2749 4911 y Fs(N)2742 4962 y Fv(mean)2930 4941 y Fo(:)1040 5122 y Fu(`)1075 5088 y Fv(mean)1233 5122 y Fz(\()p Fu(u;)14 b(v)s Fz(\))19 b Fx(\000)f Fu(\025)1575 5088 y Fv(mean)1575 5143 y(1)1747 5122 y Fx(h)p Fu(u;)c(v)s Fx(i)1939 5147 y Fs(L)1985 5130 y Fr(2)2017 5147 y Fv(\(\012)2090 5130 y Fr(0)2123 5147 y Fv(\))2176 5122 y Fz(=)23 b Fx(h)p Fu(f)t(;)14 b(v)s Fx(i)2453 5147 y Fs(L)2499 5130 y Fr(2)2531 5147 y Fv(\(\012)2604 5130 y Fr(0)2637 5147 y Fv(\))2681 5122 y Fu(:)430 b Fz(\(4.9\))440 5313 y(In)36 b(the)f(case)g(when)h(the)g (second)e(mem)n(b)r(er)i(of)f(\(4.8\))g(has)g(the)h(form)f Fu(P)12 b(u)2796 5283 y Fv(mean)2796 5334 y(1)2989 5313 y Fz(for)35 b(some)440 5413 y(p)r(olynomial)27 b Fu(P)12 b Fz(,)28 b(w)n(e)f(can)g(explicit)h(the)g(solution)f(as)g(follo)n(ws)f (:)1831 5662 y(10)p eop %%Page: 11 11 11 10 bop 440 585 a FA(Lemma)30 b(4.5.)40 b Fo(L)l(et)29 b Fu(P)35 b Fz(=)1327 482 y Fs(n)1288 507 y Fp(X)1287 685 y Fs(k)q Fv(=0)1422 585 y Fu(p)1464 597 y Fs(k)1505 585 y Fu(t)1535 551 y Fs(k)1605 585 y Fo(b)l(e)30 b(a)g(p)l(olynomial)i (of)f(de)l(gr)l(e)l(e)f Fu(n)f Fo(such)h(that)g(:)1557 818 y Fs(n)1518 843 y Fp(X)1517 1022 y Fs(k)q Fv(=0)1638 922 y Fz(\()1670 849 y Fx(p)p 1739 849 42 4 v 73 x Fz(3\))1813 888 y Fs(k)1877 922 y Fu(k)s Fz(!)23 b Fu(p)2011 934 y Fs(k)2075 922 y Fz(=)f(0)p Fu(:)440 1198 y Fo(We)30 b(de\014ne)g(the)g(p)l(olynomial)i(of)e(de)l(gr)l(e)l(e)g Fu(n)p Fo(,)1857 1177 y Fz(~)1838 1198 y Fu(P)35 b Fx(2)24 b(P)7 b Fz(\()p Fw(R)2156 1168 y Fv(+)2217 1198 y Fz(\))p Fo(,)30 b(by)g(its)g(c)l(o)l(e\016cients)37 b Fz(~)-49 b Fu(p)2994 1210 y Fs(k)3065 1198 y Fo(:)745 1310 y Fp(8)745 1385 y(>)745 1410 y(>)745 1435 y(>)745 1459 y(>)745 1484 y(<)745 1634 y(>)745 1659 y(>)745 1684 y(>)745 1709 y(>)745 1733 y(:)871 1382 y Fz(~)g Fu(p)906 1394 y Fv(1)1026 1382 y Fz(=)83 b Fx(\000)1249 1345 y Fs(p)1283 1353 y Fr(0)p 1248 1363 67 4 v 1265 1410 a Fv(2)1325 1382 y Fu(;)867 1507 y Fz(~)-49 b Fu(p)902 1519 y Fs(k)1026 1507 y Fz(=)1218 1474 y Fv(1)p 1184 1488 103 4 v 1184 1536 a(2)p Fs(k)1253 1519 y Fr(2)1310 1415 y Fp(\020)1396 1474 y Fv(2)p 1369 1488 88 4 v 1369 1497 a Ft(p)p 1424 1497 34 3 v 48 x Fv(3)1467 1507 y Fz(\()p Fu(k)21 b Fx(\000)d Fz(1\))7 b(~)-49 b Fu(p)1762 1519 y Fs(k)q Ft(\000)p Fv(1)1906 1507 y Fx(\000)18 b Fu(p)2031 1519 y Fs(k)q Ft(\000)p Fv(1)2157 1415 y Fp(\021)2220 1507 y Fu(;)37 b Fx(8)p Fu(k)25 b Fz(=)e(2)p Fu(;)14 b(:)g(:)g(:)f(;)h(n;)871 1698 y Fz(~)-49 b Fu(p)906 1710 y Fv(0)1026 1698 y Fz(=)83 b Fx(\000)1292 1594 y Fs(n)1252 1619 y Fp(X)1253 1798 y Fs(k)q Fv(=1)1373 1698 y Fz(\()1405 1625 y Fx(p)p 1474 1625 42 4 v 73 x Fz(3\))1548 1664 y Fs(k)1619 1698 y Fu(k)s Fz(!)36 b(~)-49 b Fu(p)1759 1710 y Fs(k)1800 1698 y Fu(:)3092 1580 y Fz(\(4.10\))440 1976 y Fo(Then)36 b Fz(~)-48 b Fu(u)23 b Fz(=)833 1955 y(~)815 1976 y Fu(P)11 b(u)927 1946 y Fv(mean)927 1997 y(1)1115 1976 y Fo(is)30 b(the)g(unique)f(solution)h(for)h(the)f(pr)l(oblem)g(:)1092 2096 y Fp(\032)1276 2158 y Fz(\()p Fu(L)1365 2127 y Fv(mean)1541 2158 y Fx(\000)18 b Fu(\025)1672 2127 y Fv(mean)1672 2178 y(1)1831 2158 y Fz(\))5 b(~)-47 b Fu(u)82 b Fz(=)h Fu(P)12 b(u)2254 2127 y Fv(mean)2254 2178 y(1)2411 2158 y Fu(;)1196 2199 y Fp(R)1251 2220 y Ft(1)1235 2296 y Fv(0)1341 2266 y Fz(~)-47 b Fu(u)o Fz(\()p Fu(t)p Fz(\))p 1507 2194 301 4 v 30 w Fu(u)1555 2238 y Fv(mean)1555 2288 y(1)1713 2266 y Fz(\()p Fu(t)p Fz(\))30 b Fu(dt)83 b Fz(=)g(0)p Fu(:)3092 2213 y Fz(\(4.11\))440 2445 y Fo(Pr)l(o)l(of)31 b(:)39 b Fz(W)-7 b(e)28 b(\014rst)f(notice)g(that)h (:)1133 2670 y Fx(8)p Fu(k)d Fx(2)f Fw(N)t Fu(;)1447 2557 y Fp(Z)1530 2578 y Ft(1)1493 2746 y Fv(0)1614 2670 y Fu(t)1644 2636 y Fs(k)1685 2670 y Fx(j)p Fu(u)1756 2636 y Fv(mean)1756 2691 y(1)1913 2670 y Fz(\()p Fu(t)p Fz(\))p Fx(j)2030 2636 y Fv(2)2096 2670 y Fu(dt)f Fz(=)g Fu(k)s Fz(!)g(\()2404 2597 y Fx(p)p 2473 2597 42 4 v 73 x Fz(3\))2547 2636 y Fs(k)2588 2670 y Fu(:)481 b Fz(\(4.12\))440 2895 y(W)-7 b(e)28 b(deduce)g(immediately)g(that)f(:)917 3036 y Fs(n)877 3060 y Fp(X)877 3239 y Fs(k)q Fv(=0)998 3139 y Fz(\()1030 3067 y Fx(p)p 1099 3067 V 72 x Fz(3\))1173 3105 y Fs(k)1237 3139 y Fu(k)s Fz(!)c Fu(q)1366 3151 y Fs(k)1430 3139 y Fz(=)f(0)h Fx(\()-14 b(\))1757 3026 y Fp(Z)1840 3047 y Ft(1)1803 3215 y Fv(0)1964 3036 y Fs(n)1925 3060 y Fp(X)1924 3239 y Fs(k)q Fv(=0)2059 3139 y Fu(q)2096 3151 y Fs(k)2137 3139 y Fu(t)2167 3105 y Fs(k)2208 3139 y Fx(j)p Fu(u)2279 3105 y Fv(mean)2279 3160 y(1)2436 3139 y Fz(\()p Fu(t)p Fz(\))p Fx(j)2553 3105 y Fv(2)2619 3139 y Fu(dt)23 b Fz(=)g(0)p Fu(:)224 b Fz(\(4.13\))440 3401 y(Relation)41 b(\(4.13\))f(applied)h(with)h Fu(q)1584 3413 y Fs(k)1670 3401 y Fz(=)52 b(~)-49 b Fu(p)1822 3413 y Fs(k)1904 3401 y Fz(determined)41 b(in)g(\(4.10\))g(sho)n(ws)e (that)47 b(~)-47 b Fu(u)40 b Fz(is)440 3501 y(orthogonal)25 b(to)j Fu(u)1005 3471 y Fv(mean)1005 3521 y(1)1190 3501 y Fz(and)f(so)g(the)h(second)f(condition)h(of)f(\(4.11\))g(holds.)440 3600 y(According)j(to)h(\(4.13\))f(with)i Fu(q)1416 3612 y Fs(k)1486 3600 y Fz(=)c Fu(p)1621 3612 y Fs(k)1662 3600 y Fz(,)k(w)n(e)e(see)h(that)h Fu(P)12 b(u)2277 3570 y Fv(mean)2277 3621 y(1)2465 3600 y Fz(is)31 b(orthogonal)e(to)i Fu(u)3124 3570 y Fv(mean)3124 3621 y(1)3281 3600 y Fz(.)440 3700 y(Lemma)c(4.4)g(establishes)g(that)h(the)g(problem)f(:)1108 3815 y Fp(\032)1260 3877 y Fz(\()p Fu(L)1349 3847 y Fv(mean)1525 3877 y Fx(\000)18 b Fu(\025)1656 3847 y Fv(mean)1656 3898 y(1)1815 3877 y Fz(\))p Fu(u)83 b Fz(=)f Fu(P)12 b(u)2238 3847 y Fv(mean)2238 3898 y(1)2395 3877 y Fu(;)1212 3919 y Fp(R)1267 3940 y Ft(1)1251 4015 y Fv(0)1352 3986 y Fu(u)p Fz(\()p Fu(t)p Fz(\))p 1494 3914 301 4 v Fu(u)1542 3957 y Fv(mean)1542 4008 y(1)1699 3986 y Fz(\()p Fu(t)p Fz(\))28 b Fu(dt)84 b Fz(=)e(0)p Fu(;)440 4184 y Fz(has)30 b(a)h(unique)g(solution.)46 b(So)31 b(it)g(is)g(enough)g(to)f(pro)n(v)n (e)g(that)h(the)g(function)37 b(~)-47 b Fu(u)28 b Fz(=)3053 4163 y(~)3034 4184 y Fu(P)12 b(u)3147 4153 y Fv(mean)3147 4204 y(1)440 4283 y Fz(satis\014es)26 b(conditions)g(\(4.11\).)35 b(With)27 b(the)g(expression)e(of)h Fu(u)2307 4253 y Fv(mean)2307 4304 y(1)2491 4283 y Fz(giv)n(en)g(in)h(\(4.4\),)f(w)n(e)g (get)g(:)683 4516 y(\()p Fu(L)772 4481 y Fv(mean)948 4516 y Fx(\000)18 b Fu(\025)1079 4481 y Fv(mean)1079 4536 y(1)1237 4516 y Fz(\)\()1320 4495 y(~)1301 4516 y Fu(P)13 b(u)1415 4481 y Fv(mean)1415 4536 y(1)1572 4516 y Fz(\))23 b(=)g Fu(u)1763 4481 y Fv(mean)1763 4536 y(1)1934 4399 y Fp(\022)1996 4516 y Fx(\000)p Fz(2)p Fu(@)2147 4528 y Fs(t)2194 4495 y Fz(~)2175 4516 y Fu(P)30 b Fz(+)2386 4460 y(2)p 2351 4497 111 4 v 2351 4513 a Fx(p)p 2420 4513 42 4 v 69 x Fz(3)2472 4516 y Fu(t@)2546 4528 y Fs(t)2594 4495 y Fz(~)2575 4516 y Fu(P)g Fx(\000)18 b Fz(2)p Fu(t@)2862 4481 y Fv(2)2857 4536 y Fs(t)2917 4495 y Fz(~)2899 4516 y Fu(P)2964 4399 y Fp(\023)3039 4516 y Fu(:)440 4762 y Fz(According)j(to)g(\(4.10\),)h(the)g(constan)n (t)f(co)r(e\016cien)n(t)g(of)g Fx(\000)p Fz(2)p Fu(@)2268 4774 y Fs(t)2315 4741 y Fz(~)2297 4762 y Fu(P)d Fz(+)2476 4730 y Fv(2)p 2449 4744 88 4 v 2449 4752 a Ft(p)p 2503 4752 34 3 v 2503 4800 a Fv(3)2546 4762 y Fu(t@)2620 4774 y Fs(t)2668 4741 y Fz(~)2650 4762 y Fu(P)g Fx(\000)6 b Fz(2)p Fu(t@)2913 4732 y Fv(2)2908 4783 y Fs(t)2968 4741 y Fz(~)2949 4762 y Fu(P)33 b Fz(is)22 b(equal)440 4886 y(to)27 b Fx(\000)p Fz(2)7 b(~)-49 b Fu(p)690 4898 y Fv(1)750 4886 y Fz(=)22 b 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eop %%Page: 12 12 12 11 bop 440 531 a Fl(4.3)112 b(Study)38 b(of)g Fh(\000)p Fk(@)1283 495 y Fj(2)1278 556 y Fi(\021)440 698 y Fz(An)29 b(elemen)n(tary)e(study)h(of)g(the)h(Neumann)f(realization)f(of)h Fx(\000)p Fu(@)2462 668 y Fv(2)2457 719 y Fs(\021)2527 698 y Fz(leads)g(to)g(the)g(follo)n(wing)440 798 y(lemma)g(:)440 954 y FA(Lemma)j(4.6.)40 b Fo(L)l(et)31 b Fu(f)i Fx(2)25 b(P)h(\012)18 b(S)6 b Fz(\()p 1524 884 116 4 v Fw(R)1579 930 y Fv(+)1640 954 y Fz(\))31 b Fo(such)g(that)f(for)i(al)t(l)g Fu(t)24 b Fx(2)p 2451 884 V 25 w Fw(R)2505 930 y Fv(+)2566 954 y Fo(,)2623 887 y Fp(R)2688 883 y Fr(1)p 2688 892 29 3 v 2688 925 a(2)2662 988 y Ft(\000)2724 965 y Fr(1)p 2724 974 V 2724 1008 a(2)2780 954 y Fu(f)9 b Fz(\()p Fu(t;)14 b(\021)s Fz(\))31 b Fu(d\021)d Fz(=)c(0)p Fo(.)440 1093 y(Then)30 b(ther)l(e)g(exists)g(a)g(unique)k Fz(~)-47 b Fu(u)23 b Fx(2)g(P)i(\012)18 b(S)6 b Fz(\()p 1833 1023 116 4 v Fw(R)1888 1069 y Fv(+)1949 1093 y Fz(\))30 b Fo(such)g(that)g(:)947 1201 y Fp(\()1405 1262 y Fx(\000)p Fu(@)1519 1232 y Fv(2)1514 1283 y Fs(\021)1561 1262 y Fz(~)-47 b Fu(u)119 b Fz(=)f(2)p Fu(tf)38 b(on)30 b Fz(\012)2237 1232 y Fv(0)2274 1262 y Fu(;)1056 1391 y(@)1100 1403 y Fs(\021)1146 1391 y Fz(~)-48 b Fu(u)1188 1406 y Ft(j)1208 1429 y Fe(\021)q Fr(=)p Fq(\000)1338 1413 y Fr(1)p 1338 1420 29 3 v 1338 1451 a(2)1376 1429 y Fe(;)1405 1413 y Fr(1)p 1405 1420 V 1405 1451 a(2)1474 1391 y Fz(=)23 b(0)83 b Fu(and)1906 1325 y Fp(R)1972 1321 y Fr(1)p 1972 1330 V 1972 1363 a(2)1946 1425 y Ft(\000)2008 1403 y Fr(1)p 2008 1412 V 2008 1445 a(2)2069 1391 y Fz(~)-47 b Fu(u)o Fz(\()p Fu(t;)14 b(\021)s Fz(\))31 b Fu(d\021)26 b Fz(=)d(0)p Fu(:)3092 1343 y Fz(\(4.14\))440 1600 y Fo(Then,)31 b(for)f(al)t(l)h Fu(v)c Fx(2)c(V)1136 1570 y Fs(N)1199 1600 y Fo(,)30 b(we)g(have)h(:)1207 1752 y Fz(1)p 1207 1789 42 4 v 1207 1865 a(2)1272 1695 y Fp(Z)1318 1884 y Fv(\012)1365 1867 y Fr(0)1426 1752 y Fz(1)p 1426 1789 V 1432 1865 a Fu(t)1477 1808 y(@)1521 1820 y Fs(\021)1562 1808 y Fu(u@)1654 1820 y Fs(\021)p 1694 1762 44 4 v 1694 1808 a Fu(v)i(d\021)g(dt)23 b Fz(=)2068 1695 y Fp(Z)2114 1884 y Fv(\012)2161 1867 y Fr(0)2212 1808 y Fu(f)p 2262 1762 V 9 w(v)32 b(d\021)h(dt:)545 b Fz(\(4.15\))440 2025 y(As)28 b(ab)r(o)n(v)n(e,)e(w)n(e)h(ha)n(v)n(e)g(an)g(explicit)h (solution)f(of)g(problem)h(\(4.14\))e(:)440 2247 y FA(Lemma)j(4.7.)40 b Fo(L)l(et)29 b Fu(P)12 b Fz(\()p Fu(t;)i(\021)s Fz(\))24 b(=)1502 2144 y Fs(n)1463 2168 y Fp(X)1462 2347 y Fs(k)q Fv(=0)1583 2247 y Fz(\()p Fu(p)1657 2213 y Fs(e)1657 2268 y(k)1698 2247 y Fz(\()p Fu(t)p Fz(\)\()p Fu(i\021)s Fz(\))1929 2213 y Fv(2)p Fs(k)2022 2247 y Fz(+)18 b Fu(p)2147 2213 y Fs(o)2147 2268 y(k)2187 2247 y Fz(\()p Fu(t)p Fz(\)\()p Fu(i\021)s Fz(\))2418 2213 y Fv(2)p Fs(k)q Fv(+1)2577 2247 y Fz(\))30 b Fo(b)l(e)g(p)l(olynomial)h(in)f Fu(\021)440 2430 y Fo(such)g(that)g Fu(p)841 2400 y Fs(e)841 2453 y(k)904 2430 y Fx(2)24 b(S)6 b Fz(\()p Fw(R)1125 2400 y Fv(+)1187 2430 y Fz(\))p Fo(,)30 b Fu(p)1316 2400 y Fs(o)1316 2453 y(k)1380 2430 y Fx(2)23 b(S)6 b Fz(\()p Fw(R)1601 2400 y Fv(+)1662 2430 y Fz(\))30 b Fo(and)g(:)1206 2672 y Fx(8)p Fu(t)22 b Fx(2)h Fw(R)1437 2638 y Fv(+)1498 2672 y Fu(;)1598 2569 y Fs(n)1559 2594 y Fp(X)1558 2772 y Fs(k)q Fv(=0)1762 2616 y Fu(p)1804 2586 y Fs(e)1804 2640 y(k)1844 2616 y Fz(\()p Fu(t)p Fz(\))p 1703 2653 295 4 v 1703 2729 a(\(2)p Fu(k)e Fz(+)d(1\))2022 2555 y Fp(\022)2099 2616 y Fu(i)p 2093 2653 42 4 v 2093 2729 a Fz(2)2144 2555 y Fp(\023)2205 2573 y Fv(2)p Fs(k)q Fv(+1)2386 2672 y Fz(=)23 b(0)p Fu(:)553 b Fz(\(4.16\))440 2917 y Fo(Then)30 b(the)g(p)l(olynomial)i Fu(S)j Fo(de\014ne)l(d)30 b(by)g(:)620 3149 y Fu(S)5 b Fz(\()p Fu(t;)14 b(\021)s Fz(\))83 b(=)g(2)p Fu(t)1207 3045 y Fs(n)1167 3070 y Fp(X)1168 3249 y Fs(k)q Fv(=0)1518 3093 y Fu(p)1560 3063 y Fs(e)1560 3116 y(k)1600 3093 y Fz(\()p Fu(t)p Fz(\))p 1312 3130 590 4 v 1312 3206 a(\(2)p Fu(k)21 b Fz(+)d(1\)\(2)p Fu(k)j Fz(+)d(2\))1925 3032 y Fp(\022)1986 3149 y Fz(\()p Fu(i\021)s Fz(\))2123 3115 y Fv(2)p Fs(k)q Fv(+2)2300 3149 y Fx(\000)2547 3093 y Fu(i)2576 3063 y Fv(2)p Fs(k)q Fv(+2)p 2393 3130 495 4 v 2393 3206 a Fz(\(2)p Fu(k)j Fz(+)d(3\)2)2730 3182 y Fv(2)p Fs(k)q Fv(+2)2897 3032 y Fp(\023)1082 3394 y Fz(+2)p Fu(t)1271 3291 y Fs(n)1232 3316 y Fp(X)1233 3494 y Fs(k)q Fv(=0)1403 3338 y Fu(p)1445 3308 y Fs(o)1445 3362 y(k)1485 3338 y Fz(\()p Fu(t)p Fz(\))p 1376 3375 231 4 v 1376 3451 a(2)p Fu(k)j Fz(+)d(2)1630 3277 y Fp(\022)1701 3338 y Fz(\()p Fu(i\021)s Fz(\))1838 3308 y Fv(2)p Fs(k)q Fv(+3)p 1701 3375 296 4 v 1734 3451 a Fz(2)p Fu(k)j Fz(+)d(3)2025 3394 y Fx(\000)2118 3338 y Fu(i)2147 3308 y Fv(2)p Fs(k)q Fv(+3)2305 3338 y Fu(\021)p 2118 3375 231 4 v 2134 3451 a Fz(2)2176 3427 y Fv(2)p Fs(k)q Fv(+2)2359 3277 y Fp(\023)2434 3394 y Fu(;)3092 3278 y Fz(\(4.17\))440 3639 y Fo(is)30 b(the)g(unique)f(solution)h(for) h(the)f(pr)l(oblem)h(:)944 3742 y Fp(\()1398 3804 y Fx(\000)p Fu(@)1512 3774 y Fv(2)1507 3824 y Fs(\021)1548 3804 y Fu(S)121 b Fz(=)116 b(2)p Fu(tP)41 b Fo(on)30 b Fz(\012)2246 3774 y Fv(0)2283 3804 y Fu(;)1053 3933 y(@)1097 3945 y Fs(\021)1137 3933 y Fu(S)1188 3948 y Ft(j)1208 3971 y Fe(\021)q Fr(=)p Fq(\000)1338 3955 y Fr(1)p 1338 3962 29 3 v 1338 3993 a(2)1376 3971 y Fe(;)1405 3955 y Fr(1)p 1405 3962 V 1405 3993 a(2)1475 3933 y Fz(=)22 b(0)83 b Fo(and)1901 3866 y Fp(R)1967 3862 y Fr(1)p 1967 3871 V 1967 3905 a(2)1941 3967 y Ft(\000)2002 3945 y Fr(1)p 2002 3954 V 2002 3987 a(2)2059 3933 y Fu(S)5 b Fz(\()p Fu(t;)14 b(\021)s Fz(\))30 b Fu(d\021)c Fz(=)d(0)p Fu(:)3092 3884 y Fz(\(4.18\))440 4134 y FA(Remark)37 b(4.8.)43 b Fo(The)36 b(expr)l(ession)f(\(4.17\))i(shows)e(that)g Fu(S)40 b Fo(has)35 b(the)g(same)g(p)l(arity)h(as)f Fu(P)12 b Fo(.)440 4234 y(F)-6 b(urthermor)l(e,)28 b(if)f Fu(p)1062 4204 y Fs(e)1062 4257 y(k)1129 4234 y Fo(and)h Fu(p)1330 4204 y Fs(o)1330 4257 y(k)1397 4234 y Fo(ar)l(e)f(r)l(e)l(al,)h(then)e Fu(S)32 b Fo(c)l(an)26 b(b)l(e)h(written)f(as)h(a)g(p)l(olynomial)i(in) e(the)440 4333 y(variable)32 b Fz(\()p Fu(i\021)s Fz(\))e 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Fz(\()p Fu(t;)14 b(\021)s Fz(\))26 b Fu(d\021)g Fz(=)d(0)h(for)h(ev)n (ery)e Fu(t)h Fx(2)f Fw(R)2244 5263 y Fv(+)2305 5293 y Fz(.)36 b(It)25 b(is)g(easy)f(to)h(estimate)g Fu(@)3185 5305 y Fs(\021)3226 5293 y Fu(S)5 b Fz(,)440 5413 y(v)n(erify)27 b(the)h(Neumann)g(condition)f(and)h(compute)f(the)h(second)f(deriv)-5 b(ativ)n(e)27 b(of)h Fu(S)5 b Fz(.)222 b Fd(\003)1831 5662 y Fz(12)p eop %%Page: 13 13 13 12 bop 440 531 a Fy(5)135 b(Construction)45 b(of)g(a)g(formal)h (solution)440 730 y Fl(5.1)112 b(First)37 b(terms)f(of)i(the)f (construction)440 883 y Fz(W)-7 b(e)34 b(lo)r(ok)f(for)g(t)n(w)n(o)g (sequences)g(\()p Fu(m)1559 895 y Fs(k)1600 883 y Fz(\))1632 895 y Fs(k)q Ft(2)p Fn(N)1793 883 y Fx(2)g Fw(R)1935 853 y Fn(N)2021 883 y Fz(and)g(\()p Fu(u)2268 895 y Fs(k)2309 883 y Fz(\))2341 895 y Fs(k)q Ft(2)p Fn(N)2502 883 y Fx(2)g Fz(\()p Fx(P)c(\012)22 b(S)6 b Fz(\()p 2884 813 116 4 v Fw(R)2939 859 y Fv(+)3000 883 y Fz(\)\))3064 853 y Fn(N)3145 883 y Fz(suc)n(h)440 1049 y(that,)28 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Fs(\021)1903 2098 y Fu(u)1951 2110 y Fv(0)1988 2098 y Fu(@)2032 2110 y Fs(\021)p 2073 2052 44 4 v 2073 2098 a Fu(v)30 b(dt)e(d\021)f Fz(=)22 b(0)p Fu(:)627 b Fz(\(5.1\))440 2318 y(The)33 b(unique)f(condition)g(coming)g (from)g(this)h(relation)f(is)g(that)h Fu(u)2549 2330 y Fv(0)2618 2318 y Fz(dep)r(ends)g(only)f(on)g Fu(t)p Fz(.)440 2418 y(The)c(v)-5 b(anishing)27 b(of)g(the)h(co)r(e\016cien)n (t)g(of)f Fu(\013)1750 2388 y Fv(2)p Fs(k)1825 2418 y Fz(,)g(for)h Fu(k)d Fx(\025)e Fz(0,)k(giv)n(es)g(:)518 2692 y Fu(`)p Fz(\()p Fu(u)633 2704 y Fs(k)674 2692 y Fu(;)14 b(v)s Fz(\))19 b(+)898 2635 y(1)p 898 2672 42 4 v 898 2748 a(2)963 2579 y Fp(Z)1009 2767 y Fv(\012)1056 2751 y Fr(0)1117 2635 y Fz(1)p 1117 2672 V 1123 2748 a Fu(t)1168 2692 y(@)1212 2704 y Fs(\021)1253 2692 y Fu(u)1301 2704 y Fs(k)q Fv(+1)1425 2692 y Fu(@)1469 2704 y Fs(\021)p 1510 2646 44 4 v 1510 2692 a Fu(v)31 b(dt)c(d\021)g Fz(=)1880 2579 y Fp(Z)1926 2767 y Fv(\012)1973 2751 y Fr(0)2023 2525 y Fp(0)2023 2674 y(@)2138 2588 y Fs(k)2096 2613 y Fp(X)2099 2789 y Fs(j)s Fv(=0)2230 2692 y Fu(m)2303 2704 y Fs(j)2338 2692 y Fu(u)2386 2704 y Fs(k)q Ft(\000)p Fs(j)2509 2525 y Fp(1)2509 2674 y(A)p 2595 2646 V 2595 2692 a Fu(v)k(dt)d(d\021)s(;)14 b Fx(8)p Fu(v)26 b Fx(2)d(V)3140 2657 y Fs(N)3203 2692 y Fu(:)3092 2891 y Fz(\(5)p Fu(:)p Fz(2\))3263 2903 y Fb(k)440 2990 y Fz(Let)28 b(us)f(sho)n(w)g(that)h (this)g(determines)f Fu(u)1711 3002 y Fs(k)1779 2990 y Fz(and)h Fu(m)2014 3002 y Fs(k)2082 2990 y Fz(recursiv)n(ely)-7 b(.)440 3222 y Fl(5.2)112 b(Algorithm)35 b(for)i(the)g(determination)f (of)i(the)f(co)s(e\016cien)m(ts)440 3375 y Fz(This)31 b(metho)r(d)g(can)f(b)r(e)h(applied)g(at)g(ev)n(ery)e(order)g(to)i (determine)g(the)g(other)f(co)r(e\016cien)n(ts)440 3475 y(and)d(w)n(e)h(will)f(indeed)h(pro)n(v)n(e)e(the)i(follo)n(wing)f (prop)r(osition)f(:)440 3638 y FA(Prop)s(osition)37 b(5.1.)43 b Fo(We)35 b(determine)g(r)l(e)l(cursively)h(the)f(c)l(o)l(e\016cients) g Fu(u)2740 3650 y Fs(j)2807 3638 y Fx(2)e(P)28 b(\012)22 b(S)6 b Fz(\()p 3156 3568 116 4 v Fw(R)3211 3614 y Fv(+)3272 3638 y Fz(\))440 3737 y Fo(and)30 b Fu(m)674 3749 y Fs(j)732 3737 y Fx(2)24 b Fw(R)35 b Fo(such)30 b(that,)g(formal)t(ly)i(:)997 4018 y Fu(P)1050 4030 y Fs(\013)1111 3851 y Fp(0)1111 4001 y(@)1211 3915 y Ft(1)1184 3939 y Fp(X)1187 4116 y Fs(j)s Fv(=0)1318 4018 y Fu(\013)1371 3984 y Fv(2)p Fs(j)1439 4018 y Fu(u)1487 4030 y Fs(j)1522 3851 y Fp(1)1522 4001 y(A)1618 4018 y Fx(\021)1705 3851 y Fp(0)1705 4001 y(@)1805 3915 y Ft(1)1778 3939 y Fp(X)1780 4116 y Fs(j)s Fv(=0)1912 4018 y Fu(\013)1965 3984 y Fv(2)p Fs(j)2033 4018 y Fu(m)2106 4030 y Fs(j)2141 3851 y Fp(1)2141 4001 y(A)2227 3851 y(0)2227 4001 y(@)2327 3915 y Ft(1)2300 3939 y Fp(X)2303 4116 y Fs(j)s Fv(=0)2434 4018 y Fu(\013)2487 3984 y Fv(2)p Fs(j)2555 4018 y Fu(u)2603 4030 y Fs(j)2638 3851 y Fp(1)2638 4001 y(A)2724 4018 y Fu(:)387 b Fz(\(5.3\))440 4311 y Fo(F)-6 b(or)39 b(al)t(l)i Fu(j)k Fx(\025)40 b Fz(0)p Fo(,)h(we)f(c)l(an)f(cho)l(ose)h Fu(m)1665 4323 y Fs(j)1739 4311 y Fo(and)g Fu(u)1958 4323 y Fs(j)2033 4311 y Fz(=)g Fu(u)2186 4281 y Fv(0)2186 4332 y Fs(j)2248 4311 y Fz(+)30 b(~)-47 b Fu(u)2386 4323 y Fs(j)2420 4311 y Fo(,)42 b(with)e Fu(u)2725 4281 y Fv(0)2725 4332 y Fs(j)2802 4311 y Fx(2)g(S)6 b Fz(\()p 2985 4241 V Fw(R)3040 4287 y Fv(+)3101 4311 y Fz(\))40 b Fo(and)445 4428 y Fz(~)-47 b Fu(u)488 4440 y Fs(j)546 4428 y Fx(2)23 b(P)i(\012)18 b(S)6 b Fz(\()p 878 4359 V Fw(R)933 4404 y Fv(+)994 4428 y Fz(\))30 b Fo(uniquely)g(determine)l(d)g(by)g(the)g(r)l(elations)h(:) 1647 4607 y Fu(m)1720 4619 y Fv(0)1780 4607 y Fz(=)23 b Fu(\025)1916 4573 y Fv(mean)1916 4627 y(1)2074 4607 y Fu(;)990 b Fz(\(5)p Fu(:)p Fz(4\))3258 4619 y Fb(0)1415 4785 y Fu(m)1488 4797 y Fs(j)1545 4785 y Fz(=)23 b Fx(h)6 b Fz(~)-48 b Fu(u)1713 4797 y Fs(j)1748 4785 y Fu(;)14 b(Lu)1890 4751 y Fv(mean)1890 4806 y(1)2047 4785 y Fx(i)2079 4813 y Fs(L)2125 4797 y Fr(2)2157 4813 y Fv(\(\012)2230 4797 y Fr(0)2263 4813 y Fv(\))2307 4785 y Fu(:)770 b Fz(\(5)p Fu(:)p Fz(4\))3271 4797 y Fb(j)1747 4946 y Fz(~)-47 b Fu(u)1790 4958 y Fv(0)1850 4946 y Fz(=)23 b(0)p Fu(;)1084 b Fz(\(5)p Fu(:)p Fz(5\))3258 4958 y Fb(0)718 5039 y Fp(8)718 5114 y(>)718 5139 y(>)718 5164 y(<)718 5313 y(>)718 5338 y(>)718 5363 y(:)1178 5176 y Fx(\000)p Fu(@)1292 5146 y Fv(2)1287 5197 y Fs(\021)1334 5176 y Fz(~)-47 b Fu(u)1377 5188 y Fs(j)1531 5176 y Fz(=)118 b(2)p Fu(t)1800 5034 y Fp( )1868 5069 y Fs(j)s Ft(\000)p Fv(1)1866 5098 y Fp(X)1872 5274 y Fs(i)p Fv(=0)1999 5176 y Fu(m)2072 5188 y Fs(j)s Ft(\000)p Fv(1)p Ft(\000)p Fs(i)2268 5176 y Fu(u)2316 5188 y Fs(i)2361 5176 y Fx(\000)18 b Fu(Lu)2549 5188 y Fs(j)s Ft(\000)p Fv(1)2668 5034 y Fp(!)2748 5176 y Fu(;)833 5380 y(@)877 5392 y Fs(\021)923 5380 y Fz(~)-48 b Fu(u)965 5392 y Fs(j)992 5403 y Fq(j)1011 5432 y Fe(\021)q Fr(=)p Fq(\000)1141 5416 y Fr(1)p 1141 5423 29 3 v 1141 5455 a(2)1180 5432 y Fe(;)1209 5416 y Fr(1)p 1209 5423 V 1209 5455 a(2)1282 5380 y Fz(=)23 b(0)83 b Fu(and)1714 5313 y Fp(R)1780 5309 y Fr(1)p 1780 5318 V 1780 5352 a(2)1754 5414 y Ft(\000)1816 5392 y Fr(1)p 1816 5401 V 1816 5434 a(2)1877 5380 y Fz(~)-47 b Fu(u)1920 5392 y Fs(j)1984 5380 y Fu(d\021)26 b Fz(=)d(0)p Fu(:)3100 5259 y Fz(\(5)p Fu(:)p Fz(5\))3271 5271 y Fb(j)1831 5662 y Fz(13)p eop %%Page: 14 14 14 13 bop 1660 531 a Fu(u)1708 497 y Fv(0)1708 552 y(0)1768 531 y Fz(=)23 b Fu(u)1904 497 y Fv(mean)1904 552 y(1)2061 531 y Fu(;)1003 b Fz(\(5)p Fu(:)p Fz(6\))3258 543 y Fb(0)691 603 y Fp(8)691 678 y(>)691 703 y(<)691 852 y(>)691 877 y(:)886 745 y Fz(\()p Fu(L)975 715 y Fv(mean)1152 745 y Fx(\000)18 b Fu(\025)1283 715 y Fv(mean)1283 766 y(1)1441 745 y Fz(\))p Fu(u)1521 715 y Fv(0)1521 767 y Fs(j)1641 745 y Fz(=)83 b Fx(\000)1868 679 y Fp(R)1932 675 y Fr(1)p 1932 684 29 3 v 1932 717 a(2)1906 779 y Ft(\000)1968 757 y Fr(1)p 1968 766 V 1968 799 a(2)2024 745 y Fu(L)5 b Fz(~)-47 b Fu(u)2129 757 y Fs(j)2193 745 y Fu(d\021)22 b Fz(+)2426 638 y Fs(j)2382 667 y Fp(X)2388 843 y Fs(i)p Fv(=1)2516 745 y Fu(m)2589 757 y Fs(i)2616 745 y Fu(u)2664 711 y Fv(0)2664 766 y Fs(j)s Ft(\000)p Fs(i)2774 745 y Fu(;)807 864 y Fp(R)862 884 y Ft(1)846 960 y Fv(0)946 931 y Fu(u)994 900 y Fv(0)994 952 y Fs(j)1031 931 y Fz(\()p Fu(t)p Fz(\))p 1155 858 301 4 v 30 w Fu(u)1203 902 y Fv(mean)1203 953 y(1)1361 931 y Fz(\()p Fu(t)p Fz(\))30 b Fu(dt)83 b Fz(=)g(0)p Fu(:)3100 798 y Fz(\(5)p Fu(:)p Fz(6\))3271 810 y Fb(j)440 1080 y Fo(Pr)l(o)l(of)28 b(:)37 b Fz(T)-7 b(o)24 b(pro)n(v)n(e)f(Prop)r(osition)g(5.1,)h(w)n(e)g (expand)g(in)h(p)r(o)n(w)n(er)e(of)h Fu(\013)2541 1050 y Fv(2)2603 1080 y Fz(Relation)g(\(5.3\),)h(can-)440 1180 y(cel)h(the)g(co)r(e\016cien)n(ts)f(of)h Fu(\013)1268 1150 y Fv(2)p Fs(k)1368 1180 y Fz(for)f Fu(k)h Fx(\025)d(\000)p Fz(1.)35 b(W)-7 b(e)27 b(ha)n(v)n(e)d(already)g(studied)j(the)f(case)f Fu(k)h Fz(=)c Fx(\000)p Fz(1)440 1280 y(and)29 b(deduce)35 b(~)-47 b Fu(u)930 1292 y Fv(0)993 1280 y Fz(=)25 b(0.)42 b(W)-7 b(e)30 b(no)n(w)f(lo)r(ok)g(at)g(Relations)g(\(5)p Fu(:)p Fz(2\))2334 1292 y Fs(k)2404 1280 y Fz(for)g Fu(k)g Fx(\025)c Fz(0.)42 b(W)-7 b(e)30 b(denote)g(b)n(y)440 1379 y Fx(P)7 b Fz(\()p Fu(k)s Fz(\),)28 b(for)f Fu(k)f Fx(2)d Fw(N)t Fz(,)34 b(the)28 b(prop)r(ert)n(y)e(:)490 1479 y Fx(P)7 b Fz(\()p Fu(k)s Fz(\))27 b(:)100 b(\\the)30 b(cancellation)e(of)i(the)g(co)r(e\016cien)n(t)f Fu(\013)2144 1449 y Fv(2)p Fs(k)2218 1479 y Fz(,)i(giv)n(en)d(b)n(y)i(Relation)f (\(5)p Fu(:)p Fz(2\))3114 1491 y Fs(k)3154 1479 y Fz(,)815 1578 y(determines)21 b(the)g(functions)h Fu(u)1765 1548 y Fv(0)1765 1602 y Fs(k)1805 1578 y Fz(,)28 b(~)-47 b Fu(u)1899 1590 y Fs(k)q Fv(+1)2044 1578 y Fz(and)21 b(the)g(real)f Fu(m)2563 1590 y Fs(k)2625 1578 y Fz(whic)n(h)h(are)f(giv)n(en)815 1678 y(b)n(y)27 b(solving)g(\(5)p Fu(:)p Fz(4\))1381 1690 y Fs(k)1421 1678 y Fz(,)h(\(5)p Fu(:)p Fz(6\))1643 1690 y Fs(k)1711 1678 y Fz(and)g(\(5)p Fu(:)p Fz(5\))2044 1690 y Fs(k)q Fv(+1)2168 1678 y Fz(".)440 1778 y(W)-7 b(e)28 b(no)n(w)f(pro)n(v)n(e)f(that)i Fx(P)7 b Fz(\()p Fu(k)s Fz(\))27 b(holds)h(for)f(ev)n(ery)f Fu(k)g Fx(2)d Fw(N)t Fz(.)565 1978 y(W)-7 b(e)23 b(lo)r(ok)f(at)h(the)g(relation)f (coming)g(from)h(the)g(v)-5 b(anishing)22 b(of)h(the)g(constan)n(t)g (co)r(e\016cien)n(t)440 2077 y(giv)n(en)c(b)n(y)h(Relation)f(\(5)p Fu(:)p Fz(2\))1253 2089 y Fv(0)1290 2077 y Fz(.)34 b(W)-7 b(e)20 b(restrict)g(Relation)f(\(5)p Fu(:)p Fz(2\))2258 2089 y Fv(0)2315 2077 y Fz(to)g(functions)i(only)e(dep)r(ending)440 2177 y(on)27 b Fu(t)h Fz(and)f(obtain)h(a)f(new)h(relation)e(:)1072 2328 y Fx(8)p Fu(v)f Fx(2)e(V)1320 2293 y Fs(N)1313 2348 y Fv(mean)1471 2328 y Fu(;)42 b(`)1571 2293 y Fv(mean)1728 2328 y Fz(\()p Fu(u)1808 2340 y Fv(0)1845 2328 y Fu(;)14 b(v)s Fz(\))24 b(=)f Fu(m)2142 2340 y Fv(0)2193 2328 y Fx(h)p Fu(u)2273 2340 y Fv(0)2310 2328 y Fu(;)14 b(v)s Fx(i)2422 2352 y Fs(L)2468 2336 y Fr(2)2500 2352 y Fv(\(\012)2573 2336 y Fr(0)2606 2352 y Fv(\))2650 2328 y Fu(:)461 b Fz(\(5.7\))440 2478 y(So)31 b(w)n(e)f(determine)i Fu(u)1124 2490 y Fv(0)1192 2478 y Fz(and)e Fu(m)1429 2490 y Fv(0)1498 2478 y Fz(b)n(y)g(solving)g(the)i(sp)r(ectral)e(problem)h Fu(L)2745 2448 y Fv(mean)2902 2478 y Fu(u)2950 2490 y Fv(0)3016 2478 y Fz(=)d Fu(m)3182 2490 y Fv(0)3220 2478 y Fu(u)3268 2490 y Fv(0)440 2578 y Fz(and)f(w)n(e)h(c)n(ho)r(ose)e(the) i(fundamen)n(tal)g(state)f(\()p Fu(\025)1893 2548 y Fv(mean)1893 2599 y(1)2052 2578 y Fu(;)14 b(u)2137 2548 y Fv(mean)2137 2599 y(1)2294 2578 y Fz(\).)440 2678 y(W)-7 b(e)28 b(return)f(to)h(the) g(initial)f(Relation)h(\(5)p Fu(:)p Fz(2\))1826 2690 y Fv(0)1890 2678 y Fz(where)f(the)h(only)f(unkno)n(wn)h(is)f Fu(u)2942 2690 y Fv(1)3007 2678 y Fz(:)493 2871 y Fx(8)p Fu(v)e Fx(2)f(V)742 2836 y Fs(N)804 2871 y Fu(;)879 2814 y Fz(1)p 879 2852 42 4 v 879 2928 a(2)944 2758 y Fp(Z)990 2946 y Fv(\012)1037 2930 y Fr(0)1088 2871 y Fu(@)1132 2883 y Fs(\021)1173 2871 y Fu(u)1221 2883 y Fv(1)1257 2871 y Fu(@)1301 2883 y Fs(\021)p 1342 2825 44 4 v 1342 2871 a Fu(v)1423 2814 y Fz(1)p 1423 2852 42 4 v 1429 2928 a Fu(t)1502 2871 y(dt)k(d\021)e Fz(=)d Fx(h)p Fu(\025)1881 2836 y Fv(mean)1881 2891 y(1)2040 2871 y Fu(u)2088 2836 y Fv(mean)2088 2891 y(1)2245 2871 y Fu(;)14 b(v)s Fx(i)2358 2896 y Fs(L)2404 2879 y Fr(2)2436 2896 y Fv(\(\012)2509 2879 y Fr(0)2541 2896 y Fv(\))2590 2871 y Fx(\000)k Fu(`)p Fz(\()p Fu(u)2788 2836 y Fv(mean)2788 2891 y(1)2945 2871 y Fu(;)c(v)s Fz(\))p Fu(:)54 b Fz(\(5.8\))440 3082 y(Since)31 b Fu(u)708 3052 y Fv(mean)708 3103 y(1)894 3082 y Fx(2)f(S)6 b Fz(\()p 1067 3012 116 4 v Fw(R)1121 3058 y Fv(+)1182 3082 y Fz(\),)33 b(the)e(in)n(tegration)f(b)n(y)h(parts)f Fu(`)p Fz(\()p Fu(u)2292 3052 y Fv(mean)2292 3103 y(1)2449 3082 y Fu(;)14 b(v)s Fz(\))29 b(=)g Fx(h)p Fu(Lu)2821 3052 y Fv(mean)2821 3103 y(1)2978 3082 y Fu(;)14 b(v)s Fx(i)3090 3107 y Fs(L)3136 3091 y Fr(2)3168 3107 y Fv(\(\012)3241 3091 y Fr(0)3274 3107 y Fv(\))440 3200 y Fz(holds)28 b(for)f(an)n(y)g Fu(v)g Fx(2)d(V)1146 3170 y Fs(N)1139 3223 y(`)1237 3200 y Fz(and)j(so)h(w)n(e)f(de\014ne)i(the)f(function)g Fu(v)2372 3212 y Fv(0)2419 3200 y Fz(:)52 b(=)23 b(\(\()p Fu(\025)2694 3170 y Fv(mean)2694 3220 y(1)2871 3200 y Fx(\000)c Fu(L)p Fz(\))p Fu(u)3092 3170 y Fv(mean)3092 3220 y(1)3249 3200 y Fz(\).)440 3308 y(W)-7 b(e)28 b(can)g(c)n(ho)r (ose)e(a)i(function)g Fu(u)1442 3320 y Fv(1)1507 3308 y Fz(suc)n(h)f(that)i Fu(u)1923 3320 y Fv(1)1983 3308 y Fz(=)23 b Fu(u)2119 3278 y Fv(0)2119 3329 y(1)2174 3308 y Fz(+)h(~)-47 b Fu(u)2306 3320 y Fv(1)2370 3308 y Fz(with)29 b Fu(u)2608 3278 y Fv(0)2608 3329 y(1)2668 3308 y Fx(2)24 b(S)6 b Fz(\()p 2835 3238 V Fw(R)2890 3284 y Fv(+)2951 3308 y Fz(\))28 b(free)g(and)445 3408 y(~)-47 b Fu(u)488 3420 y Fv(1)552 3408 y Fz(only)28 b(determined)f(b)n(y)h(Lemma)f(4.6)g(with)h Fu(f)k Fz(=)22 b Fu(v)2103 3420 y Fv(0)2141 3408 y Fz(.)37 b(So)27 b Fx(P)7 b Fz(\(0\))27 b(holds.)565 3607 y(Let)36 b Fu(k)k Fx(2)e Fw(N)t Fz(.)68 b(W)-7 b(e)37 b(assume)e(that)i(prop)r(ert)n(y)e Fx(P)7 b Fz(\()p Fu(j)e Fz(\))36 b(holds)g(for)g(an)n(y)f Fu(j)42 b Fx(\024)37 b Fu(k)s Fz(.)63 b(So,)38 b(b)n(y)440 3707 y(induction,)43 b(w)n(e)c(ha)n(v)n(e)f(determined)i(the)g (functions)45 b(~)-47 b Fu(u)2203 3719 y Fs(j)2237 3707 y Fz(,)43 b Fu(u)2351 3677 y Fv(0)2351 3728 y Fs(j)2388 3707 y Fz(,)f(the)e(reals)f Fu(m)2888 3719 y Fs(j)2962 3707 y Fz(satisfying)440 3806 y(Relations)23 b(\(5)p Fu(:)p Fz(5\))973 3818 y Fs(j)1008 3806 y Fz(,)h(\(5)p Fu(:)p Fz(6\))1226 3818 y Fs(j)1284 3806 y Fz(and)g(\(5)p Fu(:)p Fz(4\))1613 3818 y Fs(j)1671 3806 y Fz(for)f Fu(j)28 b Fz(=)23 b(0)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)26 b Fz(and)d(the)h (function)30 b(~)-48 b Fu(u)2904 3818 y Fs(k)q Fv(+1)3052 3806 y Fz(solving)440 3906 y(\(5)p Fu(:)p Fz(5\))611 3918 y Fs(k)q Fv(+1)736 3906 y Fz(.)37 b(W)-7 b(e)28 b(v)-5 b(anish)27 b(the)h(co)r(e\016cien)n(t)f(of)h Fu(\013)1873 3876 y Fv(2)p Fs(k)q Fv(+2)2059 3906 y Fz(giv)n(en)f(b)n(y)g(\(5)p Fu(:)p Fz(2\))2562 3918 y Fs(k)q Fv(+1)2687 3906 y Fz(.)440 4006 y(W)-7 b(e)20 b(c)n(ho)r(ose)e Fu(v)27 b Fz(=)22 b Fu(u)1032 3975 y Fv(mean)1032 4026 y(1)1210 4006 y Fz(in)d(\(5)p Fu(:)p Fz(2\))1469 4018 y Fs(k)q Fv(+1)1594 4006 y Fz(.)34 b(By)20 b(assumptions)e(on)25 b(~)-47 b Fu(u)2393 4018 y Fs(j)2427 4006 y Fz(,)22 b Fu(u)2520 3975 y Fv(0)2520 4027 y Fs(j)2556 4006 y Fz(,)g Fu(m)2674 4018 y Fs(j)2728 4006 y Fz(and)e(particularly)440 4105 y(due)26 b(to)g(the)h(expression)d(of)32 b(~)-48 b Fu(u)1376 4117 y Fs(k)q Fv(+1)1527 4105 y Fz(giv)n(en)25 b(b)n(y)h(\(5)p Fu(:)p Fz(4\))2027 4117 y Fs(k)q Fv(+1)2152 4105 y Fz(,)g(w)n(e)g (determine)g Fu(m)2781 4117 y Fs(k)q Fv(+1)2932 4105 y Fz(exactly)g(b)n(y)440 4205 y(\(5)p Fu(:)p Fz(5\))611 4217 y Fs(k)q Fv(+1)763 4205 y Fz(:)1325 4304 y Fu(m)1398 4316 y Fs(k)q Fv(+1)1545 4304 y Fz(=)d Fx(h)6 b Fz(~)-48 b Fu(u)1713 4316 y Fs(k)q Fv(+1)1838 4304 y Fu(;)14 b(Lu)1980 4270 y Fv(mean)1980 4325 y(1)2137 4304 y Fx(i)2169 4329 y Fs(L)2215 4313 y Fr(2)2247 4329 y Fv(\(\012)2320 4313 y Fr(0)2353 4329 y Fv(\))2397 4304 y Fu(:)714 b Fz(\(5.9\))440 4480 y(W)-7 b(e)28 b(no)n(w)g(restrict)f(Relation)h(\(5)p Fu(:)p Fz(2\))1549 4492 y Fs(k)q Fv(+1)1702 4480 y Fz(to)f(functions)i Fu(v)e Fx(2)d(V)2366 4450 y Fs(N)2359 4501 y Fv(mean)2517 4480 y Fz(.)38 b(Since)2795 4414 y Fp(R)2861 4410 y Fr(1)p 2861 4419 29 3 v 2861 4452 a(2)2835 4514 y Ft(\000)2896 