Content-Type: multipart/mixed; boundary="-------------0211261039767" This is a multi-part message in MIME format. ---------------0211261039767 Content-Type: text/plain; name="02-490.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-490.keywords" density, atomic eigenfunctions, quantum mechanics ---------------0211261039767 Content-Type: application/postscript; name="analytic.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="analytic.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: analytic.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o analytic.ps analytic %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2002.11.26:1619 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup 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 /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image N /rw ch-width 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 dup 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 dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 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}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /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 TeXDict begin 39158280 55380996 1000 300 300 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DI<487EA41203A21207A2120F123FB5FCA2EA0F80ABEB8180A5EB8300EA07C3EA03 FEEA00F811207F9F16>I<39FFE01FE0A2391F800700000F1306EBC00E0007130C13E000 035BA26C6C5AA26C6C5AA2EB7CC0A2137F6D5AA26DC7FCA2130EA21B167F951E>118 D<3AFFE3FF07F8A23A1F007800C09038807C01000F1580A23A07C07E030014DE5D3903E1 DF06148FD801F1138CEBF307A2D800FF13D8EBFE0315F890387C01F0A2013C5BEB3800A2 25167F9528>I<39FFE07FC0A2390F801C006C6C5A6C6C5AEBF0606C6C5A3800F980137F 6DC7FC7F80497E1337EB63E0EBC1F03801C0F848487E3807007E000E133E39FF80FFE0A2 1B167F951E>I<39FFE01FE0A2391F800700000F1306EBC00E0007130C13E000035BA26C 6C5AA26C6C5AA2EB7CC0A2137F6D5AA26DC7FCA2130EA2130CA25B1278EAFC3813305BEA 69C0EA7F80001FC8FC1B207F951E>I<387FFFF0A2387C03E0387007C0EA600F38E01F80 00C01300133E137EC65A5B485A00031330EA07E013C0380F8070121F383F0060003E13E0 EA7C03B5FCA214167E9519>I E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 261 345 a Fy(ANAL)-5 b(YTICITY)25 b(OF)g(THE)f(DENSITY)g(OF)g (ELECTR)n(ONIC)722 404 y(W)-6 b(A)g(VEFUNCTIONS)242 528 y Fx(S.)17 b(F)o(OURNAIS,)h(M.)g(HOFFMANN-OSTENHOF,)h(T.)f (HOFFMANN-OSTENHOF,)616 578 y(AND)g(T.)g(\037STER)o(GAARD)f (S\037RENSEN)377 698 y Fw(Abstra)o(ct.)j Fx(W)m(e)15 b(pro)o(v)o(e)g(that)h(the)f(electronic)i(densities)f(of)e(atomic)g (and)377 748 y(molecular)19 b(eigenfunctions)h(are)h(real)f(analytic)f (in)h Fv(R)1265 