4492 y Fr(1)p 2896 4501 V 2896 4534 a(2)2958 4480 y Fz(~)-47 b Fu(u)3001 4492 y Fs(j)3063 4480 y Fu(d\021)27 b Fz(=)d(0)440 4610 y(for)j(an)n(y)g Fu(j)h Fz(=)23 b(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k) 21 b Fz(+)d(1,)27 b(w)n(e)g(ha)n(v)n(e)g(to)g(\014nd)h Fu(u)1968 4580 y Fv(0)1968 4634 y Fs(k)q Fv(+1)2120 4610 y Fz(indep)r(enden)n(t)h(of)e Fu(\021)k Fz(suc)n(h)d(that)f(:)482 4870 y Fx(8)p Fu(v)e Fx(2)e(V)730 4836 y Fs(N)723 4891 y Fv(mean)881 4870 y Fu(;)38 b(`)977 4836 y Fv(mean)1134 4870 y Fz(\()p Fu(u)1214 4836 y Fv(0)1214 4891 y Fs(k)q Fv(+1)1339 4870 y Fu(;)14 b(v)s Fz(\))23 b(=)g Fx(\000)p Fu(`)p Fz(\()5 b(~)-47 b Fu(u)1742 4882 y Fs(k)q Fv(+1)1866 4870 y Fu(;)14 b(v)s Fz(\))c(+)2063 4757 y Fp(Z)2147 4778 y Ft(1)2110 4946 y Fv(0)2231 4703 y Fp(0)2231 4853 y(@)2303 4766 y Fs(k)q Fv(+1)2304 4791 y Fp(X)2306 4968 y Fs(j)s Fv(=0)2438 4870 y Fu(m)2511 4882 y Fs(j)2546 4870 y Fu(u)2594 4836 y Fv(0)2594 4891 y Fs(k)q Fv(+1)p Ft(\000)p Fs(j)2801 4703 y Fp(1)2801 4853 y(A)p 2888 4824 44 4 v 2888 4870 a Fu(v)26 b(dt:)42 b Fz(\(5.10\))440 5211 y(W)-7 b(e)26 b(de\014ne)g(the)g(function)g Fu(f)18 b Fz(:)50 b(=)23 b Fx(\000)1582 5144 y Fp(R)1647 5140 y Fr(1)p 1647 5149 29 3 v 1647 5182 a(2)1621 5244 y Ft(\000)1683 5222 y Fr(1)p 1683 5231 V 1683 5264 a(2)1739 5211 y Fu(L)5 b Fz(~)-47 b Fu(u)1844 5223 y Fs(k)q Fv(+1)1993 5211 y Fu(d\021)18 b Fz(+)2216 5107 y Fs(k)2174 5132 y Fp(X)2177 5309 y Fs(j)s Fv(=0)2308 5211 y Fu(m)2381 5223 y Fs(k)q Fv(+1)p Ft(\000)p Fs(j)2588 5211 y Fu(u)2636 5176 y Fv(0)2636 5231 y Fs(j)2673 5211 y Fz(.)36 b(By)26 b(regularit)n(y)d(of)445 5413 y(~)-47 b Fu(u)488 5425 y Fs(k)q Fv(+1)637 5413 y Fz(and)25 b(assumptions)f(on)g Fu(u)1425 5383 y Fv(0)1425 5435 y Fs(j)1486 5413 y Fz(for)h Fu(j)j Fz(=)22 b(0)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s Fz(,)25 b(it)g(is)g(easy)f(to)g(pro)n(v)n (e)f(that)i Fu(f)32 b Fx(2)23 b(S)6 b Fz(\()p 3156 5343 116 4 v Fw(R)3211 5389 y Fv(+)3272 5413 y Fz(\))1831 5662 y(14)p eop %%Page: 15 15 15 14 bop 440 531 a Fz(and)22 b Fu(f)31 b Fz(is)23 b(orthogonal)d(to)i Fu(u)1301 501 y Fv(mean)1301 552 y(1)1458 531 y Fz(.)36 b(W)-7 b(e)22 b(apply)h(Lemma)f(4.4)f(and)h(so)g(obtain)g(a)g(unique)h Fu(u)3180 501 y Fv(0)3180 555 y Fs(k)q Fv(+1)440 631 y Fz(satisfying)28 b(the)i(sp)r(ectral)e(problem)h(\(4.8\))f(and)h(so)g (Relation)f(\(5.10\).)41 b(W)-7 b(e)29 b(see)g(that)g Fu(u)3180 601 y Fv(0)3180 654 y Fs(k)q Fv(+1)440 731 y Fz(solv)n(es)d(\(5)p Fu(:)p Fz(6\))847 743 y Fs(k)q Fv(+1)972 731 y Fz(.)440 830 y(W)-7 b(e)28 b(no)n(w)f(come)g(bac)n(k)g (to)g(Relation)h(\(5)p Fu(:)p Fz(2\))1768 842 y Fs(k)q Fv(+1)1920 830 y Fz(to)f(determine)33 b(~)-47 b Fu(u)2457 842 y Fs(k)q Fv(+2)2609 830 y Fz(suc)n(h)27 b(that)h(:)552 1089 y Fx(8)p Fu(v)d Fx(2)f(V)801 1054 y Fs(N)863 1089 y Fu(;)928 976 y Fp(Z)974 1164 y Fv(\012)1021 1148 y Fr(0)1082 1032 y Fz(1)p 1082 1069 42 4 v 1088 1146 a Fu(t)1133 1089 y(@)1177 1101 y Fs(\021)1218 1089 y Fu(u)1266 1101 y Fs(k)q Fv(+2)1390 1089 y Fu(@)1434 1101 y Fs(\021)p 1475 1043 44 4 v 1475 1089 a Fu(v)31 b(dt)c(d\021)g Fz(=)c(2)1901 947 y Fp(*)1967 985 y Fs(k)q Fv(+1)1967 1010 y Fp(X)1970 1187 y Fs(j)s Fv(=0)2101 1089 y Fu(m)2174 1101 y Fs(j)2209 1089 y Fu(u)2257 1101 y Fs(k)q Fv(+1)p Ft(\000)p Fs(j)2483 1089 y Fx(\000)18 b Fu(Lu)2671 1101 y Fs(k)q Fv(+1)2795 1089 y Fu(;)c(v)2875 947 y Fp(+)2942 1210 y Fs(L)2988 1194 y Fr(2)3020 1210 y Fv(\(\012)3093 1194 y Fr(0)3125 1210 y Fv(\))3169 1089 y Fu(:)440 1462 y Fz(W)-7 b(e)31 b(apply)f(Lemma)h(4.6)f(with)h Fu(f)36 b Fz(=)1616 1359 y Fs(k)q Fv(+1)1616 1384 y Fp(X)1619 1560 y Fs(j)s Fv(=0)1751 1462 y Fu(m)1824 1474 y Fs(j)1858 1462 y Fu(u)1906 1474 y Fs(k)q Fv(+1)p Ft(\000)p Fs(j)2134 1462 y Fx(\000)20 b Fu(Lu)2324 1474 y Fs(k)q Fv(+1)2448 1462 y Fz(.)46 b(Then)31 b Fu(f)37 b Fx(2)28 b(P)f(\012)20 b(S)6 b Fz(\()p 3156 1393 116 4 v Fw(R)3211 1438 y Fv(+)3272 1462 y Fz(\))440 1686 y(and)27 b(it)g(is)f(easy)g(to)h(sho)n(w)f(that)1432 1619 y Fp(R)1497 1615 y Fr(1)p 1497 1624 29 3 v 1497 1657 a(2)1471 1720 y Ft(\000)1533 1697 y Fr(1)p 1533 1706 V 1533 1740 a(2)1589 1686 y Fu(f)9 b Fz(\()p Fu(t;)14 b(\021)s Fz(\))27 b Fu(d\021)f Fz(=)d(0,)k Fx(8)p Fu(t)22 b Fx(2)h Fw(R)2362 1656 y Fv(+)2450 1686 y Fz(\(using)2699 1619 y Fp(R)2764 1615 y Fr(1)p 2764 1624 V 2764 1657 a(2)2738 1720 y Ft(\000)2800 1697 y Fr(1)p 2800 1706 V 2800 1740 a(2)2861 1686 y Fz(~)-47 b Fu(u)2904 1698 y Fs(j)2938 1686 y Fu(d\021)27 b Fz(=)c(0)j(for)440 1816 y Fu(j)33 b Fz(=)27 b(0)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)24 b Fz(+)c(1)30 b(and)g(the)h(expression)e(of)h(\()p Fu(L)1948 1785 y Fv(mean)2127 1816 y Fx(\000)20 b Fu(\025)2260 1785 y Fv(mean)2260 1836 y(1)2418 1816 y Fz(\))p Fu(u)2498 1785 y Fv(0)2498 1839 y Fs(k)q Fv(+1)2623 1816 y Fz(\).)46 b(Then)30 b(w)n(e)h(c)n(ho)r(ose)445 1915 y(~)-47 b Fu(u)488 1927 y Fs(k)q Fv(+2)640 1915 y Fz(uniquely)28 b(determined)g(b)n(y)f (Relation)g(\(5)p Fu(:)p Fz(5\))2030 1927 y Fs(k)q Fv(+2)2155 1915 y Fz(.)440 2015 y(So)g(w)n(e)h(ha)n(v)n(e)e(established)h(that)h Fx(P)7 b Fz(\()p Fu(k)21 b Fz(+)d(1\))28 b(holds.)440 2114 y(Th)n(us)f Fx(P)7 b Fz(\()p Fu(k)s Fz(\))28 b(holds)f(for)g(ev)n (ery)g(in)n(teger)f Fu(k)g Fx(\025)d Fz(0)k(and)g(Prop)r(osition)f(5.1) h(is)h(established.)76 b Fd(\003)440 2346 y Fl(5.3)112 b(P)m(articular)36 b(form)g(of)i(the)f(co)s(e\016cien)m(ts)g(in)g(the)g (asymptotics)440 2499 y Fz(W)-7 b(e)28 b(can)g(determine)g(an)f (explicit)i(expression)d(for)h(ev)n(ery)g(co)r(e\016cien)n(t)h(giv)n (en)f(b)n(y)g(Prop)r(osi-)440 2599 y(tion)j(5.1)f(and)h(\(5)p Fu(:)p Fz(4\))1084 2611 y Fs(j)1118 2599 y Fz(,)h(\(5)p Fu(:)p Fz(5\))1343 2611 y Fs(j)1377 2599 y Fz(,)g(\(5)p Fu(:)p Fz(6\))1602 2611 y Fs(j)1637 2599 y Fz(.)43 b(Just)30 b(b)r(efore,)g(w)n(e)g(mak)n(e)f(an)g(easy)g(remark)f(ab)r(out)440 2698 y Fu(u)488 2668 y Fv(mean)488 2719 y(1)646 2698 y Fz(.)440 2859 y FA(Remark)j(5.2.)40 b Fo(The)31 b(applic)l(ation)h(:) 1333 3036 y Fz(\010)9 b(:)28 b Fu(P)35 b Fx(7!)23 b Fz(\()p Fu(u)1727 3002 y Fv(mean)1727 3056 y(1)1885 3036 y Fz(\))1917 2994 y Ft(\000)p Fv(1)2020 3036 y Fu(L)p Fz(\()p Fu(P)12 b(u)2222 3002 y Fv(mean)2222 3056 y(1)2379 3036 y Fz(\))440 3212 y Fo(is)30 b(wel)t(l)h(de\014ne)l(d)f(fr)l(om)g Fx(P)1254 3145 y Fp(\000)1292 3212 y Fw(R)1346 3182 y Fv(+)1425 3212 y Fx(\002)1508 3145 y Fp(\003)1543 3212 y Fx(\000)1618 3180 y Fv(1)p 1617 3194 34 4 v 1617 3241 a(2)1660 3212 y Fu(;)1707 3180 y Fv(1)p 1707 3194 V 1707 3241 a(2)1750 3145 y Fp(\002\001)1853 3212 y Fo(onto)f Fx(P)2120 3145 y Fp(\000)2158 3212 y Fw(R)2212 3182 y Fv(+)2291 3212 y Fx(\002)2374 3145 y Fp(\003)2409 3212 y Fx(\000)2484 3180 y Fv(1)p 2484 3194 V 2484 3241 a(2)2527 3212 y Fu(;)2574 3180 y Fv(1)p 2574 3194 V 2574 3241 a(2)2617 3145 y Fp(\002)o(\001)2689 3212 y Fo(.)440 3387 y(If)34 b Fu(P)41 b Fz(=)759 3283 y Fs(n)720 3308 y Fp(X)719 3487 y Fs(k)q Fv(=0)854 3387 y Fu(p)896 3399 y Fs(k)937 3387 y Fz(\()p Fu(t)p Fz(\)\()p Fu(i\021)s Fz(\))1168 3353 y Fs(k)1243 3387 y Fo(with)34 b Fu(p)1469 3399 y Fs(k)1539 3387 y Fx(2)c(P)7 b Fz(\()p Fw(R)1775 3357 y Fv(+)1836 3387 y Fz(\))p Fo(,)35 b(then)e Fz(\010\()p Fu(P)12 b Fz(\))30 b(=)2469 3283 y Fs(n)2430 3308 y Fp(X)2429 3487 y Fs(k)q Fv(=0)2571 3387 y Fz(~)-49 b Fu(p)2606 3399 y Fs(k)2647 3387 y Fz(\()p Fu(t)p Fz(\)\()p Fu(i\021)s Fz(\))2878 3353 y Fs(k)2953 3387 y Fo(with)41 b Fz(~)-49 b Fu(p)3179 3399 y Fs(k)3249 3387 y Fx(2)440 3569 y(P)7 b Fz(\()p Fw(R)591 3539 y Fv(+)652 3569 y Fz(\))30 b Fo(such)g(that)g(for)g Fu(k)c Fz(=)d(0)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)29 b Fo(:)528 3796 y Fz(~)-49 b Fu(p)563 3808 y Fs(k)627 3796 y Fz(=)715 3679 y Fp(\022)820 3740 y Fz(1)p 786 3777 111 4 v 786 3793 a Fx(p)p 855 3793 42 4 v 69 x Fz(3)925 3796 y Fx(\000)1023 3740 y Fu(t)p 1018 3777 V 1018 3853 a Fz(6)1069 3679 y Fp(\023)1144 3796 y Fu(p)1186 3808 y Fs(k)1245 3796 y Fz(+)1358 3740 y(2)p Fu(t)p 1338 3777 111 4 v 1338 3793 a Fx(p)p 1407 3793 42 4 v 69 x Fz(3)1459 3796 y Fu(p)1501 3762 y 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(4.5)g(and)g(4.7)g(easily)g(lead)g(to)g(the)h(prop)r(osition)f(:)440 4439 y FA(Prop)s(osition)44 b(5.3.)k Fo(F)-6 b(or)41 b Fu(k)47 b Fx(2)d Fw(N)t Fo(,)50 b(we)42 b(c)l(onsider)47 b Fz(~)-47 b Fu(u)2174 4451 y Fs(k)2256 4439 y Fo(and)41 b Fu(u)2476 4409 y Fv(0)2476 4463 y Fs(k)2558 4439 y Fo(determine)l(d)h(by)f(R)l(ela-)440 4548 y(tions)21 b Fz(\(5)p Fu(:)p Fz(5\))808 4560 y Fs(k)870 4548 y Fo(and)h Fz(\(5)p Fu(:)p Fz(6\))1194 4560 y Fs(k)1234 4548 y Fo(.)36 b(Then)22 b(ther)l(e)f(exist)g(two)h(p)l(olynomials)2495 4527 y Fz(~)2476 4548 y Fu(P)2529 4560 y Fs(k)2593 4548 y Fx(2)i(P)2750 4480 y Fp(\000)2788 4548 y Fw(R)2842 4517 y Fv(+)2922 4548 y Fx(\002)3005 4480 y Fp(\003)3039 4548 y Fx(\000)3114 4515 y Fv(1)p 3114 4529 34 4 v 3114 4576 a(2)3157 4548 y Fu(;)3204 4515 y Fv(1)p 3204 4529 V 3204 4576 a(2)3247 4480 y Fp(\002)o(\001)440 4656 y Fo(and)30 b Fu(P)666 4625 y Fv(0)654 4679 y Fs(k)726 4656 y Fx(2)24 b(P)883 4588 y Fp(\000\003)956 4656 y Fx(\000)1031 4623 y Fv(1)p 1030 4637 V 1030 4684 a(2)1073 4656 y Fu(;)1120 4623 y Fv(1)p 1120 4637 V 1120 4684 a(2)1163 4588 y Fp(\002\001)1266 4656 y Fo(such)30 b(that)f(:)1268 4846 y Fz(~)-48 b Fu(u)1310 4858 y Fs(k)1374 4846 y Fz(=)1480 4825 y(~)1462 4846 y Fu(P)1515 4858 y Fs(k)1556 4846 y Fu(u)1604 4812 y Fv(mean)1604 4867 y(1)1791 4846 y Fo(and)30 b Fu(u)2000 4812 y Fv(0)2000 4867 y Fs(k)2064 4846 y Fz(=)22 b Fu(P)2216 4812 y Fv(0)2204 4867 y Fs(k)2253 4846 y Fu(u)2301 4812 y Fv(mean)2301 4867 y(1)2459 4846 y Fu(:)440 5039 y Fo(F)-6 b(urthermor)l(e)938 5018 y Fz(~)920 5039 y Fu(P)973 5051 y Fs(k)1043 5039 y Fo(has)31 b(the)f(form)1549 5018 y Fz(~)1530 5039 y Fu(P)1583 5051 y Fs(k)1648 5039 y Fz(=)1735 4960 y Fp(X)1737 5138 y Fs(j)s Ft(\025)p Fv(0)1876 5039 y Fz(~)-49 b Fu(p)1911 5051 y Fs(j)1946 5039 y Fz(\()p Fu(t)p Fz(\)\()p Fu(i\021)s Fz(\))2177 5005 y Fs(j)2242 5039 y Fo(with)38 b Fz(~)-49 b Fu(p)2465 5051 y Fs(j)2523 5039 y Fx(2)23 b(P)7 b Fz(\()p Fw(R)2752 5009 y Fv(+)2813 5039 y Fz(\))30 b Fo(r)l(e)l(al.)565 5295 y Fz(Prop)r(osition)36 b(5.1)h(giv)n(es)g(an)h(expression)f(of)h (all)g(the)g(co)r(e\016cien)n(ts)g Fu(m)2805 5307 y Fs(j)2840 5295 y Fz(,)46 b(~)-47 b Fu(u)2952 5307 y Fs(j)3024 5295 y Fz(and)38 b Fu(u)3244 5265 y Fv(0)3244 5317 y Fs(j)3281 5295 y Fz(.)440 5413 y(According)26 b(to)h(Prop)r(osition)f(5.3,)g(w)n (e)h(kno)n(w)g(that)g(there)g(exist)g(p)r(olynomials)2944 5392 y(~)2926 5413 y Fu(P)2979 5425 y Fs(j)3041 5413 y Fz(and)g Fu(P)3267 5383 y Fv(0)3255 5435 y Fs(j)1831 5662 y Fz(15)p eop %%Page: 16 16 16 15 bop 440 531 a Fz(suc)n(h)29 b(that)35 b(~)-48 b Fu(u)858 543 y Fs(j)918 531 y Fz(=)1027 510 y(~)1008 531 y Fu(P)1061 543 y Fs(j)1097 531 y Fu(u)1145 543 y Fv(0)1211 531 y Fz(and)29 b Fu(u)1422 501 y Fv(0)1422 553 y Fs(j)1484 531 y Fz(=)c Fu(P)1639 501 y Fv(0)1627 553 y Fs(j)1676 531 y Fu(u)1724 543 y Fv(0)1761 531 y Fz(.)42 b(So)28 b(w)n(e)h(deduce)g(an)g(algorithm)f(to)h(determine)459 628 y(~)440 649 y Fu(P)493 661 y Fs(j)528 649 y Fz(,)f Fu(P)644 619 y Fv(0)632 671 y Fs(j)709 649 y Fz(and)f Fu(m)943 661 y Fs(j)1006 649 y Fz(recursiv)n(ely)e(with)k(formal)d (computations.)440 749 y(A)n(t)i(the)g(\014rst)f(order,)g(w)n(e)g(ha)n (v)n(e)f(:)1207 907 y Fu(m)1280 919 y Fv(0)1340 907 y Fz(=)1465 874 y Fv(1)p 1438 888 88 4 v 1438 897 a Ft(p)p 1492 897 34 3 v 1492 945 a Fv(3)1535 907 y Fu(;)83 b(P)1706 877 y Fv(0)1694 928 y(0)1743 907 y Fz(\()p Fu(t)p Fz(\))24 b(=)f(1)p Fu(;)2115 886 y Fz(~)2096 907 y Fu(P)2149 919 y Fv(0)2187 907 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))23 b(=)g(0)p Fu(:)440 1076 y Fz(F)-7 b(or)27 b(the)h(second)f(term)h(:)647 1258 y Fu(m)720 1270 y Fv(1)841 1258 y Fz(=)82 b Fx(\000)1118 1202 y Fz(23)p 1063 1239 194 4 v 1063 1324 a(35)1147 1255 y Fx(p)p 1215 1255 42 4 v 1215 1324 a Fz(3)1266 1258 y Fu(;)561 1478 y(P)626 1444 y Fv(0)614 1498 y(1)663 1478 y Fz(\()p Fu(t)p Fz(\))i(=)e Fx(\000)1063 1422 y Fz(19)p 1063 1459 84 4 v 1063 1535 a(42)1174 1478 y(+)1323 1422 y(23)p 1267 1459 194 4 v 1267 1544 a(70)1351 1475 y Fx(p)p 1419 1475 42 4 v 1419 1544 a Fz(3)1471 1478 y Fu(t)18 b Fz(+)1633 1422 y(11)p 1612 1459 125 4 v 1612 1535 a(210)1747 1478 y Fu(t)1777 1444 y Fv(2)1832 1478 y Fx(\000)2022 1422 y Fz(2)p 1925 1459 236 4 v 1925 1544 a(189)2051 1475 y Fx(p)p 2119 1475 42 4 v 2119 1544 a Fz(3)2171 1478 y Fu(t)2201 1444 y Fv(3)2238 1478 y Fu(;)510 1691 y Fz(~)491 1712 y Fu(P)544 1724 y Fv(1)582 1712 y Fz(\()p Fu(t;)c(\021)s Fz(\))84 b(=)1040 1656 y(7)p 998 1693 125 4 v 998 1769 a(720)1133 1712 y Fu(t)1163 1678 y Fv(2)1218 1712 y Fz(+)18 b Fu(i\021)1390 1656 y(t)p 1384 1693 42 4 v 1384 1769 a Fz(2)1449 1595 y Fp(\022)1511 1712 y Fx(\000)p Fz(1)f(+)1768 1656 y Fu(t)p 1728 1693 111 4 v 1728 1710 a Fx(p)p 1797 1710 42 4 v 68 x Fz(3)1849 1595 y Fp(\023)1928 1712 y Fx(\000)h Fu(\021)2055 1678 y Fv(2)2103 1656 y Fu(t)2133 1626 y Fv(2)p 2103 1693 68 4 v 2116 1769 a Fz(6)2198 1712 y(+)g Fu(i\021)2354 1678 y Fv(3)2405 1595 y Fp(\022)2476 1656 y Fz(2)p 2476 1693 42 4 v 2476 1769 a(3)2528 1712 y Fu(t)g Fx(\000)2725 1656 y Fz(2)p 2669 1693 153 4 v 2669 1778 a(3)2711 1710 y Fx(p)p 2780 1710 42 4 v 68 x Fz(3)2831 1712 y Fu(t)2861 1678 y Fv(2)2899 1595 y Fp(\023)2978 1712 y Fz(+)g Fu(\021)3105 1678 y Fv(4)3153 1656 y Fu(t)3183 1626 y Fv(2)p 3153 1693 68 4 v 3165 1769 a Fz(3)3230 1712 y Fu(:)440 1921 y Fz(The)28 b(same)f(algorithm)f(w)n(orks)g(for)h(an)n(y)g(\014nite)h (expansion.)440 2192 y Fy(6)135 b(Upp)t(er)44 b(b)t(ound)g(for)h(the)g (asymptotics)i(of)e Fc(\026)p Fm(\()p Fc(\013)q Fm(\))440 2374 y FA(Prop)s(osition)35 b(6.1.)43 b Fo(L)l(et)33 b Fu(n)h Fo(b)l(e)g(a)g(p)l(ositive)i(inte)l(ger)d(and)i Fu(m)2388 2386 y Fs(k)2462 2374 y Fo(b)l(e)f(the)g(r)l(e)l(als)g (determine)l(d)440 2474 y(by)c(Pr)l(op)l(osition)i(5.1,)f(for)g Fu(k)26 b Fx(\024)c Fu(n)p Fo(.)39 b(We)30 b(de\014ne)f(:)1544 2699 y Fu(\026)1594 2665 y Fv(\()p Fs(n)p Fv(\))1714 2699 y Fz(=)1842 2595 y Fs(n)1802 2620 y Fp(X)1802 2799 y Fs(k)q Fv(=0)1936 2699 y Fu(\013)1989 2665 y Fv(2)p Fs(k)2064 2699 y Fu(m)2137 2711 y Fs(k)2177 2699 y Fu(:)934 b Fz(\(6.1\))440 2937 y Fo(Then)30 b(ther)l(e)g(exists)g Fu(\013)1144 2949 y Fv(0)1204 2937 y Fu(>)23 b Fz(0)29 b Fo(and)h(a)g(p)l(ositive)h(c)l(onstant)e Fu(c)h Fo(such)g(that)g(:) 1218 3101 y Fx(8)p Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)1551 3113 y Fv(0)1588 3101 y Fz([)p Fu(;)37 b(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\024)f Fu(\013\026)2053 3067 y Fv(\()p Fs(n)p Fv(\))2168 3101 y Fz(+)c Fu(c\013)2341 3067 y Fv(2)p Fs(n)p Fv(+3)2503 3101 y Fu(:)440 3265 y Fo(F)-6 b(urthermor)l(e,)30 b(ther)l(e)g(exists)f(an)h(eigenvalue)38 b Fz(~)-49 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))31 b Fo(of)f Fu(P)2247 3277 y Ft(A)2300 3285 y Fr(0)2333 3277 y Fs(;)p Fv(\012)2400 3285 y Fe(\013)2476 3265 y Fo(such)g(that)g(:)1169 3438 y Fz(~)-48 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))24 b(=)f Fu(\013\026)1545 3404 y Fv(\()p Fs(n)p Fv(\))1661 3438 y Fz(+)18 b Fx(O)r Fz(\()p Fu(\013)1897 3404 y Fv(2)p Fs(n)p Fv(+1)2060 3438 y Fz(\))30 b Fo(when)g Fu(\013)24 b Fx(!)f Fz(0)p Fu(:)440 3602 y Fo(Pr)l(o)l(of)31 b(:)39 b Fz(W)-7 b(e)28 b(consider)e(the)i(functions)g Fu(u)1742 3614 y Fs(k)1811 3602 y Fz(and)f(the)h(reals)f Fu(m)2383 3614 y Fs(k)2451 3602 y Fz(determined)h(b)n(y)f(Prop)r(osi-)440 3702 y(tion)h(5.1)e(and)i(de\014ne)g(for)f(ev)n(ery)f(in)n(teger)h Fu(n)c Fx(\025)f Fz(1)27 b(:)1489 3927 y Fu(U)1555 3893 y Fv(\()p Fs(n)p Fv(\))1735 3927 y Fz(=)1922 3824 y Fs(n)1883 3848 y Fp(X)1882 4027 y Fs(k)q Fv(=0)2017 3927 y Fu(\013)2070 3893 y Fv(2)p Fs(k)2144 3927 y Fu(u)2192 3939 y Fs(k)2232 3927 y Fu(;)879 b Fz(\(6.2\))440 4181 y(W)-7 b(e)23 b(w)n(an)n(t)f(to)g (estimate)h(the)g(Ra)n(yleigh)e(quotien)n(t)h(for)h Fu(U)2186 4151 y Fv(\()p Fs(n)p Fv(\))2282 4181 y Fz(.)36 b(Let)22 b Fu(v)27 b Fx(2)c(V)2687 4151 y Fs(N)2750 4181 y Fz(,)g(then,)h(b)n(y) f(using)440 4281 y(Prop)r(osition)e(5.1,)j(the)f(v)-5 b(anishing)23 b(of)g(the)g(co)r(e\016cien)n(t)g(of)g Fu(\013)2291 4251 y Fv(2)p Fs(k)2389 4281 y Fz(for)g Fu(k)i Fx(\024)e Fu(n)p Fz(,)h(giv)n(en)e(in)i(\(5)p Fu(:)p Fz(2\))3241 4293 y Fs(k)3281 4281 y Fz(,)440 4381 y(and)j(the)h(de\014nition)g(of)g Fu(u)1256 4393 y Fv(0)1293 4381 y Fz(,)f(w)n(e)h(ha)n(v)n(e)e(:)665 4639 y Fu(a)709 4651 y Fs(\013)757 4639 y Fz(\()p Fu(U)855 4605 y Fv(\()p Fs(n)p Fv(\))952 4639 y Fu(;)14 b(v)s Fz(\))83 b(=)1295 4536 y Fs(n)p Ft(\000)p Fv(1)1298 4560 y Fp(X)1297 4739 y Fs(k)q Fv(=0)1435 4639 y Fu(\013)1488 4605 y Fv(2)p Fs(k)1576 4526 y Fp(Z)1622 4715 y Fv(\012)1669 4698 y Fr(0)1720 4472 y Fp(0)1720 4622 y(@)1834 4536 y Fs(k)1792 4560 y Fp(X)1795 4737 y Fs(j)s Fv(=0)1926 4639 y Fu(m)1999 4651 y Fs(j)2034 4639 y Fu(u)2082 4651 y Fs(k)q Ft(\000)p Fs(j)2205 4472 y Fp(1)2205 4622 y(A)p 2292 4594 44 4 v 2292 4639 a Fu(v)30 b(dt)e(d\021)22 b Fz(+)c Fu(\013)2705 4605 y Fv(2)p Fs(n)2784 4639 y Fu(`)p Fz(\()p Fu(u)2899 4651 y Fs(n)2943 4639 y Fu(;)c(v)s Fz(\))p Fu(:)56 b Fz(\(6.3\))440 4993 y(In)28 b(\(6.3\),)f(w)n(e)g(rewrite)g(the)h(sum) 1530 4889 y Fs(k)1488 4914 y Fp(X)1490 5091 y Fs(j)s Fv(=0)1622 4993 y Fu(m)1695 5005 y Fs(j)1730 4993 y Fu(u)1778 5005 y Fs(k)q Ft(\000)p Fs(j)1928 4993 y Fz(:)619 5231 y Fs(n)579 5256 y Fp(X)579 5435 y Fs(k)q Fv(=0)714 5335 y Fu(\013)767 5301 y Fv(2)p Fs(k)896 5231 y(k)855 5256 y Fp(X)857 5433 y Fs(j)s Fv(=0)988 5335 y Fu(m)1061 5347 y Fs(j)1096 5335 y Fu(u)1144 5347 y Fs(k)q Ft(\000)p Fs(j)1291 5335 y Fz(=)22 b Fu(\026)1428 5301 y Fv(\()p Fs(n)p Fv(\))1525 5335 y Fu(U)1591 5301 y Fv(\()p Fs(n)p Fv(\))1707 5335 y Fx(\000)c Fu(\013)1843 5301 y Fv(2)p Fs(n)p Fv(+2)2019 5231 y Fs(n)p Ft(\000)p Fv(1)2022 5256 y Fp(X)2022 5435 y Fs(k)q Fv(=0)2159 5335 y Fu(\013)2212 5301 y Fv(2)p Fs(k)2381 5231 y(n)2342 5256 y Fp(X)2300 5435 y Fs(j)s Fv(=)p Fs(k)q Fv(+1)2517 5335 y Fu(m)2590 5347 y Fs(k)q Fv(+)p Fs(n)p Fv(+1)p Ft(\000)p Fs(j)2889 5335 y Fu(u)2937 5347 y Fs(j)2972 5335 y Fu(:)139 b Fz(\(6.4\))1831 5662 y(16)p eop %%Page: 17 17 17 16 bop 440 531 a Fz(Th)n(us,)27 b(there)h(exists)f(a)g(constan)n(t)g Fu(C)34 b Fz(suc)n(h)27 b(that)h(for)f Fu(\013)d(<)e Fz(1)28 b(and)f(for)g(ev)n(ery)f Fu(v)h Fx(2)c(V)3050 501 y Fs(N)3140 531 y Fz(:)440 652 y Fp(\014)440 702 y(\014)440 751 y(\014)440 801 y(\014)468 772 y Fu(a)512 784 y Fs(\013)559 772 y Fz(\()p Fu(U)657 738 y Fv(\()p Fs(n)p Fv(\))754 772 y Fu(;)14 b(v)s Fz(\))19 b Fx(\000)f Fu(\026)1018 738 y Fv(\()p Fs(n)p Fv(\))1129 680 y Fp(D)1180 772 y Fu(U)1246 738 y Fv(\()p Fs(n)p Fv(\))1342 772 y Fu(;)c(v)1422 680 y Fp(E)1473 830 y Fs(L)1519 814 y Fr(2)1551 830 y Fv(\(\012)1624 814 y Fr(0)1657 830 y Fv(\))1687 652 y Fp(\014)1687 702 y(\014)1687 751 y(\014)1687 801 y(\014)1798 772 y Fx(\024)82 b Fu(\013)1998 738 y Fv(2)p Fs(n)2091 602 y Fp(\014)2091 652 y(\014)2091 702 y(\014)2091 751 y(\014)2091 801 y(\014)2091 851 y(\014)2119 772 y Fu(`)p Fz(\()p Fu(u)2234 784 y Fs(n)2278 772 y Fu(;)14 b(v)s Fz(\))19 b Fx(\000)2531 668 y Fs(n)2492 693 y Fp(X)2494 870 y Fs(j)s Fv(=0)2626 772 y Fu(m)2699 784 y Fs(j)2747 772 y Fx(h)q Fu(u)2828 784 y Fs(n)p Ft(\000)p Fs(j)2955 772 y Fu(;)14 b(v)s Fx(i)3068 800 y Fs(L)3114 783 y Fr(2)3146 800 y Fv(\(\012)3219 783 y Fr(0)3251 800 y Fv(\))3281 602 y Fp(\014)3281 652 y(\014)3281 702 y(\014)3281 751 y(\014)3281 801 y(\014)3281 851 y(\014)1945 1006 y Fz(+)p Fu(C)6 b(\013)2128 971 y Fv(2)p Fs(n)p Fv(+2)2291 1006 y Fx(jj)p Fu(v)s Fx(jj)2426 1021 y Fs(L)2472 1004 y Fr(2)2504 1021 y Fv(\(\012)2577 1004 y Fr(0)2610 1021 y Fv(\))2640 1006 y Fu(:)489 b Fz(\(6.5\))440 1163 y(W)-7 b(e)28 b(b)r(ound)g(from)f (b)r(elo)n(w)g(the)h(norm)f Fx(jj)p Fu(U)1742 1133 y Fv(\()p Fs(n)p Fv(\))1839 1163 y Fx(jj)1885 1178 y Fs(L)1931 1162 y Fr(2)1963 1178 y Fv(\(\012)2036 1162 y Fr(0)2069 1178 y Fv(\))2127 1163 y Fz(as)g(follo)n(ws)g(:)1038 1371 y Fx(jj)p Fu(U)1150 1337 y Fv(\()p Fs(n)p Fv(\))1247 1371 y Fx(jj)1293 1386 y Fs(L)1339 1370 y Fr(2)1371 1386 y Fv(\(\012)1444 1370 y Fr(0)1477 1386 y Fv(\))1530 1371 y Fx(\025)c Fz(1)18 b Fx(\000)85 b Fz(sup)1761 1442 y Fs(k)q Fv(=0)p Fs(;:::)o(;n)2035 1371 y Fx(jj)p Fu(u)2129 1383 y Fs(k)2170 1371 y Fx(jj)2216 1386 y Fs(L)2262 1370 y Fr(2)2294 1386 y Fv(\(\012)2367 1370 y Fr(0)2399 1386 y Fv(\))2511 1315 y Fu(\013)2564 1285 y Fv(2)p 2439 1352 234 4 v 2439 1428 a Fz(1)18 b Fx(\000)g Fu(\013)2635 1404 y Fv(2)2683 1371 y Fu(:)440 1578 y Fz(So)27 b(there)h(exists)f Fu(\013)1050 1590 y Fv(0)1110 1578 y Fu(<)c Fz(1)k(and)h(a)f(constan)n (t)g Fu(C)1892 1590 y Fv(0)1952 1578 y Fu(>)c Fz(0)k(suc)n(h)g(that)h (for)f(ev)n(ery)g Fu(\013)c Fx(2)p Fz(]0)p Fu(;)14 b(\013)3109 1590 y Fv(0)3146 1578 y Fz([)28 b(:)1446 1724 y Fx(jj)p Fu(U)1558 1689 y Fv(\()p Fs(n)p Fv(\))1655 1724 y Fx(jj)1701 1739 y Fs(L)1747 1722 y Fr(2)1779 1739 y Fv(\(\012)1852 1722 y Fr(0)1885 1739 y Fv(\))1938 1724 y Fx(\025)23 b Fu(C)2085 1736 y Fv(0)2146 1724 y Fu(>)f Fz(0)p Fu(:)836 b Fz(\(6.6\))440 1869 y(Due)30 b(to)e(the)i(Cauc)n(h)n(y-Sc)n(h)n(w)n (arz)c(inequalit)n(y)i(and)h(the)g(lo)n(w)n(er)f(b)r(ound)h(\(6.6\),)h (the)f(sp)r(ectral)440 1969 y(theorem)e(sho)n(ws)g(that)g(:)1419 2069 y Fu(d)p Fz(\()p Fu(\026)1544 2034 y Fv(\()p Fs(n)p Fv(\))1642 2069 y Fu(;)14 b(\033)s Fz(\()p Fu(P)1814 2081 y Fs(\013)1862 2069 y Fz(\)\))24 b Fx(\024)e(O)r Fz(\()p Fu(\013)2190 2034 y Fv(2)p Fs(n)2270 2069 y Fz(\))p Fu(:)440 2196 y Fz(By)27 b(c)n(hange)g(of)g(v)-5 b(ariables,)27 b(there)g(exists)g(an)h(eigen)n(v)-5 b(alue)33 b(~)-49 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))29 b(of)f Fu(P)2604 2208 y Ft(A)2657 2216 y Fr(0)2690 2208 y Fs(;)p Fv(\012)2757 2216 y Fe(\013)2831 2196 y Fz(with)g(:)1373 2352 y Fx(j)p Fu(\013\026)1499 2318 y Fv(\()p Fs(n)p Fv(\))1615 2352 y Fx(\000)c Fz(~)-48 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))p Fx(j)24 b(\024)f(O)r Fz(\()p Fu(\013)2153 2318 y Fv(2)p Fs(n)p Fv(+1)2316 2352 y Fz(\))p Fu(:)440 2498 y Fz(F)-7 b(or)37 b(using)h(the)g(min-max)g(principle,)i(w)n(e)e(c)n(ho)r(ose)f Fu(v)43 b Fz(=)d Fu(U)2377 2468 y Fv(\()p Fs(n)p Fv(\))2512 2498 y Fz(in)e(\(6.5\))g(and)f(estimate)440 2598 y Fu(`)p Fz(\()p Fu(u)555 2610 y Fs(n)600 2598 y Fu(;)14 b(U)703 2567 y Fv(\()p Fs(n)p Fv(\))799 2598 y Fz(\))29 b(using)e(\(5)p Fu(:)p Fz(4\))1248 2610 y Fs(n)1293 2598 y Fz(,)h(the)h(orthogonalit)n (y)c(b)r(et)n(w)n(een)j Fu(u)2369 2567 y Fv(0)2369 2618 y Fs(n)2442 2598 y Fz(and)g Fu(u)2652 2567 y Fv(mean)2652 2618 y(1)2837 2598 y Fz(and)g(the)h(rela-)440 2697 y(tion)f Fu(`)p Fz(\()5 b(~)-47 b Fu(u)726 2709 y Fs(n)770 2697 y Fu(;)14 b(u)855 2667 y Fv(mean)855 2718 y(1)1013 2697 y Fz(\))23 b(=)g Fx(h)5 b Fz(~)-47 b Fu(u)1236 2709 y Fs(n)1281 2697 y Fu(;)14 b(Lu)1423 2667 y Fv(mean)1423 2718 y(1)1580 2697 y Fx(i)1612 2722 y Fs(L)1658 2706 y Fr(2)1690 2722 y Fv(\(\012)1763 2706 y Fr(0)1796 2722 y Fv(\))1826 2697 y Fz(,)27 b(then)h(:)597 2928 y Fu(`)p Fz(\()p Fu(u)712 2940 y Fs(n)757 2928 y Fu(;)14 b(U)860 2894 y Fv(\()p Fs(n)p Fv(\))957 2928 y Fz(\))23 b(=)g Fu(`)p Fz(\()p Fu(u)1215 2940 y Fs(n)1259 2928 y Fu(;)14 b(u)1344 2894 y Fv(mean)1344 2949 y(1)1502 2928 y Fz(\))k(+)1675 2824 y Fs(n)1636 2849 y Fp(X)1635 3028 y Fs(k)q Fv(=1)1770 2928 y Fu(\013)1823 2894 y Fv(2)p Fs(k)1897 2928 y Fu(`)p Fz(\()p Fu(u)2012 2940 y Fs(n)2057 2928 y Fu(;)c(u)2142 2940 y Fs(k)2182 2928 y Fz(\))24 b(=)e Fu(m)2398 2940 y Fs(n)2462 2928 y Fz(+)2585 2824 y Fs(n)2545 2849 y Fp(X)2545 3028 y Fs(k)q Fv(=1)2679 2928 y Fu(\013)2732 2894 y Fv(2)p Fs(k)2807 2928 y Fu(`)p Fz(\()p Fu(u)2922 2940 y Fs(n)2966 2928 y Fu(;)14 b(u)3051 2940 y Fs(k)3092 2928 y Fz(\))p Fu(;)440 3149 y Fz(No)n(w,)27 b(since)h Fu(u)904 3118 y Fv(mean)904 3169 y(1)1089 3149 y Fz(is)f(normalized,)g (w)n(e)g(deduce)h(:)1132 3361 y Fu(`)p Fz(\()p Fu(u)1247 3373 y Fs(n)1291 3361 y Fu(;)14 b(U)1394 3326 y Fv(\()p Fs(n)p Fv(\))1491 3361 y Fz(\))19 b Fx(\000)1664 3257 y Fs(n)1625 3282 y Fp(X)1627 3459 y Fs(j)s Fv(=0)1759 3361 y Fu(m)1832 3373 y Fs(j)1880 3269 y Fp(D)1931 3361 y Fu(u)1979 3373 y Fs(n)p Ft(\000)p Fs(j)2107 3361 y Fu(;)14 b(U)2210 3326 y Fv(\()p Fs(n)p Fv(\))2306 3269 y Fp(E)2357 3419 y Fs(L)2403 3402 y Fr(2)2435 3419 y Fv(\(\012)2508 3402 y Fr(0)2541 3419 y Fv(\))984 3682 y Fz(=)1172 3578 y Fs(n)1132 3603 y Fp(X)1132 3782 y Fs(k)q Fv(=1)1266 3682 y Fu(\013)1319 3647 y Fv(2)p Fs(k)1408 3515 y Fp(0)1408 3664 y(@)1480 3682 y Fu(`)p Fz(\()p Fu(u)1595 3694 y Fs(n)1640 3682 y Fu(;)g(u)1725 3694 y Fs(k)1765 3682 y Fz(\))19 b Fx(\000)1938 3578 y Fs(n)1899 3603 y Fp(X)1901 3780 y Fs(j)s Fv(=0)2033 3682 y Fu(m)2106 3694 y Fs(j)2154 3682 y Fx(h)q Fu(u)2235 3694 y Fs(n)p Ft(\000)p Fs(j)2362 3682 y Fu(;)14 b(u)2447 3694 y Fs(k)2487 3682 y Fx(i)2520 3710 y Fs(L)2566 3693 y Fr(2)2598 3710 y Fv(\(\012)2671 3693 y Fr(0)2703 3710 y Fv(\))2733 3515 y Fp(1)2733 3664 y(A)2820 3682 y Fu(:)440 3922 y Fz(According)32 b(to)h(the)g(expression)f(of)h Fu(u)1649 3934 y Fs(k)1723 3922 y Fz(giv)n(en)f(in)h(Prop)r(osition)f(5.3)g(and)h(Remarks)f(4.12) 440 4022 y(and)e(5.2,)g(w)n(e)g(deduce)h(that)f(there)g(exists)g(a)g (constan)n(t)g(only)f(dep)r(enden)n(t)i(on)f Fu(n)p Fz(,)h(denoted)440 4122 y(b)n(y)c Fu(c)591 4134 y Fs(n)664 4122 y Fz(suc)n(h)g(that)h(:) 972 4192 y Fp(\014)972 4242 y(\014)972 4292 y(\014)972 4342 y(\014)972 4392 y(\014)972 4441 y(\014)1000 4363 y Fu(`)p Fz(\()p Fu(u)1115 4375 y Fs(n)1159 4363 y Fu(;)14 b(U)1262 4328 y Fv(\()p Fs(n)p Fv(\))1359 4363 y Fz(\))19 b Fx(\000)1532 4259 y Fs(n)1493 4284 y Fp(X)1495 4460 y Fs(j)s Fv(=0)1627 4363 y Fu(m)1700 4375 y Fs(j)1748 4270 y Fp(D)1799 4363 y Fu(u)1847 4375 y Fs(n)p Ft(\000)p Fs(j)1975 4363 y Fu(;)14 b(U)2078 4328 y Fv(\()p Fs(n)p Fv(\))2174 4270 y Fp(E)2225 4421 y Fs(L)2271 4404 y Fr(2)2303 4421 y Fv(\(\012)2376 4404 y Fr(0)2409 4421 y Fv(\))2439 4192 y Fp(\014)2439 4242 y(\014)2439 4292 y(\014)2439 4342 y(\014)2439 4392 y(\014)2439 4441 y(\014)2490 4363 y Fx(\024)22 b Fu(c)2613 4375 y Fs(n)2658 4363 y Fu(\013)2711 4328 y Fv(2)2749 4363 y Fu(:)362 b Fz(\(6.7\))440 4608 y(Using)31 b(the)g(upp)r(er)g(b)r(ound)g(\(6.7\),)h(Inequalit)n(y)e (\(6.5\))h(and)f(estimate)h(of)g(the)g(norm)g(\(6.6\),)440 4708 y(w)n(e)c(get)h(a)f(constan)n(t)g Fu(c)c(>)f Fz(0)27 b(suc)n(h)h(that)g(:)1081 4852 y Fu(a)1125 4864 y Fs(\013)1172 4852 y Fz(\()p Fu(U)1270 4822 y Fv(\()p Fs(n)p Fv(\))1367 4852 y Fu(;)14 b(U)1470 4822 y Fv(\()p Fs(n)p Fv(\))1567 4852 y Fz(\))p 1081 4889 519 4 v 1106 4967 a Fx(jj)p Fu(U)1218 4943 y Fv(\()p Fs(n)p Fv(\))1315 4967 y Fx(jj)1361 4938 y Fv(2)1361 4995 y Fs(L)1407 4978 y Fr(2)1439 4995 y Fv(\(\012)1512 4978 y Fr(0)1545 4995 y Fv(\))1632 4908 y Fx(\024)23 b Fu(\026)1770 4874 y Fv(\()p Fs(n)p Fv(\))1886 4908 y Fz(+)18 b Fu(c\013)2058 4874 y Fv(2)p Fs(n)p Fv(+2)2220 4908 y Fu(;)c Fx(8)p Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)2590 4920 y Fv(0)2627 4908 y Fz([)p Fu(:)461 b Fz(\(6.8\))440 5121 y(The)28 b(min-max)f(principle)g(and)h(a)f(c)n (hange)g(of)g(v)-5 b(ariables)26 b(lead)i(to)f(the)h(upp)r(er)g(b)r (ound)g(:)780 5267 y Fx(8)p Fu(n)22 b Fx(2)h Fw(N)t Fu(;)14 b Fx(9)p Fu(\013)1167 5279 y Fv(0)1210 5267 y Fu(;)g Fx(9)p Fu(c)23 b(>)g Fz(0)k Fu(s:t:)h Fx(8)p Fu(\013)22 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)1984 5279 y Fv(0)2022 5267 y Fz([)p Fu(;)41 b(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\024)f Fu(\013\026)2491 5233 y Fv(\()p Fs(n)p Fv(\))2607 5267 y Fz(+)18 b Fu(c\013)2779 5233 y Fv(2)p Fs(n)p Fv(+3)2942 5267 y Fu(:)3240 5413 y Fd(\003)1831 5662 y Fz(17)p eop %%Page: 18 18 18 17 bop 440 531 a Fy(7)135 b(Some)45 b(estimates)i(of)e (eigenfunctions)440 730 y Fl(7.1)112 b(Agmon's)37 b(tec)m(hniques)440 883 y Fz(W)-7 b(e)22 b(giv)n(e)e(a)h(priori)g(estimates)g(on)g(the)h (deca)n(y)f(of)g(the)h(eigenfunctions.)35 b(By)21 b(using)g(Agmon's)440 983 y(pap)r(er)28 b([1)o(],)g(w)n(e)g(prop)r(ose)f(some)g(estimates)h (for)f(the)h(lo)r(calization)f(of)h(the)g(eigenfunctions.)440 1082 y(Let)g(us)f(recall)g(principles)g(of)h(the)g(Agmon's)f (estimates.)440 1182 y(Let)35 b Fu(\013)649 1194 y Fv(1)720 1182 y Fz(small)f(enough)g(suc)n(h)g(that)h Fu(\026)1668 1194 y Fs(k)1709 1182 y Fz(\()p Fu(\013)p Fz(\))g Fu(<)f Fz(\002)2025 1194 y Fv(0)2096 1182 y Fz(for)g(ev)n(ery)f Fu(\013)i Fx(2)p Fz(]0)p Fu(;)14 b(\013)2754 1194 y Fv(1)2791 1182 y Fz(])35 b(where)e Fu(\026)3145 1194 y Fs(k)3186 1182 y Fz(\()p Fu(\013)p Fz(\))440 1282 y(is)i(the)h Fu(k)s Fz(-th)g(elemen)n(t)f(of)g(the)h(sp)r(ectrum)g(of)f Fu(P)1963 1294 y Fs(\013)2011 1282 y Fz(.)60 b(W)-7 b(e)36 b(denote)f(b)n(y)h Fu(u)2692 1294 y Fs(k)q(;\013)2830 1282 y Fz(a)f(normalized)440 1381 y(eigenfunction)28 b(asso)r(ciated)e(to)h Fu(\026)1496 1393 y Fs(k)1537 1381 y Fz(\()p Fu(\013)p Fz(\).)440 1481 y(Let)h Fu(\036)g Fz(b)r(e)g(uniformaly)f(lipsc)n(hitzian)g(on)g(\012)1810 1493 y Fs(\013)1858 1481 y Fz(,)h(then,)g(b)n(y)f(assumptions)g(on)g Fu(u)2871 1493 y Fs(k)q(;\013)3002 1481 y Fz(:)583 1596 y Fp(Z)629 1785 y Fv(\012)676 1793 y Fe(\013)737 1642 y Fp(\000)775 1709 y Fu(\026)825 1721 y Fs(k)866 1709 y Fz(\()p Fu(\013)p Fz(\))19 b(+)f Fx(jr)p Fu(\036)p Fx(j)1249 1675 y Fv(2)1287 1642 y Fp(\001)1367 1709 y Fu(e)1406 1675 y Fv(2)p Fs(\036)1483 1709 y Fx(j)p Fu(u)1554 1721 y Fs(k)q(;\013)1657 1709 y Fx(j)1680 1675 y Fv(2)1745 1709 y Fu(dx)24 b Fz(=)1946 1596 y Fp(Z)1992 1785 y Fv(\012)2039 1793 y Fe(\013)2100 1709 y Fx(j)p Fz(\()p Fx(r)19 b(\000)f Fu(i)p Fx(A)2421 1721 y Fv(0)2458 1709 y Fz(\)\()p Fu(e)2561 1675 y Fs(\036)2605 1709 y Fu(u)2653 1721 y Fs(k)q(;\013)2756 1709 y Fz(\))p Fx(j)2811 1675 y Fv(2)2877 1709 y Fu(dx:)144 b Fz(\(7.1\))440 1957 y(Let)30 b Fu(\037)643 1969 y Fv(1)710 1957 y Fz(and)g Fu(\037)926 1969 y Fv(2)993 1957 y Fz(b)r(e)g(real,)g (p)r(ositiv)n(e,)g(regular)e(functions)i(on)p 2390 1891 108 4 v 29 w(\012)2450 1969 y Fs(\013)2498 1957 y Fz(,)g(with)h(supp)r (ort)e(resp)r(ec-)440 2057 y(tiv)n(ely)h(included)g(in)p 1105 1990 V 31 w(\012)1165 2069 y Fs(\013)1232 2057 y Fx(\\)21 b(B)s Fz(\(0)p Fu(;)14 b Fz(2\))28 b(and)p 1743 1990 V 30 w(\012)1803 2069 y Fs(\013)1871 2057 y Fx(n)19 b(B)s Fz(\(0)p Fu(;)14 b Fz(1\))29 b(suc)n(h)h(that)g Fx(j)p Fu(\037)2651 2069 y Fv(1)2688 2057 y Fx(j)2711 2027 y Fv(2)2769 2057 y Fz(+)19 b Fx(j)p Fu(\037)2928 2069 y Fv(2)2966 2057 y Fx(j)2989 2027 y Fv(2)3053 2057 y Fz(=)27 b(1)j(on)p 440 2090 V 440 2157 a(\012)500 2169 y Fs(\013)547 2157 y Fz(.)37 b(W)-7 b(e)28 b(de\014ne)g Fu(\037)1042 2127 y Fs(R)1042 2178 y(j)1106 2157 y Fz(:)51 b(=)22 b Fu(\037)1319 2169 y Fs(j)1368 2089 y Fp(\000)1431 2124 y Fs(:)p 1416 2138 51 4 v 1416 2185 a(R)1476 2089 y Fp(\001)1542 2157 y Fz(on)27 b(\012)1717 2169 y Fs(\013)1764 2157 y Fz(,)h(then)g(:)469 2430 y Fu(q)506 2442 y Ft(A)559 2450 y Fr(0)592 2442 y Fs(;)p Fv(\012)659 2450 y Fe(\013)705 2430 y Fz(\()p Fu(e)776 2396 y Fs(\036)820 2430 y Fu(u)868 2442 y Fs(k)q(;\013)972 2430 y Fz(\))23 b(=)1158 2327 y Fv(2)1115 2352 y Fp(X)1117 2528 y Fs(j)s Fv(=1)1248 2430 y Fu(q)1285 2442 y Ft(A)1338 2450 y Fr(0)1371 2442 y Fs(;)p Fv(\012)1438 2450 y Fe(\013)1484 2430 y Fz(\()p Fu(\037)1568 2396 y Fs(R)1568 2451 y(j)1623 2430 y Fu(e)1662 2396 y Fs(\036)1706 2430 y Fu(u)1754 2442 y Fs(k)q(;\013)1857 2430 y Fz(\))c Fx(\000)2031 2374 y Fz(1)p 2001 2411 101 4 v 2001 2487 a Fu(R)2065 2463 y Fv(2)2169 2327 y(2)2126 2352 y Fp(X)2128 2528 y Fs(j)s Fv(=1)2259 2430 y Fx(jj)p Fu(e)2344 2396 y Fs(\036)2388 2430 y Fu(u)2436 2442 y Fs(k)q(;\013)2567 2430 y Fx(jr)p Fu(\037)2711 2396 y Fs(R)2711 2451 y(j)2766 2430 y Fx(j)27 b(jj)2862 2396 y Fv(2)2862 2453 y Fs(L)2908 2436 y Fr(2)2941 2453 y Fv(\(\012)3014 2461 y Fe(\013)3056 2453 y Fv(\))3086 2430 y Fu(:)e Fz(\(7.2\))440 2709 y(W)-7 b(e)28 b(use)f(the)h (assumptions)e(on)h Fu(\037)1507 2679 y Fs(R)1507 2731 y(j)1589 2709 y Fz(and)h(rep)r(ort)e(\(7.2\))h(on)g(\(7.1\),)h(then)f (the)h(p)r(ositivit)n(y)f(of)440 2821 y Fu(q)477 2833 y Ft(A)530 2841 y Fr(0)563 2833 y Fs(;)p Fv(\012)630 2841 y Fe(\013)676 2821 y Fz(\()p Fu(\037)760 2791 y Fs(R)760 2842 y Fv(1)815 2821 y Fu(e)854 2791 y Fs(\036)898 2821 y Fu(u)946 2833 y Fs(k)q(;\013)1049 2821 y Fz(\))h(leads)f(to)g (the)h(upp)r(er)g(b)r(ound)g(:)692 3056 y Fu(q)729 3068 y Ft(A)782 3076 y Fr(0)815 3068 y Fs(;)p Fv(\012)882 3076 y Fe(\013)928 3056 y Fz(\()p Fu(\037)1012 3022 y Fs(R)1012 3077 y Fv(2)1067 3056 y Fu(e)1106 3022 y Fs(\036)1150 3056 y Fu(u)1198 3068 y Fs(k)q(;\013)1301 3056 y Fz(\))83 b Fx(\024)1564 2943 y Fp(Z)1610 3132 y Fv(\012)1657 3140 y Fe(\013)1717 2939 y Fp(\022)1778 3056 y Fu(\026)1828 3068 y Fs(k)1869 3056 y Fz(\()p Fu(\013)p Fz(\))20 b(+)e Fx(jr)p Fu(\036)p Fx(j)2253 3022 y Fv(2)2309 3056 y Fz(+)2420 3000 y Fu(C)p 2402 3037 V 2402 3113 a(R)2466 3089 y Fv(2)2513 2939 y Fp(\023)2588 3056 y Fu(e)2627 3022 y Fv(2)p Fs(\036)2704 3056 y Fx(j)p Fu(u)2775 3068 y Fs(k)q(;\013)2878 3056 y Fx(j)2901 3022 y Fv(2)2939 3056 y Fu(dx:)82 b Fz(\(7.3\))440 3294 y(Let)28 b(us)f(assume)g(that)h(there)f(exists)h Fu(\026)p Fz(\()p Fu(\013;)14 b(R)q Fz(\))23 b Fu(>)g Fz(0)k(suc)n(h)g(that)h(:)1022 3477 y Fu(q)1059 3489 y Ft(A)1112 3497 y Fr(0)1145 3489 y Fs(;)p Fv(\012)1212 3497 y Fe(\013)1259 3477 y Fz(\()p Fu(\037)1343 3442 y Fs(R)1343 3497 y Fv(2)1397 3477 y Fu(e)1436 3442 y Fs(\036)1480 3477 y Fu(u)1528 3489 y Fs(k)q(;\013)1631 3477 y Fz(\))c Fx(\025)e Fu(\026)p Fz(\()p Fu(\013;)14 b(R)q Fz(\))p Fx(jj)p Fu(\037)2140 3442 y Fs(R)2140 3497 y Fv(2)2195 3477 y Fu(e)2234 3442 y Fs(\036)2278 3477 y Fu(u)2326 3489 y Fs(k)q(;\013)2429 3477 y Fx(jj)2475 3442 y Fv(2)2475 3499 y Fs(L)2521 3483 y Fr(2)2553 3499 y Fv(\(\012)2626 3507 y Fe(\013)2669 3499 y Fv(\))2699 3477 y Fu(:)412 b Fz(\(7.4\))440 3675 y(By)18 b(assumption)g(on)g(the)h (supp)r(ort)f(of)g Fu(\037)1666 3645 y Fs(R)1666 3696 y Fv(2)1721 3675 y Fz(,)i(w)n(e)e(can)g(b)r(ound)h(from)f(b)r(elo)n(w)g Fx(jj)p Fu(\037)2778 3645 y Fs(R)2778 3696 y Fv(2)2832 3675 y Fu(e)2871 3645 y Fs(\036)2915 3675 y Fu(u)2963 3687 y Fs(k)q(;\013)3066 3675 y Fx(jj)3112 3645 y Fv(2)3112 3702 y Fs(L)3158 3686 y Fr(2)3190 3702 y Fv(\(\012)3263 3710 y Fe(\013)3306 3702 y Fv(\))440 3796 y Fz(b)n(y)27 b Fx(jj)p Fu(e)640 3765 y Fs(\036)684 3796 y Fu(u)732 3808 y Fs(k)q(;\013)835 3796 y Fx(jj)881 3765 y Fv(2)881 3822 y Fs(L)927 3806 y Fr(2)960 3822 y Fv(\(\012)1033 3830 y Fe(\013)1075 3822 y Ft(nB)q Fv(\(0)p Fs(;)p Fv(2)p Fs(R)p Fv(\)\))1372 3796 y Fz(,)h(and)g(so)e(deduce)i(from)f(\(7.4\))h (and)f(\(7.3\))h(that)f(:)906 3936 y Fp(Z)952 4124 y Fv(\012)999 4132 y Fe(\013)1042 4124 y Ft(nB)q Fv(\(0)p Fs(;)p Fv(2)p Fs(R)p Fv(\))1327 3931 y Fp(\022)1388 4049 y Fu(\026)p Fz(\()p Fu(\013;)14 b(R)q Fz(\))19 b Fx(\000)f Fu(\026)1808 4061 y Fs(k)1849 4049 y Fz(\()p Fu(\013)p Fz(\))h Fx(\000)f(jr)p Fu(\036)p Fx(j)2232 4014 y Fv(2)2289 4049 y Fx(\000)2400 3992 y Fu(C)p 2382 4029 V 2382 4105 a(R)2446 4081 y Fv(2)2493 3931 y Fp(\023)2568 4049 y Fu(e)2607 4014 y Fv(2)p Fs(\036)2684 4049 y Fx(j)p Fu(u)2755 4061 y Fs(k)q(;\013)2858 4049 y Fz(\))p Fx(j)2913 4014 y Fv(2)2978 4049 y Fu(dx)758 4292 y Fx(\024)906 4179 y Fp(Z)952 4368 y Fv(\012)999 4376 y Fe(\013)1042 4368 y Ft(\\B)q Fv(\(0)p Fs(;)p Fv(2)p Fs(R)p Fv(\))1338 4175 y Fp(\022)1399 4292 y Fu(\026)1449 4304 y Fs(k)1490 4292 y Fz(\()p Fu(\013)p Fz(\))h(+)f Fx(jr)p Fu(\036)p Fx(j)1873 4258 y Fv(2)1930 4292 y Fz(+)2040 4236 y Fu(C)p 2023 4273 V 2023 4349 a(R)2087 4325 y Fv(2)2133 4175 y Fp(\023)2208 4292 y Fu(e)2247 4258 y Fv(2)p Fs(\036)2324 4292 y Fx(j)p Fu(u)2395 4304 y Fs(k)q(;\013)2499 4292 y Fx(j)2522 4258 y Fv(2)2559 4292 y 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b Fu(<)f Fz(\002)2511 697 y Fv(0)2578 685 y Fo(for)31 b(every)g Fu(\013)25 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)3219 697 y Fv(1)3256 685 y Fz(])p Fo(.)440 784 y(We)35 b(design)h(by)g Fu(u)1018 796 y Fs(\013)1100 784 y Fo(a)f(normalize)l(d)i(eigenfunction)f(asso)l (ciate)l(d)h(to)e Fu(\026)p Fz(\()p Fu(\013)p Fz(\))p Fo(.)56 b(F)-6 b(or)35 b Fu(\013)e Fx(\024)g Fu(\013)3267 796 y Fv(1)440 959 y Fo(and)d(for)h(al)t(l)g Fu(\017)22 b Fx(2)988 817 y Fp(#)1036 959 y Fz(0)p Fu(;)14 b Fz(\002)1180 971 y Fv(0)1235 959 y Fx(\000)82 b Fz(sup)1318 1032 y Fs(\013)p Ft(2)p Fv(]0)p Fs(;\013)1521 1040 y Fr(1)1553 1032 y Fv(])1586 959 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))1753 817 y Fp(")1802 959 y Fo(,)30 b(ther)l(e)g(exists)f(a)i(c)l(onstant)e Fu(C)2754 971 y Fs(\017;\013)2878 959 y Fo(such)h(that)g(:)1130 1150 y Fp(Z)1176 1339 y Fv(\012)1223 1347 y Fe(\013)1284 1263 y Fu(e)1323 1229 y Fv(2)1356 1173 y Fx(p)p 1425 1173 351 4 v 56 x Fv(\002)1476 1237 y Fr(0)1508 1229 y Ft(\000)p Fs(\026)p Fv(\()p Fs(\013)p Fv(\))p Ft(\000)p Fs(\017)g Ft(j)p Fs(x)p Ft(j)1916 1263 y Fx(j)p Fu(u)1987 1275 y Fs(\013)2034 1263 y Fz(\()p Fu(x)p Fz(\))p Fx(j)2168 1229 y Fv(2)2236 1263 y Fu(dx)23 b Fx(\024)g Fu(C)2496 1275 y Fs(\017;\013)2591 1263 y Fu(:)520 b Fz(\(7.7\))440 1497 y Fo(Pr)l(o)l(of)42 b(:)62 b Fz(By)40 b(de\014nition)h(\(2.3\))f (of)g(\006\()p Fx(\000)p Fz(\001)1830 1509 y Ft(A)1883 1517 y Fr(0)1920 1497 y Fu(;)14 b(r)r Fz(\),)44 b(the)c(assumption)g (\(7.4\))g(holds)g(with)440 1597 y Fu(\026)p Fz(\()p Fu(\013;)14 b(r)r Fz(\))25 b(=)d(\006\()p Fx(\000)p Fz(\001)1021 1609 y Ft(A)1074 1617 y Fr(0)1111 1597 y Fu(;)14 b(r)r Fz(\).)38 b(F)-7 b(urthermore,)25 b(\(2.10\))h(giv)n(es)f(a)h(lo)n(w)n (er)f(b)r(ound)i(of)f(\006\()p Fx(\000)p Fz(\001)3082 1609 y Ft(A)3135 1617 y Fr(0)3172 1597 y Fu(;)14 b(r)r Fz(\).)440 1696 y(W)-7 b(e)27 b(de\014ne)f Fu(\036)h Fz(on)e(\012)1069 1708 y Fs(\013)1143 1696 y Fz(b)n(y)h Fu(\036)p Fz(\()p Fu(x)p Fz(\))9 b(:)52 b(=)1589 1625 y Fp(p)p 1672 1625 548 4 v 71 x Fz(\002)1737 1708 y Fv(0)1792 1696 y Fx(\000)18 b Fu(\026)1925 1708 y Fs(k)1966 1696 y Fz(\()p Fu(\013)p Fz(\))h Fx(\000)f Fu(\017)26 b Fx(j)p Fu(x)p Fx(j)p Fz(,)i(then)e(\(7.5\))g(com)n(bined)g(with)440 1796 y(\(2.10\))h(leads)g(to)g(:)1190 1912 y Fp(Z)1236 2101 y Fv(\012)1283 2109 y Fe(\013)1326 2101 y Ft(nB)q Fv(\(0)p Fs(;)p Fv(2)p Fs(R)p Fv(\))1611 1908 y Fp(\022)1672 2025 y Fu(\017)18 b Fx(\000)1895 1969 y Fu(c)p 1817 2006 192 4 v 1817 2082 a(\013)1870 2058 y Fv(2)1908 2082 y Fu(R)1972 2058 y Fv(2)2037 2025 y Fx(\000)2148 1969 y Fu(C)p 2130 2006 101 4 v 2130 2082 a(R)2194 2058 y Fv(2)2241 1908 y Fp(\023)2316 2025 y Fu(e)2355 1991 y Fv(2)p Fs(\036)2432 2025 y Fx(j)p Fu(u)2503 2037 y Fs(k)q(;\013)2606 2025 y Fx(j)2629 1991 y Fv(2)2694 2025 y Fu(dx)1042 2269 y Fx(\024)1190 2156 y Fp(Z)1236 2344 y Fv(\012)1283 2352 y Fe(\013)1326 2344 y Ft(\\B)q Fv(\(0)p Fs(;)p Fv(2)p Fs(R)p Fv(\))1622 2152 y Fp(\022)1683 2269 y Fz(\002)1748 2281 y Fv(0)1803 2269 y Fx(\000)g Fu(\017)g Fz(+)2049 2213 y Fu(C)p 2031 2250 V 2031 2326 a(R)2095 2302 y Fv(2)2142 2152 y Fp(\023)2217 2269 y Fu(e)2256 2234 y Fv(2)p Fs(\036)2333 2269 y Fx(j)p Fu(u)2404 2281 y Fs(k)q(;\013)2507 2269 y Fx(j)2530 2234 y Fv(2)2568 2269 y Fu(dx:)453 b Fz(\(7.8\))440 2520 y(If)27 b(w)n(e)e(c)n(ho)r(ose)g Fu(R)f Fz(=)1089 2486 y Fs(C)1137 2494 y Fr(1)p 1089 2501 81 4 v 1107 2549 a Fs(\013)1205 2520 y Fz(big)i(enough)f(for)h(that)h Fu(\017)15 b Fx(\000)2134 2488 y Fs(c)p 2070 2502 159 4 v 2070 2549 a(\013)2113 2532 y Fr(2)2146 2549 y Fs(R)2196 2532 y Fr(2)2254 2520 y Fx(\000)2360 2488 y Fs(C)p 2344 2502 83 4 v 2344 2549 a(R)2394 2532 y Fr(2)2460 2520 y Fx(\025)2560 2488 y Fs(\017)p 2557 2502 34 4 v 2557 2549 a Fv(2)2627 2520 y Fz(and)26 b(\002)2852 2532 y Fv(0)2904 2520 y Fx(\000)15 b Fu(\017)h Fz(+)3139 2488 y Fs(C)p 3124 2502 83 4 v 3124 2549 a(R)3174 2532 y Fr(2)3240 2520 y Fx(\024)440 2620 y Fz(\002)505 2632 y Fv(0)560 2620 y Fx(\000)656 2587 y Fs(\017)p 653 2601 34 4 v 653 2649 a Fv(2)696 2620 y Fz(,)28 b(then)g(\(7.8\))f(allo)n(ws)g(us)g(to)h (conclude)f(that)h(:)1173 2737 y Fp(Z)1219 2926 y Fv(\012)1266 2934 y Fe(\013)1326 2850 y Fu(e)1365 2816 y Fv(2)p Fs(\036)1442 2850 y Fx(j)p Fu(u)1513 2862 y Fs(k)q(;\013)1616 2850 y Fx(j)1639 2816 y Fv(2)1704 2850 y Fu(dx)c Fx(\024)f Fu(c)1942 2862 y Fs(\017)1973 2850 y Fu(e)2012 2816 y Fv(4)p Fs(C)2093 2824 y Fr(1)2135 2742 y Ft(p)p 2190 2742 345 3 v 37 x Fr(\002)2233 2791 y(0)2265 2779 y Fq(\000)p Fe(\026)2345 2794 y(k)2381 2779 y Fr(\()p Fe(\013)p Fr(\))p Fq(\000)p Fe(\017)p 2135 2803 400 3 v 2316 2836 a(\013)2549 2850 y Fu(:)3240 3078 y Fd(\003)440 3241 y FA(Theorem)33 b(7.2.)41 b Fo(L)l(et)32 b Fu(\017)1219 3253 y Fv(0)1282 3241 y Fu(>)27 b Fz(0)p Fo(.)44 b(We)32 b(de\014ne)g(by)g Fu(\026)2042 3211 y Fs(N)6 b(D)r(N)2220 3241 y Fz(\()p Fu(\013;)14 b(\017)2376 3253 y Fv(0)2413 3241 y Fz(\))32 b Fo(the)g(b)l(ottom)g(of)g(the)g(sp)l(e)l(c-)440 3341 y(trum)j(of)i Fx(\000r)889 3311 y Fv(2)889 3364 y Ft(A)942 3372 y Fr(0)1015 3341 y Fo(on)g 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Fu(:)557 b Fz(\(7.9\))440 3899 y Fo(L)l(et)27 b Fu(u)629 3911 y Fs(k)q(;\013)759 3899 y Fo(b)l(e)g(a)h(normalize)l(d)h(eigenve)l(ctor)f(asso)l(ciate)l (d)h(to)e(the)g Fu(k)s Fo(-th)g(eigenvalue)h Fu(\026)3015 3911 y Fs(k)3056 3899 y Fz(\()p Fu(\013)p Fz(\))g Fo(for)440 3998 y Fu(P)493 4010 y Ft(A)546 4018 y Fr(0)579 4010 y Fs(;)p Fv(\012)646 4018 y Fe(\013)692 3998 y Fo(.)39 b(Then,)31 b(ther)l(e)f(exists)f Fu(\013)1485 4010 y Fv(1)1546 3998 y Fx(\024)22 b Fu(\013)1686 4010 y Fv(0)1753 3998 y Fo(such)30 b(that)g(for)g(every)h Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)2750 4010 y Fv(1)2788 3998 y Fz(])29 b Fo(:)933 4110 y Fp(Z)979 4298 y Fv(\012)1026 4306 y Fe(\013)1086 4223 y Fu(e)1125 4132 y Fx(p)p 1194 4132 336 4 v 56 x Fv(2)p Fs(\026)1267 4172 y Fe(N)5 b(D)q(N)1418 4188 y Fv(\()p Fs(\017)1472 4196 y Fr(0)1504 4188 y Fv(\))p Ft(j)p Fs(x)p Ft(j)1611 4223 y Fx(j)p Fu(u)1682 4235 y Fs(k)q(;\013)1785 4223 y Fz(\()p Fu(x)p Fz(\))p Fx(j)1919 4188 y Fv(2)1987 4223 y Fu(dx)24 b Fx(\024)f 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b(F)-7 b(urthermore,)35 b Fu(\026)g Fz(is)f(p)r(ositiv)n(e)g(and)h(con)n(tin)n(uous)e(on)h Fw(R)p Fz(,)440 5313 y(so)d(there)h(exists)g(a)g(lo)n(w)n(er)e(b)r (ound)j Fu(\026)p 1553 5343 51 4 v 31 w(>)d Fz(0)h(of)i Fu(\026)p Fz(\()p Fu(\030)2020 5325 y Fv(1)2057 5313 y Fz(\))g(for)f(ev)n(ery)e Fu(\030)2513 5325 y Fv(1)2582 5313 y Fx(2)h Fw(R)p Fz(.)56 b(Consequen)n(tly)-7 b(,)440 5413 y Fu(\026)p Fz(\(\012\()p Fu(\017)648 5425 y Fv(0)686 5413 y Fz(\)\))23 b Fx(\025)g Fu(\026)p 861 5442 V 23 w(>)g Fz(0)k(and)g(a)g(fortiori,)g(recalling)f(\(7.14\),)h(inf)21 b Fu(\033)2381 5425 y Fs(ess)2480 5413 y Fu(P)2556 5423 y Fv(~)2533 5438 y Ft(A)p Fs(;)p Fv(\012\()p Fs(\017)2707 5446 y Fr(0)2739 5438 y Fv(\))2793 5413 y Fx(\025)h Fu(\026)p 2880 5442 V 23 w(>)h Fz(0)p Fu(:)1831 5662 y Fz(20)p eop %%Page: 21 21 21 20 bop 440 531 a Fz(Assume)28 b(that)h(the)f(smallest)g(eigen)n(v)-5 b(alue)27 b(of)h Fu(P)1960 541 y Fv(~)1937 556 y Ft(A)1990 557 y Fs(;)p Fv(\012\()p Fs(\017)2111 565 y Fr(0)2144 557 y Fv(\))2202 531 y Fz(is)g(equal)f(to)h(0,)g(then)h(there)f(exists) 440 649 y(a)f(normalized)g Fu(u)22 b Fx(2)i(D)1149 619 y Fs(N)1212 649 y Fz(\()p Fu(P)1320 659 y Fv(~)1297 674 y Ft(A)q Fs(;)p Fv(\012\()p Fs(\017)1472 682 y Fr(0)1504 674 y Fv(\))1534 649 y Fz(\))k(suc)n(h)f(that)h(:)1137 829 y Fu(D)1206 841 y Fs(x)1244 849 y Fr(1)1280 829 y Fu(u)23 b Fz(=)f(0)p Fu(;)14 b Fz(\()p Fu(D)1618 841 y Fs(x)1656 849 y Fr(2)1710 829 y Fx(\000)k Fu(x)1840 841 y Fv(1)1878 829 y Fz(\))p Fu(u)23 b Fz(=)f(0)28 b(in)g Fx(D)2301 794 y Ft(0)2324 829 y Fz(\(\012\()p Fu(\017)2482 841 y Fv(0)2520 829 y Fz(\)\))p Fu(:)440 993 y Fz(W)-7 b(e)26 b(deduce)f(b)n(y)h(these)f(relations)f(that)i Fu(u)d Fz(=)f(0)j(in)h Fx(D)2078 963 y Ft(0)2102 993 y Fz(\(\012\()p Fu(\017)2260 1005 y Fv(0)2297 993 y Fz(\)\),)h(whic)n (h)e(is)g(not)h(p)r(ossible.)36 b(So)440 1093 y(the)28 b(smallest)f(eigen)n(v)-5 b(alue)27 b(of)g Fu(P)1469 1103 y Fv(~)1446 1118 y Ft(A)q Fs(;)p Fv(\012\()p Fs(\017)1621 1126 y Fr(0)1653 1118 y Fv(\))1711 1093 y Fz(is)g(strictly)h(p)r (ositiv)n(e,)f(th)n(us)h Fu(\025)p Fz(\(\012\()p Fu(\017)2798 1105 y Fv(0)2836 1093 y Fz(\)\))23 b Fu(>)g Fz(0.)164 b Fd(\003)440 1203 y Fz(Let)28 b(us)f(use)h(Lemma)f(7.3)g(to)g(pro)n(v) n(e)f(\(7.9\).)440 1303 y(W)-7 b(e)42 b(assume)g(that)g(\(7.9\))g(do)r (es)g(not)g(hold)g(and)f(so)h(there)g(exists)f(a)h(sequence)g Fu(v)3114 1315 y Fs(\013)3157 1323 y Fe(n)3249 1303 y Fx(2)440 1403 y Fu(H)516 1372 y Fv(1)532 1424 y(~)509 1439 y Ft(A)567 1403 y Fz(\(\012)659 1372 y Fs(N)6 b(D)r(N)659 1423 y(\013)702 1431 y Fe(n)743 1423 y Fs(;\017)791 1431 y Fr(0)837 1403 y Fz(\))28 b(suc)n(h)f(that)h Fx(jj)p Fu(v)1350 1415 y Fs(\013)1393 1423 y Fe(n)1438 1403 y Fx(jj)1484 1419 y Fs(L)1530 1402 y Fr(2)1562 1419 y Fv(\(\012)1635 1402 y Fe(N)5 b(D)q(N)1635 1435 y(\013)1673 1443 y(n)1714 1435 y(;\017)1758 1447 y Fr(0)1795 1419 y Fv(\))1848 1403 y Fz(=)23 b(1)k(and)62 b(lim)2167 1454 y Fs(\013)2210 1462 y Fe(n)2250 1454 y Ft(!)p Fv(0)2364 1403 y Fu(q)s Fz(\()p Fu(v)2476 1415 y Fs(\013)2519 1423 y Fe(n)2564 1403 y Fz(\))24 b(=)e(0.)37 b(So)27 b(:)1001 1657 y Fu(q)s Fz(\()p Fu(v)1113 1669 y Fs(\013)1156 1677 y Fe(n)1202 1657 y Fz(\))c Fx(\025)1345 1544 y Fp(Z)1391 1732 y Fv(\012\()p Fs(\017)1492 1740 y Fr(0)1524 1732 y Fv(\))1568 1657 y Fx(j)p Fu(D)1660 1669 y Fs(x)1698 1677 y Fr(1)1734 1657 y Fu(v)1774 1669 y Fs(\013)1817 1677 y Fe(n)1862 1657 y Fx(j)1885 1622 y Fv(2)1941 1657 y Fz(+)18 b Fx(j)p Fz(\()p Fu(D)2148 1669 y Fs(x)2186 1677 y Fr(2)2241 1657 y Fx(\000)g Fu(x)2371 1669 y Fv(1)2409 1657 y Fz(\))p Fu(v)2481 1669 y Fs(\013)2524 1677 y Fe(n)2569 1657 y Fx(j)2592 1622 y Fv(2)2629 1657 y Fu(dx:)350 b Fz(\(7.17\))440 1882 y(Lemma)27 b(7.3)g(and)h(the)g(min-max)f(principle)g(b)r(ound)h (from)g(b)r(elo)n(w)f Fu(q)s Fz(\()p Fu(v)2664 1894 y Fs(\013)2707 1902 y Fe(n)2752 1882 y Fz(\))h(b)n(y)g(:)535 2089 y Fu(q)s Fz(\()p Fu(v)647 2101 y Fs(\013)690 2109 y Fe(n)735 2089 y Fz(\))c Fx(\025)878 1976 y Fp(Z)924 2165 y Fv(\012\()p Fs(\017)1025 2173 y Fr(0)1058 2165 y Fv(\))1102 2089 y Fx(j)p Fu(D)1194 2101 y Fs(x)1232 2109 y Fr(1)1268 2089 y Fu(v)1308 2101 y Fs(\013)1351 2109 y Fe(n)1396 2089 y Fx(j)1419 2055 y Fv(2)1475 2089 y Fz(+)18 b Fx(j)p Fz(\()p Fu(D)1682 2101 y Fs(x)1720 2109 y Fr(2)1774 2089 y Fx(\000)g Fu(x)1904 2101 y Fv(1)1942 2089 y Fz(\))p Fu(v)2014 2101 y Fs(\013)2057 2109 y Fe(n)2103 2089 y Fx(j)2126 2055 y Fv(2)2163 2089 y Fu(dx)24 b Fx(\025)e Fu(\025)p Fz(\(\012\()p Fu(\017)2570 2101 y Fv(0)2608 2089 y Fz(\)\))p Fx(jj)p Fu(v)2758 2101 y Fs(\013)2801 2109 y Fe(n)2847 2089 y Fx(jj)2893 2055 y Fv(2)2893 2112 y Fs(L)2939 2095 y Fr(2)2971 2112 y Fv(\(\012\()p Fs(\017)3098 2120 y Fr(0)3131 2112 y Fv(\)\))3187 2089 y Fu(:)3092 2250 y Fz(\(7.18\))440 2350 y(By)d(assumptions,)53 b(lim)1050 2401 y Fs(\013)1093 2409 y Fe(n)1133 2401 y Ft(!)p Fv(0)1247 2350 y Fu(q)s Fz(\()p Fu(v)1359 2362 y Fs(\013)1402 2370 y Fe(n)1447 2350 y Fz(\))23 b(=)g(0)c(so)52 b(lim)1744 2401 y Fs(\013)1787 2409 y Fe(n)1828 2401 y Ft(!)p Fv(0)1941 2350 y Fx(jj)p Fu(v)2027 2362 y Fs(\013)2070 2370 y Fe(n)2115 2350 y Fx(jj)2161 2365 y Fs(L)2207 2348 y Fr(2)2239 2365 y Fv(\(\012\()p Fs(\017)2366 2373 y Fr(0)2399 2365 y Fv(\)\))2478 2350 y Fz(=)22 b(0)d(since)g Fu(\025)p Fz(\(\012\()p Fu(\017)3027 2362 y Fv(0)3064 2350 y Fz(\)\))24 b Fu(>)f Fz(0.)440 2484 y(Let)28 b Fu(\037)641 2496 y Fv(1)706 2484 y Fz(b)r(e)g(a)f(regular)f(cut-o\013)h(function)h(de\014ned)g(on)g Fw(R)33 b Fz(with)28 b(0)23 b Fx(\024)g Fu(\037)2638 2496 y Fv(1)2698 2484 y Fx(\024)g Fz(1)k(and)g(:)1310 2693 y Fu(\037)1362 2705 y Fv(1)1400 2693 y Fz(\()p Fu(x)1479 2705 y Fv(2)1517 2693 y Fz(\))c(=)1660 2576 y Fp(\032)1763 2643 y Fz(1)83 b(if)g Fu(x)2066 2655 y Fv(2)2127 2643 y Fx(\025)2225 2609 y Fs(\017)2253 2617 y Fr(0)p 2225 2624 61 4 v 2238 2671 a Fv(4)2295 2643 y Fu(;)1763 2742 y Fz(0)g(if)g Fu(x)2066 2754 y Fv(2)2127 2742 y Fx(\024)23 b(\000)2290 2708 y Fs(\017)2318 2716 y Fr(0)p 2289 2723 V 2303 2771 a Fv(4)2359 2742 y Fu(:)3092 2693 y Fz(\(7.19\))440 2924 y(W)-7 b(e)28 b(also)f(de\014ne)g Fu(\037)1041 2936 y Fv(2)1102 2924 y Fz(=)1189 2852 y Fp(p)p 1272 2852 233 4 v 72 x Fz(1)18 b Fx(\000)g Fu(\037)1467 2896 y Fv(2)1467 2947 y(1)1532 2924 y Fz(and)28 b(easily)e(deduce)i(:)440 3089 y Fu(q)s Fz(\()p Fu(v)552 3101 y Fs(\013)595 3109 y Fe(n)641 3089 y Fz(\))23 b(=)g Fu(q)s Fz(\()p Fu(\037)908 3101 y Fv(1)945 3089 y Fu(v)985 3101 y Fs(\013)1028 3109 y Fe(n)1073 3089 y Fz(\))5 b(+)g Fu(q)s Fz(\()p Fu(\037)1304 3101 y Fv(2)1342 3089 y Fu(v)1382 3101 y Fs(\013)1425 3109 y Fe(n)1470 3089 y Fz(\))g Fx(\000)g(jj)p Fu(D)1692 3101 y Fs(x)1730 3109 y Fr(2)1766 3089 y Fu(\037)1818 3101 y Fv(1)1876 3089 y Fu(v)1916 3101 y Fs(\013)1959 3109 y Fe(n)2005 3089 y Fx(jj)2051 3055 y Fv(2)2051 3113 y Fs(L)2097 3096 y Fr(2)2129 3113 y Fv(\(\012)2202 3096 y Fe(N)g(D)q(N)2202 3130 y(\013)2240 3138 y(n)2281 3130 y(;\017)2325 3142 y Fr(0)2362 3113 y Fv(\))2397 3089 y Fx(\000)g(jj)p Fu(D)2582 3101 y Fs(x)2620 3109 y Fr(2)2656 3089 y Fu(\037)2708 3101 y Fv(2)2766 3089 y Fu(v)2806 3101 y Fs(\013)2849 3109 y Fe(n)2894 3089 y Fx(jj)2940 3055 y Fv(2)2940 3113 y Fs(L)2986 3096 y Fr(2)3018 3113 y Fv(\(\012)3091 3096 y Fe(N)g(D)q(N)3091 3130 y(\013)3129 3138 y(n)3170 3130 y(;\017)3214 3142 y Fr(0)3251 3113 y Fv(\))3281 3089 y Fu(:)3092 3213 y Fz(\(7.20\))440 3312 y(Lo)r(oking)17 b(at)h(the)h(supp)r(ort)f(of)h(\()p Fu(D)1461 3324 y Fs(x)1499 3332 y Fr(2)1535 3312 y Fu(\037)1587 3324 y Fs(j)1622 3312 y Fz(\))1654 3324 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)1826 3312 y Fz(,)h(w)n(e)e(de\014ne)h Fu(a)9 b Fz(:)51 b(=)2437 3278 y Fs(\017)2465 3286 y Fr(0)p 2437 3293 61 4 v 2451 3341 a Fv(4)2521 3312 y Fz(tan)2665 3278 y Fs(\013)2708 3286 y Fe(n)p 2665 3293 84 4 v 2690 3341 a Fv(2)2777 3312 y Fz(and)19 b(\012)2990 3324 y Fv(2)3027 3312 y Fz(\()p Fu(\017)3093 3324 y Fv(0)3130 3312 y Fz(\))9 b(:)51 b(=)440 3412 y(\012)500 3382 y Fs(N)6 b(D)r(N)500 3432 y(\013)543 3440 y Fe(n)584 3432 y Fs(;\017)632 3440 y Fr(0)678 3412 y Fx(\\)p Fz(])19 b Fx(\000)f Fu(a;)c Fz(0])k Fx(\002)1105 3345 y Fp(\003)1139 3412 y Fx(\000)1214 3378 y Fs(\017)1242 3386 y Fr(0)p 1214 3393 61 4 v 1227 3440 a Fv(4)1284 3412 y Fu(;)1331 3378 y Fs(\017)1359 3386 y Fr(0)p 1331 3393 V 1344 3440 a Fv(4)1401 3345 y Fp(\002)1463 3412 y Fz(and)27 b(so)g(there)h(exists) f(a)g(constan)n(t)g Fu(c)c(>)g Fz(0)k(suc)n(h)g(that)h(:)702 3609 y Fx(jj)p Fu(D)817 3621 y Fs(x)855 3629 y Fr(2)891 3609 y Fu(\037)943 3621 y Fs(j)1006 3609 y Fu(v)1046 3621 y Fs(\013)1089 3629 y Fe(n)1134 3609 y Fx(jj)1180 3575 y Fv(2)1180 3633 y Fs(L)1226 3616 y Fr(2)1258 3633 y Fv(\(\012)1331 3616 y Fe(N)5 b(D)q(N)1331 3649 y(\013)1369 3657 y(n)1410 3649 y(;\017)1454 3661 y Fr(0)1491 3633 y Fv(\))1544 3609 y Fx(\024)1659 3553 y Fu(c)p 1642 3590 71 4 v 1642 3666 a(\017)1676 3637 y Fv(2)1676 3688 y(0)1723 3609 y Fx(jj)p Fu(v)1809 3621 y Fs(\013)1852 3629 y Fe(n)1897 3609 y Fx(jj)1943 3575 y Fv(2)1943 3631 y Fs(L)1989 3615 y Fr(2)2021 3631 y Fv(\(\012\()p Fs(\017)2148 3639 y Fr(0)2180 3631 y Fv(\))p Ft([)p Fv(\012)2298 3639 y Fr(2)2331 3631 y Fv(\()p Fs(\017)2385 3639 y Fr(0)2417 3631 y Fv(\)\))2473 3609 y Fu(;)42 b(j)28 b Fz(=)22 b(1)p Fu(;)14 b Fz(2)p Fu(:)261 b Fz(\(7.21\))440 3829 y(W)-7 b(e)19 b(kno)n(w)e(that)53 b(lim)952 3880 y Fs(\013)995 3888 y Fe(n)1036 3880 y Ft(!)p Fv(0)1149 3829 y Fx(jj)p Fu(v)1235 3841 y Fs(\013)1278 3849 y Fe(n)1323 3829 y Fx(jj)1369 3844 y Fs(L)1415 3827 y Fr(2)1447 3844 y Fv(\(\012\()p Fs(\017)1574 3852 y Fr(0)1607 3844 y Fv(\)\))1686 3829 y Fz(=)23 b(0,)c(so)f(it)h(is)f (enough)g(to)g(estimate)g Fx(jj)p Fu(v)2877 3841 y Fs(\013)2920 3849 y Fe(n)2966 3829 y Fx(jj)3012 3799 y Fv(2)3012 3856 y Fs(L)3058 3839 y Fr(2)3090 3856 y Fv(\(\012)3163 3864 y Fr(2)3196 3856 y Fv(\()p Fs(\017)3250 3864 y Fr(0)3282 3856 y Fv(\)\))3338 3829 y Fz(.)440 3963 y(Since)k Fu(v)691 3975 y Fs(\013)734 3983 y Fe(n)800 3963 y Fz(v)-5 b(anishes)21 b(on)g(the)h(curv)n(e)e(of)i(the)g(b)r(oundary)-7 b(,)22 b(w)n(e)f(can)g(use)g(some)g(P)n(oincar)n(\023)-39 b(e)18 b(tec)n(h-)440 4062 y(niques)27 b(to)h(establish)f(:)1064 4152 y Fp(Z)1110 4340 y Fv(\012)1157 4348 y Fr(2)1189 4340 y Fv(\()p Fs(\017)1243 4348 y Fr(0)1275 4340 y Fv(\))1319 4265 y Fx(j)p Fu(v)1382 4277 y Fs(\013)1425 4285 y Fe(n)1471 4265 y Fz(\()p Fu(x)1550 4277 y Fv(1)1588 4265 y Fu(;)14 b(x)1672 4277 y Fv(2)1709 4265 y Fz(\))p Fx(j)1764 4230 y Fv(2)1830 4265 y Fu(dx)1920 4277 y Fv(1)1985 4265 y Fu(dx)2075 4277 y Fv(2)2196 4265 y Fx(\024)83 b Fu(a)2388 4230 y Fv(2)2425 4265 y Fu(q)s Fz(\()p Fu(v)2537 4277 y Fs(\013)2580 4285 y Fe(n)2625 4265 y Fz(\))p Fu(:)412 b Fz(\(7.22\))440 4490 y(Relations)26 b(\(7.18\))h(and)f(\(7.22\))h (pro)n(v)n(e)e(the)i(existence)g(of)g Fu(\013)2310 4502 y Fv(0)2374 4490 y Fz(and)g Fu(C)6 b Fz(\()p Fu(\017)2666 4502 y Fv(0)2704 4490 y Fz(\))23 b Fu(>)g Fz(0)j(suc)n(h)h(that)g(:) 1022 4655 y Fx(jj)p Fu(v)1108 4667 y Fs(\013)1151 4675 y Fe(n)1196 4655 y Fx(jj)1242 4621 y Fv(2)1242 4678 y Fs(L)1288 4661 y Fr(2)1321 4678 y Fv(\(\012\()p Fs(\017)1448 4686 y Fr(0)1480 4678 y Fv(\))p Ft([)p Fv(\012)1598 4686 y Fr(2)1630 4678 y Fv(\()p Fs(\017)1684 4686 y Fr(0)1716 4678 y Fv(\)\))1795 4655 y Fx(\024)c Fu(C)6 b Fz(\()p Fu(\017)2014 4667 y Fv(0)2052 4655 y Fz(\))p Fu(q)s Fz(\()p Fu(v)2196 4667 y Fs(\013)2239 4675 y Fe(n)2285 4655 y Fz(\))p Fu(;)14 b Fx(8)p Fu(\013)2454 4667 y Fs(n)2521 4655 y Fx(\024)23 b Fu(\013)2662 4667 y Fv(0)2699 4655 y Fu(:)370 b Fz(\(7.23\))440 4820 y(Let)28 b Fu(\017)23 b(>)f Fz(0,)27 b(\(7.20\),)g(\(7.21\))g(and)h(\(7.23\))e(sho)n(w)h (that)h(there)f(exists)36 b(~)-50 b Fu(\013)2610 4832 y Fv(0)2670 4820 y Fx(\024)23 b Fu(\013)2811 4832 y Fv(0)2876 4820 y Fz(suc)n(h)k(that)h(:)1043 4984 y Fx(8)p Fu(\013)1143 4996 y Fs(n)1211 4984 y Fx(\024)j Fz(~)-50 b Fu(\013)1352 4996 y Fv(0)1389 4984 y Fu(;)41 b(q)s Fz(\()p Fu(v)1565 4996 y Fs(\013)1608 5004 y Fe(n)1654 4984 y Fz(\))23 b Fx(\025)g Fu(q)s Fz(\()p Fu(\037)1921 4996 y Fv(1)1959 4984 y Fu(v)1999 4996 y Fs(\013)2042 5004 y Fe(n)2087 4984 y Fz(\))c(+)f Fu(q)s Fz(\()p Fu(\037)2345 4996 y Fv(2)2382 4984 y Fu(v)2422 4996 y Fs(\013)2465 5004 y Fe(n)2510 4984 y Fz(\))h Fx(\000)f Fu(\017:)391 b Fz(\(7.24\))440 5149 y(By)31 b(comparison)f(with)h(the)h(op)r(erator)e Fu(P)1771 5159 y Fv(~)1748 5174 y Ft(A)p Fs(;)p Fn(R)p Ft(\002)p Fn(R)1963 5157 y Fr(+)2008 5149 y Fz(,)i(using)f(gauge)f (transform)g(and)h(in)n(v)-5 b(ari-)440 5248 y(ance)27 b(under)h(domain's)f(translation,)f(w)n(e)h(deduce)h(:)734 5413 y Fu(q)s Fz(\()p Fu(\037)858 5425 y Fv(1)895 5413 y Fu(v)935 5425 y Fs(\013)978 5433 y Fe(n)1023 5413 y Fz(\))c Fx(\025)e Fz(\002)1231 5425 y Fv(0)1268 5413 y Fx(jj)p Fu(\037)1366 5425 y Fv(1)1403 5413 y Fu(v)1443 5425 y Fs(\013)1486 5433 y Fe(n)1532 5413 y Fx(jj)1578 5379 y Fv(2)1578 5435 y Fs(L)1624 5419 y Fr(2)1688 5413 y Fz(and)27 b Fu(q)s Fz(\()p Fu(\037)1973 5425 y Fv(2)2011 5413 y Fu(v)2051 5425 y Fs(\013)2094 5433 y Fe(n)2139 5413 y Fz(\))c Fx(\025)g Fz(\002)2347 5425 y Fv(0)2384 5413 y Fx(jj)p Fu(\037)2482 5425 y Fv(2)2519 5413 y Fu(v)2559 5425 y Fs(\013)2602 5433 y Fe(n)2647 5413 y Fx(jj)2693 5379 y Fv(2)2693 5435 y Fs(L)2739 5419 y Fr(2)2775 5413 y Fu(:)294 b Fz(\(7.25\))1831 5662 y(21)p eop %%Page: 22 22 22 21 bop 440 531 a Fz(Th)n(us)27 b(:)1290 631 y Fu(q)s Fz(\()p Fu(v)1402 643 y Fs(\013)1445 651 y Fe(n)1490 631 y Fz(\))c Fx(\025)g Fz(\002)1698 643 y Fv(0)1735 631 y Fx(jj)p Fu(v)1821 643 y Fs(\013)1864 651 y Fe(n)1909 631 y Fx(jj)1955 597 y Fv(2)1955 655 y Fs(L)2001 638 y Fr(2)2033 655 y Fv(\(\012)2106 638 y Fe(N)5 b(D)q(N)2106 672 y(\013)2144 680 y(n)2185 672 y(;\017)2229 684 y Fr(0)2266 655 y Fv(\))2315 631 y Fx(\000)18 b Fu(\017:)637 b Fz(\(7.26\))440 782 y(This)28 b(is)f(imp)r(ossible)h(since)f Fu(v)1361 794 y Fs(\013)1404 802 y Fe(n)1477 782 y Fz(is)g(normalized.)1259 b Fd(\003)440 925 y FA(Corollary)37 b(7.4.)42 b Fo(We)33 b(denote)h(by)f Fu(u)1644 937 y Fs(k)q(;\013)1780 925 y Fo(the)g Fu(k)s Fo(-th)g(normalize)l(d)h(for)g Fu(P)2713 937 y Fs(\013)2760 925 y Fo(.)49 b(Then,)34 b(for)g(al)t(l)440 1025 y Fu(\017)23 b(>)f Fz(0)p Fo(,)30 b(ther)l(e)g(exists)f Fu(\016)d(>)d Fz(0)29 b Fo(and)h Fu(\013)1551 1037 y Fv(0)1612 1025 y Fu(>)23 b Fz(0)29 b Fo(such)h(that)f(:)884 1217 y Fx(8)p Fu(\013)23 b Fx(\024)f Fu(\013)1147 1229 y Fv(0)1185 1217 y Fu(;)43 b Fx(8)p Fu(T)34 b Fx(\025)1488 1161 y Fu(\017)p 1478 1198 54 4 v 1478 1274 a(\013)1542 1217 y(;)1579 1104 y Fp(Z)1625 1292 y Fv(])p Fs(T)5 b(;)p Fv(+)p Ft(1)p Fv([)p Ft(\002)p Fv(])p Ft(\000)1976 1270 y Fr(1)p 1976 1279 29 3 v 1976 1312 a(2)2014 1292 y Fs(;)2044 1270 y Fr(1)p 2044 1279 V 2044 1312 a(2)2082 1292 y Fv([)2119 1217 y Fx(j)p Fu(u)2190 1229 y Fs(k)q(;\013)2293 1217 y Fx(j)2316 1182 y Fv(2)2383 1217 y Fu(dt)30 b(d\021)d Fx(\024)22 b Fu(e)2723 1182 y Ft(\000)2790 1160 y Fe(\016)p 2785 1169 38 3 v 2785 1202 a(\013)2837 1217 y Fu(:)232 b Fz(\(7.27\))440 1524 y Fy(8)135 b(First)45 b(term)g(of)g(the)h (asymptotic)g(expansion)440 1722 y Fl(8.1)112 b(W)-9 b(eak)38 b(lo)m(w)m(er)e(b)s(ound)440 1875 y FA(Theorem)30 b(8.1.)40 b Fo(F)-6 b(or)30 b(every)h Fu(\017)23 b(>)f Fz(0)p Fo(,)30 b(ther)l(e)g(exists)f Fu(\013)2138 1887 y Fv(0)2199 1875 y Fu(>)22 b Fz(0)30 b Fo(such)f(that)h(:)1378 2046 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\025)1695 1990 y Fu(\013)p 1667 2027 111 4 v 1667 2044 a Fx(p)p 1736 2044 42 4 v 68 x Fz(3)1806 2046 y Fx(\000)18 b Fu(\017\013;)44 b Fx(8)p Fu(\013)22 b Fx(\024)h Fu(\013)2306 2058 y Fv(0)2343 2046 y Fu(:)440 2243 y Fo(Pr)l(o)l(of)35 b(:)46 b Fz(The)31 b(pro)r(of)g(of)h(Theorem)f(8.1)f(consists)h(of)h(comparing)e(the)i (eigen)n(v)-5 b(alue)30 b(of)i(the)440 2343 y(op)r(erator)27 b Fu(P)829 2355 y Fs(\013)905 2343 y Fz(with)h(the)h(\014rst)f(eigen)n (v)-5 b(alue)27 b(of)i Fu(L)1960 2313 y Fv(mean)2145 2343 y Fz(b)n(y)f(using)g(min-max)g(principle)h(and)440 2443 y(estimates)e(of)h(the)g(deca)n(y)e(of)i(the)g(\014rst)f(eigen)n (v)n(ector)f(of)h Fu(P)2261 2455 y Fs(\013)2309 2443 y Fz(.)440 