733 y Fu(3)1301 748 y Fx(a)o(w)o(a)o(y)f(from)f(the)377 798 y(n)o(uclei.)385 1016 y Ft(1.)27 b Fs(Intr)o(oduction)19 b(and)f(st)m(a)m(tement)g(of)g(the)h(resul)m(ts)278 1103 y Ft(W)l(e)f(consider)g(an)h Fr(N)5 b Ft(-electron)18 b(molecule)d(with)j Fr(L)h Ft(\014xed)f(n)o(uclei.)26 b(The)18 b(non-)228 1161 y(relativistic)c(Hamiltonian)g(of)j(the)f (molecule)d(is)j(giv)o(en)g(b)o(y)269 1309 y Fr(H)i Ft(=)c Fr(H)419 1316 y Fq(N)r(;L)485 1309 y Ft(\()p Fy(R)p Fr(;)8 b Fy(Z)p Ft(\))14 b(=)707 1247 y Fq(N)686 1262 y Fp(X)692 1367 y Fq(j)r Fo(=1)767 1224 y Fp( )817 1309 y Fn(\000)d Ft(\001)908 1316 y Fq(j)937 1309 y Fn(\000)1011 1247 y Fq(L)987 1262 y Fp(X)995 1368 y Fq(l)p Fo(=1)1141 1275 y Fr(Z)1174 1282 y Fq(l)p 1072 1298 185 2 v 1072 1343 a Fn(j)p Fr(x)1114 1350 y Fq(j)1143 1343 y Fn(\000)g Fr(R)1230 1350 y Fq(l)1243 1343 y Fn(j)1262 1224 y Fp(!)726 1465 y Ft(+)819 1417 y Fp(X)775 1523 y Fo(1)p Fm(\024)p Fq(i)d Ft(0)j(suc)o(h)f(that) g(for)h(all)f Fr(u)d Fn(2)h Fr(W)1256 2492 y Fo(1)p Fq(;)p Fo(2)1303 2510 y Ft(\()p Fl(R)1361 2492 y Fo(3)p Fq(N)1409 2510 y Ft(\),)660 2588 y Fn(k)p Fr(V)d(u)p Fn(k)777 2598 y Fq(L)801 2588 y Fg(2)818 2598 y Fo(\()p Fh(R)859 2588 y Fg(3)p Ff(N)900 2598 y Fo(\))930 2588 y Fn(\024)i Fr(C)t Fn(k)p Fr(u)p Fn(k)1099 2598 y Fq(W)1137 2588 y Fg(1)p Ff(;)p Fg(2)1179 2598 y Fo(\()p Fh(R)1219 2588 y Fg(3)p Ff(N)1260 2598 y Fo(\))1276 2588 y Fr(:)308 b Ft(\(1.11\))p eop %%Page: 5 5 5 4 bop 300 118 a Fj(ANAL)m(YTICITY)14 b(OF)j(THE)f(DENSITY)h(OF)f (ELECTR)o(ONIC)g(W)l(A)l(VEFUNCTIONS)52 b(5)253 213 y Ft(\(2\))21 b(There)c(exists)g(a)g(constan)o(t)h Fr(L)d(>)h Ft(0)h(\(dep)q(ending)g(on)h Fr(")p Ft(\))f(suc)o(h)g(that)g(for)h(all) 336 271 y Fr(\013)c Fn(2)h Fl(N)464 253 y Fo(3)484 271 y Ft(,)h(w)o(e)g(ha)o(v)o(e)527 339 y Fq(N)506 354 y Fp(X)512 459 y Fq(j)r Fo(=1)587 401 y Fn(k)p Fr(@)641 381 y Fq(\013)665 401 y Fr(V)693 408 y Fq(j)711 401 y Fn(k)11 b Ft(+)861 339 y Fq(N)841 354 y Fp(X)796 460 y Fq(j;k)q Fo(=1)p Fq(;j)r Fm(6)p Fo(=)p Fq(k)966 401 y Fn(k)p Fr(@)1020 381 y Fq(\013)1044 401 y Fr(W)1090 408 y Fq(j;k)1135 401 y Fn(k)j(\024)g Fr(L)1260 381 y Fm(j)p Fq(\013)p Fm(j)p Fo(+1)1349 401 y Fn(j)p Fr(\013)p Fn(j)p Ft(!)p Fr(;)162 b Ft(\(1.12\))336 544 y(where)16 b(the)g(norms)g(in)g(\(1.12\))h(are)f(in)g Fr(L)1077 526 y Fm(1)1123 504 y Fp(\000)o(\010)1175 544 y Fr(x)d Fn(2)h Fl(R)1302 526 y Fo(3)1327 502 y Fp(\014)1327 532 y(\014)1352 544 y Fn(j)p Fr(x)p Fn(j)f Fr(>)h(")1496 504 y Fp(\011\001)1548 544 y Ft(.)228 617 y(The)k(\014rst)h(condition,) f(\(1.11\))q(,)g(is)h(a)f(kind)g(of)h(relativ)o(e)e(b)q(oundedness)i (assump-)228 675 y(tion.)i(The)15 b(second)g(condition,)g(\(1.12\))q(,) g(means)f(that)i Fr(V)26 b Ft(is)15 b(real)g(analytic)g(a)o(w)o(a)o(y) 228 733 y(from)i(\006)h(\(with)g(a)h(uniformit)o(y)c(at)k(in\014nit)o (y\).)25 b(Theorem)17 b(1.1)h(remains)f(true)h(for)228 791 y(an)o(y)13 b Fr(V)24 b Ft(satisfying)13 b(these)g(t)o(w)o(o)g (assumptions)g(.)20 b(F)l(or)13 b(instance,)g(replacing)g(one)g(or)228 849 y(more)c(of)j(the)e(Coulom)o(b)g(p)q(oten)o(tials)h(in)g Fr(V)22 b Ft(b)o(y)10 b(the)h(Y)l(uk)m(a)o(w)o(a)g(p)q(oten)o(tial,)g Fr(e)1551 831 y Fm(\000)p Fq(\013)p Fm(j)p Fq(x)p Fm(j)1642 849 y Fr(=)p Fn(j)p Fr(x)p Fn(j)228 907 y Ft(\(with)19 b Fr(\013)h(>)f Ft(0\),)i(w)o(e)e(still)f(get)i(Theorem)e(1.1.)32 b(But)19 b(here)g(w)o(e)g(concen)o(trate)g(on)228 965 y(the)c(ph)o(ysically)g(imp)q(ortan)o(t)g(case)g(of)i(Coulom)o(b)d(p)q (oten)o(tials)i(and)h(do)f(not)g(striv)o(e)228 1023 y(for)g(generalit)o (y)l(.)278 1114 y Fd(Or)n(ganisation)h(of)h(the)g(p)n(ap)n(er:)228 1172 y 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y(partition)i(of)g(unit)o(y)g(of)g Fl(R)720 2026 y Fo(3)p Fq(N)787 2044 y Ft(and)h(then)f(pro)q(ceed)g(b)o (y)g(a)g(similar)e(construction)228 2102 y(as)f(in)g([1])g(whic)o(h)f (together)h(with)g(Corollary)g(3.2)h(implies)c(Lemma)h(4.3,)i(in)f (par-)228 2160 y(ticular)j(\(4.10\).)29 b(Once)18 b(Lemma)e(4.3)j(is)f (pro)o(v)o(ed,)g(Prop)q(osition)i(4.1)f(follo)o(ws)g(b)o(y)228 2218 y(easy)d(argumen)o(ts.)622 2335 y(2.)27 b Fs(Basic)20 b(f)l(a)o(cts)e(and)g(not)m(a)m(tion)278 2422 y Ft(Remem)n(b)q(er)13 b(that)k(for)f(m)o(ultiindice)o(s)e Fr(\013)g Ft(=)g(\()p Fr(a)1117 2429 y Fo(1)1136 2422 y Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(a)1280 2429 y Fo(3)p Fq(M)1336 2422 y Ft(\))14 b Fn(2)g Fl(N)1451 2404 y Fo(3)p Fq(M)1509 2422 y Ft(,)844 2555 y Fn(j)p Fr(\013)p Fn(j)g Ft(=)977 2493 y Fo(3)p Fq(M)968 2508 y Fp(X)974 2613 y Fq(j)r Fo(=1)1049 2555 y Fr(a)1075 2562 y Fq(j)1093 2555 y Fr(:)p eop %%Page: 6 6 6 5 bop 228 120 a Fj(6)70 b(S.)17 b(F)o(OURNAIS,)f(M.)g(AND)h(T.)f 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y(\014)949 372 y Fn(j)p Fr(x)991 379 y Fq(j)1009 372 y Fn(j)f Fr(>)h(")j Fd(for)g Fr(j)g Fn(2)d Fr(P)1321 379 y Fq(s)1340 372 y Fr(;)866 481 y Fn(j)p Fr(x)908 488 y Fq(j)937 481 y Fn(\000)d Fr(x)1015 488 y Fq(k)1036 481 y Fn(j)i Fr(>)h(")j Fd(for)g Fr(j)g Fn(2)d Fr(P)1348 488 y Fq(s)1367 481 y Fr(;)8 b(k)16 b Fn(2)e Fr(Q)1516 488 y Fq(s)1534 425 y Fp(o)1567 481 y Fr(:)41 b Ft(\(3.1\))228 625 y Fd(Denote)696 757 y Fr(U)729 764 y Fq(P)751 769 y Fg(1)769 764 y Fq(;:::)5 b(;P)846 770 y Ff(M)882 757 y Ft(\()p Fr(")p Ft(\))14 b(=)f Fn(\\)1041 737 y Fq(M)1041 770 y(s)p Fo(=1)1105 757 y Fr(U)1138 764 y Fq(P)1160 768 y Ff(s)1179 757 y Ft(\()p Fr(")p Ft(\))p Fr(:)368 b Ft(\(3.2\))228 893 y Fd(Then)20 b(ther)n(e)f(exist)h Fr(C)q(;)8 b(L)19 b Fd(\(dep)n(ending)i(on)f Fr(")p Fd(\))f(such)g(that)h(for)f(al)r(l)h (multiindic)n(es,)228 951 y Fr(\013)14 b Ft(=)g(\()p Fr(\013)375 958 y Fo(1)394 951 y Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(\013)543 958 y Fq(M)583 951 y Ft(\))13 b Fn(2)h Fl(N)697 933 y Fo(3)p 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y(Let)i(us)g(in)o(tro)q(duce)f(the)h (follo)o(wing)f(practical)g(notation.)27 b(F)l(or)18 b(a)h(m)o(ultiinde)o(x)228 1444 y Fr(\013)f Ft(=)f(\()p Fr(\013)382 1451 y Fo(1)402 1444 y Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(\013)551 1451 y Fq(M)590 1444 y Ft(\))18 b Fn(2)g Fl(N)713 1426 y Fo(3)p Fq(M)789 1444 y Ft(and)h(giv)o(en)f Fr(P)1047 1451 y Fo(1)1067 1444 y Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(P)1216 1451 y Fq(M)1274 1444 y Ft(as)19 b(in)f(Lemma)f(3.1,)i (w)o(e)228 1502 y(de\014ne)d Fr(@)398 1484 y Fq(\013)395 1515 y(x)415 1521 y Fc(P)458 1502 y Ft(and)h Fr(U)586 1509 y Fb(P)616 1502 y Ft(\()p Fr(")p Ft(\))f(b)o(y)488 1636 y Fr(@)517 1615 y Fq(\013)514 1648 y(x)534 1654 y Fc(P)575 1636 y Ft(=)e Fr(@)656 1615 y Fq(\013)679 1620 y Fg(1)653 1648 y Fq(x)673 1654 y Ff(P)692 1661 y Fg(1)722 1636 y Fn(\001)8 b(\001)g(\001)g Fr(@)817 1615 y Fq(\013)840 1621 y Ff(M)814 1648 y Fq(x)834 1654 y Ff(P)853 1662 y(M)891 1636 y Fr(;)107 b(U)1045 1643 y Fb(P)1075 1636 y Ft(\()p Fr(")p Ft(\))14 b(=)g Fr(U)1235 1643 y Fq(P)1257 1648 y Fg(1)1275 1643 y Fq(;:::)5 b(;P)1352 1649 y Ff(M)1388 1636 y Ft(\()p Fr(")p Ft(\))p Fr(:)159 b Ft(\(3.3\))278 1775 y(W)l(e)15 b(will)g(need)g(the)g(result)g(of)h (Lemma)e(3.1)i(in)f(a)h(sligh)o(tly)f(di\013eren)o(t)f(form)h(for)228 1833 y(the)k(pro)q(of)h(of)g(Theorem)e(1.1.)31 b(F)l(or)19 b(later)g(con)o(v)o(enience,)f(w)o(e)h(state)g(and)h(pro)o(v)o(e)228 1891 y(this)c(reform)o(ulation)e(here.)228 2009 y Fy(Corollary)19 b(3.2.)k Fd(L)n(et)15 b(the)h(notation)h(and)f(assumptions)f(b)n(e)h (as)g(in)g(L)n(emma)e(3.1)228 2067 y(\(using)22 b Ft(\(3.3\))q Fd(\).)g(Then)c(ther)n(e)f(exist)i(c)n(onstants)f Fr(C)1149 2074 y Fo(1)1168 2067 y Fr(;)8 b(L)1223 2074 y Fo(1)1261 2067 y Fd(such)18 b(that)527 2158 y Fp(Z)555 2271 y Fq(U)579 2277 y Fc(P)604 2271 y Fo(\()p Fq(")p Fo(\))658 2183 y Fp(\014)658 2213 y(\014)675 2226 y Fr(@)704 2205 y Fq(\013)701 2238 y(x)721 2244 y Fc(P)748 2226 y Fn(j)p Fr( )r Fn(j)810 2205 y Fo(2)829 