2542 y(Lo)r(oking)e(at)h(the)h(op)r(erator)d Fu(P)1386 2554 y Fs(\013)1434 2542 y Fz(,)i(w)n(e)g(see)g(that)h(the)f (term)2264 2510 y Fv(1)p 2264 2524 34 4 v 2268 2571 a Fs(t)2334 2542 y Fz(is)g(di\016cult)h(to)f(analyze)f(when)440 2642 y Fu(t)i Fz(b)r(ecomes)g(v)n(ery)f(large.)35 b(So,)27 b(w)n(e)f(a)n(v)n(oid)g(this)h(problem)g(b)n(y)f(dealing)h(with)g(new)g (op)r(erators)440 2742 y(de\014ned)h(on)f(a)g(b)r(ounded)h(domain)g(:) 1436 2924 y(\012)1496 2936 y Fs(T)1558 2924 y Fz(:)51 b(=]0)p Fu(;)14 b(T)e Fz([)p Fx(\002)1962 2807 y Fp(\025)2003 2924 y Fx(\000)2078 2868 y Fz(1)p 2078 2905 42 4 v 2078 2981 a(2)2129 2924 y Fu(;)2176 2868 y Fz(1)p 2176 2905 V 2176 2981 a(2)2227 2807 y Fp(\024)2285 2924 y Fu(;)440 3123 y Fz(estimate)24 b(the)h(b)r(ottom)g(of)f(the)h(sp)r(ectrum)g(for) f(these)g(new)h(op)r(erators)d(and)j(compare)e(with)440 3223 y(the)37 b(old)g(ones.)65 b(W)-7 b(e)37 b(b)r(egin)g(with)g(some)g (lemmas)f(preparing)g(the)h(pro)r(of)g(of)g(the)g(lo)n(w)n(er)440 3323 y(b)r(ound)25 b(in)g(Theorem)f(8.1.)35 b(As)25 b(w)n(e)f(will)h (see)g(in)g(the)g(follo)n(wing,)f(remainders)g(coming)g(from)440 3422 y(the)k(cut-o\013)g(are)e(v)n(ery)h(small)g(and)g(w)n(e)h(will)f (estimate)h(the)g(error.)440 3650 y Fl(8.2)112 b(The)38 b(mean)g(op)s(erator)f Fk(L)1744 3614 y Fj(mean)p Fi(;T)440 3803 y Fz(W)-7 b(e)27 b(sho)n(w)f(that)h(the)g(b)r(ottom)h(of)e(the)i (sp)r(ectrum)f(of)f Fu(L)2149 3773 y Fv(mean)2334 3803 y Fz(is)g(v)n(ery)g(close)g(to)h(the)g(b)r(ottom)440 3903 y(obtained)g(from)h(the)g(realization)e(of)h Fu(L)1678 3872 y Fv(mean)1863 3903 y Fz(on)h(a)f(b)r(ounded)h(domain.)36 b(W)-7 b(e)28 b(de\014ne)g(:)835 4056 y Fx(V)893 4022 y Fs(N)886 4077 y Fv(mean)o Fs(;T)1204 4056 y Fz(:)51 b(=)83 b Fx(f)p Fu(u)22 b Fx(2)h Fu(L)1673 4022 y Fv(2)1710 4056 y Fz(\(0)p Fu(;)14 b(T)e Fz(\))p Fx(j)1964 3985 y(p)p 2033 3985 30 4 v 71 x Fu(tD)2132 4068 y Fs(t)2161 4056 y Fu(u)22 b Fx(2)i Fu(L)2367 4022 y Fv(2)2403 4056 y Fz(\(0)p Fu(;)14 b(T)e Fz(\))p Fu(;)41 b(u)2719 4071 y Ft(j)2739 4079 y Fe(t)p Fr(=)p Fe(T)2877 4056 y Fz(=)23 b(0)p Fx(g)p Fu(;)659 4195 y(`)694 4161 y Fv(mean)o Fs(;T)920 4195 y Fz(\()p Fu(u;)14 b(v)s Fz(\))124 b(=)h Fu(`)1461 4161 y Fv(mean)1618 4195 y Fz(\()p Fu(u;)14 b(v)s Fz(\))p Fu(;)37 b Fx(8)p Fu(u;)14 b(v)25 b Fx(2)f(V)2204 4161 y Fs(N)2197 4216 y Fv(mean)p Fs(;)p Fv(T)2421 4195 y Fu(;)838 4386 y(\025)886 4347 y Fv(mean)o Fs(;T)886 4409 y Fv(1)1204 4386 y Fz(:)51 b(=)263 b(inf)1426 4449 y Fs(u)p Ft(2V)1557 4429 y Fe(N)1552 4468 y Fr(mean)p Fe(;)p Fr(T)1745 4449 y Fs(;u)p Ft(6)p Fv(=0)1911 4330 y Fu(`)1946 4300 y Fv(mean)o Fs(;T)2172 4330 y Fz(\()p Fu(u;)14 b(u)p Fz(\))p 1911 4367 457 4 v 1953 4443 a Fx(jj)p Fu(u)p Fx(jj)2093 4415 y Fv(2)2093 4472 y Fs(L)2139 4455 y Fr(2)2170 4472 y Fv(\(0)p Fs(;T)9 b Fv(\))2378 4386 y Fu(:)733 b Fz(\(8.1\))440 4606 y FA(Lemma)30 b(8.2.)40 b Fo(Ther)l(e)30 b(exists)g(a)g(p)l(ositive)h(c)l(onstant)g Fz(~)-44 b Fu(c)30 b Fo(such)f(that)h(:)1067 4778 y Fu(\025)1115 4744 y Fv(mean)1115 4799 y(1)1296 4778 y Fx(\024)23 b Fu(\025)1432 4738 y Fv(mean)o Fs(;T)1432 4800 y Fv(1)1681 4778 y Fx(\024)g Fu(\025)1817 4744 y Fv(mean)1817 4799 y(1)1994 4778 y Fz(+)d(~)-44 b Fu(ce)2152 4736 y Ft(\000)2243 4713 y Fe(T)p 2213 4722 103 3 v 2213 4764 a Fr(2)2241 4730 y Fq(p)p 2287 4730 29 3 v 34 x Fr(3)2329 4778 y Fu(;)44 b Fx(8)p Fu(T)33 b Fx(\025)23 b Fz(1)p Fu(:)456 b Fz(\(8.2\))440 4932 y Fo(Pr)l(o)l(of)39 b(:)55 b Fz(The)37 b(pro)r(of)f(is)g(v)n(ery)g(easy)-7 b(.)63 b(By)36 b(the)h(inclusion)f Fx(V)2380 4902 y Fs(N)2373 4955 y Fv(mean)p Fs(;)p Fv(T)2636 4932 y Fx(\032)i(V)2797 4902 y Fs(N)2790 4952 y Fv(mean)2947 4932 y Fz(,)h(w)n(e)e(ha)n(v)n(e)440 5054 y(immediately)24 b Fu(\025)957 5024 y Fv(mean)957 5075 y(1)1138 5054 y Fx(\024)f Fu(\025)1274 5014 y Fv(mean)o Fs(;T)1274 5076 y Fv(1)1500 5054 y Fz(.)36 b(F)-7 b(or)22 b(the)i(second)f(inequalit)n (y)-7 b(,)24 b(w)n(e)f(use)h(the)f(eigen)n(v)n(ector)440 5154 y Fu(u)488 5166 y Fv(0)563 5154 y Fz(of)37 b Fu(L)724 5124 y Fv(mean)920 5154 y Fz(whic)n(h)h(is)f(exp)r(onen)n(tially)h (decreasing)e(and)h(a)h(cut-o\013)g(function)g Fu(\037)g Fz(with)440 5272 y(supp)r(ort)33 b(in)h(\012)916 5284 y Fs(T)968 5272 y Fz(.)54 b(W)-7 b(e)34 b(estimate)f(the)h(Ra)n(yleigh) e(quotien)n(t)2372 5232 y Fs(`)2400 5206 y Fr(mean)o Fe(;T)2595 5232 y Fv(\()p Fs(\037u)2701 5240 y Fr(0)2734 5232 y Fs(;\037u)2834 5240 y Fr(0)2867 5232 y Fv(\))p 2372 5253 521 4 v 2433 5303 a Ft(jj)p Fs(\037u)2553 5311 y Fr(0)2586 5303 y Ft(jj)2626 5283 y Fr(2)2626 5333 y Fe(L)2666 5321 y Fr(2)2697 5333 y(\(\012)2759 5347 y Fe(T)2805 5333 y Fr(\))2936 5272 y Fz(to)i(ac)n(hiev)n(e)440 5413 y(the)28 b(pro)r(of.)2444 b Fd(\003)1831 5662 y Fz(22)p eop %%Page: 23 23 23 22 bop 440 531 a Fl(8.3)112 b(The)38 b(op)s(erator)g Fk(P)1457 495 y Fi(T)1443 556 y(\013)1548 531 y Fl(on)g(a)g(truncated)f (sector)440 685 y Fz(In)c(order)f(to)i(compare)e Fu(\025)p Fz(\()p Fu(\013)p Fz(\))i(and)g Fu(\025)1633 654 y Fv(mean)o Fs(;T)1859 685 y Fz(,)h(w)n(e)e(in)n(tro)r(duce)g(a)f(new)i(op)r (erator)d Fu(P)3064 697 y Fs(\013)3112 685 y Fz(.)54 b(W)-7 b(e)440 784 y(denote)37 b Fu(P)782 754 y Fs(T)770 805 y(\013)872 784 y Fz(the)g(op)r(erator)f(asso)r(ciated)g(to)h(the)h (form)f Fu(a)2288 796 y Fs(\013)2335 784 y Fz(,)j(restricted)d(to)g (\012)2949 796 y Fs(T)3001 784 y Fz(,)j(with)e(a)440 884 y(Diric)n(hlet)28 b(b)r(oundary)f(condition)g(at)g Fu(t)c Fz(=)g Fu(T)39 b Fz(de\014ned)28 b(on)f(:)540 1109 y Fx(V)598 1075 y Fs(N)591 1130 y(T)5 b(;)p Fv(0)702 1109 y Fz(:)51 b(=)863 992 y Fp(\032)925 1109 y Fu(u)23 b Fx(2)g Fu(L)1131 1075 y Fv(2)1168 1109 y Fz(\(\012)1260 1121 y Fs(T)1313 1109 y Fz(\))1345 989 y Fp(\014)1345 1039 y(\014)1345 1089 y(\014)1345 1139 y(\014)1439 1053 y Fz(1)p 1410 1090 100 4 v 1410 1107 a Fx(p)p 1480 1107 30 4 v 1480 1174 a Fu(t)1519 1109 y(@)1563 1121 y Fs(\021)1604 1109 y Fu(u)g Fx(2)g Fu(L)1810 1075 y Fv(2)1847 1109 y Fz(\(\012)1939 1121 y Fs(T)1991 1109 y Fz(\))p Fu(;)2088 1038 y Fx(p)p 2157 1038 V 71 x Fu(tD)2256 1121 y Fs(t)2285 1109 y Fu(u)g Fx(2)g Fu(L)2491 1075 y Fv(2)2528 1109 y Fz(\(\012)2620 1121 y Fs(T)2673 1109 y Fz(\))p Fu(;)41 b(u)2817 1124 y Ft(j)2837 1132 y Fe(t)p Fr(=)p Fe(T)2976 1109 y Fz(=)22 b(0)3105 992 y Fp(\033)3181 1109 y Fu(:)3134 1259 y Fz(\(8.3\))440 1359 y(W)-7 b(e)28 b(denote)g(b)n(y)f Fu(\025)1014 1328 y Fs(T)1067 1359 y Fz(\()p Fu(\013)p Fz(\))h(the)g(b)r(ottom)g(of)g(the)g(sp)r(ectrum)f(of)h Fu(P)2408 1328 y Fs(T)2396 1379 y(\013)2488 1359 y Fz(:)649 1594 y Fu(\025)697 1560 y Fs(T)750 1594 y Fz(\()p Fu(\013)p Fz(\))c(=)152 b(inf)979 1657 y Fs(u)p Ft(2V)1110 1637 y Fe(N)1105 1676 y(T)t(;)p Fr(0)1195 1657 y Fs(;u)p Ft(6)p Fv(=0)1443 1538 y Fu(q)1483 1508 y Fs(T)1480 1559 y(\013)1535 1538 y Fz(\()p Fu(u)p Fz(\))p 1362 1575 368 4 v 1362 1651 a Fx(jj)p Fu(u)p Fx(jj)1502 1623 y Fv(2)1502 1680 y Fs(L)1548 1663 y Fr(2)1579 1680 y Fv(\(\012)1652 1688 y Fe(T)1699 1680 y Fv(\))1766 1594 y Fz(with)28 b Fu(q)1995 1560 y Fs(T)1992 1615 y(\013)2048 1594 y Fz(\()p Fu(u)p Fz(\))23 b(=)2270 1481 y Fp(Z)2317 1670 y Fv(\012)2364 1678 y Fe(T)2428 1594 y Fx(jr)2520 1606 y Ft(A)2578 1594 y Fu(u)p Fx(j)2649 1560 y Fv(2)2713 1594 y Fu(dt)28 b(d\021)s(:)210 b Fz(\(8.4\))440 1841 y FA(Lemma)30 b(8.3.)40 b Fo(F)-6 b(or)30 b(every)g Fu(\017)23 b(>)g Fz(0)p Fo(,)29 b(ther)l(e)h(exists)f Fu(\016)e(>)22 b Fz(0)29 b Fo(and)i Fu(\013)2455 1853 y Fv(0)2515 1841 y Fu(>)23 b Fz(0)29 b Fo(such)h(that)f(:)985 2038 y Fx(8)p Fu(\013)23 b Fx(\024)f Fu(\013)1248 2050 y Fv(0)1286 2038 y Fu(;)43 b Fx(8)p Fu(T)34 b Fx(\025)1589 1982 y Fu(\017)p 1579 2019 54 4 v 1579 2095 a(\013)1643 2038 y(;)43 b(\025)p Fz(\()p Fu(\013)p Fz(\))25 b Fx(\024)d Fu(\025)2034 2004 y Fs(T)2087 2038 y Fz(\()p Fu(\013)p Fz(\))i Fx(\024)f Fu(\025)p Fz(\()p Fu(\013)p Fz(\))d(+)e Fu(e)2623 2004 y Ft(\000)2689 1981 y Fe(\016)p 2684 1990 38 3 v 2684 2024 a(\013)2736 2038 y Fu(:)375 b Fz(\(8.5\))440 2242 y Fo(Pr)l(o)l(of)31 b(:)39 b Fz(By)27 b(inclusion)g(of)h(the)g (form)f(domain,)g(w)n(e)g(deduce)h(the)g(lo)n(w)n(er)e(b)r(ound)i(:) 1613 2422 y Fu(\025)1661 2388 y Fs(T)1713 2422 y Fz(\()p Fu(\013)p Fz(\))c Fx(\025)f Fu(\025)p Fz(\()p Fu(\013)p Fz(\))p Fu(:)1004 b Fz(\(8.6\))440 2603 y(T)-7 b(o)28 b(establish)g(an)g(upp)r(er)g(b)r(ound,)h(w)n(e)f(m)n(ultiply)h(the)f (functions)h(in)f(the)h(domains)f Fx(V)3126 2572 y Fs(N)3217 2603 y Fz(b)n(y)440 2702 y(a)37 b(cut-o\013)g(function,)j(so)d(they)g (b)r(elong)g(to)g(the)h(form)f(domain)g Fx(V)2565 2672 y Fs(N)2558 2725 y(T)5 b(;)p Fv(0)2696 2702 y Fz(and)38 b(w)n(e)e(estimate)440 2802 y(the)29 b(error)d(coming)h(from)h(a)g (cutting.)39 b(W)-7 b(e)28 b(consider)f(a)h(regular)e(real)h(cut-o\013) i(function)f Fu(\037)p Fz(,)440 2901 y(de\014ned)33 b(on)f Fw(R)905 2871 y Fv(+)966 2901 y Fz(,)i(with)f(supp)r(ort)f(in)g([0)p Fu(;)14 b Fz(1])32 b(and)g(equal)g(to)g(1)g(on)g([0)p Fu(;)2631 2869 y Fv(1)p 2631 2883 34 4 v 2631 2930 a(2)2674 2901 y Fz(].)51 b(F)-7 b(or)32 b Fu(T)42 b(>)31 b Fz(0,)i(w)n(e)440 3001 y(de\014ne)28 b Fu(\037)732 3013 y Fs(T)794 3001 y Fz(:)51 b(=)23 b Fu(\037)1022 2934 y Fp(\000)1084 2968 y Fs(:)p 1070 2982 49 4 v 1070 3030 a(T)1128 2934 y Fp(\001)1194 3001 y Fz(on)k Fw(R)1363 2971 y Fv(+)1452 3001 y Fz(and)h Fu(v)1657 2971 y Fs(T)1654 3022 y(\013)1719 3001 y Fz(:)51 b(=)23 b Fu(\037)1933 3013 y Fs(T)2013 3001 y Fu(u)2061 3013 y Fs(\013)2136 3001 y Fz(on)k(\012)2311 3013 y Fs(T)2391 3001 y Fz(where)h Fu(u)2680 3013 y Fs(\013)2754 3001 y Fz(is)g(a)f(normalized)440 3101 y(eigen)n(v)n(ector)f(for)h Fu(\025)p Fz(\()p Fu(\013)p Fz(\).)38 b(W)-7 b(e)28 b(remark)e(that)i Fu(v)1875 3071 y Fs(T)1872 3121 y(\013)1951 3101 y Fx(2)23 b(V)2087 3071 y Fs(N)2080 3124 y(T)5 b(;)p Fv(0)2209 3101 y Fz(and)27 b(easily)g(pro)n(v)n(e)f(that)i(:)776 3298 y Fu(q)816 3264 y Fs(T)813 3318 y(\013)869 3298 y Fz(\()p Fu(v)944 3264 y Fs(T)941 3318 y(\013)996 3298 y Fz(\))84 b(=)105 b(Re)1417 3231 y Fp(\012)1456 3298 y Fu(\037)1508 3264 y Fv(2)1508 3318 y Fs(T)1588 3298 y Fu(P)1641 3310 y Fs(\013)1689 3298 y Fu(u)1737 3310 y Fs(\013)1784 3298 y Fu(;)14 b(u)1869 3310 y Fs(\013)1916 3231 y Fp(\013)1955 3331 y Fs(L)2001 3315 y Fr(2)2033 3331 y Fv(\(\012)2106 3339 y Fe(T)2152 3331 y Fv(\))2200 3298 y Fz(+)k Fx(jj)28 b(jr)p Fu(\037)2501 3310 y Fs(T)2554 3298 y Fx(j)p Fu(u)2625 3310 y Fs(\013)2672 3298 y Fx(jj)2718 3264 y Fv(2)2718 3321 y Fs(L)2764 3304 y Fr(2)2796 3321 y Fv(\(\012)2869 3329 y Fe(T)2915 3321 y Fv(\))2945 3298 y Fu(:)166 b Fz(\(8.7\))440 3492 y(By)26 b(assumption)g(on)f Fu(u)1168 3504 y Fs(\013)1241 3492 y Fz(and)h(construction)g(of)g Fu(\037)2021 3504 y Fs(T)2073 3492 y Fz(,)h(there)f(exists)f(a)h(p)r (ositiv)n(e)g(constan)n(t)f Fu(c)440 3592 y Fz(indep)r(enden)n(t)j(of)g Fu(\013)g Fz(and)g Fu(T)38 b Fz(suc)n(h)28 b(that)f(:)865 3789 y Fu(q)905 3755 y Fs(T)902 3810 y(\013)957 3789 y Fz(\()p Fu(v)1032 3755 y Fs(T)1029 3810 y(\013)1085 3789 y Fz(\))83 b Fx(\024)g Fu(\025)p Fz(\()p Fu(\013)p Fz(\))p Fx(jj)p Fu(v)1602 3755 y Fs(T)1599 3810 y(\013)1656 3789 y Fx(jj)1702 3755 y Fv(2)1702 3812 y Fs(L)1748 3795 y Fr(2)1780 3812 y Fv(\(\012)1853 3820 y Fe(T)1899 3812 y Fv(\))1948 3789 y Fz(+)2053 3733 y Fu(c)p 2041 3770 61 4 v 2041 3846 a(T)2111 3789 y Fx(jj)p Fu(u)2205 3801 y Fs(\013)2252 3789 y Fx(jj)2298 3755 y Fv(2)2298 3819 y Fs(L)2344 3802 y Fr(2)2376 3819 y Fv(\(])2431 3797 y Fe(T)p 2431 3806 42 3 v 2438 3839 a Fr(2)2482 3819 y Fs(;T)9 b Fv([)p Ft(\002)p Fv(])p Ft(\000)2701 3797 y Fr(1)p 2701 3806 29 3 v 2701 3839 a(2)2740 3819 y Fs(;)2769 3797 y Fr(1)p 2769 3806 V 2769 3839 a(2)2808 3819 y Fv([\))2856 3789 y Fu(:)255 b Fz(\(8.8\))440 4003 y(W)-7 b(e)28 b(estimate)g Fx(jj)p Fu(u)1010 4015 y Fs(\013)1057 4003 y Fx(jj)1103 3973 y Fv(2)1103 4037 y Fs(L)1149 4020 y Fr(2)1181 4037 y Fv(\(])1236 4014 y Fe(T)p 1236 4023 42 3 v 1243 4057 a Fr(2)1288 4037 y Fs(;T)9 b Fv([)p Ft(\002)p Fv(])p Ft(\000)1507 4014 y Fr(1)p 1507 4023 29 3 v 1507 4057 a(2)1545 4037 y Fs(;)1575 4014 y Fr(1)p 1575 4023 V 1575 4057 a(2)1613 4037 y Fv([\))1690 4003 y Fz(with)28 b(Corollary)e(7.4;)h (so)h(there)f(exists)h Fu(\016)f(>)c Fz(0)28 b(and)440 4123 y Fu(\013)493 4135 y Fv(0)553 4123 y Fu(>)23 b Fz(0)k(suc)n(h)h (that)f(:)996 4316 y Fx(8)p Fu(\013)c Fx(\024)f Fu(\013)1259 4328 y Fv(0)1297 4316 y Fu(;)41 b Fx(8)p Fu(T)34 b Fx(\025)1598 4260 y Fu(\017)p 1588 4297 54 4 v 1588 4373 a(\013)1652 4316 y(;)41 b Fx(jj)p Fu(u)1810 4328 y Fs(\013)1857 4316 y Fx(jj)1903 4282 y Fv(2)1903 4346 y Fs(L)1949 4329 y Fr(2)1981 4346 y Fv(\(])2036 4323 y Fe(T)p 2036 4332 42 3 v 2043 4366 a Fr(2)2088 4346 y Fs(;T)9 b Fv([)p Ft(\002)p Fv(])p Ft(\000)2307 4323 y Fr(1)p 2307 4332 29 3 v 2307 4366 a(2)2345 4346 y Fs(;)2375 4323 y Fr(1)p 2375 4332 V 2375 4366 a(2)2413 4346 y Fv([\))2485 4316 y Fx(\024)22 b Fu(e)2611 4282 y Ft(\000)2678 4259 y Fe(\016)p 2673 4268 38 3 v 2673 4302 a(\013)2725 4316 y Fu(:)386 b Fz(\(8.9\))440 4532 y(W)-7 b(e)28 b(also)f(estimate)g Fx(jj)p Fu(v)1171 4502 y Fs(T)1168 4552 y(\013)1224 4532 y Fx(jj)1270 4502 y Fv(2)1270 4559 y Fs(L)1316 4542 y Fr(2)1348 4559 y Fv(\(\012)1421 4567 y Fe(T)1467 4559 y Fv(\))1525 4532 y Fz(:)1374 4748 y(1)18 b Fx(\000)g Fu(e)1556 4713 y Ft(\000)1622 4691 y Fe(\016)p 1617 4700 V 1617 4733 a(\013)1692 4748 y Fx(\024)23 b(jj)p Fu(v)1869 4713 y Fs(T)1866 4768 y(\013)1922 4748 y Fx(jj)1968 4713 y Fv(2)1968 4770 y Fs(L)2014 4754 y Fr(2)2046 4770 y Fv(\(\012)2119 4778 y Fe(T)2165 4770 y Fv(\))2218 4748 y Fx(\024)g Fz(1)p Fu(:)721 b Fz(\(8.10\))440 4928 y(The)28 b(min-max)f(principle)g(concludes)h(to)f(\(8.5\))g(\(with)i(a)e (smaller)g Fu(\016)s Fz(\).)593 b Fd(\003)440 5160 y Fl(8.4)112 b(The)38 b(regularized)e(op)s(erator)i Fk(P)2035 5124 y Fi(T)6 b(;\032)440 5313 y Fz(T)-7 b(o)28 b(a)n(v)n(oid)f(the)i (singularit)n(y)e(on)i Fu(P)1509 5325 y Fs(\013)1585 5313 y Fz(with)g(the)g(term)2128 5281 y Fv(1)p 2128 5295 34 4 v 2132 5342 a Fs(t)2200 5313 y Fz(whic)n(h)f(tends)h(to)g(0)f (when)g Fu(t)h Fz(tends)440 5413 y(to)c(in\014nit)n(y)-7 b(,)26 b(w)n(e)f(deal)g(with)h(the)g(op)r(erator)e Fu(P)1858 5383 y Fs(T)1846 5434 y(\013)1935 5413 y Fz(and)h(w)n(e)g(can)g(b)r (ound)h(from)f(b)r(elo)n(w)3055 5380 y Fv(1)p 3055 5394 V 3059 5442 a Fs(t)3123 5413 y Fz(b)n(y)3254 5380 y Fv(1)p 3246 5394 49 4 v 3246 5442 a Fs(T)1831 5662 y Fz(23)p eop %%Page: 24 24 24 23 bop 440 531 a Fz(whic)n(h)23 b(is)g(a)g(constan)n(t.)34 b(So)23 b(it)h(is)f(quite)g(natural)f(to)h(in)n(tro)r(duce)g(the)g (self-adjoin)n(t)g(extension)440 631 y Fu(P)505 601 y Fs(T)5 b(;\032)635 631 y Fz(to)28 b(the)g(form)f Fu(a)1120 601 y Fs(T)5 b(;\032)1251 631 y Fz(de\014ned)28 b(on)f Fx(V)1710 591 y Fs(N)s(;D)1703 655 y(T)1872 631 y Fz(as)g(follo)n(ws)g (:)523 807 y Fx(V)581 767 y Fs(N)s(;D)574 831 y(T)725 807 y Fz(:)51 b(=)887 739 y Fp(\010)935 807 y Fu(u)23 b Fx(2)g Fu(L)1141 776 y Fv(2)1178 807 y Fz(\(\012)1270 819 y Fs(T)1322 807 y Fz(\))p Fx(j)28 b Fz(\()p Fu(D)1506 819 y Fs(t)1554 807 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(u)23 b Fx(2)h Fu(L)1920 776 y Fv(2)1956 807 y Fz(\(\012)2048 819 y Fs(T)2101 807 y Fz(\))p Fu(;)42 b(D)2267 819 y Fs(\021)2307 807 y Fu(u)22 b Fx(2)i Fu(L)2513 776 y Fv(2)2550 807 y Fz(\(\012)2642 819 y Fs(T)2694 807 y Fz(\))p Fu(;)42 b(u)2839 822 y Ft(j)2859 830 y Fe(t)p Fr(=)p Fe(T)2997 807 y Fz(=)23 b(0)3127 739 y Fp(\011)3189 807 y Fu(;)747 965 y(a)791 935 y Fs(T)5 b(;\032)893 965 y Fz(\()p Fu(u;)14 b(v)s Fz(\))9 b(:)52 b(=)1256 898 y Fp(R)1295 994 y Fv(\012)1342 1002 y Fe(T)1407 873 y Fp(\020)1456 965 y Fz(2)p Fu(t)p Fz(\()p Fu(D)1629 977 y Fs(t)1676 965 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(u)p 1883 893 352 4 v Fz(\()p Fu(D)1984 977 y Fs(t)2032 965 y Fx(\000)g Fu(\021)s Fz(\))p Fu(v)k Fz(+)2346 928 y Fs(\032)p 2346 946 35 4 v 2347 993 a Fv(2)2391 965 y Fu(D)2460 977 y Fs(\021)2500 965 y Fu(u)p 2548 898 153 4 v(D)2617 977 y Fs(\021)2657 965 y Fu(v)2700 873 y Fp(\021)2791 965 y Fu(dt)28 b(d\021)s(:)95 b Fz(\(8.11\))440 1160 y(W)-7 b(e)19 b(denote)f(b)n(y)g Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))19 b(the)f(b)r(ottom)h(of)f(the)h(sp)r (ectrum)g(of)f Fu(P)2343 1130 y Fs(T)5 b(;\032)2464 1160 y Fz(de\014ned)19 b(on)f Fx(D)2913 1130 y Fs(N)s(;D)3049 1160 y Fz(\()p Fu(P)3146 1130 y Fs(T)5 b(;\032)3249 1160 y Fz(\))18 b(:)440 1348 y Fx(D)506 1314 y Fs(N)s(;D)642 1348 y Fz(\()p Fu(P)739 1314 y Fs(T)5 b(;\032)842 1348 y Fz(\))92 b(:)51 b(=)1187 1256 y Fp(n)1243 1348 y Fu(u)22 b Fx(2)i(V)1450 1308 y Fs(N)s(;D)1443 1372 y(T)1585 1348 y Fx(j)j Fz(\()p Fu(D)1736 1360 y Fs(t)1784 1348 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(t)p Fz(\()p Fu(D)2074 1360 y Fs(t)2122 1348 y Fx(\000)g Fu(\021)s Fz(\))p Fu(u)23 b Fx(2)h Fu(L)2488 1314 y Fv(2)2525 1348 y Fz(\(\012)2617 1360 y Fs(T)2669 1348 y Fz(\))p Fu(;)42 b(D)2837 1314 y Fv(2)2835 1368 y Fs(\021)2875 1348 y Fu(u)23 b Fx(2)g Fu(L)3081 1314 y Fv(2)3118 1348 y Fz(\(\012)3210 1360 y Fs(T)3262 1348 y Fz(\))p Fu(;)1197 1531 y(tD)1296 1543 y Fs(t)1325 1531 y Fu(u)1373 1546 y Ft(j)1393 1554 y Fe(t)p Fr(=0)1518 1531 y Fz(=)g(0)p Fu(;)41 b(D)1781 1543 y Fs(\021)1821 1531 y Fu(u)1869 1551 y Ft(j)p Fs(\021)r Fv(=)p Ft(\006)2038 1529 y Fr(1)p 2038 1538 29 3 v 2038 1571 a(2)2103 1531 y Fz(=)23 b(0)2233 1438 y Fp(o)2301 1531 y Fu(;)706 1710 y(P)771 1675 y Fs(T)5 b(;\032)966 1710 y Fz(:)51 b(=)82 b(2\()p Fu(D)1330 1722 y Fs(t)1378 1710 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(t)p Fz(\()p Fu(D)1668 1722 y Fs(t)1716 1710 y Fx(\000)g Fu(\021)s Fz(\))h(+)1987 1653 y Fu(\032)p 1987 1691 43 4 v 1988 1767 a Fz(2)2040 1710 y Fu(D)2111 1675 y Fv(2)2109 1730 y Fs(\021)2149 1710 y Fu(:)971 b Fz(\(8.12\))440 1911 y FA(Lemma)31 b(8.4.)41 b Fo(We)32 b(assume)f(that)g Fu(T)37 b(>)25 b Fz(1)31 b Fo(and)2008 1878 y Fs(T)p 2008 1892 49 4 v 2015 1939 a(\032)2098 1911 y Fo(is)g(uniformly)h(b)l(ounde)l(d)g(fr)l(om)g(ab)l(ove.)440 2037 y(Then)e(ther)l(e)g(exists)1109 2016 y Fz(^)1091 2037 y Fu(C)36 b Fo(such)29 b(that)h(:)1202 2264 y Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))23 b Fx(\024)g Fu(\025)1607 2225 y Fv(mean)o Fs(;T)1607 2287 y Fv(1)1856 2264 y Fx(\024)g Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))k(+)2310 2243 y(^)2291 2264 y Fu(C)2356 2116 y Fp(s)p 2439 2116 80 4 v 2449 2208 a Fu(T)p 2449 2245 61 4 v 2458 2321 a(\032)2519 2264 y(:)550 b Fz(\(8.13\))440 2512 y Fo(Pr)l(o)l(of)31 b(:)39 b Fz(By)27 b(the)h(inclusion)f Fx(V)1404 2482 y Fs(N)1397 2535 y Fv(mean)o Fs(;T)1646 2512 y Fx(\032)c(V)1792 2473 y Fs(N)s(;D)1785 2537 y(T)1926 2512 y Fz(,)28 b(w)n(e)f (immediately)h(see)f(that)h(:)1311 2698 y Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))24 b Fx(\024)e Fu(\025)1716 2658 y Fv(mean)p Fs(;T)1716 2720 y Fv(1)1943 2698 y Fu(;)36 b Fx(8)p Fu(\032)23 b Fx(\025)f Fz(0)p Fu(:)825 b Fz(\(8.14\))440 2861 y(W)-7 b(e)31 b(denote)g(b)n(y)f Fu(u)1023 2873 y Fs(T)5 b(;\032)1157 2861 y Fz(a)30 b(normalized)g(eigen)n(v)n(ector)e (asso)r(ciated)i(to)g Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\).)47 b(Com)n(bining)440 2960 y(\(8.2\))27 b(and)h(\(8.14\),)f(w)n(e)g(obtain)g(the)h(upp)r(er)g(b)r(ound)g(:)1392 3123 y Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))23 b Fx(\024)g Fu(\025)1797 3089 y Fv(mean)1797 3144 y(1)1974 3123 y Fz(+)d(~)-44 b Fu(c)23 b Fz(=)9 b(:)27 b Fu(C)2305 3089 y Ft(0)2329 3123 y Fu(:)440 3286 y Fz(But,)h(b)n(y)f(de\014nition) h(of)g Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))28 b(:)703 3491 y Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))23 b(=)1060 3378 y Fp(Z)1106 3567 y Fv(\012)1153 3575 y Fe(T)1217 3399 y Fp(\020)1267 3491 y Fu(t)p Fx(j)p Fz(2\()p Fu(D)1463 3503 y Fs(t)1510 3491 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(u)1717 3503 y Fs(T)5 b(;\032)1820 3491 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))p Fx(j)2018 3457 y Fv(2)2074 3491 y Fz(+)2167 3435 y Fu(\032)p 2167 3472 43 4 v 2168 3548 a Fz(2)2220 3491 y Fx(j)p Fu(D)2312 3503 y Fs(\021)2352 3491 y Fu(u)2400 3503 y Fs(T)5 b(;\032)2503 3491 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))p Fx(j)2701 3457 y Fv(2)2739 3399 y Fp(\021)2830 3491 y Fu(dt)28 b(d\021)s(;)440 3705 y Fz(w)n(e)f(deduce)h(:)1398 3842 y Fx(jj)p Fu(D)1513 3854 y Fs(\021)1554 3842 y Fu(u)1602 3854 y Fs(T)5 b(;\032)1704 3842 y Fx(jj)1750 3857 y Fs(L)1796 3840 y Fr(2)1828 3857 y Fv(\(\012)1901 3865 y Fe(T)1948 3857 y Fv(\))2001 3842 y Fx(\024)2088 3720 y Fp(r)p 2171 3720 63 4 v 2182 3786 a Fz(2)p 2181 3823 43 4 v 2181 3899 a Fu(\032)2234 3842 y(C)2299 3808 y Ft(0)2323 3842 y Fu(:)746 b Fz(\(8.15\))440 4019 y(As)33 b(in)g(the)h(case)e(of)h(the)g(upp)r(er)g(b)r(ound,)i(w)n (e)e(use)g(the)g(con)n(tin)n(uous,)g(b)r(ounded)h(and)f(self-)440 4119 y(adjoin)n(t)27 b(pro)5 b(jector)26 b(\005)1141 4131 y Fv(0)1206 4119 y Fz(:)989 4348 y(\005)1051 4360 y Fv(0)1098 4348 y Fz(:)h Fu(L)1205 4314 y Fv(2)1242 4348 y Fz(\(\012)1334 4360 y Fs(T)1387 4348 y Fz(\))c Fx(!)g Fu(L)1605 4314 y Fv(2)1642 4348 y Fz(\(\012)1734 4360 y Fs(T)1786 4348 y Fz(\))p Fu(;)42 b(f)32 b Fx(7!)2062 4235 y Fp(Z)2155 4233 y Fr(1)p 2155 4242 29 3 v 2155 4276 a(2)2108 4424 y Ft(\000)2170 4402 y Fr(1)p 2170 4411 V 2170 4444 a(2)2226 4348 y Fu(f)9 b Fz(\()p Fu(t;)14 b(\021)s Fz(\))28 b Fu(d\021)s(;)503 b Fz(\(8.16\))440 4583 y(and)27 b(w)n(e)h(write)f Fu(u)984 4595 y Fs(T)5 b(;\032)1114 4583 y Fz(as)27 b(a)g(sum)h(of)g(t)n(w)n(o)e(functions)i Fu(u)2118 4553 y Fv(0)2118 4606 y Fs(T)5 b(;\032)2248 4583 y Fz(and)28 b Fu(u)2458 4553 y Fv(1)2458 4606 y Fs(T)5 b(;\032)2588 4583 y Fz(suc)n(h)27 b(that)h(:)1091 4760 y Fu(u)1139 4726 y Fv(0)1139 4781 y Fs(T)5 b(;\032)1265 4760 y Fz(=)22 b(\005)1414 4772 y Fv(0)1452 4760 y Fz(\()p Fu(u)1532 4772 y Fs(T)5 b(;\032)1635 4760 y Fz(\))83 b(and)g Fu(u)2015 4726 y Fv(1)2015 4781 y Fs(T)5 b(;\032)2140 4760 y Fz(=)23 b Fu(u)2276 4772 y Fs(T)5 b(;\032)2397 4760 y Fx(\000)18 b Fu(u)2528 4726 y Fv(0)2528 4781 y Fs(T)5 b(;\032)2630 4760 y Fu(:)439 b Fz(\(8.17\))440 4923 y(W)-7 b(e)32 b(iden)n(tify)f(the)h(range)d(of)i(\005)1428 4935 y Fv(0)1497 4923 y Fz(with)h Fu(L)1747 4893 y Fv(2)1784 4923 y Fz(\(0)p Fu(;)14 b(T)e Fz(\))30 b(:)44 b Fu(L)2142 4893 y Fv(2)2178 4923 y Fz(\(0)p Fu(;)14 b(T)e Fz(\))31 b(is)g(injected)h(in)f Fu(L)2972 4893 y Fv(2)3009 4923 y Fz(\(\012)3101 4935 y Fs(T)3153 4923 y Fz(\))h(b)n(y)440 5023 y(de\014ning)c Fu(g)d Fx(7!)e Fu(i)p Fz(\()p Fu(g)s Fz(\))g(=)g Fu(g)s Fz(.)36 b(No)n(w,)28 b(w)n(e)f(omit)g(tak)n(e)g Fo(i)8 b Fz(.)440 5123 y(By)27 b(de\014nition)h(of)g Fu(u)1082 5092 y Fv(0)1082 5145 y Fs(T)5 b(;\032)1212 5123 y Fz(and)27 b Fu(u)1421 5092 y Fv(1)1421 5145 y Fs(T)5 b(;\032)1524 5123 y Fz(,)28 b(it)g(is)f(easy)g(to)g(see)g(that)h (:)531 5364 y Fx(jj)p Fu(D)646 5376 y Fs(\021)687 5364 y Fu(u)735 5330 y Fv(1)735 5385 y Fs(T)5 b(;\032)837 5364 y Fx(jj)883 5379 y Fs(L)929 5363 y Fr(2)961 5379 y Fv(\(\012)1034 5387 y Fe(T)1080 5379 y Fv(\))1133 5364 y Fx(\024)1231 5240 y(p)p 1300 5240 42 4 v 68 x Fz(2)p Fu(C)1407 5278 y Ft(0)p 1231 5345 200 4 v 1275 5370 a Fx(p)p 1344 5370 43 4 v 51 x Fu(\032)1440 5364 y(;)1500 5251 y Fp(Z)1583 5272 y Fv(1)p Fs(=)p Fv(2)1546 5440 y Ft(\000)p Fv(1)p Fs(=)p Fv(2)1716 5364 y Fu(u)1764 5330 y Fv(1)1764 5385 y Fs(T)g(;\032)1867 5364 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))28 b Fu(d\021)f Fz(=)22 b(0)p Fu(;)37 b(D)2439 5376 y Fs(\021)2479 5364 y Fu(u)2527 5330 y Fv(1)2527 5385 y Fs(T)5 b(;\032)2629 5400 y Ft(j)2649 5423 y Fe(\021)q Fr(=)p Fq(\006)2779 5407 y Fr(1)p 2779 5414 29 3 v 2779 5445 a(2)2849 5364 y Fz(=)22 b(0)p Fu(:)91 b Fz(\(8.18\))1831 5662 y(24)p eop %%Page: 25 25 25 24 bop 440 531 a Fz(Prop)r(erties)29 b(\(8.18\))g(sho)n(w)h(that)g Fu(u)1525 501 y Fv(1)1525 554 y Fs(T)5 b(;\032)1658 531 y Fz(is)30 b(orthogonal)e(to)i(constan)n(t)g(functions)g(and)h(since) 440 644 y Fu(\031)490 614 y Fv(2)555 644 y Fz(is)d(the)g(smallest)f(p)r (ositiv)n(e)g(eigen)n(v)-5 b(alue)27 b(for)g Fx(\000)p Fu(@)2046 614 y Fv(2)2041 664 y Fs(\021)2110 644 y Fz(on)g(])19 b Fx(\000)2360 611 y Fv(1)p 2360 625 34 4 v 2360 673 a(2)2403 644 y Fu(;)2450 611 y Fv(1)p 2450 625 V 2450 673 a(2)2493 644 y Fz([,)28 b(w)n(e)f(deduce)h(that)f(:)1196 821 y Fx(jj)p Fu(D)1311 833 y Fs(\021)1351 821 y Fu(u)1399 787 y Fv(1)1399 842 y Fs(T)5 b(;\032)1501 821 y Fx(jj)1547 836 y Fs(L)1593 820 y Fr(2)1625 836 y Fv(\(\012)1698 844 y Fe(T)1745 836 y Fv(\))1858 821 y Fx(\025)82 b Fu(\031)s Fx(jj)p Fu(u)2149 787 y Fv(1)2149 842 y Fs(T)5 b(;\032)2252 821 y Fx(jj)2298 836 y Fs(L)2344 820 y Fr(2)2376 836 y Fv(\(\012)2449 844 y Fe(T)2496 836 y Fv(\))2526 821 y Fu(:)543 b Fz(\(8.19\))440 986 y(By)30 b(de\014nition)h(of)g Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\),)31 b Fu(u)1391 956 y Fv(1)1391 1009 y Fs(T)5 b(;\032)1494 986 y Fz(,)31 b(Relation)g(\(8.19\))e(and)i(the)g(upp)r(er)g(b)r(ound)g (of)f Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\),)440 1086 y(w)n(e)27 b(establish)g(:)1068 1318 y Fx(jj)p Fu(u)1162 1284 y Fv(1)1162 1339 y Fs(T)5 b(;\032)1264 1318 y Fx(jj)1310 1333 y Fs(L)1356 1316 y Fr(2)1389 1333 y Fv(\(\012)1462 1341 y Fe(T)1508 1333 y Fv(\))1561 1318 y Fx(\024)1649 1166 y Fp(s)p 1732 1166 267 4 v 1742 1262 a Fu(\027)g Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))p 1742 1299 247 4 v 1799 1375 a Fu(\032\031)1892 1351 y Fv(2)2021 1318 y Fx(\024)2118 1180 y Fp(p)p 2201 1180 200 4 v 13 x Fx(p)p 2271 1193 42 4 v 2271 1262 a Fz(2)o Fu(C)2377 1238 y Ft(0)p 2118 1299 283 4 v 2178 1375 a Fu(\031)2228 1323 y Fx(p)p 2298 1323 43 4 v 2298 1375 a Fu(\032)2434 1318 y Fz(=)2533 1262 y Fu(C)2598 1232 y Ft(00)p 2531 1299 113 4 v 2531 1323 a Fx(p)p 2600 1323 43 4 v 52 x Fu(\032)2653 1318 y(:)416 b Fz(\(8.20\))440 1554 y(Since)28 b Fx(jj)p Fu(u)751 1523 y Fv(0)751 1576 y Fs(T)5 b(;\032)853 1554 y Fx(jj)899 1523 y Fv(2)899 1580 y Fs(L)945 1564 y Fr(2)977 1580 y Fv(\(\012)1050 1588 y Fe(T)1097 1580 y Fv(\))1150 1554 y Fz(=)22 b(1)c Fx(\000)g(jj)p Fu(u)1474 1523 y Fv(1)1474 1576 y Fs(T)5 b(;\032)1577 1554 y Fx(jj)1623 1523 y Fv(2)1623 1580 y Fs(L)1669 1564 y Fr(2)1701 1580 y Fv(\(\012)1774 1588 y Fe(T)1820 1580 y Fv(\))1850 1554 y Fz(,)28 b(Relation)f(\(8.20\))g(leads)g(to)h(:)1359 1801 y(1)18 b Fx(\000)1531 1724 y Fz(~)1512 1745 y Fu(C)1577 1715 y Ft(00)p 1512 1782 108 4 v 1544 1858 a Fu(\032)1653 1801 y Fx(\024)k(jj)p Fu(u)1834 1767 y Fv(0)1834 1822 y Fs(T)5 b(;\032)1937 1801 y Fx(jj)1983 1816 y Fs(L)2029 1800 y Fr(2)2061 1816 y Fv(\(\012)2134 1824 y Fe(T)2180 1816 y Fv(\))2233 1801 y Fx(\024)23 b Fz(1)p Fu(:)706 b Fz(\(8.21\))440 2006 y(W)-7 b(e)28 b(kno)n(w)f(that)h(:)976 2171 y(2\()p Fu(D)1119 2183 y Fs(t)1166 2171 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(t)p Fz(\()p Fu(D)1456 2183 y Fs(t)1504 2171 y Fx(\000)g Fu(\021)s Fz(\))p Fu(u)1711 2183 y Fs(T)5 b(;\032)1833 2171 y Fz(+)1926 2115 y Fu(\032)p 1926 2152 43 4 v 1927 2228 a Fz(2)1979 2171 y Fu(D)2050 2136 y Fv(2)2048 2191 y Fs(\021)2088 2171 y Fu(u)2136 2183 y Fs(T)g(;\032)2261 2171 y Fz(=)23 b Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))p Fu(u)2643 2183 y Fs(T)5 b(;\032)2745 2171 y Fu(:)324 b Fz(\(8.22\))440 2364 y(W)-7 b(e)28 b(apply)f(the)h(pro)5 b(jector)26 b(\005)1374 2376 y Fv(0)1439 2364 y Fz(to)i(Relation)f(\(8.22\))g(and)g(so)g(:)781 2600 y Fu(L)838 2566 y Fv(mean)p Fs(;)p Fv(T)1063 2600 y Fu(u)1111 2566 y Fv(0)1111 2620 y Fs(T)5 b(;\032)1232 2600 y Fz(+)1315 2487 y Fp(Z)1408 2485 y Fr(1)p 1408 2494 29 3 v 1408 2527 a(2)1361 2675 y Ft(\000)1423 2653 y Fr(1)p 1423 2662 V 1423 2695 a(2)1479 2600 y Fz(2\()p Fu(D)1622 2612 y Fs(t)1669 2600 y Fx(\000)18 b Fu(\021)s Fz(\))p Fu(t)p Fz(\()p Fu(D)1959 2612 y Fs(t)2008 2600 y Fx(\000)g Fu(\021)s Fz(\))p Fu(u)2215 2566 y Fv(1)2215 2620 y Fs(T)5 b(;\032)2345 2600 y Fu(d\021)27 b Fz(=)22 b Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))p Fu(u)2837 2566 y Fv(0)2837 2620 y Fs(T)5 b(;\032)2940 2600 y Fu(:)440 2837 y Fz(W)-7 b(e)28 b(tak)n(e)f(the)h(scalar)e(pro)r(duct)h(with)h Fu(u)1690 2807 y Fv(0)1690 2860 y Fs(T)5 b(;\032)1821 2837 y Fz(and)27 b(obtain,)g(after)h(simpli\014cations)f(:)1022 2951 y Fp(\012)1061 3019 y Fu(L)1118 2984 y Fv(mean)n Fs(;T)1343 3019 y Fu(u)1391 2984 y Fv(0)1391 3039 y Fs(T)5 b(;\032)1494 3019 y Fu(;)14 b(u)1579 2984 y Fv(0)1579 3039 y Fs(T)5 b(;\032)1681 2951 y Fp(\013)1720 3055 y Fs(L)1766 3038 y Fr(2)1798 3055 y Fv(\(0)p Fs(;T)k Fv(\))874 3215 y Fz(=)83 b Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))p Fx(jj)p Fu(u)1362 3181 y Fv(0)1362 3236 y Fs(T)5 b(;\032)1464 3215 y Fx(jj)1510 3181 y Fv(2)1510 3238 y Fs(L)1556 3221 y Fr(2)1588 3238 y Fv(\(0)p Fs(;T)k Fv(\))1764 3215 y Fx(\000)18 b Fz(2)p Fu(i)1932 3102 y Fp(Z)1977 3291 y Fv(\012)2024 3299 y Fe(T)2088 3215 y Fu(\021)s(u)2180 3181 y Fv(1)2180 3236 y Fs(T)5 b(;\032)p 2310 3139 151 4 v 2310 3215 a Fu(u)2358 3187 y Fv(0)2358 3240 y Fs(T)g(;\032)2489 3215 y Fu(dt)27 b(d\021)1022 3446 y Fz(+2)1143 3333 y Fp(Z)1188 3521 y Fv(\012)1235 3529 y