2226 y Ft(\()p Fy(x)p Ft(\))897 2183 y Fp(\014)897 2213 y(\014)930 2226 y Fr(d)p Fy(x)c Fn(\024)f Fr(C)1086 2233 y Fo(1)1106 2226 y Fr(L)1139 2200 y Fm(j)p Fq(\013)p Fm(j)1139 2238 y Fo(1)1183 2226 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)e Ft(+)g(1\))1364 2205 y Fm(j)p Fq(\013)p Fm(j)1409 2226 y Fr(:)228 2377 y Fd(Pr)n(o)n(of.)19 b Ft(By)c(Leibniz')g(rule,)g(w)o(e)h(ha)o(v)o(e)641 2538 y Fr(@)670 2517 y Fq(\013)667 2550 y(x)687 2556 y Fc(P)714 2538 y Fn(j)p Fr( )r Fn(j)776 2517 y Fo(2)809 2538 y Ft(=)860 2490 y Fp(X)860 2596 y Fq(\014)r Fm(\024)p Fq(\013)941 2467 y Fp(\022)977 2504 y Fr(\013)978 2572 y(\014)1009 2467 y Fp(\023)p 1045 2485 107 2 v 1045 2538 a Fr(@)1074 2514 y Fq(\014)1071 2543 y(x)1091 2549 y Fc(P)1118 2538 y Fr( )10 b(@)1189 2517 y Fq(\013)p Fm(\000)p Fq(\014)1186 2550 y(x)1206 2556 y Fc(P)1262 2538 y Fr( )r(:)312 b Ft(\(3.4\))p eop %%Page: 8 8 8 7 bop 228 120 a Fj(8)70 b(S.)17 b(F)o(OURNAIS,)f(M.)g(AND)h(T.)f (HOFFMANN-OSTENHOF,)f(T.)h(\037.)k(S\037RENSEN)228 228 y Ft(Applying)14 b(Cauc)o(h)o(y-Sc)o(h)o(w)o(arz)g(and)h(Lemma)e(3.1)i (to)h(b)q(oth)p 1316 175 107 2 v 16 w Fr(@)1345 204 y Fq(\014)1342 233 y(x)1362 239 y Fc(P)1389 228 y Fr( )g Ft(and)g Fr(@)1560 210 y Fq(\013)p Fm(\000)p Fq(\014)1557 240 y(x)1577 246 y Fc(P)1633 228 y Fr( )h Ft(in)228 287 y(\(3.4\),)f(w)o(e)g(\014nd)g(using)h(Prop)q(osition)g(2.1)g(for)f(the) g(equalit)o(y)f(b)q(elo)o(w,)398 340 y Fp(Z)425 453 y Fq(U)449 459 y Fc(P)475 453 y Fo(\()p Fq(")p Fo(\))529 365 y Fp(\014)529 395 y(\014)545 408 y Fr(@)574 387 y Fq(\013)571 420 y(x)591 426 y Fc(P)618 408 y Fn(j)p Fr( )r Fn(j)680 387 y Fo(2)699 408 y Ft(\()p Fy(x)p Ft(\))767 365 y Fp(\014)767 395 y(\014)800 408 y Fr(d)p Fy(x)534 552 y Fn(\024)f Fr(C)626 531 y Fo(2)645 552 y Fr(L)678 531 y Fm(j)p Fq(\013)p Fm(j)731 505 y Fp(X)731 611 y Fq(\014)r Fm(\024)p Fq(\013)811 482 y Fp(\022)848 518 y Fr(\013)848 586 y(\014)879 482 y Fp(\023)916 552 y Ft(\()p Fn(j)p Fr(\014)s Fn(j)c Ft(+)h(1\))1096 531 y Fm(j)p Fq(\014)r Fm(j)1139 552 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)g(\000)g(j)p Fr(\014)s Fn(j)f Ft(+)h(1\))1439 531 y Fm(j)p Fq(\013)p Fm(j\000j)p Fq(\014)r Fm(j)534 737 y Ft(=)j Fr(C)625 716 y Fo(2)644 737 y Fr(L)677 716 y Fm(j)p Fq(\013)p Fm(j)745 671 y(j)p Fq(\013)p Fm(j)730 689 y Fp(X)736 796 y Fq(b)p Fo(=0)810 667 y Fp(\022)847 703 y Fn(j)p Fr(\013)p Fn(j)866 771 y Fr(b)906 667 y Fp(\023)943 737 y Ft(\()p Fr(b)c Ft(+)h(1\))1085 716 y Fq(b)1103 737 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)g(\000)g Fr(b)g Ft(+)g(1\))1366 716 y Fm(j)p Fq(\013)p Fm(j\000)p Fq(b)534 865 y Fn(\024)j Fr(C)626 844 y Fo(2)645 865 y Ft(\(2)p Fr(L)p Ft(\))740 844 y Fm(j)p Fq(\013)p Fm(j)785 865 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)d Ft(+)g(1\))966 844 y Fm(j)p Fq(\013)p Fm(j)1011 865 y Fr(:)228 964 y Ft(Th)o(us,)16 b(Corollary)g(3.2)h(holds)g(with)f Fr(C)933 971 y Fo(1)966 964 y Ft(=)e Fr(C)1057 946 y Fo(2)1076 964 y Ft(,)i Fr(L)1139 971 y Fo(1)1173 964 y Ft(=)d(2)p Fr(L)p Ft(.)p 1687 964 2 33 v 1689 933 30 2 v 1689 964 V 1718 964 2 33 v 584 1108 a(4.)27 b Fs(Differentia)m(ting)17 b(the)i(density)278 1195 y Ft(Fix)g(an)i(arbitrary)g Fr(")g(>)g Ft(0.)34 b(W)l(e)20 b(will)g(alw)o(a)o(ys)g(study)h Fr(\032)p Ft(\()p Fr(x)1384 1202 y Fo(1)1403 1195 y Ft(\))g(in)f(the)g (region)228 1253 y Fn(fj)p Fr(x)295 1260 y Fo(1)314 1253 y Fn(j)14 b Fr(>)f(")p Fn(g)p Ft(.)21 b(W)l(e)16 b(will)f(pro)o(v)o(e)h (the)g(follo)o(wing)g(estimate:)228 1349 y Fy(Prop)r(osition)i(4.1.)23 b Fd(L)n(et)i Fr(")j(>)h Ft(0)d Fd(b)n(e)f(given.)48 b(Then)26 b(ther)n(e)f(exist)h(c)n(onstants)228 1407 y Fr(C)q(;)8 b(L)13 b(>)h Ft(0)p Fd(,)k(such)g(that)f(for)g(al)r(l)i Fn(j)p Fr(x)p Fn(j)13 b Fr(>)h(")j Fd(and)h(al)r(l)h Fr(\013)14 b Fn(2)g Fl(N)1252 1389 y Fo(3)1273 1407 y Fd(,)j Fr(\032)h Fd(satis\014es)678 1506 y Fn(j)p Fr(@)721 1485 y Fq(\013)745 1506 y Fr(\032)p Ft(\()p Fr(x)p Ft(\))p Fn(j)c(\024)f Fr(C)t(L)988 1485 y Fm(j)p Fq(\013)p Fm(j)1032 1506 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)e Ft(+)g(1\))1213 1485 y Fm(j)p Fq(\013)p Fm(j)1258 1506 y Fr(:)350 b Ft(\(4.1\))228 1602 y Fy(Remark)16 b(4.2.)24 b Ft(It)16 b(is)g(clear)f(that)i(Prop)q (osition)g(4.1)g(implies)c(Theorem)i(1.1.)228 1698 y Fd(Pr)n(o)n(of)h(of)h(Pr)n(op)n(osition)f(4.1.)k Ft(Cho)q(ose)e Fr(\037)976 1705 y Fo(1)995 1698 y Fr(;)8 b(\037)1048 1705 y Fo(2)1081 1698 y Fn(2)14 b Fr(C)1167 1680 y Fm(1)1204 1698 y Ft(\()p Fl(R)1262 1680 y Fo(3)1279 1698 y Ft(\),)i(satisfying) 268 1792 y Fr(\037)299 1799 y Fo(1)330 1792 y Ft(+)11 b Fr(\037)410 1799 y Fo(2)443 1792 y Ft(=)j(1)p Fr(;)58 b(\037)622 1799 y Fo(1)655 1792 y Fn(\021)13 b Ft(1)k(on)g Fr(B)s Ft(\(0)p Fr(;)8 b("=)p Ft(\(4)p Fr(N)d Ft(\)\))p Fr(;)58 b Ft(supp)9 b Fr(\037)1305 1799 y Fo(1)1338 1792 y Fn(\032)14 b Fr(B)s Ft(\(0)p Fr(;)8 b("=)p Ft(\(2)p Fr(N)d Ft(\)\))p Fr(;)228 1887 y Ft(and)16 b(let)f(further)h Fr(\037)587 1894 y Fo(1)606 1887 y Fr(;)8 b(\037)659 1894 y Fo(2)694 1887 y Ft(b)q(e)17 b(radially)e(symmetri)o(c)e (functions.)21 b(Using)16 b(this)f(par-)228 1945 y(tition)h(of)g(unit)o (y)f(and)i(the)f(notation)582 2052 y Fn(M)e Ft(=)707 1997 y Fp(n)741 2052 y Ft(\()p Fr(j;)8 b(k)r Ft(\))13 b