Fe(T)1299 3446 y Fu(\021)s(tD)1442 3458 y Fs(t)1471 3446 y Fu(u)1519 3411 y Fv(1)1519 3466 y Fs(T)5 b(;\032)p 1649 3370 V 1649 3446 a Fu(u)1697 3417 y Fv(0)1697 3470 y Fs(T)g(;\032)1800 3446 y Fu(dt)27 b(d\021)22 b Fx(\000)c Fz(2)2145 3333 y Fp(Z)2191 3521 y Fv(\012)2238 3529 y Fe(T)2302 3446 y Fu(t\021)2376 3411 y Fv(2)2413 3446 y Fu(u)2461 3411 y Fv(1)2461 3466 y Fs(T)5 b(;\032)p 2592 3370 V 2592 3446 a Fu(u)2640 3417 y Fv(0)2640 3470 y Fs(T)g(;\032)2742 3446 y Fu(dt)28 b(d\021)s(:)139 b Fz(\(8.23\))440 3666 y(W)-7 b(e)28 b(no)n(w)f(estimate)h(eac)n(h)e (term)i(of)f(\(8.23\))g(:)565 3843 y Fx(\017)648 3747 y Fp(\014)648 3797 y(\014)648 3847 y(\014)675 3776 y(R)714 3872 y Fv(\012)761 3880 y Fe(T)826 3843 y Fu(\021)s(u)918 3812 y Fv(1)918 3865 y Fs(T)5 b(;\032)p 1048 3767 V 1048 3843 a Fu(u)1096 3814 y Fv(0)1096 3867 y Fs(T)g(;\032)1226 3843 y Fu(dt)28 b(d\021)1414 3747 y Fp(\014)1414 3797 y(\014)1414 3847 y(\014)1465 3843 y Fx(\024)1563 3810 y Fv(1)p 1563 3824 34 4 v 1563 3871 a(2)1616 3810 y Fs(C)1668 3785 y Fq(00)p 1616 3824 93 4 v 1618 3835 a Ft(p)p 1672 3835 35 3 v 1672 3871 a Fs(\032)1746 3843 y Fz(according)e(to)h (\(8.20\))g(and)g(\(8.21\).)565 4062 y Fx(\017)648 3967 y Fp(\014)648 4017 y(\014)648 4066 y(\014)675 3995 y(R)714 4092 y Fv(\012)761 4100 y Fe(T)826 4062 y Fu(\021)870 4032 y Fv(2)907 4062 y Fu(tu)985 4032 y Fv(1)985 4085 y Fs(T)5 b(;\032)p 1115 3986 151 4 v 1115 4062 a Fu(u)1163 4034 y Fv(0)1163 4087 y Fs(T)g(;\032)1293 4062 y Fu(dt)28 b(d\021)1481 3967 y Fp(\014)1481 4017 y(\014)1481 4066 y(\014)1532 4062 y Fx(\024)1630 4030 y Fs(C)1682 4005 y Fq(00)p 1630 4044 93 4 v 1660 4091 a Fv(4)1732 3971 y Fp(q)p 1815 3971 69 4 v 1825 4030 a Fs(T)p 1825 4044 49 4 v 1832 4091 a(\032)1883 4062 y Fx(jj)1929 3995 y(p)p 1999 3995 30 4 v 1999 4062 a Fu(tu)2077 4032 y Fv(0)2077 4085 y Fs(T)5 b(;\032)2179 4062 y Fx(jj)2225 4077 y Fs(L)2271 4061 y Fr(2)2303 4077 y Fv(\(\012)2376 4085 y Fe(T)2422 4077 y Fv(\))2453 4062 y Fz(.)565 4242 y Fx(\017)41 b Fz(F)-7 b(or)27 b(the)h(third)f(term,)h(w)n(e)f(use)h(an)f(in)n (tegration)f(b)n(y)h(parts)g(and)h(then)g(:)737 4332 y Fp(\014)737 4382 y(\014)737 4432 y(\014)737 4481 y(\014)765 4339 y(Z)811 4528 y Fv(\012)858 4536 y Fe(T)922 4452 y Fu(\021)s(tD)1065 4464 y Fs(t)1095 4452 y Fu(u)1143 4418 y Fv(1)1143 4473 y Fs(T)5 b(;\032)p 1273 4376 151 4 v 1273 4452 a Fu(u)1321 4424 y Fv(0)1321 4477 y Fs(T)g(;\032)1451 4452 y Fu(dt)28 b(d\021)1639 4332 y Fp(\014)1639 4382 y(\014)1639 4432 y(\014)1639 4481 y(\014)1690 4452 y Fx(\024)1810 4396 y Fu(C)1875 4366 y Ft(00)p 1787 4433 154 4 v 1787 4509 a Fz(2)1829 4458 y Fx(p)p 1898 4458 43 4 v 51 x Fu(\032)1969 4452 y Fz(+)2062 4396 y Fu(C)2127 4366 y Ft(00)2170 4326 y Fx(p)p 2239 4326 61 4 v 70 x Fu(T)p 2062 4433 237 4 v 2104 4509 a Fz(2)2146 4458 y Fx(p)p 2215 4458 43 4 v 51 x Fu(\032)2309 4452 y Fx(jj)2355 4381 y(p)p 2425 4381 30 4 v 2425 4452 a Fu(tD)2524 4464 y Fs(t)2552 4452 y Fu(u)2600 4418 y Fv(0)2600 4473 y Fs(T)5 b(;\032)2703 4452 y Fx(jj)2749 4467 y Fs(L)2795 4451 y Fr(2)2827 4467 y Fv(\(0)p Fs(;T)k Fv(\))2984 4452 y Fu(:)440 4655 y Fz(W)-7 b(e)28 b(de\014ne)g Fu(C)6 b Fz(\()p Fu(T)h(;)14 b(\032)p Fz(\))9 b(:)51 b(=)1259 4588 y Fp(\012)1298 4655 y Fu(L)1355 4625 y Fv(mean)n Fs(;T)1580 4655 y Fu(u)1628 4625 y Fv(0)1628 4678 y Fs(T)5 b(;\032)1731 4655 y Fu(;)14 b(u)1816 4625 y Fv(0)1816 4678 y Fs(T)5 b(;\032)1918 4588 y Fp(\013)1957 4694 y Fs(L)2003 4677 y Fr(2)2035 4694 y Fv(\(0)p Fs(;T)k Fv(\))2220 4655 y Fz(and)27 b(remark)f(that)i(:)626 4864 y Fx(jj)672 4792 y(p)p 741 4792 V 72 x Fu(tu)819 4829 y Fv(0)819 4884 y Fs(T)5 b(;\032)922 4864 y Fx(jj)968 4879 y Fs(L)1014 4862 y Fr(2)1046 4879 y Fv(\(0)p Fs(;T)k Fv(\))1226 4864 y Fx(\024)1314 4791 y(p)p 1383 4791 42 4 v 73 x Fz(6)1424 4789 y Fp(p)p 1507 4789 266 4 v 75 x Fu(C)d Fz(\()p Fu(T)h(;)14 b(\032)p Fz(\))28 b(and)g Fx(jj)2008 4792 y(p)p 2077 4792 30 4 v 72 x Fu(tD)2176 4876 y Fs(t)2205 4864 y Fu(u)2253 4829 y Fv(0)2253 4884 y Fs(T)5 b(;\032)2355 4864 y Fx(jj)2401 4879 y Fs(L)2447 4862 y Fr(2)2479 4879 y Fv(\(0)p Fs(;T)k Fv(\))2659 4864 y Fx(\024)2747 4789 y Fp(p)p 2830 4789 266 4 v 75 x Fu(C)d Fz(\()p Fu(T)h(;)14 b(\032)p Fz(\))p Fu(:)440 5029 y Fz(Then)37 b(a)g(upp)r(er)g(b)r(ound)h(coming)e(from)h (Relation)g(\(8.23\))g(and)g(previous)f(estimates)g(of)440 5128 y(eac)n(h)27 b(term)g(lead)h(to)f(the)h(b)r(ound)g(:)1033 5358 y Fu(C)6 b Fz(\()p Fu(T)h(;)14 b(\032)p Fz(\))24 b Fx(\024)e Fu(\027)5 b Fz(\()p Fu(T)i(;)14 b(\032)p Fz(\))19 b(+)1767 5302 y(3)p Fu(C)1874 5271 y Ft(00)p 1767 5339 150 4 v 1785 5363 a Fx(p)p 1855 5363 43 4 v 1855 5415 a Fu(\032)1944 5358 y Fz(+)f(4)p Fu(C)2134 5323 y Ft(00)2177 5209 y Fp(s)p 2260 5209 80 4 v 2270 5302 a Fu(T)p 2270 5339 61 4 v 2279 5415 a(\032)2340 5283 y Fp(p)p 2423 5283 266 4 v 75 x Fu(C)6 b Fz(\()p Fu(T)h(;)14 b(\032)p Fz(\))p Fu(:)1831 5662 y Fz(25)p eop %%Page: 26 26 26 25 bop 440 531 a Fz(W)-7 b(e)31 b(assume)e(that)i Fu(T)38 b Fx(\025)27 b Fz(1)j(and)g(that)h(there)f(exists)f(a)h (constan)n(t)g Fu(M)39 b Fz(suc)n(h)30 b(that)3014 499 y Fs(T)p 3014 513 49 4 v 3021 560 a(\032)3100 531 y Fx(\024)d Fu(M)9 b Fz(.)440 642 y(Then)30 b Fu(C)6 b Fz(\()p Fu(T)h(;)14 b(\032)p Fz(\))30 b(is)g(\014nite)g(and)g(there)f(exists)h(a)f(constan) n(t)g Fu(C)6 b Fz(\()p Fx(1)p Fz(\))28 b Fu(>)e Fz(0)k(indep)r(enden)n (t)g(of)g Fu(T)440 741 y Fz(and)f Fu(\032)g Fz(suc)n(h)f(that)h(:)40 b Fu(C)6 b Fz(\()p Fu(T)h(;)14 b(\032)p Fz(\))25 b Fx(\024)g Fu(C)6 b Fz(\()p Fx(1)p Fz(\))26 b Fu(<)f Fz(+)p Fx(1)p Fz(.)40 b(This)29 b(com)n(bined)g(with)g(\(8.21\))f(leads)g(to)440 850 y(the)g(b)r(ound)g(\(8.13\))f(of)g Fu(\025)1221 810 y Fv(mean)p Fs(;T)1221 872 y Fv(1)1448 850 y Fz(.)1769 b Fd(\003)440 1083 y Fl(8.5)112 b(Pro)s(of)37 b(of)h(the)f(lo)m(w)m(er) g(b)s(ound)h(on)g(Theorem)f(8.1)440 1236 y Fz(Let)28 b Fu(\021)e(>)d Fz(0.)36 b(W)-7 b(e)28 b(apply)f(Lemmas)h(8.2)e(and)i (8.4)e(with)j Fu(\017)22 b Fz(=)2342 1199 y Fs(\021)p 2318 1217 85 4 v 2318 1275 a Fv(2)2366 1260 y(^)2351 1275 y Fs(C)2440 1236 y Fz(and)28 b Fu(T)34 b Fz(=)2790 1203 y Fs(\017)p 2782 1217 44 4 v 2782 1264 a(\013)2836 1236 y Fz(,)27 b(then)h(:)1314 1487 y Fu(\027)1374 1370 y Fp(\022)1435 1487 y Fu(T)7 b(;)1592 1431 y Fz(1)p 1537 1468 151 4 v 1537 1544 a Fu(\013)1590 1520 y Fv(2)1628 1544 y Fu(T)1698 1370 y Fp(\023)1782 1487 y Fx(\025)23 b Fu(\025)1918 1452 y Fv(mean)1918 1507 y(1)2094 1487 y Fx(\000)2187 1431 y Fu(\021)p 2187 1468 45 4 v 2188 1544 a Fz(2)2242 1487 y Fu(:)827 b Fz(\(8.24\))440 1728 y(Since)27 b Fu(\025)704 1697 y Fs(T)756 1728 y Fz(\()p Fu(\013)p Fz(\))d Fx(\025)f Fu(\027)1045 1660 y Fp(\000)1083 1728 y Fu(T)7 b(;)1231 1695 y Fv(1)p 1185 1709 124 4 v 1185 1756 a Fs(\013)1228 1740 y Fr(2)1261 1756 y Fs(T)1319 1660 y Fp(\001)1357 1728 y Fz(,)27 b(Lemma)f(8.3)f(sho)n(ws)h(that)g (with)h(the)g(previous)e(c)n(hoice)g(of)i Fu(\017)p Fz(,)440 1827 y(there)g(exists)h Fu(\013)935 1839 y Fv(0)995 1827 y Fu(>)23 b Fz(0)k(and)g Fu(\016)f(>)d Fz(0)k(suc)n(h)g(that)h(:)1061 2051 y Fu(\025)p Fz(\()p Fu(\013)p Fz(\))c Fx(\025)f Fu(\027)1398 1934 y Fp(\022)1459 2051 y Fu(T)7 b(;)1616 1995 y Fz(1)p 1561 2032 151 4 v 1561 2108 a Fu(\013)1614 2084 y Fv(2)1652 2108 y Fu(T)1722 1934 y Fp(\023)1802 2051 y Fx(\000)18 b Fu(e)1924 2017 y Ft(\000)1990 1995 y Fe(\016)p 1985 2004 38 3 v 1985 2037 a(\013)2037 2051 y Fu(;)42 b Fx(8)p Fu(\013)22 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)2434 2063 y Fv(0)2471 2051 y Fz(])p Fu(:)575 b Fz(\(8.25\))440 2303 y(So)28 b(there)f(exists)h Fu(\013)1051 2315 y Fv(1)1112 2303 y Fx(\024)23 b Fu(\013)1253 2315 y Fv(0)1318 2303 y Fz(suc)n(h)28 b(that)g(for)g(an)n(y)f Fu(\013)d Fx(2)p Fz(]0)p Fu(;)14 b(\013)2258 2315 y Fv(1)2295 2303 y Fz(],)28 b Fu(e)2408 2273 y Ft(\000)2474 2250 y Fe(\016)p 2470 2259 V 2470 2293 a(\013)2545 2303 y Fx(\024)2643 2266 y Fs(\021)p 2643 2284 37 4 v 2645 2331 a Fv(2)2690 2303 y Fz(;)g(then)g(\(8.24\))g(and)440 2402 y(\(8.25\))f(lead)g(to)h(:)1463 2585 y Fu(\025)p Fz(\()p Fu(\013)p Fz(\))c Fx(\025)f Fu(\025)1788 2551 y Fv(mean)1788 2606 y(1)1965 2585 y Fx(\000)18 b Fu(\021)s(:)977 b Fz(\(8.26\))440 2768 y(Since)28 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))c(=)f Fu(\013\025)p Fz(\()p Fu(\013)p Fz(\),)29 b(w)n(e)e(can)h(also)e(ac)n (hiev)n(e)g(the)i(pro)r(of)f(of)h(Theorem)f(8.1.)369 b Fd(\003)440 3042 y Fy(9)135 b(Beha)l(vior)45 b(of)h(eigen)l(v)l (ectors)440 3224 y Fz(It)32 b(is)g(easy)f(to)g(deduce)h(from)g(the)g (pro)r(of)f(of)h(Theorem)f(8.1)g(the)h(b)r(eha)n(vior)e(of)i(the)g (eigen-)440 3324 y(v)n(ectors)26 b(of)i Fu(P)882 3294 y Fs(T)5 b(;\032)1012 3324 y Fz(and)27 b Fu(L)1230 3294 y Fv(mean)p Fs(;)p Fv(T)1483 3324 y Fz(as)g(follo)n(ws.)440 3490 y FA(Prop)s(osition)i(9.1.)39 b Fo(We)29 b(denote)g(by)h Fu(u)1714 3450 y Fs(T)5 b(;\032)1714 3512 y Fv(1)1845 3490 y Fo(the)29 b(\014rst)f(normalize)l(d)i(eigenve)l(ctor)g(for)g Fu(P)3202 3460 y Fs(T)5 b(;\032)440 3600 y Fo(and)32 b(de\014ne)g(the)f(function)h Fu(u)1363 3570 y Fv(0)1363 3623 y Fs(T)5 b(;\032)1474 3600 y Fz(:)55 b(=)25 b(\005)1704 3612 y Fv(0)1742 3600 y Fz(\()p Fu(u)1822 3560 y Fs(T)5 b(;\032)1822 3622 y Fv(1)1925 3600 y Fz(\))31 b Fo(on)h Fz(]0)p Fu(;)14 b(T)e Fz([)p Fo(.)42 b(We)32 b(denote)f(by)h Fu(u)2939 3560 y Fv(mean)o Fs(;T)2939 3622 y Fv(1)3196 3600 y Fo(the)440 3712 y(\014rst)26 b(normalize)l(d)h(eigenve)l(ctor)h (for)f Fu(L)1642 3682 y Fv(mean)n Fs(;T)1867 3712 y Fo(.)38 b(Then,)28 b(ther)l(e)f(exists)f Fu(T)2646 3724 y Fv(0)2705 3712 y Fu(>)d Fz(0)p Fo(,)k Fu(C)33 b Fo(such)26 b(that)440 3812 y(for)31 b Fu(T)j Fx(\025)22 b Fu(T)792 3824 y Fv(0)859 3812 y Fo(and)30 b Fu(\032)23 b(>)g Fz(0)29 b Fo(with)1435 3779 y Fs(T)p 1435 3793 49 4 v 1442 3840 a(\032)1523 3812 y Fo(uniformly)i(b)l(ounde)l(d)f(:)1043 4099 y Fx(jj)p Fu(u)1137 4064 y Fv(0)1137 4119 y Fs(T)5 b(;\032)1258 4099 y Fx(\000)18 b Fu(u)1389 4059 y Fv(mean)o Fs(;T)1389 4121 y Fv(1)1615 4099 y Fx(jj)1661 4064 y Fv(2)1661 4121 y Fs(L)1707 4105 y Fr(2)1739 4121 y Fv(\(0)p Fs(;T)9 b Fv(\))1979 4099 y Fx(\024)82 b Fu(C)2205 3957 y Fp( )2282 4042 y Fz(1)p 2281 4079 43 4 v 2281 4155 a Fu(\032)2352 4099 y Fz(+)2435 3950 y Fp(s)p 2518 3950 80 4 v 2528 4042 a Fu(T)p 2528 4079 61 4 v 2537 4155 a(\032)2598 3957 y Fp(!)2678 4099 y Fu(:)433 b Fz(\(9.1\))440 4363 y Fo(Pr)l(o)l(of)31 b(:)39 b Fz(W)-7 b(e)28 b(decomp)r(ose)e Fu(u)1335 4333 y Fv(0)1335 4386 y Fs(T)5 b(;\032)1465 4363 y Fz(according)26 b(to)i(a)f(basis)g(of)g(eigen)n(v)n(ectors)f (for)h Fu(L)2957 4333 y Fv(mean)n Fs(;T)3210 4363 y Fz(:)1362 4569 y Fu(u)1410 4534 y Fv(0)1410 4589 y Fs(T)5 b(;\032)1596 4569 y Fz(=)82 b Fu(\013)1796 4581 y Fv(0)1834 4569 y Fu(u)1882 4529 y Fv(mean)n Fs(;T)1882 4591 y Fv(1)2125 4569 y Fz(+)24 b(~)-47 b Fu(u)2257 4534 y Fv(0)2257 4589 y Fs(T)5 b(;\032)2359 4569 y Fu(;)752 b Fz(\(9.2\))440 4771 y(with)46 b(~)-48 b Fu(u)689 4741 y Fv(0)689 4794 y Fs(T)5 b(;\032)832 4771 y Fz(orthogonal)37 b(to)j Fu(u)1421 4731 y Fv(mean)n Fs(;T)1421 4793 y Fv(1)1646 4771 y Fz(.)74 b(W)-7 b(e)40 b(estimate)2243 4704 y Fp(\012)2282 4771 y Fu(L)2339 4741 y Fv(mean)o Fs(;T)2565 4771 y Fu(u)2613 4741 y Fv(0)2613 4794 y Fs(T)5 b(;\032)2715 4771 y Fu(;)14 b(u)2800 4741 y Fv(0)2800 4794 y Fs(T)5 b(;\032)2903 4704 y Fp(\013)2942 4810 y Fs(L)2988 4793 y Fr(2)3020 4810 y Fv(\(0)p Fs(;T)k Fv(\))3217 4771 y Fz(b)n(y)440 4895 y(using)41 b(\(8.23\),)i(Relation)e(\(8.14\))f(and)g(the)i (expression)d(\(9.2\).)77 b(Then)41 b(there)f(exists)h(a)440 4994 y(constan)n(t)c Fu(A)i Fz(indep)r(enden)n(t)f(of)g Fu(T)49 b Fz(and)38 b Fu(\032)g Fz(suc)n(h)g(that)g(if)h Fu(T)51 b Fx(\025)40 b Fz(1)e(and)2763 4962 y Fs(T)p 2763 4976 49 4 v 2770 5023 a(\032)2860 4994 y Fz(is)g(uniformly)440 5094 y(b)r(ounded,)28 b(then)g(:)586 5290 y Fp(\012)625 5357 y Fu(L)682 5323 y Fv(mean)n Fs(;T)907 5357 y Fu(u)955 5323 y Fv(0)955 5378 y Fs(T)5 b(;\032)1058 5357 y Fu(;)14 b(u)1143 5323 y Fv(0)1143 5378 y Fs(T)5 b(;\032)1245 5290 y Fp(\013)1284 5393 y Fs(L)1330 5376 y Fr(2)1362 5393 y Fv(\(0)p Fs(;T)k Fv(\))1542 5357 y Fx(\024)23 b Fu(\025)1678 5317 y Fv(mean)o Fs(;T)1678 5379 y Fv(1)1904 5357 y Fz(\()p Fu(\013)1989 5323 y Fv(2)1989 5378 y(0)2045 5357 y Fz(+)18 b Fx(jj)5 b Fz(~)-47 b Fu(u)2222 5323 y Fv(0)2222 5378 y Fs(T)5 b(;\032)2325 5357 y Fx(jj)2371 5323 y Fv(2)2371 5380 y Fs(L)2417 5363 y Fr(2)2449 5380 y Fv(\(0)p Fs(;T)k Fv(\))2606 5357 y Fz(\))19 b(+)f Fu(A)2802 5208 y Fp(s)p 2885 5208 80 4 v 2895 5301 a Fu(T)p 2895 5338 61 4 v 2904 5414 a(\032)2965 5357 y(:)146 b Fz(\(9.3\))1831 5662 y(26)p eop %%Page: 27 27 27 26 bop 440 531 a Fz(W)-7 b(e)23 b(denote)f(b)n(y)g Fu(\025)998 491 y Fv(mean)p Fs(;T)998 553 y Fv(2)1247 531 y Fz(the)h(second)f(eigen)n(v)-5 b(alue)22 b(for)f Fu(L)2218 501 y Fv(mean)o Fs(;T)2466 531 y Fz(and)i(apply)f(the)h (min-max)440 631 y(principle,)28 b(then)g(:)644 735 y Fp(\012)683 802 y Fu(L)740 768 y Fv(mean)o Fs(;T)966 802 y Fu(u)1014 768 y Fv(0)1014 823 y Fs(T)5 b(;\032)1116 802 y Fu(;)14 b(u)1201 768 y Fv(0)1201 823 y Fs(T)5 b(;\032)1303 735 y Fp(\013)1343 838 y Fs(L)1389 822 y Fr(2)1421 838 y Fv(\(0)p Fs(;T)k Fv(\))1601 802 y Fx(\025)22 b Fu(\025)1736 762 y Fv(mean)o Fs(;T)1736 825 y Fv(1)1962 802 y Fu(\013)2015 768 y Fv(2)2015 823 y(0)2072 802 y Fz(+)c Fu(\025)2203 762 y Fv(mean)o Fs(;T)2203 825 y Fv(2)2429 802 y Fx(jj)5 b Fz(~)-47 b Fu(u)2523 768 y Fv(0)2523 823 y Fs(T)5 b(;\032)2625 802 y Fx(jj)2671 768 y Fv(2)2671 825 y Fs(L)2717 808 y Fr(2)2749 825 y Fv(\(0)p Fs(;T)k Fv(\))2906 802 y Fu(:)205 b Fz(\(9.4\))440 995 y(F)-7 b(rom)27 b(\(9.3\))h(and)f(\(9.4\),)g(w)n (e)h(deduce)f(that)h(:)1012 1159 y Fp(\020)1062 1252 y Fu(\025)1110 1212 y Fv(mean)o Fs(;T)1110 1274 y Fv(2)1355 1252 y Fx(\000)18 b Fu(\025)1486 1212 y Fv(mean)o Fs(;T)1486 1274 y Fv(1)1712 1159 y Fp(\021)1775 1252 y Fx(jj)5 b Fz(~)-47 b Fu(u)1869 1217 y Fv(0)1869 1272 y Fs(T)5 b(;\032)1972 1252 y Fx(jj)2018 1217 y Fv(2)2018 1274 y Fs(L)2064 1258 y Fr(2)2096 1274 y Fv(\(0)p Fs(;T)k Fv(\))2336 1252 y Fx(\024)83 b Fu(A)2546 1103 y Fp(s)p 2629 1103 80 4 v 2639 1195 a Fu(T)p 2639 1233 61 4 v 2648 1309 a(\032)2709 1252 y(:)402 b Fz(\(9.5\))440 1508 y(W)-7 b(e)28 b(can)f(b)r(ound)h (from)f(b)r(elo)n(w)h Fu(\025)1471 1468 y Fv(mean)o Fs(;T)1471 1530 y Fv(2)1725 1508 y Fz(b)n(y)f Fu(\025)1888 1478 y Fv(mean)1888 1529 y(2)2074 1508 y Fz(and)g(according)f(to)i(\(8.2\),) f(w)n(e)g(ha)n(v)n(e)g(:)1280 1664 y(1)p 976 1701 651 4 v 976 1790 a Fu(\025)1024 1751 y Fv(mean)o Fs(;T)1024 1813 y Fv(2)1269 1790 y Fx(\000)18 b Fu(\025)1400 1751 y Fv(mean)o Fs(;T)1400 1813 y Fv(1)1719 1720 y Fx(\024)2290 1664 y Fz(1)p 1877 1701 869 4 v 1877 1804 a Fu(\025)1925 1775 y Fv(mean)1925 1826 y(2)2102 1804 y Fx(\000)g Fu(\025)2233 1775 y Fv(mean)2233 1826 y(1)2409 1804 y Fx(\000)i Fz(~)-44 b Fu(ce)2567 1762 y Ft(\000)2659 1739 y Fe(T)p 2629 1748 103 3 v 2629 1790 a Fr(2)2657 1756 y Fq(p)p 2703 1756 29 3 v 34 x Fr(3)2755 1720 y Fu(:)356 b Fz(\(9.6\))440 1983 y(Due)28 b(to)g(\(9.6\),)f(there)g(exists)h Fu(T)1428 1995 y Fv(0)1492 1983 y Fz(and)1675 1962 y(~)1653 1983 y Fu(A)g Fz(indep)r(enden)n(t)h(of)e Fu(\032)h Fz(suc)n(h)f(that)h(for) f(all)h Fu(T)34 b Fx(\025)22 b Fu(T)3208 1995 y Fv(0)3273 1983 y Fz(:)1394 2239 y Fx(jj)5 b Fz(~)-47 b Fu(u)1488 2205 y Fv(0)1488 2260 y Fs(T)5 b(;\032)1590 2239 y Fx(jj)1636 2205 y Fv(2)1636 2262 y Fs(L)1682 2245 y Fr(2)1714 2262 y Fv(\(0)p Fs(;T)k Fv(\))1955 2239 y Fx(\024)2124 2218 y Fz(~)2102 2239 y Fu(A)2164 2090 y Fp(s)p 2247 2090 80 4 v 2257 2183 a Fu(T)p 2257 2220 61 4 v 2266 2296 a(\032)2327 2239 y(:)784 b Fz(\(9.7\))440 2484 y(This)28 b(sho)n(ws)e(that)i Fx(jj)5 b Fz(~)-47 b Fu(u)1142 2453 y Fv(0)1142 2506 y Fs(T)5 b(;\032)1244 2484 y Fx(jj)1290 2453 y Fv(2)1290 2510 y Fs(L)1336 2494 y Fr(2)1369 2510 y Fv(\(0)p Fs(;T)k Fv(\))1553 2484 y Fz(tends)28 b(to)g(0)f(when)h Fu(\032)f Fz(tends)h(to)f(in\014nit)n(y)-7 b(.)440 2601 y(W)g(e)28 b(use)f(the)h(decomp)r(osition)f(of)h Fu(u)1559 2571 y Fv(0)1559 2624 y Fs(T)5 b(;\032)1661 2601 y Fz(,)28 b(Relations)f(\(8.21\))g(and)g(\(9.7\))h(to)f(obtain)h(:)1385 2871 y(1)18 b Fx(\000)1557 2793 y Fz(~)1538 2814 y Fu(C)1603 2784 y Ft(00)p 1538 2851 108 4 v 1570 2927 a Fu(\032)1674 2871 y Fx(\000)1779 2850 y Fz(~)1757 2871 y Fu(A)1820 2722 y Fp(s)p 1903 2722 80 4 v 1913 2814 a Fu(T)p 1913 2851 61 4 v 1922 2927 a(\032)2006 2871 y Fx(\024)k Fu(\013)2146 2836 y Fv(2)2146 2891 y(0)2207 2871 y Fx(\024)h Fz(1)p Fu(:)774 b Fz(\(9.8\))440 3105 y(This)28 b(giv)n(es)e(:)574 3357 y Fx(jj)p Fu(u)668 3323 y Fv(0)668 3377 y Fs(T)5 b(;\032)789 3357 y Fx(\000)18 b Fu(u)920 3323 y Fv(mean)n Fs(;T)1145 3357 y Fx(jj)1191 3323 y Fv(2)1191 3379 y Fs(L)1237 3363 y Fr(2)1269 3379 y Fv(\(0)p Fs(;T)9 b Fv(\))1449 3357 y Fz(=)23 b(\(1)18 b Fx(\000)g Fu(\013)1765 3369 y Fv(0)1803 3357 y Fz(\))1835 3323 y Fv(2)1891 3357 y Fz(+)g Fx(jj)5 b Fz(~)-47 b Fu(u)2068 3323 y Fv(0)2068 3377 y Fs(T)5 b(;\032)2170 3357 y Fx(jj)2216 3323 y Fv(2)2216 3379 y Fs(L)2262 3363 y Fr(2)2294 3379 y Fv(\(0)p Fs(;T)k Fv(\))2474 3357 y Fx(\024)2572 3301 y Fu(C)p 2572 3338 66 4 v 2583 3414 a(\032)2666 3357 y Fz(+)18 b Fu(C)2814 3208 y Fp(s)p 2897 3208 80 4 v 2907 3301 a Fu(T)p 2907 3338 61 4 v 2916 3414 a(\032)2977 3357 y(:)134 b Fz(\(9.9\))3240 3590 y Fd(\003)440 3689 y Fz(W)-7 b(e)28 b(also)f(can)g(compare)f(the)i (\014rst)f(eigen)n(v)n(ectors)f(for)h Fu(L)2197 3659 y Fv(mean)2382 3689 y Fz(and)g Fu(L)2600 3659 y Fv(mean)o Fs(;T)2853 3689 y Fz(:)440 3846 y FA(Prop)s(osition)34 b(9.2.)42 b Fo(L)l(et)33 b Fu(u)1347 3816 y Fv(mean)1347 3867 y(1)1538 3846 y Fo(b)l(e)g(the)g(normalize)l(d)h(p)l(ositive)h (eigenve)l(ctor)f(asso)l(ciate)l(d)440 3955 y(to)g Fu(\025)592 3925 y Fv(mean)592 3976 y(1)751 3955 y Fo(,)h Fu(u)859 3915 y Fv(mean)o Fs(;T)859 3977 y Fv(1)1119 3955 y Fo(b)l(e)f(the)h (normalize)l(d)g(p)l(ositive)h(eigenve)l(ctor)f(asso)l(ciate)l(d)h(to)e Fu(\025)3078 3915 y Fv(mean)o Fs(;T)3078 3977 y Fv(1)440 4065 y Fo(and)i Fz(~)-48 b Fu(u)649 4025 y Fv(mean)o Fs(;T)649 4087 y Fv(1)904 4065 y Fo(its)30 b(extension)f(by)i(0)f(on)g Fz([)p Fu(T)7 b(;)14 b Fz(+)p Fx(1)p Fz([)p Fo(.)37 b(Then)31 b(:)912 4263 y Fx(jj)p Fu(u)1006 4228 y Fv(mean)1006 4283 y(1)1182 4263 y Fx(\000)24 b Fz(~)-48 b Fu(u)1313 4223 y Fv(mean)o Fs(;T)1313 4285 y Fv(1)1539 4263 y Fx(jj)1585 4278 y Fs(L)1631 4262 y Fr(2)1663 4278 y Fv(\()p Fn(R)1736 4262 y Fr(+)1777 4278 y Fv(\))1831 4263 y Fz(=)22 b Fx(O)2000 4171 y Fp(\020)2050 4263 y Fu(e)2089 4220 y Ft(\000)2181 4198 y Fe(T)p 2151 4207 103 3 v 2151 4249 a Fr(4)2179 4215 y Fq(p)p 2224 4215 29 3 v 2224 4249 a Fr(3)2267 4171 y Fp(\021)2360 4263 y Fu(as)29 b(T)34 b Fx(!)24 b Fz(+)p Fx(1)p Fu(:)259 b Fz(\(9.10\))440 4476 y Fo(Pr)l(o)l(of)36 b(:)48 b Fz(W)-7 b(e)33 b(decomp)r(ose)38 b(~)-48 b Fu(u)1360 4437 y Fv(mean)o Fs(;T)1360 4499 y Fv(1)1617 4476 y Fz(=)32 b Fu(a)1758 4488 y Fv(0)1795 4476 y Fu(u)1843 4446 y Fv(mean)1843 4497 y(1)2022 4476 y Fz(+)22 b Fu(v)2149 4488 y Fv(1)2186 4476 y Fz(,)35 b(with)e Fu(v)2478 4488 y Fv(1)2548 4476 y Fz(orthogonal)e(to)i Fu(u)3124 4446 y Fv(mean)3124 4497 y(1)3281 4476 y Fz(,)440 4576 y(then)28 b(:)1147 4676 y Fx(jj)5 b Fz(~)-47 b Fu(u)1241 4636 y Fv(mean)o Fs(;T)1241 4698 y Fv(1)1467 4676 y Fx(jj)1513 4641 y Fv(2)1513 4699 y Fs(L)1559 4682 y Fr(2)1591 4699 y Fv(\()p Fn(R)1664 4682 y Fr(+)1706 4699 y Fv(\))1759 4676 y Fz(=)22 b(1)h(=)g Fu(a)2043 4641 y Fv(2)2043 4696 y(0)2098 4676 y Fz(+)18 b Fx(jj)p Fu(v)2267 4688 y Fv(1)2305 4676 y Fx(jj)2351 4641 y Fv(2)2351 4699 y Fs(L)2397 4682 y Fr(2)2429 4699 y Fv(\()p Fn(R)2502 4682 y Fr(+)2544 4699 y Fv(\))2574 4676 y Fu(:)495 b Fz(\(9.11\))440 4818 y(F)-7 b(urthermore)27 b(:)558 4990 y Fu(\025)606 4950 y Fv(mean)o Fs(;T)606 5012 y Fv(1)855 4990 y Fz(=)943 4898 y Fp(D)993 4990 y Fu(L)1050 4955 y Fv(mean)1213 4990 y Fz(~)-47 b Fu(u)1256 4950 y Fv(mean)n Fs(;T)1256 5012 y Fv(1)1481 4990 y Fu(;)19 b Fz(~)-47 b Fu(u)1566 4950 y Fv(mean)n Fs(;T)1566 5012 y Fv(1)1791 4898 y Fp(E)1842 5048 y Fs(L)1888 5031 y Fr(2)1920 5048 y Fv(\()p Fn(R)1993 5031 y Fr(+)2035 5048 y Fv(\))2088 4990 y Fx(\025)23 b Fu(a)2220 4955 y Fv(2)2220 5010 y(0)2257 4990 y Fu(\025)2305 4955 y Fv(mean)2305 5010 y(1)2482 4990 y Fz(+)18 b Fu(\025)2613 4955 y Fv(mean)2613 5010 y(2)2771 4990 y Fx(jj)p Fu(v)2857 5002 y Fv(1)2895 4990 y Fx(jj)2941 4955 y Fv(2)2941 5013 y Fs(L)2987 4996 y Fr(2)3019 5013 y Fv(\()p Fn(R)3092 4996 y Fr(+)3133 5013 y Fv(\))3164 4990 y Fu(:)3092 5133 y Fz(\(9.12\))440 5233 y(Relations)27 b(\(9.11\))g(and)g(\(9.12\))g (lead)g(to)h(:)935 5413 y(\()p Fu(\025)1015 5379 y Fv(mean)1015 5434 y(2)1192 5413 y Fx(\000)18 b Fu(\025)1323 5373 y Fv(mean)o Fs(;T)1323 5435 y Fv(1)1549 5413 y Fz(\))p Fx(jj)p Fu(v)1667 5425 y Fv(1)1705 5413 y Fx(jj)1751 5379 y Fv(2)1751 5436 y Fs(L)1797 5420 y Fr(2)1829 5436 y Fv(\()p Fn(R)1902 5420 y Fr(+)1944 5436 y Fv(\))2057 5413 y Fx(\024)82 b Fu(\025)2252 5373 y Fv(mean)p Fs(;T)2252 5435 y Fv(1)2497 5413 y Fx(\000)18 b Fu(\025)2628 5379 y Fv(mean)2628 5434 y(1)2787 5413 y Fu(:)1831 5662 y Fz(27)p eop %%Page: 28 28 28 27 bop 440 531 a Fz(F)-7 b(rom)27 b(Lemma)h(8.2,)e(w)n(e)i(deduce)f (:)932 607 y Fp(\020)982 699 y Fu(\025)1030 665 y Fv(mean)1030 719 y(2)1207 699 y Fx(\000)18 b Fu(\025)1338 665 y Fv(mean)1338 719 y(1)1514 699 y Fx(\000)i Fz(~)-44 b Fu(ce)1672 656 y Ft(\000)1764 634 y Fe(T)p 1734 643 103 3 v 1734 685 a Fr(2)1762 651 y Fq(p)p 1808 651 29 3 v 34 x Fr(3)1850 607 y Fp(\021)1913 699 y Fx(jj)p Fu(v)1999 711 y Fv(1)2037 699 y Fx(jj)2083 665 y Fv(2)2083 722 y Fs(L)2129 706 y Fr(2)2161 722 y Fv(\()p Fn(R)2234 706 y Fr(+)2276 722 y Fv(\))2389 699 y Fx(\024)84 b Fz(~)-43 b Fu(c)o(e)2611 656 y Ft(\000)2703 634 y Fe(T)p 2673 643 103 3 v 2673 685 a Fr(2)2701 651 y Fq(p)p 2747 651 29 3 v 34 x Fr(3)2789 699 y Fu(:)440 902 y Fz(But)34 b(observing)e(that)h Fu(\025)1222 872 y Fv(mean)1222 923 y(2)1403 902 y Fx(\000)22 b Fu(\025)1538 872 y Fv(mean)1538 923 y(1)1729 902 y Fz(=)1864 869 y Fv(2)p 1837 883 88 4 v 1837 892 a Ft(p)p 1891 892 34 3 v 1891 940 a Fv(3)1934 902 y Fz(,)35 b(w)n(e)e(get)h Fx(jj)p Fu(v)2351 914 y Fv(1)2388 902 y Fx(jj)2434 872 y Fv(2)2434 929 y Fs(L)2480 913 y Fr(2)2512 929 y Fv(\()p Fn(R)2585 913 y Fr(+)2627 929 y Fv(\))2690 902 y Fz(=)e Fx(O)r Fz(\()p Fu(e)2926 860 y Ft(\000)3019 837 y Fe(T)p 2989 846 103 3 v 2989 888 a Fr(2)3017 854 y Fq(p)p 3062 854 29 3 v 3062 888 a Fr(3)3105 902 y Fz(\))i(and)440 1049 y Fu(a)484 1061 y Fv(0)544 1049 y Fz(=)23 b(1)18 b(+)g Fx(O)r Fz(\()p Fu(e)914 1006 y Ft(\000)1006 984 y Fe(T)p 976 993 103 3 v 976 1035 a Fr(2)1004 1001 y Fq(p)p 1050 1001 29 3 v 34 x Fr(3)1092 1049 y Fz(\))28 b(as)f Fu(T)39 b Fz(tends)28 b(to)f(in\014nit)n(y)-7 b(.)37 b(T)-7 b(o)28 b(ac)n(hiev)n(e)e(the)i(pro)r(of,)f(w)n(e)g(just)h (write)g(:)593 1205 y Fx(jj)p Fu(u)687 1171 y Fv(mean)687 1226 y(1)864 1205 y Fx(\000)23 b Fz(~)-47 b Fu(u)995 1165 y Fv(mean)n Fs(;T)995 1227 y Fv(1)1220 1205 y Fx(jj)1266 1171 y Fv(2)1266 1228 y Fs(L)1312 1212 y Fr(2)1344 1228 y Fv(\()p Fn(R)1417 1212 y Fr(+)1459 1228 y Fv(\))1572 1205 y Fz(=)82 b(\(1)19 b Fx(\000)f Fu(a)1939 1217 y Fv(0)1976 1205 y Fz(\))2008 1171 y Fv(2)2045 1205 y Fx(jj)5 b Fz(~)-47 b Fu(u)2139 1165 y Fv(mean)o Fs(;T)2139 1227 y Fv(1)2365 1205 y Fx(jj)2411 1171 y Fv(2)2411 1228 y Fs(L)2457 1212 y Fr(2)2489 1228 y Fv(\()p Fn(R)2562 1212 y Fr(+)2604 1228 y Fv(\))2652 1205 y Fz(+)18 b Fx(jj)p Fu(v)2821 1217 y Fv(1)2859 1205 y Fx(jj)2905 1171 y Fv(2)2905 1228 y Fs(L)2951 1212 y Fr(2)2983 1228 y Fv(\()p Fn(R)3056 1212 y Fr(+)3098 1228 y Fv(\))3128 1205 y Fu(:)440 1394 y Fz(So)27 b Fx(jj)p Fu(u)649 1364 y Fv(mean)649 1415 y(1)825 1394 y Fx(\000)d Fz(~)-48 b Fu(u)956 1354 y Fv(mean)o Fs(;T)956 1416 y Fv(1)1182 1394 y Fx(jj)1228 1364 y Fv(2)1228 1422 y Fs(L)1274 1405 y Fr(2)1306 1422 y Fv(\()p Fn(R)1379 1405 y Fr(+)1421 1422 y Fv(\))1474 1394 y Fz(=)22 b Fx(O)1644 1302 y Fp(\020)1693 1394 y Fu(e)1732 1352 y Ft(\000)1824 1329 y Fe(T)p 1794 1338 103 3 v 1794 1380 a Fr(2)1822 1346 y Fq(p)p 1868 1346 29 3 v 34 x Fr(3)1910 1302 y Fp(\021)1959 1394 y Fz(.)1258 b Fd(\003)440 1663 y Fy(10)135 b(Splitting)46 b(b)t(et)l(w)l(een)g(the)f(t)l(w)l(o)h(\014rst)f(eigen)l (v)-7 b(alues)440 1861 y Fl(10.1)112 b(Main)38 b(prop)s(osition)440 2014 y Fz(The)i(upp)r(er)h(b)r(ound)g(of)f(the)h(b)r(ottom)g(of)f(the)h (sp)r(ectrum)g(giv)n(en)e(in)i(Section)f(6)g(pro)n(v)n(es)440 2114 y(the)31 b(existence)f(of)h(an)f(eigen)n(v)-5 b(alue)30 b(with)h(the)g(asymptotics)f Fu(\013\026)2462 2084 y Fv(\()p Fs(n)p Fv(\))2580 2114 y Fz(+)20 b Fx(O)r Fz(\()p Fu(\013)2818 2084 y Fv(2)p Fs(n)p Fv(+3)2981 2114 y Fz(\).)46 b(In)31 b(the)440 2214 y(previous)e(section,)i(w)n(e)f(ha)n(v)n(e)f (established)h(the)g(lo)n(w)n(er)f(b)r(ound)i Fu(\026)p Fz(\()p Fu(\013)p Fz(\))d Fx(\025)2768 2181 y Fs(\013)p 2745 2195 88 4 v 2745 2204 a Ft(p)p 2800 2204 34 3 v 48 x Fv(3)2863 2214 y Fx(\000)20 b Fu(\017\013)p Fz(.)45 b(If)31 b(the)440 2325 y(splitting)h(b)r(et)n(w)n(een)f(the)h(t)n(w)n (o)f(\014rst)g(eigen)n(v)-5 b(alues)30 b(of)i Fu(P)2164 2337 y Fs(\013)2243 2325 y Fz(is)f(big)g(enough)g(or)g(if)h(w)n(e)f (obtain)440 2424 y(a)26 b(\\go)r(o)r(d")f(lo)n(w)n(er)g(b)r(ound)i(of)f (the)h(second)f(eigen)n(v)-5 b(alue,)26 b(then)h(sp)r(ectral)f(theorem) g(will)g(giv)n(e)440 2524 y(the)i(asymptotics)f(of)g Fu(\026)p Fz(\()p Fu(\013)p Fz(\).)440 2662 y FA(Prop)s(osition)e (10.1.)36 b Fo(As)26 b Fu(\013)g Fo(tends)f(to)h(0,)i(the)e(numb)l(er)f (of)h(eigenvalues)i(of)e Fu(P)2888 2674 y Ft(A)2941 2682 y Fr(0)2974 2674 y Fs(;)p Fv(\012)3041 2682 y Fe(\013)3113 2662 y Fo(b)l(elow)440 2761 y(the)e(essential)g(sp)l(e)l(ctrum)f(tends) h(to)g(in\014nity.)36 b(If)24 b(we)g(denote)h(by)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\))p Fo(,)i Fu(\026)2701 2773 y Fv(2)2739 2761 y Fz(\()p Fu(\013)p Fz(\))e Fo(the)g(two)g(\014rst)440 2861 y(eigenvalues)32 b(of)h Fu(P)1029 2873 y Ft(A)1082 2881 y Fr(0)1115 2873 y Fs(;)p Fv(\012)1182 2881 y Fe(\013)1259 2861 y Fo(and)f(by)g Fu(\025)1580 2831 y Fv(mean)1580 2882 y(1)1738 2861 y Fo(,)g Fu(\025)1843 2831 y Fv(mean)1843 2882 y(2)2033 2861 y Fo(the)f(smal)t(lest)h(eigenvalues)g(for)g Fu(L)3121 2831 y Fv(mean)3279 2861 y Fo(,)440 2961 y(then)e(for)g (every)h Fu(\017)23 b(>)f Fz(0)p Fo(,)30 b(ther)l(e)g(exists)f Fu(\013)1705 2973 y Fv(0)1772 2961 y Fo(and)h Fu(C)g(>)22 b Fz(0)30 b Fo(such)f(that)h(for)h(al)t(l)g Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)3077 2973 y Fv(0)3114 2961 y Fz(])30 b Fo(:)1430 3108 y Fu(\026)1480 3120 y Fv(2)1518 3108 y Fz(\()p Fu(\013)p Fz(\))84 b Fx(\025)e Fu(\013\025)1967 3074 y Fv(mean)1967 3129 y(2)2145 3108 y Fx(\000)18 b Fu(\017\013;)754 b Fz(\(10.1\))1151 3224 y Fu(\026)1201 3236 y Fv(2)1238 3224 y Fz(\()p Fu(\013)p Fz(\))20 b Fx(\000)e Fu(\026)p Fz(\()p Fu(\013)p Fz(\))p 1151 3261 475 4 v 1362 3337 a Fu(\013)1719 3280 y Fx(\025)82 b Fz(\()p Fu(\025)1946 3246 y Fv(mean)1946 3301 y(2)2123 3280 y Fx(\000)18 b Fu(\025)2254 3246 y Fv(mean)2254 3301 y(1)2413 3280 y Fz(\))h Fx(\000)f Fu(\017:)488 b Fz(\(10.2\))565 3451 y(T)-7 b(o)31 b(pro)n(v)n(e)f(Prop)r(osition)g (10.1,)h(w)n(e)g(will)h(use)g(the)g(same)f(op)r(erators)e(as)i(in)h (Section)g(8,)440 3551 y(that)25 b(is)g(to)g(sa)n(y)-7 b(,)24 b(the)h(op)r(erators)e Fu(L)1516 3520 y Fv(mean)n Fs(;T)1741 3551 y Fz(,)j Fu(P)1855 3520 y Fs(T)1843 3571 y(\013)1932 3551 y Fz(and)f Fu(P)2156 3520 y Fs(T)5 b(;\032)2283 3551 y Fz(de\014ned)26 b(resp)r(ectiv)n(ely)d(in)j(\(8.1\),)440 3650 y(\(8.4\),)h(\(8.12\))g(and)h(studied)g(resp)r(ectiv)n(ely)e(in)i (Subsections)g(8.2,)e(8.3)h(and)h(8.4.)440 3877 y Fl(10.2)112 b(Estimates)36 b(of)i(the)f(second)h(eigen)m(v)-6 b(alue)440 4030 y Fz(W)f(e)27 b(use)f(the)h(index)g(2)f(to)h(indicate)f(the)h (second)f(eigen)n(v)-5 b(alue)26 b(\(when)h(it)g(exists\).)37 b(W)-7 b(e)27 b(ha)n(v)n(e)440 4129 y(an)g(expression)f(of)i Fu(\025)1098 4141 y Fv(2)1136 4129 y Fz(\()p Fu(\013)p Fz(\))g(according)e(to)i(the)g(min-max)f(principle)g(:)1133 4325 y Fu(\025)1181 4337 y Fv(2)1219 4325 y Fz(\()p Fu(\013)p Fz(\))84 b(=)165 b(sup)1567 4398 y Fs(v)r Ft(2)p Fs(L)1693 4382 y Fr(2)1725 4398 y Fv(\(\012)1798 4382 y Fr(0)1831 4398 y Fv(\))1985 4325 y Fz(inf)1871 4383 y Fs(u)p Ft(2V)2002 4367 y Fe(N)2055 4383 y Fs(;u)p Ft(?)p Fs(v)2303 4268 y Fu(q)2340 4280 y Fs(\013)2388 4268 y Fz(\()p Fu(u)p Fz(\))p 2225 4306 354 4 v 2225 4382 a Fx(jj)p Fu(u)p Fx(jj)2365 4353 y Fv(2)2365 4410 y Fs(L)2411 4393 y Fr(2)2442 4410 y Fv(\(\012)2515 4393 y Fr(0)2548 4410 y Fv(\))2588 4325 y Fu(:)481 b Fz(\(10.3\))440 4560 y(By)43 b(the)g(same)g (principle,)k(w)n(e)c(ha)n(v)n(e)f(expressions)f(for)i Fu(\025)2336 4530 y Fs(T)2336 4581 y Fv(2)2388 4560 y Fz(\()p Fu(\013)p Fz(\),)48 b Fu(\025)2624 4530 y Fv(mean)2624 4581 y(2)2783 4560 y Fz(,)f Fu(\025)2901 4520 y Fv(mean)o Fs(;T)2901 4582 y Fv(2)3171 4560 y Fz(and)440 4660 y Fu(\027)481 4672 y Fv(2)518 4660 y Fz(\()p Fu(T)7 b(;)14 b(\032)p Fz(\).)37 b(So)27 b(w)n(e)h(deduce)f(that)h(:)1430 4807 y Fu(\025)1478 4773 y Fv(mean)1478 4827 y(2)1720 4807 y Fx(\025)82 b Fu(\025)1915 4819 y Fv(2)1953 4807 y Fz(\()p Fu(\013)p Fz(\))p Fu(;)999 b Fz(\(10.4\))1418 4931 y Fu(\025)1466 4897 y Fs(T)1466 4952 y Fv(2)1519 4931 y Fz(\()p Fu(\013)p Fz(\))84 b Fx(\025)e Fu(\025)1915 4943 y Fv(2)1953 4931 y Fz(\()p Fu(\013)p Fz(\))p Fu(;)999 b Fz(\(10.5\))1363 5065 y Fu(\025)1411 5025 y Fv(mean)o Fs(;T)1411 5087 y Fv(2)1720 5065 y Fx(\025)82 b Fu(\025)1915 5031 y Fv(mean)1915 5086 y(2)2074 5065 y Fu(;)995 b Fz(\(10.6\))1363 5200 y Fu(\025)1411 5160 y Fv(mean)o Fs(;T)1411 5222 y Fv(2)1720 5200 y Fx(\025)82 b Fu(\027)1908 5212 y Fv(2)1946 5200 y Fz(\()p Fu(T)7 b(;)14 b(\032)p Fz(\))p Fu(;)923 b Fz(\(10.7\))1418 5376 y Fu(\025)1466 5341 y Fs(T)1466 5396 y Fv(2)1519 5376 y Fz(\()p Fu(\013)p Fz(\))84 b Fx(\025)e Fu(\027)1908 5388 y Fv(2)1960 5259 y Fp(\022)2021 5376 y Fu(T)7 b(;)2178 5320 y Fz(1)p 2123 5357 151 4 v 2123 5433 a Fu(\013)2176 5409 y Fv(2)2214 5433 y Fu(T)2284 5259 y Fp(\023)2359 5376 y Fu(:)710 b Fz(\(10.8\))1831 5662 y(28)p eop %%Page: 29 29 29 28 bop 440 531 a Fz(Due)28 b(to)g(Relation)f(\(10.4\),)g(w)n(e)g (can)g(b)r(ound)h(from)g(ab)r(o)n(v)n(e)e Fu(\025)2322 543 y Fv(2)2360 531 y Fz(\()p Fu(\013)p Fz(\))i(b)n(y)g(a)f(p)r(ositiv) n(e)g(constan)n(t)440 631 y(and)k(so)f Fu(\026)760 643 y Fv(2)797 631 y Fz(\()p Fu(\013)p Fz(\))g(=)d Fu(\013\025)1137 643 y Fv(2)1176 631 y Fz(\()p Fu(\013)p Fz(\))k(tends)h(to)e(0)h(with)g Fu(\013)p Fz(.)47 b(Then)31 b(there)g(exists)f Fu(\013)2763 643 y Fv(1)2831 631 y Fz(suc)n(h)h(that)g(for)440 731 y(all)24 b Fu(\013)f Fx(\024)g Fu(\013)769 743 y Fv(1)806 731 y Fz(,)i Fu(\026)904 743 y Fv(2)941 731 y Fz(\()p Fu(\013)p Fz(\))f Fu(<)f Fz(\002)1235 743 y Fv(0)1272 731 y Fz(.)36 b(By)23 b(Lemma)h(2.3,)g(w)n(e)g(ha)n(v)n(e)f (established)g(that)i(the)f(b)r(ottom)g(of)440 830 y(the)30 b(essen)n(tial)f(sp)r(ectrum)h(of)g Fu(P)1433 842 y Ft(A)1486 850 y Fr(0)1519 842 y Fs(;)p Fv(\012)1586 850 y Fe(\013)1662 830 y Fz(is)g(equal)g(to)f(\002)2138 842 y Fv(0)2205 830 y Fz(and)h(so,)g Fu(\026)2547 842 y Fv(2)2584 830 y Fz(\()p Fu(\013)p Fz(\))h(is)f(an)f(eigen)n(v)-5 b(alue)440 930 y(as)27 b(so)r(on)g(as)g Fu(\026)886 942 y Fv(2)923 930 y Fz(\()p Fu(\013)p Fz(\))d Fu(<)f Fz(\002)1217 942 y Fv(0)1254 930 y Fz(.)440 1262 y Fl(10.3)112 b(Comparison)37 b(b)s(et)m(w)m(een)h Fk(\027)1860 1277 y Fj(2)1937 1262 y Fl(and)g Fk(\025)2210 1226 y Fj(mean)2210 1287 y(2)440 1415 y FA(Lemma)32 b(10.2.)41 b Fo(L)l(et)31 b Fu(\017)26 b(>)g Fz(0)p Fo(,)32 b(then)f(ther)l(e)h(exists)f Fu(\013)2092 1427 y Fv(0)2156 1415 y Fu(>)26 b Fz(0)31 b Fo(and)h Fu(c)26 b(>)g Fz(0)31 b Fo(indep)l(endent)h(of)h Fu(\017)440 1515 y Fo(such)d(that)g(for)g Fu(\013)24 b Fx(\024)e Fu(\013)1148 1527 y Fv(0)1215 1515 y Fo(and)31 b Fu(T)j Fz(=)1565 1482 y Fs(\017)p 1557 1496 44 4 v 1557 1543 a(\013)1640 1515 y Fo(:)1378 1755 y Fu(\027)1419 1767 y Fv(2)1470 1638 y Fp(\022)1531 1755 y Fu(T)7 b(;)1688 1699 y Fz(1)p 1633 1736 151 4 v 1633 1812 a Fu(\013)1686 1788 y Fv(2)1724 1812 y Fu(T)1794 1638 y Fp(\023)1878 1755 y Fx(\025)23 b Fu(\025)2014 1721 y Fv(mean)2014 1776 y(2)2191 1755 y Fx(\000)18 b Fu(c\017:)725 b Fz(\(10.9\))440 1996 y Fo(Pr)l(o)l(of)41 b(:)59 b Fz(Let)39 b Fu(u)965 2008 y Fv(1)p Fs(;T)1069 1996 y Fz(,)j Fu(u)1182 2008 y Fv(2)p Fs(;T)1325 1996 y Fz(the)d(normalized)f(eigen)n(v)n(ectors)e (asso)r(ciated)h(to)i Fu(\027)2969 1929 y Fp(\000)3007 1996 y Fu(T)7 b(;)3155 1963 y Fv(1)p 3109 1977 124 4 v 3109 2024 a Fs(\013)3152 2008 y Fr(2)3185 2024 y Fs(T)3243 1929 y Fp(\001)3281 1996 y Fz(,)440 2104 y Fu(\027)481 2116 y Fv(2)532 2037 y Fp(\000)570 2104 y Fu(T)g(;)718 2071 y Fv(1)p 672 2085 V 672 2132 a Fs(\013)715 2116 y Fr(2)748 2132 y Fs(T)806 2037 y Fp(\001)844 2104 y Fz(.)35 b(W)-7 b(e)20 b(de\014ne)h Fu(u)1318 2074 y Fv(0)1318 2127 y(1)p Fs(;T)1432 2104 y Fz(:)51 b(=)22 b(\005)1655 2116 y Fv(0)1693 2104 y Fu(u)1741 2116 y Fv(1)p Fs(;T)1845 2104 y Fz(,)g Fu(u)1938 2074 y Fv(0)1938 2127 y(2)p Fs(;T)2052 2104 y Fz(:)51 b(=)23 b(\005)2276 2116 y Fv(0)2313 2104 y Fu(u)2361 2116 y Fv(2)p Fs(;T)2486 2104 y Fz(and)d Fu(u)2688 2074 y Fv(1)2688 2127 y(1)p Fs(;T)2816 2104 y Fz(=)j Fu(u)2952 2116 y Fv(1)p Fs(;T)3060 2104 y Fx(\000)t Fu(u)3177 2074 y Fv(0)3177 2127 y(1)p Fs(;T)3281 2104 y Fz(,)440 2216 y Fu(u)488 2186 y Fv(1)488 2239 y(2)p Fs(;T)626 2216 y Fz(=)32 b Fu(u)771 2228 y Fv(2)p Fs(;T)898 2216 y Fx(\000)22 b Fu(u)1033 2186 y Fv(0)1033 2239 y(2)p Fs(;T)1138 2216 y Fz(.)55 b(According)33 b(to)g(\(8.21\))g(with)h Fu(\032)f Fz(=)2394 2184 y Fv(1)p 2349 2198 V 2349 2245 a Fs(\013)2392 2228 y Fr(2)2424 2245 y Fs(T)2515 2216 y Fz(=)2642 2184 y Fv(1)p 2623 2198 71 4 v 2623 2245 a Fs(\013\017)2704 2216 y Fz(,)i(there)e(exists)h Fu(C)3281 2186 y Ft(0)440 2316 y Fz(indep)r(enden)n(t)28 b(of)g Fu(\013)g Fz(and)g Fu(T)38 b Fz(suc)n(h)28 b(that)f(:)802 2499 y Fx(jj)p Fu(u)896 2464 y Fv(1)896 2519 y(1)p Fs(;T)1001 2499 y Fx(jj)1047 2514 y Fs(L)1093 2497 y Fr(2)1125 2514 y Fv(\(0)p Fs(;T)9 b Fv(\))1305 2499 y Fx(\024)23 b Fu(C)1458 2464 y Ft(0)1482 2435 y Fx(p)p 1551 2435 34 4 v 64 x Fu(\017)1584 2435 y Fx(p)p 1654 2435 54 4 v 1654 2499 a Fu(\013;)166 b Fx(jj)p Fu(u)1990 2464 y Fv(1)1990 2519 y(2)p Fs(;T)2095 2499 y Fx(jj)2141 2514 y Fs(L)2187 2497 y Fr(2)2219 2514 y Fv(\(0)p Fs(;T)9 b Fv(\))2399 2499 y Fx(\024)23 b Fu(C)2552 2464 y Ft(0)2575 2435 y Fx(p)p 2644 2435 34 4 v 64 x Fu(\017)2678 2435 y Fx(p)p 2747 2435 54 4 v 64 x Fu(\013)q(;)227 b Fz(\(10.10\))646 2635 y(1)18 b Fx(\000)g Fu(C)854 2601 y Ft(0)877 2635 y Fu(\017\013)23 b Fx(\024)g(jj)p Fu(u)1169 2601 y Fv(0)1169 2656 y(1)p Fs(;T)1274 2635 y Fx(jj)1320 2650 y Fs(L)1366 2634 y Fr(2)1398 2650 y Fv(\(0)p Fs(;T)9 b Fv(\))1578 2635 y Fx(\024)22 b Fz(1)p Fu(;)166 b Fz(1)18 b Fx(\000)g Fu(C)2104 2601 y Ft(0)2128 2635 y Fu(\017\013)23 b Fx(\024)f(jj)p Fu(u)2419 2601 y Fv(0)2419 2656 y(2)p Fs(;T)2524 2635 y Fx(jj)2570 2650 y Fs(L)2616 2634 y Fr(2)2648 2650 y Fv(\(0)p Fs(;T)9 b Fv(\))2828 2635 y Fx(\024)23 b Fz(1)p Fu(:)70 b Fz(\(10.11\))440 2838 y(W)-7 b(e)30 b(decomp)r(ose)g Fu(u)1054 2808 y Fv(0)1054 2861 y(2)p Fs(;T)1186 2838 y Fz(=)c Fu(a)1321 2850 y Fv(1)1359 2838 y Fu(u)1407 2798 y Fv(mean)n Fs(;T)1407 2860 y Fv(1)1652 2838 y Fz(+)f(~)-47 b Fu(u)1785 2808 y Fv(0)1785 2861 y(2)p Fs(;T)1919 2838 y Fz(with)36 b(~)-47 b Fu(u)2159 2808 y Fv(0)2159 2861 y(2)p Fs(;T)2294 2838 y Fz(orthogonal)28 b(to)i Fu(u)2864 2798 y Fv(mean)n Fs(;T)2864 2860 y Fv(1)3089 2838 y Fz(,)h(then)440 2948 y Fu(a)484 2918 y Fv(2)484 2969 y(1)537 2948 y Fz(+)16 b Fx(jj)5 b Fz(~)-47 b Fu(u)712 2918 y Fv(0)712 2971 y(2)p Fs(;T)817 2948 y Fx(jj)863 2918 y Fv(2)863 2975 y Fs(L)909 2958 y Fr(2)941 2975 y Fv(\(]0)p Fs(;T)9 b Fv([\))1158 2948 y Fx(\024)23 b Fz(1,)j(so)g Fx(j)p Fu(a)1505 2960 y Fv(1)1542 2948 y Fx(j)d(\024)g Fz(1.)36 b(Since)27 b Fx(h)p Fu(u)2073 2960 y Fv(2)p Fs(;T)2178 2948 y Fu(;)14 b(u)2263 2960 y Fv(1)p Fs(;T)2367 2948 y Fx(i)2399 2976 y Fs(L)2445 2960 y Fr(2)2477 2976 y Fv(\(\012)2550 2984 y Fe(T)2597 2976 y Fv(\))2650 2948 y Fz(=)22 b(0,)27 b(w)n(e)f(estimate)g(:)440 3086 y Fp(\012)479 3154 y Fu(u)527 3119 y Fv(0)527 3174 y(2)p Fs(;T)632 3154 y Fu(;)14 b(u)717 3119 y Fv(0)717 3174 y(1)p Fs(;T)821 3086 y Fp(\013)861 3190 y Fs(L)907 3173 y Fr(2)939 3190 y Fv(\(0)p Fs(;T)9 b Fv(\))1119 3154 y Fz(=)1206 3086 y Fp(\012)1245 3154 y Fu(u)1293 3119 y Fv(1)1293 3174 y(2)p Fs(;T)1398 3154 y Fu(;)14 b(u)1483 3119 y Fv(1)1483 3174 y(1)p Fs(;T)1588 3086 y Fp(\013)1627 3190 y Fs(L)1673 3173 y Fr(2)1705 3190 y Fv(\(\012)1778 3198 y Fe(T)1824 3190 y Fv(\))1864 3154 y Fx(\000)1939 3086 y Fp(\012)1978 3154 y Fu(u)2026 3119 y Fv(1)2026 3174 y(2)p Fs(;T)2131 3154 y Fu(;)g(u)2216 3166 y Fv(1)p Fs(;T)2321 3086 y Fp(\013)2360 3190 y Fs(L)2406 3173 y Fr(2)2438 3190 y Fv(\(\012)2511 3198 y Fe(T)2557 3190 y Fv(\))2597 3154 y Fx(\000)2672 3086 y Fp(\012)2712 3154 y Fu(u)2760 3166 y Fv(2)p Fs(;T)2864 3154 y Fu(;)g(u)2949 3119 y Fv(1)2949 3174 y Fs(T)3001 3086 y Fp(\013)3040 3187 y Fs(L)3086 3170 y Fr(2)3118 3187 y Fv(\(\012)3191 3195 y Fe(T)3237 3187 y Fv(\))3281 3154 y Fu(:)3051 3275 y Fz(\(10.12\))440 3383 y(The)27 b(decomp)r(osition)g(of)g Fu(u)1299 3353 y Fv(0)1299 3406 y(2)p Fs(;T)1403 3383 y Fz(,)h(the)f(normalization)f (of)h Fu(u)2264 3344 y Fv(mean)n Fs(;T)2264 3406 y Fv(1)2516 3383 y Fz(and)g(the)h(orthogonalit)n(y)440 3506 y(b)r(et)n(w)n(een)33 b(~)-47 b Fu(u)809 3475 y Fv(0)809 3528 y(2)p Fs(;T)941 3506 y Fz(and)27 b Fu(u)1150 3466 y Fv(mean)o Fs(;T)1150 3528 y Fv(1)1403 3506 y Fz(giv)n(e)g(an)g(other)g(expression)f(for)i (the)g(scalar)d(pro)r(duct)j(:)498 3659 y Fp(\012)537 3727 y Fu(u)585 3692 y Fv(0)585 3747 y(2)p Fs(;T)690 3727 y Fu(;)14 b(u)775 3692 y Fv(0)775 3747 y(1)p Fs(;T)879 3659 y Fp(\013)918 3762 y Fs(L)964 3746 y Fr(2)996 3762 y Fv(\(0)p Fs(;T)9 b Fv(\))1176 3727 y Fz(=)23 b Fu(a)1308 3739 y Fv(1)1364 3727 y Fz(+)1447 3634 y Fp(D)1497 3727 y Fu(a)1541 3739 y Fv(1)1578 3727 y Fu(u)1626 3687 y Fv(mean)o Fs(;T)1626 3749 y Fv(1)1870 3727 y Fz(+)h(~)-48 b Fu(u)2001 3692 y Fv(0)2001 3747 y(2)p Fs(;T)2106 3727 y Fu(;)14 b(u)2191 3692 y Fv(0)2191 3747 y(1)p Fs(;T)2314 3727 y Fx(\000)k Fu(u)2445 3687 y Fv(mean)n Fs(;T)2445 3749 y Fv(1)2670 3634 y Fp(E)2721 3785 y Fs(L)2767 3768 y Fr(2)2799 3785 y Fv(\(0)p Fs(;T)9 b Fv(\))2970 3727 y Fu(:)58 b Fz(\(10.13\))440 3970 y(Relation)35 b(\(9.1\))f(sho)n(ws)g (that)h Fx(jj)p Fu(u)1513 3940 y Fv(0)1513 3993 y(1)p Fs(;T)1642 3970 y Fx(\000)22 b Fu(u)1777 3930 y Fv(mean)o Fs(;T)1777 3992 y Fv(1)2003 3970 y Fx(jj)2049 3985 y Fs(L)2095 3968 y Fr(2)2127 3985 y Fv(\(0)p Fs(;T)9 b Fv(\))2319 3970 y Fx(\024)2438 3949 y Fz(~)2419 3970 y Fu(C)d(\017)p Fz(.)59 b(F)-7 b(urthermore,)36 b(Rela-)440 4108 y(tions)25 b(\(10.10\))f(b)r(ound)i(from)f(ab)r(o)n(v)n(e)1600 4013 y Fp(\014)1600 4062 y(\014)1600 4112 y(\014)1628 4041 y(\012)1667 4108 y Fu(u)1715 4078 y Fv(1)1715 4131 y(2)p Fs(;T)1820 4108 y Fu(;)14 b(u)1905 4120 y Fv(1)p Fs(;T)2010 4041 y Fp(\013)2049 4146 y Fs(L)2095 4130 y Fr(2)2127 4146 y Fv(\(\012)2200 4154 y Fe(T)2246 4146 y Fv(\))2276 4013 y Fp(\014)2276 4062 y(\014)2276 4112 y(\014)2329 4108 y Fz(and)2488 4013 y Fp(\014)2488 4062 y(\014)2488 4112 y(\014)2516 4041 y(\012)2555 4108 y Fu(u)2603 4120 y Fv(2)p Fs(;T)2708 4108 y Fu(;)g(u)2793 4078 y Fv(1)2793 4131 y(1)p Fs(;T)2897 4041 y Fp(\013)2936 4146 y Fs(L)2982 4130 y Fr(2)3014 4146 y Fv(\(\012)3087 4154 y Fe(T)3134 4146 y Fv(\))3164 4013 y Fp(\014)3164 4062 y(\014)3164 4112 y(\014)3217 4108 y Fz(b)n(y)440 4266 y Fu(C)505 4236 y Ft(0)529 4206 y Fx(p)p 598 4206 34 4 v 60 x Fu(\017)631 4206 y Fx(p)p 701 4206 54 4 v 701 4266 a Fu(\013)31 b Fz(and)f(also)1119 4170 y Fp(\014)1119 4220 y(\014)1119 4270 y(\014)1147 4199 y(\012)1186 4266 y Fu(u)1234 4236 y Fv(1)1234 4289 y(2)p Fs(;T)1339 4266 y Fu(;)14 b(u)1424 4236 y Fv(1)1424 4289 y(1)p Fs(;T)1528 4199 y Fp(\013)1567 4304 y Fs(L)1613 4287 y Fr(2)1645 4304 y Fv(\(\012)1718 4312 y Fe(T)1765 4304 y Fv(\))1795 4170 y Fp(\014)1795 4220 y(\014)1795 4270 y(\014)1853 4266 y Fz(b)n(y)31 b Fu(C)2037 4236 y Ft(0)p Fv(2)2093 4266 y Fu(\017\013)p Fz(.)47 b(W)-7 b(e)31 b(rep)r(ort)f(these)h (estimates)f(in)440 4390 y(\(10.12\))c(and)i(\(10.13\))e(kno)n(wing)h (that)h Fx(j)p Fu(a)1739 4402 y Fv(1)1776 4390 y Fx(j)23 b(\024)f Fz(1,)28 b(then)g(:)1205 4573 y Fx(j)p Fu(a)1272 4585 y Fv(1)1309 4573 y Fx(j)23 b(\024)g Fz(2)1503 4552 y(~)1485 4573 y Fu(C)5 b(\017)19 b Fz(+)f(2)p Fu(C)1792 4539 y Ft(0)1815 4509 y Fx(p)p 1884 4509 34 4 v 64 x Fu(\017)1918 4509 y Fx(p)p 1987 4509 54 4 v 64 x Fu(\013)h Fz(+)f Fu(C)2207 4539 y Ft(0)p Fv(2)2263 4573 y Fu(\017\013:)678 b Fz(\(10.14\))440 4756 y(So)27 b(there)h(exists)f Fu(\013)1050 4768 y Fv(0)1110 4756 y Fu(>)c Fz(0)k(and)h Fu(C)1488 4768 y Fv(1)1553 4756 y Fz(suc)n(h)f(that)h(for)f Fu(\013)d Fx(\024)e Fu(\013)2264 4768 y Fv(0)2329 4756 y Fz(:)1617 4938 y Fx(j)p Fu(a)1684 4950 y Fv(1)1721 4938 y Fx(j)83 b(\024)f Fu(C)2033 4950 y Fv(1)2071 4938 y Fu(\017:)923 b Fz(\(10.15\))440 5121 y(T)-7 b(aking)27 b(again)f(\(10.11\))h(and)g (\(10.15\),)f(there)i(exists)2146 5100 y(^)2127 5121 y Fu(C)34 b Fz(with)28 b(:)861 5304 y(1)18 b Fx(\000)1023 5283 y Fz(^)1004 5304 y Fu(C)1069 5269 y Fv(2)1107 5304 y Fu(\017)1141 5269 y Fv(2)1201 5304 y Fx(\024)23 b(jj)5 b Fz(~)-47 b Fu(u)1383 5269 y Fv(0)1383 5324 y(2)p Fs(;T)1487 5304 y Fx(jj)1533 5269 y Fv(2)1533 5326 y Fs(L)1579 5310 y Fr(2)1611 5326 y Fv(\(0)p Fs(;T)9 b Fv(\))1791 5304 y Fz(=)23 b Fx(jj)p Fu(u)1973 5269 y Fv(0)1973 5324 y(2)p Fs(;T)2078 5304 y Fx(jj)2124 5269 y Fv(2)2124 5326 y Fs(L)2170 5310 y Fr(2)2202 5326 y Fv(\(0)p Fs(;T)9 b Fv(\))2377 5304 y Fx(\000)18 b Fu(a)2504 5269 y Fv(2)2504 5324 y(1)2565 5304 y Fx(\024)k Fz(1)p Fu(:)334 b Fz(\(10.16\))1831 5662 y(29)p eop %%Page: 30 30 30 29 bop 440 531 a Fz(As)28 b(w)n(e)f(made)g(for)1033 464 y Fp(\012)1073 531 y Fu(L)1130 501 y Fv(mean)n Fs(;T)1355 531 y Fu(u)1403 501 y Fv(0)1403 554 y Fs(T)1455 531 y Fu(;)14 b(u)1540 501 y Fv(0)1540 554 y Fs(T)1591 464 y Fp(\013)1631 565 y Fs(L)1677 548 y Fr(2)1709 565 y Fv(\(0)p Fs(;T)9 b Fv(\))1866 531 y Fz(,)27 b(w)n(e)h(b)r(ound)2295 464 y Fp(\012)2334 531 y Fu(L)2391 501 y Fv(mean)n Fs(;T)2616 531 y Fu(u)2664 501 y Fv(0)2664 554 y(2)p Fs(;T)2769 531 y Fu(;)14 b(u)2854 501 y Fv(0)2854 554 y(2)p Fs(;T)2958 464 y Fp(\013)2998 570 y Fs(L)3044 553 y Fr(2)3075 570 y Fv(\(0)p Fs(;T)9 b Fv(\))3260 531 y Fz(:)624 724 y Fp(\012)663 791 y Fu(L)720 757 y Fv(mean)n Fs(;T)945 791 y Fu(u)993 757 y Fv(0)993 812 y(2)p Fs(;T)1098 791 y Fu(;)14 b(u)1183 757 y Fv(0)1183 812 y(2)p Fs(;T)1288 724 y Fp(\013)1327 827 y Fs(L)1373 810 y Fr(2)1405 827 y Fv(\(0)p Fs(;T)9 b Fv(\))1585 791 y Fx(\024)22 b Fu(\027)1713 803 y Fv(2)1764 674 y Fp(\022)1826 791 y Fu(T)7 b(;)1982 735 y Fz(1)p 1928 772 151 4 v 1928 848 a Fu(\013)1981 824 y Fv(2)2019 848 y Fu(T)2089 674 y Fp(\023)2163 791 y Fx(jj)p Fu(u)2257 757 y Fv(0)2257 812 y(2)p Fs(;T)2362 791 y Fx(jj)2408 757 y Fv(2)2408 814 y Fs(L)2454 797 y Fr(2)2486 814 y Fv(\(0)p Fs(;T)i Fv(\))2662 791 y Fz(+)18 b Fu(C)6 b(\017;)184 b Fz(\(10.17\))440 1014 y(and)27 b(also)g(:)604 1125 y Fp(\012)643 1192 y Fu(L)700 1158 y Fv(mean)o Fs(;T)926 1192 y Fu(u)974 1158 y Fv(0)974 1213 y(2)p Fs(;T)1078 1192 y Fu(;)14 b(u)1163 1158 y Fv(0)1163 1213 y(2)p Fs(;T)1268 1125 y Fp(\013)1307 1228 y Fs(L)1353 1211 y Fr(2)1385 1228 y Fv(\(0)p Fs(;T)9 b Fv(\))1565 1192 y Fx(\025)23 b Fu(a)1697 1158 y Fv(2)1697 1213 y(1)1734 1192 y Fu(\025)1782 1152 y Fv(mean)o Fs(;T)1782 1214 y Fv(1)2026 1192 y Fz(+)18 b Fu(\025)2157 1152 y Fv(mean)p Fs(;T)2157 1214 y Fv(2)2384 1192 y Fx(jj)5 b Fz(~)-47 b Fu(u)2478 1158 y Fv(0)2478 1213 y(2)p Fs(;T)2582 1192 y Fx(jj)2628 1158 y Fv(2)2628 1215 y Fs(L)2674 1198 y Fr(2)2707 1215 y Fv(\(0)p Fs(;T)9 b Fv(\))2864 1192 y Fu(:)164 b Fz(\(10.18\))440 1391 y(W)-7 b(e)20 b(lo)r(ok)f(at)h (Relations)f(\(10.17\),)h(\(10.18\);)h(w)n(e)e(use)h(\(10.11\),)g (\(10.15\),)g(\(10.16\))f(and)g(\(10.6\),)440 1491 y(then)28 b(:)1318 1698 y Fu(\027)1359 1710 y Fv(2)1410 1581 y Fp(\022)1471 1698 y Fu(T)7 b(;)1628 1642 y Fz(1)p 1573 1679 V 1573 1755 a Fu(\013)1626 1731 y Fv(2)1664 1755 y Fu(T)1734 1581 y Fp(\023)1878 1698 y Fx(\025)83 b Fu(\025)2074 1664 y Fv(mean)2074 1718 y(2)2251 1698 y Fx(\000)18 b Fu(c\017:)624 b Fz(\(10.19\))3240 1919 y Fd(\003)440 2151 y Fl(10.4)112 b(Comparison)37 b(b)s(et)m(w)m(een)h Fk(\025)1869 2166 y Fj(2)1908 2151 y FF(\()p Fk(\013)q FF(\))f Fl(and)i Fk(\025)2358 2115 y Fi(T)2358 2176 y Fj(2)2413 2151 y FF(\()p Fk(\013)q FF(\))440 2333 y FA(Lemma)c(10.3.)42 b Fo(F)-6 b(or)34 b(every)h Fu(\017)30 b Fx(2)1556 2241 y Fp(i)1595 2333 y Fz(0)p Fu(;)1711 2301 y Fv(2)p 1684 2315 88 4 v 1684 2323 a Ft(p)p 1738 2323 34 3 v 1738 2371 a Fv(3)1781 2241 y Fp(h)1821 2333 y Fo(,)35 b(ther)l(e)f(exists)f Fu(\013)2376 2345 y Fv(0)2413 2333 y Fo(,)j Fu(\016)d(>)d Fz(0)j Fo(and)h Fu(C)j(>)30 b Fz(0)j Fo(such)440 2458 y(that)d(for)g Fu(\013)24 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)1029 2470 y Fv(0)1066 2458 y Fz([)30 b Fo(and)g Fu(T)k Fz(=)1468 2425 y Fs(\017)p 1461 2439 44 4 v 1461 2486 a(\013)1544 2458 y Fo(:)1358 2658 y Fz(0)23 b Fx(\024)g Fu(\025)1559 2624 y Fs(T)1559 2679 y Fv(2)1611 2658 y Fz(\()p Fu(\013)p Fz(\))d Fx(\000)e Fu(\025)1879 2670 y Fv(2)1916 2658 y Fz(\()p Fu(\013)p Fz(\))24 b Fx(\024)f Fu(C)6 b(e)2249 2624 y Ft(\000)2316 2602 y Fe(\016)p 2311 2611 38 3 v 2311 2644 a(\013)2363 2658 y Fu(:)665 b Fz(\(10.20\))440 2836 y Fo(Pr)l(o)l(of)23 b(:)35 b Fz(As)19 b(for)g(the)h(\014rst)f (eigen)n(v)-5 b(alue,)20 b(w)n(e)f(sho)n(w)f(that)i Fu(\025)2190 2848 y Fv(2)2227 2836 y Fz(\()p Fu(\013)p Fz(\))h(is)e(closed)g(to)g Fu(\025)2818 2806 y Fs(T)2818 2857 y Fv(2)2870 2836 y Fz(\()p Fu(\013)p Fz(\))i(b)n(y)e(using)440 2936 y(sp)r(ectal)j (theorem.)34 b(W)-7 b(e)23 b(de\014ne)f Fu(v)1482 2906 y Fs(T)1479 2956 y(\013)1558 2936 y Fz(=)g Fu(\037)1697 2948 y Fs(T)1749 2936 y Fu(u)1797 2948 y Fv(2)p Fs(;\013)1919 2936 y Fz(and)g(estimate)g Fx(jj)p Fz(\()p Fu(P)2545 2906 y Fs(T)2533 2956 y(\013)2605 2936 y Fx(\000)7 b Fu(\025)2725 2948 y Fv(2)2762 2936 y Fz(\()p Fu(\013)p Fz(\)\))p Fu(v)2954 2906 y Fs(T)2951 2956 y(\013)3008 2936 y Fx(jj)3054 2951 y Fs(L)3100 2934 y Fr(2)3132 2951 y Fv(\(\012)3205 2959 y Fe(T)3251 2951 y Fv(\))3281 2936 y Fz(.)440 3036 y(F)-7 b(or)27 b(\()p Fu(t;)14 b(\021)s Fz(\))24 b Fx(2)f Fz(\012)926 3048 y Fs(T)1006 3036 y Fz(:)749 3213 y Fu(P)814 3179 y Fs(T)802 3234 y(\013)867 3213 y Fu(v)910 3179 y Fs(T)907 3234 y(\013)962 3213 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))24 b(=)e Fu(P)1301 3225 y Fs(\013)1349 3213 y Fu(v)1392 3179 y Fs(T)1389 3234 y(\013)1445 3213 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))23 b(=)g Fu(\037)1783 3225 y Fs(T)1835 3213 y Fu(P)1888 3225 y Fs(\013)1936 3213 y Fu(u)1984 3225 y Fv(2)p Fs(;\013)2083 3213 y Fz(\()p Fu(t;)14 b(\021)s Fz(\))20 b(+)e([)p Fu(P)2437 3225 y Fs(\013)2484 3213 y Fu(;)c(\037)2573 3225 y Fs(T)2625 3213 y Fz(])p Fu(u)2696 3225 y Fv(2)p Fs(;\013)2796 3213 y Fz(\()p Fu(t;)g(\021)s Fz(\))p Fu(:)440 3391 y Fz(By)24 b(assumption,)g(there)f(exists)h Fu(C)30 b Fz(suc)n(h)23 b(that)h Fx(j)p Fu(D)2000 3403 y Fs(t)2029 3391 y Fu(\037)2081 3403 y Fs(T)2134 3391 y Fx(j)f(\024)2292 3359 y Fs(C)p 2277 3373 81 4 v 2277 3420 a(T)2325 3403 y Fr(2)2392 3391 y Fz(and)h Fx(j)p Fu(D)2644 3361 y Fv(2)2642 3412 y Fs(t)2681 3391 y Fu(\037)2733 3403 y Fs(T)2785 3391 y Fx(j)f(\024)2943 3359 y Fs(C)p 2929 3373 V 2929 3420 a(T)2977 3403 y Fr(4)3019 3391 y Fz(.)36 b(Corol-)440 3491 y(lary)d(7.4)h(pro)n(v)n(es)e(the)j(existence)f(of)g Fu(\016)j(>)d Fz(0)f(and)h Fu(\013)2099 3503 y Fv(0)2171 3491 y Fu(>)g Fz(0)f(suc)n(h)h(that)h(for)f Fu(\013)g Fx(\024)g Fu(\013)3099 3503 y Fv(0)3171 3491 y Fz(and)440 3591 y Fu(T)g Fz(=)628 3558 y Fs(\017)p 621 3572 44 4 v 621 3619 a(\013)702 3591 y Fz(:)1330 3713 y Fx(jj)p Fu(u)1424 3725 y Fv(2)p Fs(;\013)1524 3713 y Fx(jj)1570 3735 y Fs(L)1616 3718 y Fr(2)1648 3735 y Fv(\(])1702 3713 y Fe(T)p 1702 3722 42 3 v 1709 3755 a Fr(2)1754 3735 y Fs(;T)9 b Fv([)p Ft(\002)p Fv(])p Ft(\000)1973 3713 y Fr(1)p 1973 3722 29 3 v 1973 3755 a(2)2011 3735 y Fs(;)2041 3713 y Fr(1)p 2041 3722 V 2041 3755 a(2)2079 3735 y Fv([\))2151 3713 y Fx(\024)23 b Fu(e)2278 3679 y Ft(\000)2344 3656 y Fe(\016)p 2339 3665 38 3 v 2339 3699 a(\013)2391 3713 y Fu(:)637 b Fz(\(10.21\))440 3873 y(So,)23 b(with)f(an)f(in)n(tegration)g(b)n(y)g(parts)g(for)h(the)g (term)g Fx(jj)p Fu(tD)2196 3885 y Fs(t)2225 3873 y Fu(u)2273 3885 y Fv(2)p Fs(;\013)2372 3873 y Fu(D)2441 3885 y Fs(t)2470 3873 y Fu(\037)2522 3885 y Fs(T)2574 3873 y Fx(jj)2620 3888 y Fs(L)2666 3871 y Fr(2)2699 3888 y Fv(\(\012)2772 3896 y Fe(T)2818 3888 y Fv(\))2848 3873 y Fz(,)h(w)n(e)e(deduce)h(:) 1245 4074 y Fx(jj)p Fz(\()p Fu(P)1388 4040 y Fs(T)1376 4095 y(\013)1459 4074 y Fx(\000)c Fu(\025)1590 4086 y Fv(2)1628 4074 y Fz(\()p Fu(\013)p Fz(\)\))p Fu(v)1820 4040 y Fs(T)1817 4095 y(\013)1874 4074 y Fx(jj)1920 4089 y Fs(L)1966 4072 y Fr(2)1998 4089 y Fv(\(\012)2071 4097 y Fe(T)2117 4089 y Fv(\))2170 4074 y Fx(\024)23 b Fu(C)6 b(e)2362 4040 y Ft(\000)2428 4017 y Fe(\016)p 2424 4026 V 2424 4060 a(\013)2476 4074 y Fu(:)552 b Fz(\(10.22\))440 4252 y(With)33 b Fx(jj)p Fu(v)748 4222 y Fs(T)745 4272 y(\013)801 4252 y Fx(jj)847 4222 y Fv(2)847 4279 y Fs(L)893 4262 y Fr(2)925 4279 y Fv(\(\012)998 4287 y Fe(T)1044 4279 y Fv(\))1105 4252 y Fx(\025)d(jj)p Fu(u)1294 4222 y Fs(T)1294 4272 y Fv(2)p Fs(;\013)1394 4252 y Fx(jj)1440 4222 y Fv(2)1440 4279 y Fs(L)1486 4262 y Fr(2)1518 4279 y Fv(\(\012)1601 4287 y Fe(T)p 1601 4294 42 3 v 1608 4325 a Fr(2)1657 4279 y Fv(\))1718 4252 y Fz(=)g(1)21 b Fx(\000)g(jj)p Fu(u)2056 4222 y Fs(T)2056 4272 y Fv(2)p Fs(;\013)2156 4252 y Fx(jj)2202 4222 y Fv(2)2202 4286 y Fs(L)2248 4269 y Fr(2)2280 4286 y Fv(\(])2334 4263 y Fe(T)p 2334 4272 V 2341 4306 a Fr(2)2386 4286 y Fs(;)p Fv(+)p Ft(1)p Fv([)p Ft(\002)p Fv(])p Ft(\000)2674 4263 y Fr(1)p 2674 4272 29 3 v 2674 4306 a(2)2713 4286 y Fs(;)2742 4263 y Fr(1)p 2742 4272 V 2742 4306 a(2)2780 4286 y Fv([\))2862 4252 y Fz(and)32 b(\(10.21\),)440 4391 y(there)27 b(is)h Fu(c)g Fz(suc)n(h)f(that)h(:)1372 4569 y Fx(jj)p Fu(v)1461 4535 y Fs(T)1458 4589 y(\013)1514 4569 y Fx(jj)1560 4584 y Fs(L)1606 4567 y Fr(2)1638 4584 y Fv(\(\012)1711 4592 y Fe(T)1757 4584 y Fv(\))1870 4569 y Fx(\025)83 b Fz(1)18 b Fx(\000)g Fu(ce)2236 4535 y Ft(\000)2302 4512 y Fe(\016)p 2297 4521 38 3 v 2297 4555 a(\013)2349 4569 y Fu(:)679 b Fz(\(10.23\))440 4747 y(W)-7 b(e)36 b(apply)f(the)h(sp)r(ectral)f (Theorem)g(and)g(Relations)g(\(10.22\))f(and)i(\(10.23\).)59 b(So)36 b(there)440 4846 y(exists)27 b Fu(\013)722 4858 y Fv(1)783 4846 y Fx(\024)22 b Fu(\013)923 4858 y Fv(0)961 4846 y Fz(,)29 b(~)-44 b Fu(c)28 b Fz(and)f(for)g(all)h Fu(T)34 b Fz(=)1667 4814 y Fs(\017)p 1660 4828 44 4 v 1660 4875 a(\013)1713 4846 y Fz(,)27 b Fu(\025)1811 4816 y Fs(T)1811 4870 y(k)1864 4846 y Fz(\()p Fu(\013)p Fz(\))d Fx(2)g Fu(\033)s Fz(\()p Fu(P)2231 4816 y Fs(T)2219 4867 y(\013)2283 4846 y Fz(\))k(suc)n(h)g(that)f(for)h Fu(\013)23 b Fx(\024)g Fu(\013)3055 4858 y Fv(1)3120 4846 y Fz(:)1366 5047 y Fx(j)p Fu(\025)1437 5013 y Fs(T)1437 5068 y(k)1490 5047 y Fz(\()p Fu(\013)p Fz(\))c Fx(\000)f Fu(\025)1757 5059 y Fv(2)1795 5047 y Fz(\()p Fu(\013)p Fz(\))p Fx(j)84 b(\024)h Fz(~)-44 b Fu(ce)2242 5013 y Ft(\000)2308 4990 y Fe(\016)p 2303 4999 38 3 v 2303 5033 a(\013)2355 5047 y Fu(:)673 b Fz(\(10.24\))440 5225 y(W)-7 b(e)26 b(sho)n(w)f(that)h Fu(k)g Fx(\025)d Fz(2)i(b)n(y)g(con)n(tradiction.)35 b(W)-7 b(e)26 b(assume)f Fu(k)h Fz(=)d(1.)36 b(It)26 b(is)f(easy)g(to)h(see)f(that)h(:)1361 5413 y Fu(\025)1409 5425 y Fv(2)1447 5413 y Fz(\()p Fu(\013)p Fz(\))19 b Fx(\000)g Fu(\025)1715 5373 y Fv(mean)o Fs(;T)1715 5435 y Fv(1)2024 5413 y Fx(\024)84 b Fz(~)-44 b Fu(ce)2246 5379 y Ft(\000)2313 5356 y Fe(\016)p 2308 5365 V 2308 5399 a(\013)2360 5413 y Fu(:)1831 5662 y Fz(30)p eop %%Page: 31 31 31 30 bop 440 531 a Fz(But,)28 b(due)g(to)f(\(10.8\),)g(w)n(e)g(deduce) h(:)1202 757 y Fu(\027)1243 769 y Fv(2)1294 640 y Fp(\022)1355 757 y Fu(T)7 b(;)1512 701 y Fz(1)p 1457 738 151 4 v 1457 814 a Fu(\013)1510 790 y Fv(2)1548 814 y Fu(T)1618 640 y Fp(\023)1697 757 y Fx(\000)18 b Fu(\025)1828 717 y Fv(mean)p Fs(;T)1828 779 y Fv(1)2078 757 y Fx(\024)24 b Fz(~)-44 b Fu(ce)2240 722 y Ft(\000)2306 700 y Fe(\016)p 2302 709 38 3 v 2302 742 a(\013)2354 757 y Fu(:)440 982 y Fz(W)-7 b(e)28 b(apply)f(Lemmas)g(10.2)g(and)g(\(8.2\))g(to)h (establish)f(the)h(upp)r(er)g(b)r(ound)g(:)604 1180 y Fu(\025)652 1146 y Fv(mean)652 1201 y(2)829 1180 y Fx(\000)18 b Fu(\025)960 1146 y Fv(mean)960 1201 y(1)1141 1180 y Fx(\024)25 b Fz(~)-44 b Fu(ce)1304 1146 y Ft(\000)1370 1124 y Fe(\016)p 1365 1133 V 1365 1166 a(\013)1436 1180 y Fz(+)18 b Fu(C)6 b(e)1623 1138 y Ft(\000)1715 1116 y Fe(T)p 1685 1125 103 3 v 1685 1166 a Fr(2)1713 1132 y Fq(p)p 1758 1132 29 3 v 1758 1166 a Fr(3)1819 1180 y Fz(+)18 b Fu(c\017)23 b Fz(=)h(~)-43 b Fu(ce)2158 1146 y Ft(\000)2224 1124 y Fe(\016)p 2219 1133 38 3 v 2219 1166 a(\013)2290 1180 y Fz(+)18 b Fu(C)6 b(e)2477 1138 y Ft(\000)2596 1116 y Fe(\017)p 2538 1125 140 3 v 2538 1166 a Fr(2)2566 1132 y Fq(p)p 2612 1132 29 3 v 34 x Fr(3)p Fe(\013)2711 1180 y Fz(+)18 b Fu(c\017:)164 b Fz(\(10.25\))440 1356 y(Since)31 b Fu(\025)708 1326 y Fv(mean)708 1377 y(2)888 1356 y Fx(\000)20 b Fu(\025)1021 1326 y Fv(mean)1021 1377 y(1)1208 1356 y Fz(=)1339 1323 y Fv(2)p 1312 1337 88 4 v 1312 1346 a Ft(p)p 1367 1346 34 3 v 48 x Fv(3)1410 1356 y Fz(,)32 b(\(10.25\))e(is)h(imp)r(ossible)g (as)f(so)r(on)h Fu(\017)e(<)2739 1323 y Fv(2)p 2712 1337 88 4 v 2712 1346 a Ft(p)p 2767 1346 34 3 v 48 x Fv(3)2810 1356 y Fz(.)47 b(Th)n(us)31 b Fu(k)h Fx(\025)d Fz(2)440 1479 y(and)e(then)i Fu(\025)839 1449 y Fs(T)839 1503 y(k)891 1479 y Fz(\()p Fu(\013)p Fz(\))24 b Fx(\025)f Fu(\025)1168 1449 y Fs(T)1168 1500 y Fv(2)1221 1479 y Fz(\()p Fu(\013)p Fz(\).)38 b(W)-7 b(e)28 b(deduce)f(\(10.20\))g(b)n(y) g(using)g(\(10.5\).)572 b Fd(\003)440 1710 y Fl(10.5)112 b(Pro)s(of)37 b(of)h(Prop)s(osition)e(10.1)440 1864 y Fz(Let)d Fu(\021)i(>)d Fz(0.)53 b(W)-7 b(e)33 b(apply)g(Lemma)g(10.2)f (with)h Fu(\017)f Fz(=)2130 1827 y Fs(\021)p 2117 1845 63 4 v 2117 1892 a Fv(2)p Fs(c)2189 1864 y Fz(.)53 b(Then,)35 b(recalling)d(\(10.6\),)i(there)440 1963 y(exists)27 b Fu(\013)722 1975 y Fv(0)787 1963 y Fz(suc)n(h)h(that)f(for)g Fu(\013)d Fx(2)p Fz(]0)p Fu(;)14 b(\013)1568 1975 y Fv(0)1605 1963 y Fz(])28 b(and)f Fu(T)34 b Fz(=)2006 1931 y Fs(\017)p 1998 1945 44 4 v 1998 1992 a(\013)2079 1963 y Fz(:)1381 2197 y Fu(\027)1422 2209 y Fv(2)1473 2080 y Fp(\022)1534 2197 y Fu(T)7 b(;)1691 2140 y Fz(1)p 1636 2178 151 4 v 1636 2254 a Fu(\013)1689 2230 y Fv(2)1727 2254 y Fu(T)1797 2080 y Fp(\023)1881 2197 y Fx(\025)23 b Fu(\025)2017 2162 y Fv(mean)2017 2217 y(2)2193 2197 y Fx(\000)2286 2140 y Fu(\021)p 2286 2178 45 4 v 2287 2254 a Fz(2)2341 2197 y Fu(:)687 b Fz(\(10.26\))440 2422 y(With)38 b(this)f(c)n(hoice)f (of)h Fu(\017)p Fz(,)i(Lemma)d(10.3)g(and)g(Relation)h(\(10.8\))f(giv)n (e)g(the)h(existence)f(of)440 2522 y Fu(\013)493 2534 y Fv(1)553 2522 y Fx(\024)23 b Fu(\013)694 2534 y Fv(0)759 2522 y Fz(suc)n(h)k(that)h(for)f Fu(\013)d Fx(2)p Fz(]0)p Fu(;)14 b(\013)1540 2534 y Fv(1)1577 2522 y Fz(])28 b(:)1322 2747 y Fu(\025)1370 2759 y Fv(2)1408 2747 y Fz(\()p Fu(\013)p Fz(\))84 b Fx(\025)e Fu(\027)1797 2759 y Fv(2)1848 2630 y Fp(\022)1910 2747 y Fu(T)7 b(;)2066 2691 y Fz(1)p 2012 2728 151 4 v 2012 2804 a Fu(\013)2065 2780 y Fv(2)2102 2804 y Fu(T)2172 2630 y Fp(\023)2252 2747 y Fx(\000)2345 2691 y Fu(\021)p 2345 2728 45 4 v 2346 2804 a Fz(2)2399 2747 y Fu(:)629 b Fz(\(10.27\))440 2973 y(W)-7 b(e)28 b(tak)n(e)f(the)h(lo)n(w)n(er)e(b)r(ounds)i(\(10.26\))e(and)h (\(10.27\),)g(then,)h(for)f Fu(\013)d Fx(2)p Fz(]0)p Fu(;)14 b(\013)2785 2985 y Fv(1)2822 2973 y Fz(])27 b(:)1527 3148 y Fu(\025)1575 3160 y Fv(2)1613 3148 y Fz(\()p Fu(\013)p Fz(\))d Fx(\025)f Fu(\025)1890 3114 y Fv(mean)1890 3169 y(2)2067 3148 y Fx(\000)18 b Fu(\021)s(:)834 b Fz(\(10.28\))440 3324 y(Since)28 b Fu(\025)p Fz(\()p Fu(\013)p Fz(\))c Fx(\024)f Fu(\025)982 3294 y Fv(mean)982 3344 y(1)1140 3324 y Fz(,)28 b(w)n(e)f(deduce)h(:)1207 3499 y Fu(\025)1255 3511 y Fv(2)1293 3499 y Fz(\()p Fu(\013)p Fz(\))19 b Fx(\000)f Fu(\025)p Fz(\()p Fu(\013)p Fz(\))25 b Fx(\025)d Fz(\()p Fu(\025)1869 3465 y Fv(mean)1869 3520 y(2)2047 3499 y Fx(\000)c Fu(\025)2178 3465 y Fv(mean)2178 3520 y(1)2336 3499 y Fz(\))h Fx(\000)f Fu(\021)s(:)514 b Fz(\(10.29\))3240 3675 y Fd(\003)440 3775 y Fz(W)-7 b(e)28 b(no)n(w)f(use)g(Prop)r (osition)f(10.1)h(to)g(justify)i(the)f(asymptotics)e(of)i Fu(\026)p Fz(\()p Fu(\013)p Fz(\).)440 4006 y Fl(10.6)112 b(Asymptotics)36 b(of)h Fk(\026)p FF(\()p Fk(\013)q FF(\))440 4159 y FA(Theorem)20 b(10.4.)32 b Fo(L)l(et)21 b Fu(n)g Fo(a)h(p)l(ositive)h(inte)l(ger.)36 b(We)22 b(c)l(onsider)g Fz(\()p Fu(m)2499 4171 y Fs(j)2535 4159 y Fz(\))2567 4171 y Fs(j)s Ft(\024)p Fs(n)2716 4159 y Fo(given)g(r)l(e)l(cursively) 440 4259 y(by)30 b(Pr)l(op)l(osition)i(5.1,)f(then)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\))g Fo(has)h(the)f(asymptotics)h(:)1020 4501 y Fu(\026)p Fz(\()p Fu(\013)p Fz(\))84 b(=)e Fu(\013)1525 4397 y Fs(n)1486 4422 y Fp(X)1488 4599 y Fs(j)s Fv(=0)1619 4501 y Fu(m)1692 4513 y Fs(j)1727 4501 y Fu(\013)1780 4466 y Fv(2)p Fs(j)1867 4501 y Fz(+)18 b Fx(O)2016 4513 y Fs(n)2061 4501 y Fz(\()p Fu(\013)2146 4466 y Fv(2)p Fs(n)p Fv(+3)2309 4501 y Fz(\))30 b Fo(as)h Fu(\013)23 b Fx(!)g Fz(0)p Fu(:)326 b Fz(\(10.30\))440 4776 y Fo(Pr)l(o)l(of)34 b(:)46 b Fz(W)-7 b(e)32 b(tak)n(e)f(again)f Fu(U)1359 4746 y Fv(\()p Fs(n)p Fv(\))1488 4776 y Fz(and)h Fu(\026)1703 4746 y Fv(\()p Fs(n)p Fv(\))1832 4776 y Fz(de\014ned)g(in)h(\(6.2\))f (and)h(\(6.1\).)48 b(According)30 b(to)440 4876 y(Prop)r(osition)c (6.1,)h(there)g(is)h Fu(\013)1393 4888 y Fv(0)1458 4876 y Fz(and)f(for)g Fu(\013)d