Fn(2)h(f)p Ft(1)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(N)d Fn(g)1144 2031 y Fo(2)1173 1995 y Fp(\014)1173 2025 y(\014)1173 2054 y(\014)1197 2052 y Fr(j)17 b(<)d(k)1313 1997 y Fp(o)1355 2052 y Fr(;)228 2158 y Ft(w)o(e)i(can)g(write)301 2266 y Fr(\032)p Ft(\()p Fr(x)373 2273 y Fo(1)393 2266 y Ft(\))42 b(=)533 2198 y Fp(Z)592 2266 y Fn(j)p Fr( )r Ft(\()p Fy(x)p Ft(\))p Fn(j)722 2245 y Fo(2)748 2218 y Fp(Y)749 2325 y Fq(j)r()g(")p Fn(g)228 526 y Ft(for)i(all)g Fr(I)h Fn(\032)d(M)p Ft(,)i(namely)701 619 y Fn(j)p Fr(@)744 598 y Fq(\013)768 619 y Fr(\032)793 626 y Fq(I)813 619 y Fn(j)e(\024)g Fr(C)t(L)966 598 y Fm(j)p Fq(\013)p Fm(j)1010 619 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)d Ft(+)g(1\))1191 598 y Fm(j)p Fq(\013)p Fm(j)1235 619 y Fr(:)373 b Ft(\(4.4\))278 713 y(The)17 b(estimate)f(\(4.1\))i(follo)o(ws)g(from)f(\(4.4\))h (\(with)f(a)h(di\013eren)o(t)f Fr(C)t Ft(\))g(since)g(the)228 771 y(sum)e(in)h(\(4.2\))g(is)h(\014nite.)278 829 y(The)11 b(estimate)e(\(4.4\))j(is)f(a)g(consequence)f(of)15 b(\(4.10\))d (\(with)f Fr(\036)j Ft(=)g Fr(\036)1463 836 y Fq(I)1482 829 y Ft(\))e(in)e(Lemma)228 888 y(4.3)h(b)q(elo)o(w,)h(using)f(a)h (Sob)q(olev)f(im)o(b)q(edding)f(theorem.)18 b(Since)10 b(w)o(e)h(ha)o(v)o(e)f(not)i(found)228 946 y(an)17 b(ideal)e(reference) g(w)o(e)g(include)g(the)h(follo)o(wing)g(easy)h(argumen)o(t:)278 1004 y(Let)h Fr(v)g Fn(2)g Fr(C)499 986 y Fm(1)536 1004 y Ft(\()p Fl(R)594 986 y Fo(3)610 1004 y Ft(\),)h Fr(v)r Ft(\()p Fr(x)p Ft(\))d(=)h(1)h(for)h Fn(j)p Fr(x)p Fn(j)e(\025)f Fr(")p Ft(,)j Fr(v)r Ft(\()p Fr(x)p Ft(\))d(=)h(0)h(for)h Fn(j)p Fr(x)p Fn(j)d(\024)h Fr("=)p Ft(2.)28 b(Let)228 1062 y(furthermore)15 b Fn(F)21 b Ft(denote)c(the)f(F)l(ourier)g (transformation.)22 b(W)l(e)17 b(get)g(for)f Fr(\013)f Fn(2)g Fl(N)1688 1044 y Fo(3)1709 1062 y Ft(,)228 1120 y Fn(j)p Fr(x)p Fn(j)e(\025)h Fr(\017)p Ft(,)428 1211 y Fr(@)457 1191 y Fq(\013)482 1211 y Fr(\032)p Ft(\()p Fr(x)p Ft(\))f(=)h(\()p Fr(v)r(@)712 1191 y Fq(\013)735 1211 y Fr(\032)p Ft(\)\()p Fr(x)p Ft(\))586 1317 y(=)g Fr(c)667 1249 y Fp(Z)725 1317 y Fr(e)748 1297 y Fq(ixp)800 1317 y Ft(\(1)d(+)g Fr(p)927 1297 y Fo(2)947 1317 y Ft(\))966 1297 y Fm(\000)p Fo(2)1013 1317 y Fn(F)1062 1277 y Fp(\000)1085 1317 y Ft(\(1)h Fn(\000)f Ft(\001\))1250 1297 y Fo(2)1277 1317 y Ft(\()p Fr(v)r(@)1351 1297 y Fq(\013)1375 1317 y Fr(\032)p Ft(\))1419 1277 y Fp(\001)1458 1317 y Fr(dp:)228 1432 y Ft(Therefore,)357 1523 y Fn(j)p Fr(@)400 1503 y Fq(\013)424 1523 y Fr(\032)p Ft(\()p Fr(x)p Ft(\))p Fn(j)j(\024)f Fr(c)p Fn(k)p Ft(\(1)f(+)f Fr(p)769 1503 y Fo(2)789 