Fx(\024)e Fu(\013)1963 4888 y Fv(0)2001 4876 y Fz(,)27 b Fu(\026)2101 4888 y Fs(N)2164 4876 y Fz(\()p Fu(\013)p Fz(\))d Fx(2)g Fu(\033)s Fz(\()p Fu(P)2519 4888 y Ft(A)2572 4896 y Fr(0)2606 4888 y Fs(;)p Fv(\012)2673 4896 y Fe(\013)2719 4876 y Fz(\))k(suc)n(h)f(that)h(:)1375 5062 y Fx(j)p Fu(\026)1448 5074 y Fs(N)1511 5062 y Fz(\()p Fu(\013)p Fz(\))20 b Fx(\000)e Fu(\013\026)1834 5028 y Fv(\()p Fs(n)p Fv(\))1931 5062 y Fx(j)23 b(\024)g Fu(C)6 b(\013)2183 5028 y Fv(2)p Fs(n)p Fv(+1)2346 5062 y Fu(:)682 b Fz(\(10.31\))440 5237 y(Assume)28 b(that)g Fu(N)j Fx(\025)23 b Fz(2,)k(then)h(:)1225 5413 y Fu(\026)1275 5425 y Fs(N)1338 5413 y Fz(\()p Fu(\013)p Fz(\))20 b Fx(\000)e Fu(\013\026)1661 5379 y Fv(\()p Fs(n)p Fv(\))1841 5413 y Fx(\025)83 b Fu(\026)2039 5425 y Fv(2)2076 5413 y Fz(\()p Fu(\013)p Fz(\))19 b Fx(\000)g Fu(\013\026)2399 5379 y Fv(\()p Fs(n)p Fv(\))2496 5413 y Fu(:)532 b Fz(\(10.32\))1831 5662 y(31)p eop %%Page: 32 32 32 31 bop 440 531 a Fz(W)-7 b(e)26 b(c)n(ho)r(ose)d Fu(\017)g Fx(\024)1023 499 y Fv(1)p 996 513 88 4 v 996 521 a Ft(p)p 1050 521 34 3 v 1050 569 a Fv(3)1093 531 y Fz(,)j(then)g(due)f(to)g (Prop)r(osition)f(10.1)f(and)i(construction)g(of)g Fu(\026)2974 501 y Fv(\()p Fs(n)p Fv(\))3071 531 y Fz(,)h(there)440 647 y(exists)h Fu(\013)722 659 y Fv(1)783 647 y Fx(\024)22 b Fu(\013)923 659 y Fv(0)988 647 y Fz(suc)n(h)28 b(that)f(for)h Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b(\013)1769 659 y Fv(1)1806 647 y Fz(])28 b(:)1009 851 y Fu(\026)1059 863 y Fv(2)1096 851 y Fz(\()p Fu(\013)p Fz(\))c Fx(\025)e Fu(\013\025)1425 811 y Fv(mo)n(y)1425 873 y(2)1571 851 y Fx(\000)c Fu(\013)1721 795 y(\017)p 1717 832 42 4 v 1717 908 a Fz(2)1768 851 y Fu(;)167 b(\013\026)2061 817 y Fv(\()p Fs(n)p Fv(\))2181 851 y Fx(\024)23 b Fu(\013\025)2370 811 y Fv(mo)n(y)2370 873 y(1)2515 851 y Fx(\000)18 b Fu(\013)2665 795 y(\017)p 2661 832 V 2661 908 a Fz(2)2713 851 y Fu(:)315 b Fz(\(10.33\))440 1069 y(W)-7 b(e)28 b(rep)r(ort)f(\(10.33\))f(in)i(\(10.32\),)e(then,)j(since)e Fu(\025)1981 1039 y Fv(mean)1981 1090 y(2)2158 1069 y Fx(\000)18 b Fu(\025)2289 1039 y Fv(mean)2289 1090 y(1)2470 1069 y Fz(=)2595 1037 y Fv(2)p 2568 1051 88 4 v 2568 1059 a Ft(p)p 2623 1059 34 3 v 48 x Fv(3)2693 1069 y Fz(:)948 1298 y Fu(\026)998 1310 y Fs(N)1061 1298 y Fz(\()p Fu(\013)p Fz(\))h Fx(\000)f Fu(\013\026)1383 1264 y Fv(\()p Fs(n)p Fv(\))1564 1298 y Fx(\025)82 b Fu(\013)p Fz(\()p Fu(\025)1844 1264 y Fv(mean)1844 1318 y(2)2022 1298 y Fx(\000)18 b Fu(\025)2153 1264 y Fv(mean)2153 1318 y(1)2311 1298 y Fz(\))h Fx(\000)f Fu(\013\017)23 b Fx(\025)2681 1242 y Fu(\013)p 2653 1279 111 4 v 2653 1295 a Fx(p)p 2722 1295 42 4 v 69 x Fz(3)2773 1298 y Fu(:)255 b Fz(\(10.34\))440 1529 y(This)28 b(is)f(imp)r(ossible)h(with)g(\(10.31\),)e(so)h Fu(N)32 b Fz(=)22 b(1)28 b(and)f Fu(\026)2180 1541 y Fs(N)2243 1529 y Fz(\()p Fu(\013)p Fz(\))d(=)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\).)578 b Fd(\003)440 1803 y Fy(11)135 b(Estimates)47 b(in)e(the)g(semi-classical)i(case)440 2002 y Fl(11.1)112 b(Lo)s(calization)36 b(tec)m(hniques)440 2155 y FA(11.1.1)93 b(Lo)s(calization)32 b(with)g(a)g(partition)g(of)f (unit)m(y)440 2308 y Fz(As)f(in)g([11],)g(p.)h(617-621,)c(w)n(e)j(can)g (giv)n(e)f(a)g(lo)n(w)n(er)g(b)r(ound)h(and)g(an)g(upp)r(er)g(b)r(ound) g(for)g(the)440 2408 y(fundamen)n(tal)35 b(state)f(of)h(the)g(Sc)n (hr\177)-42 b(odinger)33 b(op)r(erator)g(with)j(a)e(non)h(constan)n(t)f (magnetic)440 2507 y(\014eld)28 b(and)f(a)h(b)r(ounded)g(op)r(en)f(set) h(of)f Fw(R)1668 2477 y Fv(2)1739 2507 y Fz(whose)g(b)r(oundary)g(is)g (a)h(curvilinear)e(p)r(olygon.)440 2607 y(Our)19 b(goal)g(is)g(no)n(w)h (to)f(pro)n(v)n(e)f(Theorem)h(1.2.)34 b(First,)21 b(w)n(e)f(giv)n(e)f (a)g(lo)n(w)n(er)f(b)r(ound)i(and)g(secondly)440 2707 y(an)30 b(upp)r(er)g(b)r(ound.)44 b(The)30 b(partition)g(of)g(unit)n(y) g(pla)n(ys)f(an)g(imp)r(ortan)n(t)h(role)f(and)h(w)n(e)f(recall)440 2806 y(its)f(construction.)440 2906 y(The)34 b(idea)g(of)g(the)h(lo)r (calization)e(to)h(compare)f(the)i(mo)r(del)f(cases)f Fw(R)2616 2876 y Fv(2)2659 2906 y Fz(,)j Fw(R)25 b Fx(\002)18 b Fw(R)2934 2876 y Fv(+)3029 2906 y Fz(and)34 b(\012)3257 2918 y Fs(\013)440 3006 y Fz(comes)27 b(from)g(the)h(follo)n(wing)f (prop)r(osition)f(:)440 3172 y FA(Prop)s(osition)34 b(11.1.)41 b Fo(L)l(et)32 b Fz(0)27 b Fx(\024)h Fu(\032)g Fx(\024)f Fz(1)p Fo(.)46 b(Ther)l(e)34 b(exists)e(a)g(c)l(onstant)g(C)h(and)g(a)f (p)l(artition)440 3271 y(of)f(unity)e Fu(\037)804 3241 y Fs(h)804 3293 y(j)877 3271 y Fo(of)h Fz(\012)g Fo(satisfying)h(:)736 3491 y Fu(\037)788 3457 y Fs(h)788 3511 y(j)831 3491 y Fz(\()p Fu(x)p Fz(\))84 b(=)f Fu(\037)1226 3503 y Fs(j)1274 3399 y Fp(\020)1354 3435 y Fu(x)p 1334 3472 87 4 v 1334 3548 a(h)1382 3524 y Fs(\032)1430 3399 y Fp(\021)1494 3491 y Fu(;)43 b Fo(with)31 b Fu(\037)1793 3503 y Fs(j)1857 3491 y Fo(a)g(p)l(artition)f(of)h(unity)e(of)i Fw(R)2734 3457 y Fv(2)2777 3491 y Fu(;)631 3589 y Fp(X)676 3766 y Fs(j)765 3668 y Fx(j)p Fu(\037)840 3634 y Fs(h)840 3689 y(j)883 3668 y Fx(j)906 3634 y Fv(2)1026 3668 y Fz(=)83 b(1)p Fu(;)1853 b Fz(\(11.1\))562 3827 y Fp(X)606 4004 y Fs(j)696 3906 y Fx(jr)p Fu(\037)840 3872 y Fs(h)840 3927 y(j)883 3906 y Fx(j)906 3872 y Fv(2)1026 3906 y Fx(\024)83 b Fu(C)6 b(h)1287 3872 y Ft(\000)p Fv(2)p Fs(\032)1410 3906 y Fu(;)1659 b Fz(\(11.2\))612 4281 y(supp)q(\()p Fu(\037)868 4246 y Fs(h)868 4301 y(j)911 4281 y Fz(\))83 b Fx(\032)g(B)s Fz(\()p Fu(z)1303 4293 y Fs(j)1337 4281 y Fu(;)14 b(h)1422 4246 y Fs(\032)1460 4281 y Fz(\))30 b(s)p Fu(:)p Fz(t)p Fu(:)1677 4061 y Fp(8)1677 4135 y(>)1677 4160 y(>)1677 4185 y(<)1677 4335 y(>)1677 4360 y(>)1677 4385 y(:)1792 4127 y Fz(either)43 b(supp\()p Fu(\037)2298 4097 y Fs(h)2298 4149 y(j)2342 4127 y Fz(\))18 b Fx(\\)h Fu(@)5 b Fz(\012)23 b(=)g Fx(;)p Fu(;)1792 4230 y Fz(either)29 b Fu(z)2068 4242 y Fs(j)2126 4230 y Fx(2)24 b Fu(@)5 b Fz(\012)29 b(and)1816 4330 y(supp\()p Fu(\037)2071 4300 y Fs(h)2071 4352 y(j)2114 4330 y Fz(\))19 b Fx(\\)g(f)p Fu(S)2332 4342 y Fs(k)2372 4330 y Fu(;)14 b(k)26 b Fz(=)c(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fx(g)23 b Fz(=)f Fx(;)p Fu(;)1792 4433 y Fz(or)29 b Fu(z)1935 4445 y Fs(j)1992 4433 y Fz(=)23 b Fu(S)2131 4445 y Fs(j)2166 4433 y Fu(:)3092 4281 y Fz(\(11.3\))440 4617 y Fo(\(with)30 b(the)g(choic)l(e)i(of)e(index)g Fu(j)35 b Fo(such)30 b(that)f Fu(z)1827 4629 y Fs(j)1885 4617 y Fz(=)23 b Fu(S)2024 4629 y Fs(j)2088 4617 y Fo(for)31 b Fu(j)d Fz(=)22 b(1)p Fu(;)14 b(:)g(:)g(:)g(;)g(N)9 b Fo(\).)440 4783 y Fz(W)-7 b(e)28 b(notice)f(immediately)h(:)1556 4815 y Fp(X)1600 4992 y Fs(j)1690 4894 y Fu(\037)1742 4860 y Fs(h)1742 4915 y(j)1785 4894 y Fz(\()p Fx(r)p Fu(\037)1938 4860 y Fs(h)1938 4915 y(j)1981 4894 y Fz(\))23 b(=)g(0)p Fu(:)903 b Fz(\(11.4\))440 5138 y(T)-7 b(o)27 b(compare)g(with)h(mo)r(dels)f Fw(R)1417 5107 y Fv(2)1460 5138 y Fz(,)h Fw(R)d Fx(\002)18 b Fw(R)1727 5107 y Fv(+)1815 5138 y Fz(and)28 b(\012)2037 5150 y Fs(\013)2084 5138 y Fz(,)g(w)n(e)f(share)f(indices)i(in)g(three)f(parts)g(:)466 5320 y Fu(cor)12 b Fz(:)51 b(=)22 b Fx(f)p Fu(j)5 b Fx(j)27 b Fu(z)922 5332 y Fs(j)980 5320 y Fz(=)c Fu(S)1119 5332 y Fs(j)1154 5320 y Fx(g)p Fu(;)36 b(bd)9 b Fz(:)51 b(=)23 b Fx(f)p Fu(j)5 b Fx(j)27 b Fu(z)1675 5332 y Fs(j)1732 5320 y Fx(2)d Fu(@)5 b Fz(\012)18 b Fx(n)g(f)p Fu(S)2091 5332 y Fv(1)2127 5320 y Fu(;)c(:)g(:)g(:)g(;)g(S)2363 5332 y Fs(N)2426 5320 y Fx(gg)p Fu(;)36 b(int)9 b Fz(:)50 b(=)23 b Fx(f)p Fu(j)5 b Fx(j)27 b Fu(z)3018 5332 y Fs(j)3076 5320 y Fx(2)c Fz(\012)p Fx(g)p Fu(;)1831 5662 y Fz(32)p eop %%Page: 33 33 33 32 bop 440 531 a Fz(corresp)r(onding)32 b(resp)r(ectiv)n(ely)h(to)h (the)g(corners,)g(the)h(regular)d(p)r(oin)n(ts)i(of)g(the)g(b)r (oundary)440 631 y(and)27 b(the)h(p)r(oin)n(ts)g(of)g(the)f(in)n (terior.)36 b(W)-7 b(e)28 b(deduce)g(for)f(ev)n(ery)f Fu(u)d Fx(2)g Fu(H)2548 601 y Fv(1)2541 654 y Fs(h;)p Ft(A)2657 631 y Fz(\(\012\))29 b(that)e(:)648 844 y Fu(q)685 856 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)868 844 y Fz(\()p Fu(u)p Fz(\))83 b(=)1211 765 y Fp(X)1226 942 y Fs(int)1344 844 y Fu(q)1381 856 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)1565 844 y Fz(\()p Fu(\037)1649 810 y Fs(h)1649 865 y(j)1692 844 y Fu(u)p Fz(\))18 b(+)1873 765 y Fp(X)1901 944 y Fs(bd)2007 844 y Fu(q)2044 856 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)2227 844 y Fz(\()p Fu(\037)2311 810 y Fs(h)2311 865 y(j)2354 844 y Fu(u)p Fz(\))h(+)2536 765 y Fp(X)2548 940 y Fs(cor)2669 844 y Fu(q)2706 856 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)2890 844 y Fz(\()p Fu(\037)2974 810 y Fs(h)2974 865 y(j)3017 844 y Fu(u)p Fz(\))1211 1073 y Fx(\000)p Fu(h)1324 1039 y Fv(2)1374 994 y Fp(X)1419 1171 y Fs(j)1508 1073 y Fx(jj)28 b(jr)p Fu(\037)1726 1039 y Fs(h)1726 1093 y(j)1769 1073 y Fx(j)p Fu(u)p Fx(jj)1886 1039 y Fv(2)1886 1095 y Fs(L)1932 1079 y Fr(2)1963 1095 y Fv(\(\012\))2067 1073 y Fu(:)1002 b Fz(\(11.5\))440 1339 y(W)-7 b(e)32 b(see)e(t)n(w)n(o)h(kinds)g(of)g(errors)e(:)44 b(errors)29 b(coming)h(from)h(appro)n(ximation)e(whic)n(h)i(suggest)440 1439 y(the)41 b(c)n(hoice)e Fu(\032)i Fz(large)e(and)h(errors)e(coming) i(from)g(lo)r(calization)f(\(last)h(term)h(of)f(\(11.5\)\))440 1539 y(whic)n(h)29 b(suggest)e(to)i(c)n(ho)r(ose)f Fu(\032)g Fz(small.)41 b(So)28 b(w)n(e)g(will)i(try)e(to)h(optimize)g(b)r(et)n(w) n(een)f(these)h(t)n(w)n(o)440 1638 y(constrain)n(ts.)440 1854 y FA(11.1.2)93 b(Change)32 b(of)g(v)-5 b(ariables)440 2007 y Fz(T)e(o)30 b(compare)f(with)i(the)f(mo)r(dels)g Fw(R)25 b Fx(\002)18 b Fw(R)1735 1977 y Fv(+)1826 2007 y Fz(and)30 b(\012)2050 2019 y Fs(\013)2097 2007 y Fz(,)h(w)n(e)f(mak)n (e)g(a)f(lo)r(cal)h(c)n(hange)f(of)h(v)-5 b(ari-)440 2107 y(ables.)440 2273 y FA(Lemma)44 b(11.2.)j Fo(Ther)l(e)42 b(exists)f(p)l(ositive)h(c)l(onstants)e Fu(h)2261 2285 y Fv(0)2339 2273 y Fo(and)i Fu(C)2571 2285 y Fv(1)2650 2273 y Fo(such)f(that)g(for)h(any)440 2373 y(r)l(e)l(gular)c(p)l(oint)g Fu(z)j Fx(2)c Fu(@)5 b Fz(\012)24 b Fx(n)g(f)p Fu(S)1406 2385 y Fv(1)1443 2373 y Fu(;)14 b(:)g(:)g(:)f(;)h(S)1678 2385 y Fs(N)1741 2373 y Fx(g)p Fo(,)40 b(for)e(any)g Fu(\017)g(<)f(d)p Fz(\()p Fu(z)t(;)14 b Fx(f)p Fu(S)2577 2385 y Fv(1)2613 2373 y Fu(;)g(:)g(:)g(:)g(;)g(S)2849 2385 y Fs(N)2911 2373 y Fx(g)p Fz(\))p Fo(,)40 b(we)e(c)l(an)440 2472 y(write)32 b(for)g Fu(u)25 b Fx(2)h Fu(H)1017 2442 y Fv(1)1010 2496 y Fs(h;)p Ft(A)1126 2472 y Fz(\(\012\))32 b Fo(with)f(supp)l(ort)h(in)p 1862 2406 60 4 v 31 w Fz(\012)19 b Fx(\\)h(B)s Fz(\()p Fu(z)t(;)14 b(\017)p Fz(\))30 b Fo(the)h(form)h(in)f(the)h(new)f(domain)440 2581 y Fw(R)24 b Fx(\002)18 b Fw(R)655 2551 y Fv(+)746 2581 y Fo(by)30 b(:)443 2826 y Fu(q)480 2838 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)664 2826 y Fz(\()p Fu(v)s Fz(\))23 b(=)882 2713 y Fp(Z)928 2902 y Fn(R)p Ft(\002)p Fn(R)1070 2885 y Fr(+)1193 2747 y Fp(X)1129 2926 y Fv(1)p Ft(\024)p Fs(k)q(;l)p Ft(\024)p Fv(2)1390 2826 y Fu(g)1430 2838 y Fs(k)q(;l)1512 2826 y Fz(\()5 b(~)-47 b Fu(x)q Fz(\))1638 2709 y Fp(\022)1699 2826 y Fu(h)1779 2770 y(@)8 b Fz(~)-45 b Fu(v)p 1757 2807 137 4 v 1757 2883 a(@)10 b Fz(~)-47 b Fu(x)1853 2895 y Fs(k)1952 2826 y Fx(\000)18 b Fu(i)2085 2805 y Fz(~)2064 2826 y Fu(A)2126 2838 y Fs(k)2170 2826 y Fz(~)-46 b Fu(v)2210 2709 y Fp(\023)p 2285 2696 603 4 v 14 w(\022)2346 2826 y Fu(h)2418 2770 y(@)8 b Fz(~)-45 b Fu(v)p 2404 2807 122 4 v 2404 2883 a(@)10 b Fz(~)-47 b Fu(x)2500 2895 y Fs(l)2583 2826 y Fx(\000)18 b Fu(i)2717 2805 y Fz(~)2695 2826 y Fu(A)2757 2838 y Fs(l)2786 2826 y Fz(~)-45 b Fu(v)2826 2709 y Fp(\023)2887 2751 y(p)p 2970 2751 188 4 v 75 x Fz(det)30 b Fu(g)i(d)5 b Fz(~)-47 b Fu(x)q(;)3092 3016 y Fz(\(11.6\))440 3116 y Fo(with)36 b Fz(~)-48 b Fu(u)30 b Fo(and)887 3095 y Fz(~)859 3116 y Fx(A)g Fo(de)l(duc)l(e)l(d) g(fr)l(om)g Fu(u)g Fo(and)g Fx(A)g Fo(by)g(change)h(of)g(variables.)440 3215 y(F)-6 b(urthermor)l(e,)27 b(for)g Fu(h)c Fx(2)p Fz(]0)p Fu(;)14 b(h)1347 3227 y Fv(0)1384 3215 y Fz(])27 b Fo(and)f Fu(u)g Fo(with)h(supp)l(ort)f(in)h Fx(B)s Fz(\()p Fu(z)t(;)14 b(h)2449 3185 y Fs(\032)2486 3215 y Fz(\))d Fx(\\)g Fz(\()p 2627 3148 60 4 v(\012)g Fx(n)g(f)p Fu(S)2844 3227 y Fv(1)2880 3215 y Fu(;)j(:)g(:)g(:)g(;)g(S)3116 3227 y Fs(N)3179 3215 y Fx(g)p Fz(\))26 b Fo(:)755 3398 y Fz(\(1)18 b Fx(\000)g Fu(C)989 3410 y Fv(1)1027 3398 y Fu(h)1075 3364 y Fs(\032)1113 3398 y Fz(\))p Fu(q)1182 3423 y Fs(h;)1264 3408 y Fv(~)1241 3423 y Ft(A)p Fs(;)1321 3408 y Fv(~)1314 3423 y(\012)1366 3398 y Fz(\()5 b(~)-47 b Fu(u)o Fz(\))24 b Fx(\024)e Fu(q)1625 3410 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)1809 3398 y Fz(\()p Fu(u)p Fz(\))h Fx(\024)f Fz(\(1)d(+)f Fu(C)2266 3410 y Fv(1)2303 3398 y Fu(h)2351 3364 y Fs(\032)2390 3398 y Fz(\))p Fu(q)2459 3423 y Fs(h;)2540 3408 y Fv(~)2518 3423 y Ft(A)p Fs(;)2598 3408 y Fv(~)2591 3423 y(\012)2642 3398 y Fz(\()5 b(~)-47 b Fu(u)p Fz(\))p Fu(:)315 b Fz(\(11.7\))565 3580 y(W)-7 b(e)28 b(obtain)f(a)g(similar)g(result)g(for)g(corners.)440 3746 y FA(Lemma)44 b(11.3.)j Fo(Ther)l(e)42 b(exists)f(p)l(ositive)h(c) l(onstants)e Fu(h)2261 3758 y Fv(0)2339 3746 y Fo(and)i Fu(C)2571 3758 y Fv(1)2650 3746 y Fo(such)f(that)g(for)h(any)440 3846 y(c)l(orners)25 b Fu(S)781 3858 y Fs(j)816 3846 y Fo(,)h(for)g(any)f Fu(u)e Fx(2)g Fu(H)1374 3816 y Fv(1)1367 3870 y Fs(h;)p Ft(A)1483 3846 y Fz(\(\012\))j Fo(with)f(supp)l(ort)g (in)p 2194 3779 V 25 w Fz(\012)8 b Fx(\\)g(B)s Fz(\()p Fu(S)2466 3858 y Fs(j)2500 3846 y Fu(;)14 b(\017)p Fz(\))25 b Fo(\(with)h Fu(\017)c(<)h(d)p Fz(\()p Fu(S)3108 3858 y Fs(j)3143 3846 y Fu(;)14 b(S)3231 3858 y Fs(k)3272 3846 y Fz(\))440 3946 y Fo(for)31 b(every)f Fu(k)c Fx(6)p Fz(=)d Fu(j)5 b Fo(\),)30 b(we)g(c)l(an)g(write)g(the)g(form)g(in)g (the)g(new)g(domain)h Fz(\012)2669 3958 y Fs(\013)2746 3946 y Fo(by)f(:)490 4195 y Fu(q)527 4207 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)710 4195 y Fz(\()p Fu(v)s Fz(\))24 b(=)929 4082 y Fp(Z)975 4270 y Fv(\012)1022 4278 y Fe(\013)1146 4116 y Fp(X)1082 4295 y Fv(1)p Ft(\024)p Fs(k)q(;l)p Ft(\024)p Fv(2)1344 4195 y Fu(g)1384 4207 y Fs(k)q(;l)1465 4195 y Fz(\()5 b(~)-47 b Fu(x)q Fz(\))1591 4078 y Fp(\022)1652 4195 y Fu(h)1733 4139 y(@)8 b Fz(~)-45 b Fu(v)p 1710 4176 137 4 v 1710 4252 a(@)10 b Fz(~)-47 b Fu(x)1806 4264 y Fs(k)1905 4195 y Fx(\000)18 b Fu(i)2039 4174 y Fz(~)2017 4195 y Fu(A)2079 4207 y Fs(k)2123 4195 y Fz(~)-45 b Fu(v)2163 4078 y Fp(\023)p 2238 4064 603 4 v 14 w(\022)2299 4195 y Fu(h)2372 4139 y(@)8 b Fz(~)-45 b Fu(v)p 2357 4176 122 4 v 2357 4252 a(@)10 b Fz(~)-47 b Fu(x)2453 4264 y Fs(l)2537 4195 y Fx(\000)18 b Fu(i)2670 4174 y Fz(~)2649 4195 y Fu(A)2711 4207 y Fs(l)2739 4195 y Fz(~)-45 b Fu(v)2779 4078 y Fp(\023)2840 4120 y(p)p 2923 4120 188 4 v 75 x Fz(det)30 b Fu(g)j(d)5 b Fz(~)-47 b Fu(x;)3092 4385 y Fz(\(11.8\))440 4484 y Fo(with)36 b Fz(~)-48 b Fu(u)30 b Fo(and)887 4463 y Fz(~)859 4484 y Fx(A)g Fo(de)l(duc)l(e)l(d) g(fr)l(om)g Fu(u)g Fo(and)g Fx(A)g Fo(by)g(the)g(change)h(of)f (variables.)440 4584 y(F)-6 b(urthermor)l(e,)30 b(for)h(any)f Fu(h)23 b Fx(2)p Fz(]0)p Fu(;)14 b(h)1513 4596 y Fv(0)1550 4584 y Fz(])29 b Fo(and)i Fu(u)e Fo(with)h(supp)l(ort)g(in)p 2417 4517 60 4 v 30 w Fz(\012)18 b Fx(\\)h(B)s Fz(\()p Fu(z)t(;)14 b(h)2787 4554 y Fs(\032)2824 4584 y Fz(\))30 b Fo(:)755 4767 y Fz(\(1)18 b Fx(\000)g Fu(C)989 4779 y Fv(1)1027 4767 y Fu(h)1075 4732 y Fs(\032)1113 4767 y Fz(\))p Fu(q)1182 4792 y Fs(h;)1264 4777 y Fv(~)1241 4792 y Ft(A)p Fs(;)1321 4777 y Fv(~)1314 4792 y(\012)1366 4767 y Fz(\()5 b(~)-47 b Fu(u)o Fz(\))24 b Fx(\024)e Fu(q)1625 4779 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)1809 4767 y Fz(\()p Fu(u)p Fz(\))h Fx(\024)f Fz(\(1)d(+)f Fu(C)2266 4779 y Fv(1)2303 4767 y Fu(h)2351 4732 y Fs(\032)2390 4767 y Fz(\))p Fu(q)2459 4792 y Fs(h;)2540 4777 y Fv(~)2518 4792 y Ft(A)p Fs(;)2598 4777 y Fv(~)2591 4792 y(\012)2642 4767 y Fz(\()5 b(~)-47 b Fu(u)p Fz(\))p Fu(:)315 b Fz(\(11.9\))565 4949 y(The)26 b(main)g(di\016cult)n(y)h(in)g(Lemma)f(11.2)f(is)h(to)h (con)n(trol)e(the)h(uniformit)n(y)h(with)g(resp)r(ect)440 5049 y(to)g Fu(z)t Fz(.)37 b(Details)27 b(on)h(the)g(pro)r(ofs)e(of)i (Lemmas)f(11.2)f(and)i(11.3)e(are)h(giv)n(en)f(in)i([6].)1831 5662 y(33)p eop %%Page: 34 34 34 33 bop 440 531 a FA(11.1.3)93 b(Gauge)32 b(transform)440 685 y Fz(W)-7 b(e)29 b(assume)e(that)i(w)n(e)f(ha)n(v)n(e)f(c)n(hosen)g (the)i(co)r(ordinates)e(suc)n(h)h(that)g Fu(z)g Fz(=)c(\(0)p Fu(;)14 b Fz(0\))23 b Fx(2)i Fu(@)5 b Fz(\012.)39 b(By)440 784 y(a)23 b(gauge)e(transform,)i(w)n(e)g(can)g(assume)f(that)i Fx(A)p Fz(\(0\))f(and)g(the)h(linear)e(part)h(of)g(the)g(p)r(oten)n (tial)440 884 y Fx(A)29 b Fz(are)e(resp)r(ectiv)n(ely)g(equal)h(to)g(0) g(and)g Fx(A)1747 854 y Fs(`in)1853 884 y Fz(:)52 b(=)24 b Fu(B)t Fz(\(0\))p Fx(A)2256 896 y Fv(0)2322 884 y Fz(\(with)29 b Fx(A)2610 896 y Fv(0)2676 884 y Fz(de\014ned)f(on)g(\(1.2\)\).)440 983 y(W)-7 b(e)28 b(de\014ne)g Fx(A)889 953 y Ft(0)922 983 y Fz(:)50 b(=)23 b Fx(A)c(\000)f(A)1317 953 y Fs(`in)1414 983 y Fz(,)27 b(then)h(there)g(exists)f(a)g(p)r(ositiv)n(e)g(constan)n (t)2825 962 y(~)2806 983 y Fu(C)34 b Fz(suc)n(h)28 b(that)f(:)1396 1165 y Fx(jA)18 b(\000)h(A)1653 1131 y Fs(`in)1749 1165 y Fx(j)k Fz(=)g Fx(jA)1972 1131 y Ft(0)1996 1165 y Fx(j)g(\024)2148 1144 y Fz(~)2129 1165 y Fu(C)7 b Fx(j)p Fu(x)p Fx(j)2288 1131 y Fv(2)2326 1165 y Fu(:)702 b Fz(\(11.10\))440 1346 y(By)27 b(the)h(decomp)r(osition)f(of)h Fx(A)g Fz(as)f(the)h(sum)f Fx(A)1935 1316 y Fs(`in)2051 1346 y Fz(+)18 b Fx(A)2200 1316 y Ft(0)2223 1346 y Fz(,)28 b(w)n(e)f(write)h(:)519 1566 y Fu(q)556 1578 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)739 1566 y Fz(\()p Fu(v)s Fz(\))84 b(=)1078 1453 y Fp(Z)1124 1641 y Fv(\012)1189 1473 y Fp(\020)1239 1495 y(\014)1239 1545 y(\014)1266 1566 y Fz(\()p Fu(h@)1390 1578 y Fs(x)1428 1586 y Fr(1)1483 1566 y Fx(\000)18 b Fu(iA)1657 1531 y Fs(`in)1657 1586 y Fv(1)1754 1566 y Fz(\))p Fu(v)1829 1495 y Fp(\014)1829 1545 y(\014)1857 1511 y Fv(2)1912 1566 y Fz(+)1995 1495 y Fp(\014)1995 1545 y(\014)2023 1566 y Fz(\()p Fu(h@)2147 1578 y Fs(x)2185 1586 y Fr(2)2240 1566 y Fx(\000)g Fu(iA)2414 1531 y Fs(`in)2414 1586 y Fv(2)2510 1566 y Fz(\))p Fu(v)2585 1495 y Fp(\014)2585 1545 y(\014)2613 1511 y Fv(2)2651 1473 y Fp(\021)2742 1566 y Fu(dx)1078 1787 y Fz(+2)k(Re)h Fu(i)1371 1674 y Fp(Z)1416 1863 y Fv(\012)1482 1695 y Fp(\020)1531 1787 y Fz(\()p Fu(h@)1655 1799 y Fs(x)1693 1807 y Fr(1)1748 1787 y Fx(\000)18 b Fu(iA)1922 1753 y Fs(`in)1922 1808 y Fv(1)2018 1787 y Fz(\))p Fu(v)p 2093 1717 143 4 v 3 w(A)2155 1759 y Ft(0)2155 1810 y Fv(1)2194 1787 y Fu(v)j Fz(+)d(\()p Fu(h@)2462 1799 y Fs(x)2500 1807 y Fr(2)2555 1787 y Fx(\000)g Fu(iA)2729 1753 y Fs(`in)2729 1808 y Fv(2)2825 1787 y Fz(\))p Fu(v)p 2900 1717 V 3 w(A)2962 1759 y Ft(0)2962 1810 y Fv(2)3000 1787 y Fu(v)3044 1695 y Fp(\021)3135 1787 y Fu(dx)1078 2009 y Fz(+)1157 1896 y Fp(Z)1202 2085 y Fv(\012)1267 1942 y Fp(\000)1306 2009 y Fx(j)p Fu(A)1391 1975 y Ft(0)1391 2030 y Fv(1)1428 2009 y Fu(v)s Fx(j)1494 1975 y Fv(2)1550 2009 y Fz(+)g Fx(j)p Fu(A)1718 1975 y Ft(0)1718 2030 y Fv(2)1756 2009 y Fu(v)s Fx(j)1822 1975 y Fv(2)1859 1942 y Fp(\001)1939 2009 y Fu(dx:)999 b Fz(\(11.11\))440 2248 y(There)27 b(exists)g(a)h(constan)n(t)e Fu(C)34 b Fz(suc)n(h)28 b(that)f(if)i Fu(v)h Fz(has)e(a)f(supp)r(ort)g(in)p 2542 2182 60 4 v 28 w(\012)18 b Fx(\\)h(B)s Fz(\(0)p Fu(;)14 b(h)2911 2218 y Fs(\032)2948 2248 y Fz(\),)28 b(then)g(:)1072 2430 y Fu(q)1109 2442 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)1292 2430 y Fz(\()p Fu(v)s Fz(\))84 b Fx(\024)f Fu(q)1668 2447 y Fs(h;)p Ft(A)1780 2430 y Fe(`in)1867 2447 y Fs(;)p Fv(\012)1938 2430 y Fz(\()p Fu(v)s Fz(\))20 b(+)e Fu(C)6 b(h)2261 2395 y Fv(2)p Fs(\032)2332 2430 y Fx(jj)p Fu(v)s Fx(jj)2467 2395 y Fv(2)2467 2452 y Fs(L)2513 2436 y Fr(2)2546 2452 y Fv(\(\012\))2649 2430 y Fu(:)379 b Fz(\(11.12\))440 2662 y Fl(11.2)112 b(Lo)m(w)m(er)38 b(b)s(ound)440 2815 y FA(Prop)s(osition)29 b(11.4.)40 b Fo(Under)29 b(the)g(asumptions)g (of)h(The)l(or)l(em)h(1.2,)g(ther)l(e)e(exists)g Fu(h)3107 2827 y Fv(0)3173 2815 y Fo(and)440 2915 y(a)h(c)l(onstant)f Fu(C)36 b Fo(such)30 b(that)g(for)g(any)g Fu(h)23 b Fx(2)p Fz(]0)p Fu(;)14 b(h)1864 2927 y Fv(0)1901 2915 y Fz([)30 b Fo(:)808 3146 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\025)g Fu(h)14 b Fz(min)1496 3029 y Fp(\022)1557 3146 y Fu(b;)g Fz(\002)1695 3158 y Fv(0)1731 3146 y Fu(b)1767 3112 y Ft(0)1790 3146 y Fu(;)100 b Fz(inf)1827 3200 y Fs(j)s Fv(=1)p Fs(;:::;N)2113 3146 y Fu(\026)p Fz(\()p Fu(\013)2248 3158 y Fs(j)2284 3146 y Fz(\))p Fu(B)t Fz(\()p Fu(S)2466 3158 y Fs(j)2501 3146 y Fz(\))2533 3029 y Fp(\023)2613 3146 y Fx(\000)18 b Fu(C)6 b(h)2809 3112 y Fv(5)p Fs(=)p Fv(4)2913 3146 y Fu(:)440 3377 y Fo(Pr)l(o)l(of)31 b(:)39 b Fz(W)-7 b(e)28 b(ha)n(v)n(e)e(to)h(estimate) h(eac)n(h)f(term)g(of)h(\(11.5\).)36 b(F)-7 b(rom)27 b(\(11.2\),)g(w)n(e)g(deduce)h(:)1040 3575 y Fu(h)1088 3541 y Fv(2)1139 3496 y Fp(X)1184 3673 y Fs(j)1273 3575 y Fx(jj)g(jr)p Fu(\037)1491 3541 y Fs(h)1491 3596 y(j)1534 3575 y Fx(j)g Fu(u)p Fx(jj)1679 3541 y Fv(2)1679 3598 y Fs(L)1725 3581 y Fr(2)1756 3598 y Fv(\(\012\))1943 3575 y Fx(\024)82 b Fu(C)6 b(h)2203 3541 y Fv(2)p Ft(\000)p Fv(2)p Fs(\032)2360 3575 y Fx(jj)p Fu(u)p Fx(jj)2500 3541 y Fv(2)2500 3598 y Fs(L)2546 3581 y Fr(2)2578 3598 y Fv(\(\012\))2681 3575 y Fu(:)347 b Fz(\(11.13\))440 3862 y(This)28 b(estimate)f(sho)n(ws)g(that,)h(for)f Fu(\032)c Fz(=)1694 3829 y Fv(3)p 1694 3843 34 4 v 1694 3891 a(8)1737 3862 y Fz(,)28 b(the)g(error)e(is)h(in)h Fx(O)r Fz(\()p Fu(h)2473 3810 y Fr(5)p 2473 3819 29 3 v 2473 3852 a(4)2516 3862 y Fz(\).)440 3962 y(Observing)e(that)i([)p Fu(hD)1155 3974 y Fs(x)1193 3982 y Fr(1)1247 3962 y Fx(\000)18 b Fu(A)1392 3974 y Fv(1)1430 3962 y Fu(;)c(hD)1584 3974 y Fs(x)1622 3982 y Fr(2)1676 3962 y Fx(\000)k Fu(A)1821 3974 y Fv(2)1858 3962 y Fz(])24 b(=)e Fu(i)27 b(h)h(B)t Fz(,)g(w)n(e)f(deduce)h(the)g(estimate)f(:)1067 4107 y Fp(X)1083 4284 y Fs(int)1201 4186 y Fu(q)1238 4198 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)1421 4186 y Fz(\()p Fu(\037)1505 4151 y Fs(h)1505 4206 y(j)1549 4186 y Fu(u)p Fz(\))c Fx(\025)f Fu(h)1801 4107 y Fp(X)1816 4284 y Fs(int)1935 4073 y Fp(Z)1981 4261 y Fv(\012)2046 4186 y Fu(B)t Fz(\()p Fu(x)p Fz(\))p Fx(j)p Fu(\037)2299 4151 y Fs(h)2299 4206 y(j)2343 4186 y Fu(u)p Fz(\()p Fu(x)p Fz(\))p Fx(j)2525 4151 y Fv(2)2563 4186 y Fu(dx;)375 b Fz(\(11.14\))440 4452 y(where)27 b(w)n(e)g(ha)n(v)n(e)g(used)g(that)h(supp)14 b Fu(\037)1600 4422 y Fs(h)1600 4474 y(j)1666 4452 y Fx(\032)23 b Fz(\012)p Fu(;)k Fz(for)h Fu(j)g Fx(2)23 b Fu(int)p Fz(.)565 4652 y(W)-7 b(e)36 b(ha)n(v)n(e)f(no)n(w)g(to)h (estimate)g(the)h(terms)f(whic)n(h)g(are)f(lo)r(calized)g(at)h(the)h(b) r(oundary)-7 b(.)440 4751 y(Let)34 b Fu(j)39 b Fx(2)34 b Fu(cor)25 b Fx([)e Fu(bd)p Fz(.)56 b(By)34 b(a)f(c)n(hange)g(of)h(v) -5 b(ariables,)34 b(w)n(e)g(send)g(lo)r(cally)f(the)i(domain)e(on)n(to) 440 4851 y Fw(R)24 b Fx(\002)18 b Fw(R)655 4821 y Fv(+)745 4851 y Fz(or)27 b(\012)907 4863 y Fs(\013)955 4851 y Fz(.)39 b(W)-7 b(e)28 b(apply)g(Lemmas)g(11.2)f(and)h(11.3,)g(so)f (there)h(exists)g(a)g(constan)n(t)g Fu(C)3267 4863 y Fv(1)440 4951 y Fz(indep)r(enden)n(t)g(of)g Fu(h)g Fz(suc)n(h)f(that,)h (for)f(ev)n(ery)f Fu(v)31 b Fz(with)d(supp)r(ort)g(in)p 2481 4884 60 4 v 27 w(\012)19 b Fx(\\)g(B)s Fz(\()p Fu(z)2763 4963 y Fs(j)2797 4951 y Fu(;)14 b(h)2882 4920 y Fs(\032)2920 4951 y Fz(\),)698 5132 y(\(1)19 b Fx(\000)f Fu(C)933 5144 y Fv(1)970 5132 y Fu(h)1018 5098 y Fs(\032)1057 5132 y Fz(\))p Fu(q)1126 5157 y Fs(h;)1207 5142 y Fv(~)1185 5157 y Ft(A)p Fs(;)p Fv(\012)1305 5165 y Fe(\013)1351 5132 y Fz(\()s(~)-45 b Fu(v)t Fz(\))23 b Fx(\024)g Fu(q)1607 5144 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)1790 5132 y Fz(\()p Fu(v)s Fz(\))h Fx(\024)e Fz(\(1)d(+)f Fu(C)2243 5144 y Fv(1)2280 5132 y Fu(h)2328 5098 y Fs(\032)2367 5132 y Fz(\))p Fu(q)2436 5157 y Fs(h;)2517 5142 y Fv(~)2495 5157 y Ft(A)p Fs(;)p Fv(\012)2615 5165 y Fe(\013)2661 5132 y Fz(\()s(~)-45 b Fu(v)t Fz(\))p Fu(:)259 b Fz(\(11.15\))440 5313 y(So)18 b(it)i(is)e(enough)g(to)h(analyze)f Fu(q)1391 5339 y Fs(h;)1472 5324 y Fv(~)1450 5339 y Ft(A)p Fs(;)p Fv(\012)1570 5347 y Fe(\013)1616 5313 y Fz(\()s(~)-45 b Fu(v)t Fz(\).)34 b(W)-7 b(e)19 b(no)n(w)f(omit)h(the)g(tilda)g(due)g (do)g(the)g(c)n(hange)e(of)440 5413 y(v)-5 b(ariables.)34 b(W)-7 b(e)23 b(mak)n(e)f(a)g(gauge)f(transform)g(suc)n(h)h(that)h(the) g(linear)f(term)g(of)h(the)g(magnetic)1831 5662 y(34)p eop %%Page: 35 35 35 34 bop 440 531 a Fz(p)r(oten)n(tial)40 b(is)g(equal)f(to)h Fx(A)1310 501 y Fs(`in)1416 531 y Fz(:)75 b(=)44 b Fu(B)t Fz(\()p Fu(z)1761 543 y Fs(j)1796 531 y Fz(\))p Fx(A)1894 543 y Fv(0)1932 531 y Fz(.)74 b(W)-7 b(e)40 b(denote)g(b)n(y)f Fx(A)2657 501 y Ft(0)2721 531 y Fz(the)h(remainder)f(:)440 631 y Fx(A)506 601 y Ft(0)553 631 y Fz(=)22 b Fx(A)d(\000)f(A)874 601 y Fs(`in)971 631 y Fz(.)37 b(Relation)27 b(\(11.11\))f(giv)n(es)h (:)493 805 y Fu(q)530 817 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)714 805 y Fz(\()p Fu(v)s Fz(\))83 b Fx(\025)g Fu(q)1089 822 y Fs(h;)p Ft(A)1201 805 y Fe(`in)1289 822 y Fs(;)p Fv(\012)1360 805 y Fz(\()p Fu(v)s Fz(\))1052 979 y Fx(\000)p Fz(2)22 b(Im)1317 866 y Fp(Z)1363 1055 y Fv(\012)1429 887 y Fp(\020)1478 979 y Fz(\()p Fu(h@)1602 991 y Fs(x)1640 999 y Fr(1)1695 979 y Fx(\000)c Fu(iA)1869 945 y Fs(`in)1869 1000 y Fv(1)1966 979 y Fz(\))p Fu(v)p 2069 908 143 4 v 31 w(A)2131 950 y Ft(0)2131 1001 y Fv(1)2168 979 y Fu(v)k Fz(+)c(\()p Fu(h@)2437 991 y Fs(x)2475 999 y Fr(2)2530 979 y Fx(\000)g Fu(iA)2704 945 y Fs(`in)2704 1000 y Fv(2)2800 979 y Fz(\))p Fu(v)p 2903 908 V 31 w(A)2965 950 y Ft(0)2965 1001 y Fv(2)3003 979 y Fu(v)3046 887 y Fp(\021)3137 979 y Fu(dx:)440 1195 y Fz(A)28 b(Cauc)n(h)n(y-Sc)n(h)n(w)n(arz)c(inequalit)n(y)j(leads) g(to)h(:)440 1300 y Fp(\014)440 1349 y(\014)440 1399 y(\014)440 1449 y(\014)468 1307 y(Z)514 1496 y Fv(\012)565 1420 y Fz(\()p Fu(h@)689 1432 y Fs(x)727 1441 y Fe(k)786 1420 y Fx(\000)18 b Fu(iA)960 1386 y Fs(`in)960 1440 y(k)1056 1420 y Fz(\))p Fu(v)p 1131 1349 147 4 v 3 w(A)1193 1391 y Ft(0)1193 1445 y Fs(k)1235 1420 y Fu(v)s(dx)1368 1300 y Fp(\014)1368 1349 y(\014)1368 1399 y(\014)1368 1449 y(\014)1420 1420 y Fx(\024)1517 1364 y Fu(h)1565 1334 y Fv(2)p Fs(\022)p 1517 1401 119 4 v 1556 1477 a Fz(2)1660 1307 y Fp(Z)1706 1496 y Fv(\012)1771 1349 y Fp(\014)1771 1399 y(\014)1799 1420 y Fz(\()p Fu(h@)1923 1432 y Fs(x)1961 1441 y Fe(k)2019 1420 y Fx(\000)h Fu(iA)2194 1386 y Fs(`in)2194 1440 y(k)2290 1420 y Fz(\))p Fu(v)2365 1349 y Fp(\014)2365 1399 y(\014)2393 1366 y Fv(2)2444 1420 y Fu(dx)e Fz(+)2641 1364 y Fu(h)2689 1334 y Ft(\000)p Fv(2)p Fs(\022)p 2641 1401 171 4 v 2706 1477 a Fz(2)2836 1307 y Fp(Z)2882 1496 y Fv(\012)2947 1420 y Fx(j)p Fu(A)3032 1386 y Ft(0)3032 1440 y Fs(k)3073 1420 y Fu(v)s Fx(j)3139 1374 y Fv(2)3191 1420 y Fu(dx:)3051 1569 y Fz(\(11.16\))440 1669 y(But,)31 b(b)n(y)e(a)g(T)-7 b(a)n(ylor)28 b(expansion,)i(w)n(e)f (see)h(that)g Fx(j)p Fu(A)2022 1639 y Ft(0)2022 1693 y Fs(k)2063 1669 y Fx(j)c(\024)h Fu(C)6 b Fx(j)p Fu(x)20 b Fx(\000)g Fu(z)2483 1681 y Fs(j)2517 1669 y Fx(j)27 b(\024)f Fu(C)6 b(h)2771 1639 y Fs(\032)2840 1669 y Fz(where)29 b Fu(C)36 b Fz(is)30 b(a)440 1769 y(constan)n(t)i(indep)r(enden)n(t)h (of)g Fu(h)f Fz(and)h Fu(z)1641 1781 y Fs(j)1675 1769 y Fz(.)52 b(Using)32 b(this)h(estimate,)h(taking)e(accoun)n(t)f(of)i (the)440 1868 y(error)26 b(due)h(to)h(the)g(c)n(hange)e(of)i(v)-5 b(ariables)26 b(\(11.15\))h(and)g(c)n(ho)r(osing)f Fu(v)g Fz(=)d Fu(\037)2744 1838 y Fs(h)2744 1890 y(j)2787 1868 y Fu(u)p Fz(,)k(w)n(e)h(get)f(:)467 2097 y Fu(q)504 2109 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)687 2097 y Fz(\()p Fu(\037)771 2062 y Fs(h)771 2117 y(j)814 2097 y Fu(u)p Fz(\))c Fx(\025)g Fz(\(1)18 b Fx(\000)g Fu(h)1228 2062 y Fv(2)p Fs(\022)1317 2097 y Fx(\000)g Fu(C)1459 2109 y Fv(1)1497 2097 y Fu(h)1545 2062 y Fs(\032)1583 2097 y Fz(\))p Fu(h\026)p Fz(\()p Fu(\013)1798 2109 y Fs(j)1833 2097 y Fz(\))p Fu(B)t Fz(\()p Fu(z)2003 2109 y Fs(j)2039 2097 y Fz(\))2085 1984 y Fp(Z)2131 2172 y Fv(\012)2196 2097 y Fx(j)p Fu(\037)2271 2062 y Fs(h)2271 2117 y(j)2314 2097 y Fu(u)p Fx(j)2385 2062 y Fv(2)2450 2097 y Fu(dx)h Fx(\000)2661 2076 y Fz(~)2642 2097 y Fu(C)2707 2062 y Fv(2)2745 2097 y Fu(h)2793 2062 y Fv(4)p Fs(\032)p Ft(\000)p Fv(2)p Fs(\022)2983 2097 y Fx(jj)p Fu(\037)3081 2062 y Fs(h)3081 2117 y(j)3124 2097 y Fu(u)p Fx(jj)3218 2062 y Fv(2)3255 2097 y Fu(:)3051 2243 y Fz(\(11.17\))565 2342 y(The)34 b(estimates)f(\(11.13\),)i (\(11.14\))e(and)g(\(11.17\))g(giv)n(e)g(the)i(follo)n(wing)e(lo)n(w)n (er)f(b)r(ound)440 2442 y(for)27 b Fu(q)604 2454 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)787 2442 y Fz(\()p Fu(u)p Fz(\))h(:)708 2616 y Fu(q)745 2628 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)929 2616 y Fz(\()p Fu(u)p Fz(\))561 2786 y Fx(\025)82 b Fu(h)756 2707 y Fp(X)771 2884 y Fs(int)890 2673 y Fp(Z)987 2786 y Fu(B)t Fz(\()p Fu(x)p Fz(\))p Fx(j)p Fu(\037)1240 2751 y Fs(h)1240 2806 y(j)1284 2786 y Fu(u)p Fx(j)1355 2751 y Fv(2)1419 2786 y Fu(dx)20 b Fx(\000)e Fu(C)6 b(h)1725 2751 y Fv(2)p Ft(\000)p Fv(2)p Fs(\032)1881 2786 y Fx(jj)p Fu(u)p Fx(jj)2021 2751 y Fv(2)708 3038 y Fz(+\(1)18 b Fx(\000)g Fu(h)996 3004 y Fv(2)p Fs(\022)1085 3038 y Fx(\000)g Fu(C)6 b(h)1281 3004 y Fs(\032)1320 3038 y Fz(\))p Fu(h)p Fz(\002)1465 3050 y Fv(0)1502 2959 y Fp(X)1530 3138 y Fs(bd)1622 3038 y Fu(B)t Fz(\()p Fu(z)1760 3050 y Fs(j)1795 3038 y Fz(\))1841 2925 y Fp(Z)1887 3114 y Fv(\012)1952 3038 y Fx(j)p Fu(\037)2027 3004 y Fs(h)2027 3059 y(j)2070 3038 y Fu(u)p Fx(j)2141 3004 y Fv(2)2206 3038 y Fu(dx)19 b Fx(\000)f Fu(C)2463 3004 y Fv(2)2501 3038 y Fu(h)2549 3004 y Fv(4)p Fs(\032)p Ft(\000)p Fv(2)p Fs(\022)2738 2959 y Fp(X)2767 3138 y Fs(bd)2858 3038 y Fx(jj)p Fu(\037)2956 3004 y Fs(h)2956 3059 y(j)2999 3038 y Fu(u)p Fx(jj)3093 3004 y Fv(2)708 3293 y Fz(+\(1)g Fx(\000)g Fu(h)996 3258 y Fv(2)p Fs(\022)1085 3293 y Fx(\000)g Fu(C)6 b(h)1281 3258 y Fs(\032)1320 3293 y Fz(\))p Fu(h)1400 3214 y Fp(X)1412 3388 y Fs(cor)1520 3293 y Fu(\026)p Fz(\()p Fu(\013)1655 3305 y Fs(j)1690 3293 y Fz(\))p Fu(B)t Fz(\()p Fu(S)1872 3305 y Fs(j)1908 3293 y Fz(\))1954 3180 y Fp(Z)2000 3368 y Fv(\012)2066 3293 y Fx(j)p Fu(\037)2141 3258 y Fs(h)2141 3313 y(j)2184 3293 y Fu(u)p Fx(j)2255 3258 y Fv(2)2319 3293 y Fu(dx)19 b Fx(\000)f Fu(C)2576 3258 y Fv(2)2614 3293 y Fu(h)2662 3258 y Fv(4)p Fs(\032)p Ft(\000)p Fv(2)p Fs(\022)2852 3214 y Fp(X)2864 3388 y Fs(cor)2972 3293 y Fx(jj)p Fu(\037)3070 3258 y Fs(h)3070 3313 y(j)3113 3293 y Fu(u)p Fx(jj)3207 3258 y Fv(2)3244 3293 y Fu(:)440 3549 y Fz(W)-7 b(e)31 b(c)n(ho)r(ose)e Fu(\032)e Fz(=)1024 3516 y Fv(3)p 1024 3530 34 4 v 1024 3578 a(8)1097 3549 y Fz(and)j Fu(\022)g Fz(=)1432 3516 y Fv(1)p 1432 3530 V 1432 3578 a(8)1475 3549 y Fz(,)h(so)e(there)h(exists)g(a)g(p)r(ositiv)n(e)f(constan)n(t)h Fu(C)36 b Fz(and)30 b Fu(h)3106 3561 y Fv(0)3171 3549 y Fu(>)d Fz(0)440 3649 y(suc)n(h)g(that)h(for)f(ev)n(ery)g Fu(h)22 b Fx(2)p Fz(]0)p Fu(;)14 b(h)1429 3661 y Fv(0)1466 3649 y Fz(],)28 b(w)n(e)f(ha)n(v)n(e)g(the)h(lo)n(w)n(er)e(b)r(ound)i (for)f(an)n(y)g Fu(u)22 b Fx(2)i Fu(H)2979 3618 y Fv(1)3016 3649 y Fz(\(\012\))k(:)637 3872 y Fu(q)674 3884 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)857 3872 y Fz(\()p Fu(u)p Fz(\))c Fx(\025)1080 3755 y Fp(\022)1141 3872 y Fu(h)14 b Fz(min)1355 3755 y Fp(\022)1416 3872 y Fu(b;)g Fz(\002)1554 3884 y Fv(0)1591 3872 y Fu(b)1627 3838 y Ft(0)1650 3872 y Fu(;)99 b Fz(inf)1687 3926 y Fs(j)s Fv(=1)p Fs(;:::)o(;N)1973 3872 y Fu(\026)p Fz(\()p Fu(\013)2108 3884 y Fs(j)2143 3872 y Fz(\))p Fu(B)t Fz(\()p Fu(S)2325 3884 y Fs(j)2361 3872 y Fz(\))2393 3755 y Fp(\023)2473 3872 y Fx(\000)18 b Fu(C)2615 3884 y Fv(0)2652 3872 y Fu(h)2700 3838 y Fv(5)p Fs(=)p Fv(4)2804 3755 y Fp(\023)2907 3872 y Fx(jj)p Fu(u)p Fx(jj)3047 3838 y Fv(2)3084 3872 y Fu(:)440 4091 y Fz(W)-7 b(e)28 b(apply)f(the)h(min-max)g(principle)f(and)h(get)f(:) 665 4310 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))24 b Fx(\025)e Fu(h)14 b Fz(min)1353 4193 y Fp(\022)1414 4310 y Fu(b;)g Fz(\002)1552 4322 y Fv(0)1589 4310 y Fu(b)1625 4276 y Ft(0)1648 4310 y Fu(;)99 b Fz(inf)1685 4363 y Fs(j)s Fv(=1)p Fs(;:::)o(;N)1971 4310 y Fu(\026)p Fz(\()p Fu(\013)2106 4322 y Fs(j)2141 4310 y Fz(\))p Fu(B)t Fz(\()p Fu(S)2323 4322 y Fs(j)2359 4310 y Fz(\))2391 4193 y Fp(\023)2470 4310 y Fx(\000)18 b Fu(C)2612 4322 y Fv(0)2650 4310 y Fu(h)2698 4276 y Fv(5)p Fs(=)p Fv(4)2802 4310 y Fu(:)226 b Fz(\(11.18\))3240 4527 y Fd(\003)440 4758 y Fl(11.3)112 b(Upp)s(er)38 b(b)s(ound)440 4911 y FA(Prop)s(osition)d(11.5.)42 b Fo(Under)33 b(the)g(asumptions)h(of)g(The)l(or)l(em)h(1.2)f(and)g (assuming)f(that)440 5011 y Fx(8)p Fu(j)27 b Fx(2)d Fz(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fo(,)29 b Fu(\026)p Fz(\()p Fu(\013)1118 5023 y Fs(j)1154 5011 y Fz(\))23 b Fu(<)g Fz(\002)1362 5023 y Fv(0)1398 5011 y Fo(,)30 b(then)g(ther)l(e)f (exists)g Fu(h)2119 5023 y Fv(0)2186 5011 y Fo(and)g(a)h(c)l(onstant)f Fu(C)35 b Fo(such)30 b(that)f(for)440 5111 y(any)h Fu(h)23 b Fx(2)p Fz(]0)p Fu(;)14 b(h)875 5123 y Fv(0)912 5111 y Fz([)30 b Fo(:)808 5334 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(h)14 b Fz(min)1496 5217 y Fp(\022)1557 5334 y Fu(b;)g Fz(\002)1695 5346 y Fv(0)1731 5334 y Fu(b)1767 5300 y Ft(0)1790 5334 y Fu(;)100 b Fz(inf)1827 5388 y Fs(j)s Fv(=1)p Fs(;:::;N)2113 5334 y Fu(\026)p Fz(\()p Fu(\013)2248 5346 y Fs(j)2284 5334 y Fz(\))p Fu(B)t Fz(\()p Fu(S)2466 5346 y Fs(j)2501 5334 y Fz(\))2533 5217 y Fp(\023)2613 5334 y Fz(+)18 b Fu(C)6 b(h)2809 5300 y Fv(4)p Fs(=)p Fv(3)2913 5334 y Fu(:)1831 5662 y Fz(35)p eop %%Page: 36 36 36 35 bop 440 531 a Fo(Pr)l(o)l(of)31 b(:)39 b Fz(W)-7 b(e)28 b(establish)f(three)g(upp)r(er)h(b)r(ounds)g(:)541 689 y(1.)42 b Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(h)99 b Fz(inf)1184 743 y Fs(j)s Fv(=1)p Fs(;:::)o(;N)1469 689 y Fu(\026)p Fz(\()p Fu(\013)1604 701 y Fs(j)1640 689 y Fz(\))p Fu(B)t Fz(\()p Fu(S)1822 701 y Fs(j)1857 689 y Fz(\))19 b(+)f Fu(C)2050 701 y Fv(0)2088 689 y Fu(h)2136 655 y Fv(4)p Fs(=)p Fv(3)2240 689 y Fz(.)541 907 y(2.)42 b Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(hb)1206 877 y Ft(0)1228 907 y Fz(\002)1293 919 y Fv(0)1349 907 y Fz(+)18 b Fu(C)1491 919 y Fv(0)1528 907 y Fu(h)1576 877 y Fv(4)p Fs(=)p Fv(3)1680 907 y Fz(.)541 1069 y(3.)42 b Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(hb)18 b Fz(+)g Fu(C)1366 1081 y Fv(0)1403 1069 y Fu(h)1451 1039 y Fv(4)p Fs(=)p Fv(3)1555 1069 y Fz(.)440 1227 y Fx(\017)32 b Fz(W)-7 b(e)33 b(b)r(egin)g(with)g(lo)r(oking)f(at)h(the)g(v)n (ertices.)51 b(Let)33 b Fu(S)2189 1239 y Fs(j)2256 1227 y Fz(b)r(e)h(a)e(v)n(ertex.)51 b(By)33 b(a)f(c)n(hange)f(of)440 1327 y(v)-5 b(ariables,)36 b(w)n(e)f(send)g(lo)r(cally)g(\012)g(on)n (to)g(\012)1765 1339 y Fs(\013)1808 1347 y Fe(j)1843 1327 y Fz(.)60 b(W)-7 b(e)36 b(will)g(consider)e(a)h(function)h Fu(u)f Fz(with)h(a)440 1437 y(supp)r(ort)27 b(in)h(a)f(ball)p 1074 1371 60 4 v 27 w(\012)18 b Fx(\\)g(B)s Fz(\()p Fu(S)1366 1449 y Fs(j)1401 1437 y Fu(;)c(h)1486 1407 y Fs(\032)1524 1437 y Fz(\),)28 b(so)f(according)e(to)j(\(11.9\),)e(it)i(is)f(enough)g (to)g(estimate)440 1537 y Fu(q)477 1549 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)656 1557 y Fe(\013)694 1570 y(j)733 1537 y Fz(\()p Fu(u)p Fz(\))h(with)g(the)g(new)g(co)r(ordinates)e(and)h(the) h(error)e(is)i(in)f Fu(h)2553 1507 y Fs(\032)2592 1537 y Fz(.)440 1650 y(By)c(assumption)g Fu(\026)p Fz(\()p Fu(\013)1136 1662 y Fs(j)1171 1650 y Fz(\))g Fu(<)g Fz(\002)1379 1662 y Fv(0)1416 1650 y Fz(,)h(then)g Fu(\026)p Fz(\()p Fu(\013)1783 1662 y Fs(j)1818 1650 y Fz(\))g(is)f(an)g(eigen)n(v)-5 b(alue.)34 b(W)-7 b(e)24 b(use)f(the)g(normalized)440 1749 y(eigenfunction)h Fu(v)986 1761 y Fv(0)1048 1749 y Fz(asso)r(ciated)f(to)h(the)h(Sc)n(hr\177)-42 b(odinger)22 b(op)r(erator)g Fu(P)2513 1761 y Ft(A)2566 1769 y Fr(0)2599 1761 y Fs(;)p Fv(\012)2666 1769 y Fe(\013)2704 1782 y(j)2768 1749 y Fz(and)i(w)n(e)g(de\014ne)g(:)1032 2018 y Fx(8)p Fu(x)f Fx(2)g Fz(\012)1287 2030 y Fs(\013)1330 2038 y Fe(j)1366 2018 y Fu(;)41 b(v)1470 2030 y Fs(h)1513 2018 y Fz(\()p Fu(x)p Fz(\))25 b(=)1736 1883 y Fp(r)p 1819 1883 238 4 v 1829 1962 a Fu(B)t Fz(\()p Fu(S)1979 1974 y Fs(j)2014 1962 y Fz(\))p 1829 1999 218 4 v 1914 2075 a Fu(h)2084 2018 y(v)2124 2030 y Fv(0)2176 1876 y Fp( )2241 1883 y(r)p 2324 1883 238 4 v 2334 1962 a Fu(B)t Fz(\()p Fu(S)2484 1974 y Fs(j)2520 1962 y Fz(\))p 2334 1999 218 4 v 2419 2075 a Fu(h)2562 2018 y(x)2609 1876 y Fp(!)2689 2018 y Fu(:)440 2274 y Fz(With)e(a)e(gauge)g(transform,)h(w)n(e)f (assume)h(that)g Fx(A)h Fz(=)g Fx(A)2157 2244 y Fs(`in)2261 2274 y Fz(+)7 b Fx(A)2399 2244 y Ft(0)2444 2274 y Fz(with)22 b(the)h(same)e(notations)440 2373 y(as)27 b(in)h(Subsection)f(11.1.3.) 440 2473 y(W)-7 b(e)28 b(consider)f Fu(\037)g Fz(a)g(smo)r(oth)h (function)g(de\014ned)g(on)f Fw(R)2129 2443 y Fv(2)2200 2473 y Fz(suc)n(h)g(that)h(:)962 2646 y(supp)14 b Fu(\037)23 b Fx(\032)g(B)s Fz(\(0)p Fu(;)14 b Fz(1\))p Fu(;)81 b(\037)23 b Fz(=)g(1)k(on)g Fx(B)2075 2578 y Fp(\000)2113 2646 y Fz(0)p Fu(;)2202 2613 y Fv(1)p 2202 2627 34 4 v 2202 2674 a(2)2245 2578 y Fp(\001)2297 2646 y Fu(;)83 b Fz(0)22 b Fx(\024)h Fu(\037)g Fx(\024)g Fz(1)p Fu(:)268 b Fz(\(11.19\))440 2827 y(W)-7 b(e)28 b(denote)g(b)n(y)f Fu(\037)1018 2839 y Fs(h)1057 2822 y Fe(\032)1104 2827 y Fz(:)51 b(=)23 b Fu(\037)1332 2759 y Fp(\000)1380 2793 y Fs(:)p Ft(\000)p Fs(x)1490 2801 y Fr(0)p 1380 2808 142 4 v 1414 2855 a Fs(h)1453 2839 y Fe(\032)1531 2759 y Fp(\001)1569 2827 y Fz(.)37 b(W)-7 b(e)28 b(no)n(w)f(compute)h Fu(q)2319 2839 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)2498 2847 y Fe(\013)2536 2860 y(j)2575 2827 y Fz(\()p Fu(\037)2659 2839 y Fs(h)2698 2822 y Fe(\032)2737 2827 y Fu(v)2777 2839 y Fs(h)2820 2827 y Fz(\))g(:)550 3063 y Fu(q)587 3075 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)766 3083 y Fe(\013)804 3096 y(j)843 3063 y Fz(\()p Fu(\037)927 3075 y Fs(h)966 3059 y Fe(\032)1005 3063 y Fu(v)1045 3075 y Fs(h)1088 3063 y Fz(\))83 b(=)g Fu(q)1388 3080 y Fs(h;)p Ft(A)1500 3063 y Fe(`in)1588 3080 y Fs(;)p Fv(\012)1655 3088 y Fe(\013)1693 3101 y(j)1732 3063 y Fz(\()p Fu(\037)1816 3075 y Fs(h)1855 3059 y Fe(\032)1893 3063 y Fu(v)1933 3075 y Fs(h)1977 3063 y Fz(\))18 b(+)2110 2950 y Fp(Z)2157 3139 y Fv(\012)2204 3147 y Fe(\013)2242 3160 y(j)2295 3063 y Fx(jA)2384 3029 y Ft(0)2408 3063 y Fx(j)2431 3029 y Fv(2)2468 3063 y Fx(j)p Fu(\037)2543 3075 y Fs(h)2582 3059 y Fe(\032)2620 3063 y Fu(v)2660 3075 y Fs(h)2704 3063 y Fx(j)2727 3029 y Fv(2)2792 3063 y Fu(dx)1351 3351 y Fz(+2)k(Re)h Fu(i)1644 3238 y Fp(Z)1690 3427 y Fv(\012)1737 3435 y Fe(\013)1775 3448 y(j)1872 3248 y Fv(2)1828 3272 y Fp(X)1831 3449 y Fs(j)s Fv(=1)1948 3351 y Fz(\()p Fu(h@)2072 3363 y Fs(x)2110 3371 y Fe(j)2164 3351 y Fx(\000)18 b Fu(iA)2338 3317 y Fs(`in)2338 3372 y(j)2434 3351 y Fz(\)\()p Fu(\037)2550 3363 y Fs(h)2589 3347 y Fe(\032)2628 3351 y Fu(v)2668 3363 y Fs(h)2711 3351 y Fz(\))p 2743 3280 310 4 v Fu(A)2805 3323 y Ft(0)2805 3374 y Fs(j)2841 3351 y Fu(\037)2893 3363 y Fs(h)2932 3347 y Fe(\032)2970 3351 y Fu(v)3010 3363 y Fs(h)3081 3351 y Fu(dx:)440 3613 y Fz(By)24 b(construction)g(of)g Fu(v)1171 3625 y Fs(h)1239 3613 y Fz(and)g(assumptions)g(ab)r(out)g Fu(\037)2152 3625 y Fs(h)2191 3608 y Fe(\032)2254 3613 y Fz(and)g Fx(A)2478 3582 y Ft(0)2502 3613 y Fz(,)h(w)n(e)f(deduce)h (that)g(there)440 3712 y(exists)i(a)g(p)r(ositiv)n(e)h(constan)n(t)e Fu(C)34 b Fz(suc)n(h)28 b(that)g(:)482 3885 y Fu(q)519 3897 y Fs(h;)p Ft(A)p Fs(;)p Fv(\012)698 3905 y Fe(\013)736 3918 y(j)775 3885 y Fz(\()p Fu(\037)859 3897 y Fs(h)898 3881 y Fe(\032)936 3885 y Fu(v)976 3897 y Fs(\036)1021 3885 y Fz(\))23 b Fx(\024)g Fu(hB)t Fz(\()p Fu(S)1362 3897 y Fs(j)1397 3885 y Fz(\))p Fu(\026)p Fz(\()p Fu(\013)1564 3897 y Fs(j)1600 3885 y Fz(\))p Fx(jj)p Fu(\037)1730 3897 y Fs(h)1769 3881 y Fe(\032)1807 3885 y Fu(v)1847 3897 y Fs(h)1890 3885 y Fx(jj)1936 3851 y Fv(2)1936 3908 y Fs(L)1982 3891 y Fr(2)2015 3908 y Fv(\(\012)2088 3916 y Fe(\013)2126 3929 y(j)2161 3908 y Fv(\))2194 3885 y Fz(+)s Fu(C)6 b(h)2375 3851 y Fv(2)p Ft(\000)p Fv(2)p Fs(\032)2533 3885 y Fz(+)s Fu(C)g(h)2714 3851 y Fv(2)2753 3885 y Fz(+)s Fu(C)g(h)2944 3828 y Fr(3)p 2944 3837 29 3 v 2944 3871 a(2)2986 3885 y Fu(:)42 b Fz(\(11.20\))440 4076 y(W)-7 b(e)28 b(no)n(w)f(b)r(ound)h(from)f(b)r(elo)n(w)g(the)h (norm)f(of)h Fu(\037)1950 4088 y Fs(h)1989 4072 y Fe(\032)2027 4076 y Fu(v)2067 4088 y Fs(h)2138 4076 y Fz(:)594 4279 y Fx(jj)p Fu(\037)692 4291 y Fs(h)731 4275 y Fe(\032)770 4279 y Fu(v)810 4291 y Fs(h)853 4279 y Fx(jj)899 4245 y Fv(2)899 4302 y Fs(L)945 4285 y Fr(2)977 4302 y Fv(\(\012)1050 4310 y Fe(\013)1088 4323 y(j)1123 4302 y Fv(\))1177 4279 y Fx(\025)22 b Fz(1)c Fx(\000)g(jj)p Fu(v)1493 4291 y Fs(h)1537 4279 y Fx(jj)1583 4245 y Fv(2)1583 4311 y Fs(L)1629 4294 y Fr(2)1661 4311 y Fv(\()p Ff({)p Ft(B)q Fv(\(0)p Fs(;)1854 4289 y Fe(h)1888 4276 y(\032)p 1854 4298 69 3 v 1875 4331 a Fr(2)1933 4311 y Fv(\))p Ft(\\)p Fv(\012)2051 4319 y Fe(\013)2089 4332 y(j)2124 4311 y Fv(\))2177 4279 y Fx(\025)23 b Fz(1)18 b Fx(\000)g Fu(C)2467 4291 y Fv(1)2532 4279 y Fu(e)2571 4245 y Ft(\000)p Fs(C)2671 4253 y Fr(2)2703 4245 y Fs(h)2742 4217 y Fe(\032)p Fq(\000)2827 4201 y Fr(1)p 2827 4208 29 3 v 2827 4239 a(2)2873 4279 y Fu(;)155 b Fz(\(11.21\))440 4480 y(b)n(y)19 b(using)h(the)g(b)r(eha)n(vior)e(of) h(the)h(eigenfuction)g(men)n(tioned)g(in)g(Section)f(7)g(and)h (particularly)440 4579 y(in)28 b(Theorem)f(7.1.)36 b(Recalling)27 b(\(11.9\),)g(w)n(e)g(deduce)g(:)602 4758 y Fu(q)639 4783 y Fs(h;)720 4768 y Fv(~)698 4783 y Ft(A)p Fs(;)p Fv(\012)822 4758 y Fz(\()p Fu(\037)906 4770 y Fs(h)945 4754 y Fe(\032)984 4758 y Fu(v)1024 4770 y Fs(h)1067 4758 y Fz(\))p 602 4808 498 4 v 608 4884 a Fx(jj)p Fu(\037)706 4896 y Fs(h)745 4880 y Fe(\032)783 4884 y Fu(v)823 4896 y Fs(h)866 4884 y Fx(jj)912 4855 y Fv(2)912 4912 y Fs(L)958 4895 y Fr(2)990 4912 y Fv(\(\012\))1192 4827 y Fx(\024)83 b Fz(\(1)18 b(+)g Fu(C)1574 4839 y Fv(1)1612 4827 y Fu(h)1660 4792 y Fs(\032)1698 4827 y Fz(\))1744 4685 y Fp( )1810 4827 y Fu(hB)t Fz(\()p Fu(S)2008 4839 y Fs(j)2043 4827 y Fz(\))p Fu(\026)p Fz(\()p Fu(\013)2210 4839 y Fs(j)2246 4827 y Fz(\))g(+)g Fu(C)2462 4771 y(h)2510 4740 y Fv(2)p Ft(\000)p Fv(2)p Fs(\032)2685 4771 y Fz(+)g Fu(h)2816 4740 y Fv(2)2872 4771 y Fz(+)g Fu(h)3013 4718 y Fr(3)p 3013 4727 29 3 v 3013 4760 a(2)p 2455 4808 609 4 v 2455 4913 a Fz(1)g Fx(\000)g Fu(C)2657 4925 y Fv(1)2722 4913 y Fu(e)2761 4889 y Ft(\000)p Fs(C)2861 4897 y Fr(2)2892 4889 y Fs(h)2931 4860 y Fe(\032)p Fq(\000)3016 4844 y Fr(1)p 3016 4851 29 3 v 3016 4882 a(2)3073 4685 y Fp(!)1192 5047 y Fx(\024)83 b Fu(hB)t Fz(\()p Fu(S)1538 5059 y Fs(j)1573 5047 y Fz(\))p Fu(\026)p Fz(\()p Fu(\013)1740 5059 y Fs(j)1776 5047 y Fz(\))18 b(+)g Fu(C)6 b Fz(\()p Fu(h)2054 5013 y Fs(\032)p Fv(+1)2196 5047 y Fz(+)18 b Fu(h)2337 4990 y Fr(3)p 2337 4999 V 2337 5033 a(2)2397 5047 y Fz(+)g Fu(h)2528 5013 y Fv(2)p Ft(\000)p Fv(2)p Fs(\032)2685 5047 y Fz(\))p Fu(:)311 b Fz(\(11.22\))440 5220 y(Cho)r(osing)27 b Fu(\032)c Fz(=)966 5187 y Fv(1)p 966 5201 34 4 v 966 5248 a(3)1009 5220 y Fz(,)28 b(w)n(e)f(get)h(:)1178 5413 y Fu(\026)1228 5425 y Fv(1)1265 5413 y Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))83 b Fx(\024)g Fu(hB)t Fz(\()p Fu(S)2007 5425 y Fs(j)2042 5413 y Fz(\))p Fu(\026)p Fz(\()p Fu(\013)2209 5425 y Fs(j)2245 5413 y Fz(\))18 b(+)g Fu(C)6 b(h)2501 5356 y Fr(4)p 2501 5365 29 3 v 2501 5399 a(3)2544 5413 y Fu(:)484 b Fz(\(11.23\))1831 5662 y(36)p eop %%Page: 37 37 37 36 bop 440 531 a Fx(\017)27 b Fz(No)n(w,)g(w)n(e)h(pro)n(v)n(e)e (the)i(upp)r(er)f(b)r(ound)i(2.)36 b(Let)28 b Fu(x)2002 543 y Fv(0)2067 531 y Fz(a)f(p)r(oin)n(t)h(of)g(the)g(b)r(oundary)f (suc)n(h)g(that)440 631 y Fu(B)t Fz(\()p Fu(x)586 643 y Fv(0)624 631 y Fz(\))c(=)g Fu(b)803 601 y Ft(0)826 631 y Fz(.)37 b(Either)27 b Fu(x)1188 643 y Fv(0)1253 631 y Fz(is)g(a)g(regular)f(p)r(oin)n(t,)i(either)f Fu(x)2209 643 y Fv(0)2274 631 y Fz(is)g(a)g(v)n(ertex.)36 b(If)28 b Fu(x)2840 643 y Fv(0)2905 631 y Fz(is)f(a)g(v)n(ertex,)440 731 y(then)34 b(w)n(e)g(tak)n(e)f(accoun)n(t)g(of)h Fu(B)t Fz(\()p Fu(x)1510 743 y Fv(0)1548 731 y Fz(\))f(=)g Fu(b)1747 700 y Ft(0)1770 731 y Fz(,)j(\(11.23\),)e(the)g(assumption)f Fu(\026)p Fz(\()p Fu(\013)p Fz(\))i Fu(<)e Fz(\002)3100 743 y Fv(0)3171 731 y Fz(and)440 830 y(deduce)28 b(:)1341 982 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))83 b Fx(\024)g Fu(hb)2019 948 y Ft(0)2041 982 y Fz(\002)2106 994 y Fv(0)2162 982 y Fz(+)18 b Fu(ch)2339 926 y Fr(4)p 2338 935 29 3 v 2338 968 a(3)2381 982 y Fu(:)647 b Fz(\(11.24\))440 1134 y(Let)19 b(us)f(no)n(w)g(assume)f(that)i Fu(x)1336 1146 y Fv(0)1392 1134 y Fz(is)g(not)f(a)g(v)n(ertex.)33 b(W)-7 b(e)19 b(use)f(the)h(same)f(tec)n(hniques)g(as)g(Hel\013er-)440 1234 y(Morame)25 b([11],)i(p.)g(648.)35 b(By)26 b(a)g(c)n(hange)g(of)g (v)-5 b(ariables,)26 b(w)n(e)g(ha)n(v)n(e)g(a)g(small)g(p)r (erturbation)g(of)440 1333 y(the)32 b(sesquilinear)d(form)i(on)g(the)h (half-plane)e(and)h(w)n(e)g(use)g(the)g(follo)n(wing)g(function)g(as)g (a)440 1433 y(trial)c(function)h(:)518 1643 y Fx(8)p Fu(x)22 b Fx(2)i Fw(R)g Fx(\002)18 b Fw(R)928 1608 y Fv(+)989 1643 y Fu(;)42 b(v)s Fz(\()p Fu(x)p Fz(\))24 b(=)f Fu(h)1368 1608 y Ft(\000)1429 1586 y Fr(1)p 1429 1595 V 1429 1628 a(8)1472 1643 y Fu(e)1511 1603 y Fs(i)1534 1532 y Fa(q)p 1599 1532 69 3 v 1609 1581 a Fe(b)1635 1569 y Fq(0)p 1609 1590 49 3 v 1616 1623 a Fe(h)1668 1603 y Fs(\020)1698 1611 y Fr(0)1730 1603 y Fs(x)1768 1611 y Fr(1)1818 1526 y Fp(\022)1890 1586 y Fu(b)1926 1556 y Ft(0)p 1890 1623 59 4 v 1896 1700 a Fu(h)1958 1526 y Fp(\023)2030 1520 y Fr(1)p 2030 1529 29 3 v 2030 1563 a(4)2086 1643 y Fu(\036)2149 1501 y Fp( )2215 1510 y(r)p 2298 1510 79 4 v 2308 1586 a Fu(b)2344 1562 y Ft(0)p 2308 1623 59 4 v 2314 1700 a Fu(h)2377 1643 y(x)2424 1655 y Fv(2)2461 1501 y Fp(!)2569 1643 y Fu(\037)p Fz(\()p Fu(x)2700 1655 y Fv(2)2737 1643 y Fz(\))28 b Fu(g)2854 1550 y Fp(\020)2903 1643 y Fu(h)2951 1608 y Ft(\000)3013 1586 y Fr(1)p 3013 1595 29 3 v 3013 1628 a(4)3055 1643 y Fu(x)3102 1655 y Fv(1)3140 1550 y Fp(\021)3204 1643 y Fu(;)3051 1817 y Fz(\(11.25\))440 1917 y(with)k Fu(g)f Fx(2)e Fu(C)853 1886 y Ft(1)847 1937 y Fv(0)924 1917 y Fz(\()p Fw(R)p Fz(\))38 b Fu(L)1137 1886 y Fv(2)1173 1917 y Fz(-normalized)30 b(supp)r(orted)h(b)n(y)g(])21 b Fx(\000)2281 1884 y Fv(1)p 2281 1898 34 4 v 2281 1945 a(2)2324 1917 y Fu(;)2371 1884 y Fv(1)p 2371 1898 V 2371 1945 a(2)2414 1917 y Fz([)31 b(and)g Fu(\037)e Fx(2)g Fu(C)2863 1886 y Ft(1)2857 1937 y Fv(0)2934 1917 y Fz(\()p Fw(R)3020 1886 y Fv(+)3081 1917 y Fz(\))j(suc)n(h)440 2016 y(that)c(:)1258 2145 y Fu(\037)p Fz(\()p Fu(x)1389 2157 y Fv(2)1426 2145 y Fz(\))c(=)1569 2028 y Fp(\032)1673 2094 y Fz(1)110 b(if)i(0)22 b Fx(\024)h Fu(x)2184 2106 y Fv(2)2245 2094 y Fx(\024)2342 2060 y Fs(\017)2370 2068 y Fr(0)p 2342 2075 61 4 v 2356 2123 a Fv(2)2412 2094 y Fu(;)1673 2194 y Fz(0)110 b(if)i Fu(x)2032 2206 y Fv(2)2092 2194 y Fx(\025)23 b Fu(\017)2214 2206 y Fv(0)2251 2194 y Fu(;)440 2321 y Fz(and)k Fu(\036;)14 b(\020)723 2333 y Fv(0)789 2321 y Fz(de\014ned)28 b(b)n(y)f(Prop)r(osition)f(2.1.)36 b(W)-7 b(e)28 b(obtain)g(the)g(upp)r(er)f(b)r(ound)h(:)1281 2473 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(h)p Fz(\002)1868 2485 y Fv(0)1905 2473 y Fu(B)t Fz(\()p Fu(x)2051 2485 y Fv(0)2089 2473 y Fz(\))c(+)f Fu(C)6 b(h)2336 2439 y Fv(3)p Fs(=)p Fv(2)2440 2473 y Fu(:)588 b Fz(\(11.26\))440 2625 y(Com)n(bining)27 b(\(11.24\))f(and)i (\(11.26\),)e(w)n(e)h(deduce)h(:)1360 2777 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(h)p Fz(\002)1947 2789 y Fv(0)1984 2777 y Fu(b)2020 2743 y Ft(0)2061 2777 y Fz(+)18 b Fu(C)6 b(h)2257 2743 y Fv(4)p Fs(=)p Fv(3)2361 2777 y Fu(:)667 b Fz(\(11.27\))440 2929 y Fx(\017)30 b Fz(Let)g(us)h(analyze)e(the)i(upp)r(er)f(b)r(ound)h(3.) 44 b(Let)31 b Fu(x)2026 2941 y Fv(0)2091 2929 y Fx(2)p 2174 2862 60 4 v 28 w Fz(\012)f(suc)n(h)g(that)h Fu(B)t Fz(\()p Fu(x)2783 2941 y Fv(0)2821 2929 y Fz(\))d(=)f Fu(b)p Fz(.)45 b(Either)440 3029 y Fu(x)487 3041 y Fv(0)556 3029 y Fx(2)32 b Fz(\012,)h(either)g Fu(x)1047 3041 y Fv(0)1116 3029 y Fx(2)e Fu(@)5 b Fz(\012.)52 b(If)33 b Fu(x)1521 3041 y Fv(0)1590 3029 y Fx(2)e Fu(@)5 b Fz(\012,)34 b(then)f Fu(b)e Fz(=)f Fu(b)2234 2998 y Ft(0)2257 3029 y Fz(.)52 b(Since)33 b(w)n(e)f(ha)n(v)n(e)f(pro)n(v)n(ed)g(that)440 3128 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(hb)998 3098 y Ft(0)1021 3128 y Fz(\002)1086 3140 y Fv(0)1141 3128 y Fz(+)18 b Fu(C)6 b(h)1337 3098 y Fv(4)p Fs(=)p Fv(3)1469 3128 y Fz(and)28 b(since)f(\002)1899 3140 y Fv(0)1959 3128 y Fu(<)c Fz(1,)k(w)n(e)g(deduce)h(that)g(:)1422 3280 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))24 b Fx(\024)e Fu(hb)c Fz(+)g Fu(C)6 b(h)2194 3246 y Fv(4)p Fs(=)p Fv(3)2299 3280 y Fu(:)729 b Fz(\(11.28\))440 3459 y(Let)28 b(us)g(no)n(w)f(assume)g Fu(x)1203 3471 y Fv(0)1264 3459 y Fx(2)d Fz(\012,)k(then)g(there)f(exists)h Fu(h)2133 3471 y Fv(0)2193 3459 y Fu(>)23 b Fz(0)28 b(suc)n(h)f(that)h Fx(B)2789 3367 y Fp(\020)2839 3459 y Fu(x)2886 3471 y Fv(0)2923 3459 y Fu(;)14 b(h)3018 3388 y Fr(1)p 3018 3397 29 3 v 3018 3431 a(2)3008 3481 y Fv(0)3060 3367 y Fp(\021)3133 3459 y Fx(\032)23 b Fz(\012.)440 3584 y(Let)28 b Fu(h)23 b Fx(2)p Fz(]0)p Fu(;)14 b(h)865 3596 y Fv(0)902 3584 y Fz(].)37 b(W)-7 b(e)28 b(consider)e(the)i(function)g Fu(u)f Fz(de\014ned)h(on)g(\012)f(b)n(y)h(:)655 3807 y Fu(u)p Fz(\()p Fu(x)p Fz(\))c(=)926 3672 y Fp(r)p 1009 3672 237 4 v 1019 3750 a Fu(B)t Fz(\()p Fu(x)1165 3762 y Fv(0)1203 3750 y Fz(\))p 1019 3788 217 4 v 1057 3864 a(4)p Fu(\031)s(h)1273 3807 y Fz(exp)1413 3690 y Fp(\022)1484 3750 y Fx(\000)p Fu(B)t Fz(\()p Fu(x)1695 3762 y Fv(0)1733 3750 y Fz(\))p Fx(j)p Fu(x)19 b Fx(\000)f Fu(x)1984 3762 y Fv(0)2022 3750 y Fx(j)2045 3720 y Fv(2)p 1484 3788 598 4 v 1739 3864 a Fz(4)p Fu(h)2092 3690 y Fp(\023)2167 3807 y Fu(;)47 b Fz(~)-47 b Fu(u)22 b Fz(=)h(exp)2531 3690 y Fp(\022)2592 3807 y Fu(i)2631 3750 y(\036)p 2631 3788 50 4 v 2632 3864 a(h)2690 3690 y Fp(\023)2765 3807 y Fu(u;)215 b Fz(\(11.29\))440 4004 y(with)21 b Fu(\036)g Fz(whic)n(h)f(realizes)f(a)h(gauge)f(transform)g(suc)n(h)h(that)g(the)h (linear)f(expansion)f(of)h Fx(A)t Fz(+)t Fx(r)p Fu(\036)440 4103 y Fz(is)27 b(equal)h(to)f Fx(A)910 4115 y Fv(0)948 4103 y Fz(.)440 4203 y(W)-7 b(e)41 b(use)g(a)f(cut-o\013)g(function)i (to)e(lo)r(calize)45 b(~)-47 b Fu(u)40 b Fz(in)h Fx(B)s Fz(\()p Fu(x)2202 4215 y Fv(0)2239 4203 y Fu(;)14 b(h)2334 4151 y Fr(1)p 2334 4160 29 3 v 2334 4193 a(2)2376 4203 y Fz(\))41 b(and)g(w)n(e)f(mak)n(e)g(a)g(T)-7 b(a)n(ylor)440 4303 y(expansion)27 b(of)g Fx(A)h Fz(at)g(the)g(\014rst)f(order)f(as)h (men)n(tioned)h(in)g(\(11.12\).)35 b(W)-7 b(e)28 b(also)f(get)g(:)1044 4455 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))24 b Fx(\024)e Fu(hB)t Fz(\()p Fu(x)1712 4467 y Fv(0)1750 4455 y Fz(\))d(+)f Fu(C)1943 4467 y Fv(0)1980 4455 y Fu(h)2028 4420 y Fv(3)p Fs(=)p Fv(2)2156 4455 y Fz(=)k Fu(hb)c Fz(+)g Fu(C)2487 4467 y Fv(0)2525 4455 y Fu(h)2573 4420 y Fv(3)p Fs(=)p Fv(2)2677 4455 y Fu(;)351 b Fz(\(11.30\))440 4607 y(with)32 b Fu(C)692 4619 y Fv(0)760 4607 y Fz(a)f(p)r(ositiv)n(e) f(constan)n(t)h(indep)r(enden)n(t)h(of)f Fu(h)p Fz(.)47 b(Thanks)30 b(to)h(\(11.28\))f(and)h(\(11.30\),)440 4706 y(w)n(e)c(obtain)h(:)1407 4806 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))23 b Fx(\024)g Fu(hb)18 b Fz(+)g Fu(C)2125 4818 y Fv(0)2162 4806 y Fu(h)2210 4772 y Fv(3)p Fs(=)p Fv(2)2314 4806 y Fu(;)714 b Fz(\(11.31\))440 4937 y(T)-7 b(aking)40 b(accoun)n(t)g(of)h(Relations)g(\(11.23\),)i (\(11.27\))d(and)h(\(11.31\),)i(w)n(e)e(get)g(the)g(upp)r(er)440 5036 y(b)r(ound)28 b(:)758 5217 y Fu(\026)p Fz(\()p Fu(h;)14 b(B)t(;)g Fz(\012\))83 b Fx(\024)g Fu(h)14 b Fz(inf)1528 5100 y Fp(\022)1589 5217 y Fu(b;)g Fz(\002)1727 5229 y Fv(0)1764 5217 y Fu(b)1800 5183 y Ft(0)1823 5217 y Fu(;)102 b Fz(inf)1860 5272 y Fs(k)q Fv(=1)p Fs(;:::)o(;N)2151 5217 y Fu(\026)p Fz(\()p Fu(\013)2286 5229 y Fs(k)2328 5217 y Fz(\))p Fu(B)t Fz(\()p Fu(S)2510 5229 y Fs(k)2551 5217 y Fz(\))2583 5100 y Fp(\023)2663 5217 y Fz(+)18 b Fu(C)6 b(h)2859 5183 y Fv(4)p Fs(=)p Fv(3)2964 5217 y Fu(:)64 b Fz(\(11.32\))3240 5413 y Fd(\003)1831 5662 y Fz(37)p eop %%Page: 38 38 38 37 bop 440 531 a FA(Remark)32 b(11.6.)40 b Fo(It)29 b(se)l(ems)i(that)f(the)g(assumption)h Fu(\026)p Fz(\()p Fu(\013)p Fz(\))25 b Fu(<)e Fz(\002)2461 543 y Fv(0)2528 531 y Fo(holds)32 b(for)f(any)g(angular)440 631 y Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b(\031)s Fz([)24 b Fo(but)e(it)h(is)h(not)f (pr)l(ove)l(d)h(for)g(the)f(moment.)36 b(If)24 b(we)f(do)h(not)f(make)h (this)f(assumption,)440 731 y(the)k(r)l(esult)e(stil)t(l)i(holds)h (with)f(a)g(worse)g(r)l(emainder)g(but)f(we)g(have)i(a)f(fourth)f(c)l (ase)h(to)g(study)f(:)440 830 y(the)d(c)l(ase)g(of)g(a)g(vertex)f Fu(S)1186 842 y Fs(j)1244 830 y Fo(with)h(angle)g Fu(\013)1679 842 y Fs(j)1737 830 y Fu(<)g(\031)j Fo(and)d Fu(\026)p Fz(\()p Fu(\013)2187 842 y Fs(j)2222 830 y Fz(\))h(=)e(\002)2430 842 y Fv(0)2490 830 y Fo(as)57 b Fz(inf)2589 884 y Fs(x)p Ft(2)p Fs(@)t Fv(\012)2771 830 y Fu(B)t Fz(\()p Fu(x)p Fz(\))25 b(=)d Fu(B)t Fz(\()p Fu(S)3211 842 y Fs(j)3247 830 y Fz(\))p Fo(.)440 963 y(L)l(et)35 b(us)g(shortly)i(explain)g(how)f (we)g(c)l(an)g(tr)l(e)l(at)f(this)h(c)l(ase.)57 b(L)l(et)36 b(us)f(c)l(onsider)i Fu(T)2976 975 y Fs(j)3046 963 y Fo(a)f(p)l(oint)440 1074 y(of)j(the)f(r)l(e)l(gular)g(b)l(oundary)h (such)f(that)g Fu(d)p Fz(\()p Fu(S)1843 1086 y Fs(j)1879 1074 y Fu(;)14 b(T)1965 1086 y Fs(j)1999 1074 y Fz(\))38 b(=)g Fu(h)2230 1022 y Fr(1)p 2230 1031 29 3 v 2230 1064 a(4)2272 1074 y Fo(,)j(we)d(lo)l(c)l(alize)i(in)e(a)g(smal)t(l)h(b)l (al)t(l)440 1174 y(ar)l(ound)27 b Fu(T)765 1186 y Fs(j)825 1174 y Fo(which)h(don)-8 b('t)27 b(me)l(et)f(c)l(orner)g(or)h(other)g (pie)l(c)l(e)g(of)g(the)g(b)l(oundary.)38 b(After)26 b(change)440 1274 y(of)37 b(variables)i(ne)l(ar)d Fu(T)1139 1286 y Fs(j)1210 1274 y Fo(to)g(send)g(lo)l(c)l(al)t(ly)j Fz(\012)d Fo(onto)g Fw(R)25 b Fx(\002)18 b Fw(R)2280 1243 y Fv(+)2341 1274 y Fo(,)38 b(we)f(c)l(onsider)g(inste)l(ad)g(of)g Fu(v)440 1373 y Fo(de\014ne)l(d)30 b(by)g(\(11.25\))i(the)e(new)g (function)f(for)i Fu(x)24 b Fx(2)f Fw(R)h Fx(\002)18 b Fw(R)2252 1343 y Fv(+)2343 1373 y Fo(:)470 1593 y Fu(v)s Fz(\()p Fu(x)p Fz(\))24 b(=)f Fu(e)775 1559 y Fs(i)798 1480 y Fa(q)p 863 1480 176 3 v 873 1518 a Fe(B)r Fr(\()p Fe(T)975 1531 y(j)1006 1518 y Fr(\))p 873 1545 156 3 v 934 1579 a Fe(h)1039 1559 y Fs(\020)1069 1567 y Fr(0)1101 1559 y Fs(x)1139 1567 y Fr(1)1189 1476 y Fp(\022)1260 1537 y Fu(B)t Fz(\()p Fu(T)1408 1549 y Fs(j)1443 1537 y Fz(\))p 1260 1574 216 4 v 1344 1650 a Fu(h)1486 1476 y Fp(\023)1557 1471 y Fr(1)p 1557 1480 29 3 v 1557 1513 a(4)1613 1593 y Fu(\036)1676 1451 y Fp( )1742 1458 y(r)p 1825 1458 236 4 v 1835 1537 a Fu(B)t Fz(\()p Fu(T)1983 1549 y Fs(j)2018 1537 y Fz(\))p 1835 1574 216 4 v 1919 1650 a Fu(h)2060 1593 y(x)2107 1605 y Fv(2)2145 1451 y Fp(!)2254 1593 y Fu(g)2294 1605 y Fv(2)2345 1476 y Fp(\022)2424 1537 y Fu(x)2471 1549 y Fv(2)p 2416 1574 101 4 v 2416 1662 a Fu(h)2474 1610 y Fr(3)p 2474 1619 29 3 v 2474 1653 a(8)2526 1476 y Fp(\023)2631 1593 y Fu(h)2679 1559 y Ft(\000)2754 1536 y Fr(3)p 2740 1545 57 3 v 2740 1579 a(16)2811 1593 y Fu(g)2851 1605 y Fv(1)2902 1501 y Fp(\020)2951 1593 y Fu(h)2999 1559 y Ft(\000)3061 1536 y Fr(3)p 3061 1545 29 3 v 3061 1579 a(8)3103 1593 y Fu(x)3150 1605 y Fv(1)3188 1501 y Fp(\021)3251 1593 y Fu(;)3051 1767 y Fz(\(11.33\))440 1886 y Fo(with)30 b Fu(g)660 1898 y Fv(1)720 1886 y Fx(2)24 b Fu(C)864 1855 y Ft(1)858 1906 y Fv(0)934 1886 y Fz(\(])19 b Fx(\000)1101 1853 y Fv(1)p 1101 1867 34 4 v 1101 1914 a(2)1144 1886 y Fu(;)1191 1853 y Fv(1)p 1191 1867 V 1191 1914 a(2)1234 1886 y Fz([)p Fu(;)14 b Fw(R)p Fz(\))36 b Fu(L)1473 1855 y Fv(2)1510 1886 y Fo(-normalize)l(d)31 b(and)f Fu(g)2162 1898 y Fv(2)2222 1886 y Fx(2)23 b Fu(C)2365 1855 y Ft(1)2359 1906 y Fv(0)2436 1886 y Fz(\()p 2468 1818 60 4 v Fw(R)2528 1832 y Fv(+)2583 1886 y Fz(\))30 b Fo(such)g(that)g(:)1267 2085 y Fu(g)1307 2097 y Fv(2)1344 2085 y Fz(\()p Fu(x)1423 2097 y Fv(2)1461 2085 y Fz(\))23 b(=)1604 1968 y Fp(\032)1708 2034 y Fz(1)112 b Fo(if)i Fz(0)22 b Fx(\024)h Fu(x)g Fx(\024)2346 2002 y Fv(1)p 2346 2016 34 4 v 2346 2063 a(2)1708 2134 y Fz(0)112 b Fo(if)i Fu(x)2073 2146 y 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b Fx(2)p Fz(]0)p Fu(;)14 b(\031)s Fz([,)141 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\025)f Fz(\002)2054 5322 y Fv(0)2101 5277 y Fs(\013)p 2101 5291 44 4 v 2102 5338 a(\031)2154 5310 y Fz(,)586 b(\(4\))490 5409 y(Prop)r(osition)26 b(4.2)h(:)99 b Fx(8)p Fu(\013)23 b Fx(2)p Fz(]0)p Fu(;)14 b Fz(2)p Fu(\031)s Fz([,)99 b Fu(\026)p Fz(\()p Fu(\013)p Fz(\))24 b Fx(\024)2084 5377 y Fs(\013)p 1999 5391 215 4 v 1999 5399 a Ft(p)p 2053 5399 160 3 v 2053 5447 a Fv(3+)p Fs(\013)2180 5430 y Fr(2)2223 5409 y Fz(.)517 b(\(5\))1831 5662 y(38)p eop %%Page: 39 39 39 38 bop 677 1866 a @beginspecial 61 @llx 200 @lly 548 @urx 589 @ury 2834 @rwi 1700 @rhi @setspecial %%BeginDocument: ../../figthese/locmualphaSoeren.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /home/virginie/figthese/locmualphaSoeren.eps %%CreationDate: 11/05/2003 18:36:14 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 61 200 548 589 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/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath 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def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef 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