1523 y Ft(\))808 1503 y Fm(\000)p Fo(2)855 1523 y Fn(k)880 1532 y Fq(L)904 1522 y Fg(1)921 1532 y Fo(\()p Fh(R)962 1522 y Fg(3)976 1532 y Fo(\))992 1523 y Fn(k)p Ft(\(1)g Fn(\000)g Ft(\001\))1181 1503 y Fo(2)1200 1523 y Ft(\()p Fr(v)r(@)1274 1503 y Fq(\013)1298 1523 y Fr(\032)p Ft(\))p Fn(k)1367 1532 y Fq(L)1391 1522 y Fg(1)1408 1532 y Fo(\()p Fh(R)1449 1522 y Fg(3)1463 1532 y Fo(\))1479 1523 y Fr(:)129 b Ft(\(4.5\))228 1620 y(W)l(e)16 b(can)g(estimate)f Fn(k)p Ft(\(1)c Fn(\000)g Ft(\001\))786 1602 y Fo(2)805 1620 y Ft(\()p Fr(v)r(@)879 1602 y Fq(\013)903 1620 y Fr(\032)p Ft(\))p Fn(k)972 1628 y Fq(L)996 1619 y Fg(1)1013 1628 y Fo(\()p Fh(R)1054 1619 y Fg(3)1068 1628 y Fo(\))1100 1620 y Ft(using)17 b(\(4.10\))g(b)o(y)l(,)439 1724 y Fn(k)p Ft(\(1)11 b Fn(\000)g Ft(\001\))628 1704 y Fo(2)647 1724 y Ft(\()p Fr(v)r(@)721 1704 y Fq(\013)745 1724 y Fr(\032)p Ft(\))p Fn(k)814 1733 y Fq(L)838 1723 y Fg(1)855 1733 y Fo(\()p Fh(R)896 1723 y Fg(3)910 1733 y Fo(\))940 1724 y Fn(\024)i Fr(c)1013 1731 y Fo(1)1033 1724 y Fr(L)1066 1699 y Fm(j)p Fq(\013)p Fm(j)p Fo(+4)1066 1736 y(1)1156 1724 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)e Ft(+)g(4)g(+)g(1\))1421 1704 y Fm(j)p Fq(\013)p Fm(j)p Fo(+4)940 1813 y Fn(\024)i Fr(c)1013 1820 y Fo(2)1033 1813 y Fr(L)1066 1787 y Fm(j)p Fq(\013)p Fm(j)1066 1825 y Fo(2)1111 1813 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)e Ft(+)g(1\))1292 1792 y Fm(j)p Fq(\013)p Fm(j)1336 1813 y Fr(;)272 b Ft(\(4.6\))228 1907 y(for)12 b(some)f(constan)o(ts)i Fr(c)649 1914 y Fo(1)669 1907 y Fr(;)8 b(L)724 1914 y Fo(1)744 1907 y Fr(;)g(c)787 1914 y Fo(2)806 1907 y Fr(;)g(L)861 1914 y Fo(2)881 1907 y Ft(.)20 b(Com)o(bining)12 b(\(4.5\))g(and)h(\(4.6\))f (yields)f(\(4.4\))q(.)228 1965 y(Pro)o(ving)k(Lemma)e(4.3)j(therefore)f (\014nishes)g(the)g(pro)q(of)h(of)g(Prop)q(osition)h(4.1.)p 1687 1965 2 33 v 1689 1934 30 2 v 1689 1965 V 1718 1965 2 33 v 228 2060 a Fy(Lemm)o(a)f(4.3.)24 b Fd(L)n(et)17 b Fr(")d(>)f Ft(0)18 b Fd(b)n(e)g(given)h(and)f(let)702 2159 y Fr(\036)13 b Ft(=)848 2112 y Fp(Y)796 2218 y Fo(1)p Fm(\024)p Fq(j)r()f(")p Fn(g)h(\032)g(\\)1076 2568 y Fq(M)1076 2601 y(j)r Fo(=1)1139 2588 y Fr(U)1172 2595 y Fq(P)1194 2600 y Ff(j)1213 2588 y Ft(\()p Fr("=)p Ft(\(4)p Fr(N)5 b Ft(\)\))p Fr(:)204 b Ft(\(4.8\))p eop %%Page: 10 10 10 9 bop 228 120 a Fj(10)51 b(S.)17 b(F)o(OURNAIS,)f(M.)g(AND)h(T.)f (HOFFMANN-OSTENHOF,)f(T.)h(\037.)k(S\037RENSEN)336 213 y Fd(Then)i(ther)n(e)e(exist)i(c)n(onstants)g Fr(C)q(;)8 b(L)19 b(>)h Ft(0)h Fd(\(dep)n(ending)h(on)f Fr(")p Fd(\))g(such)g (that)336 271 y(for)c(al)r(l)i(multiindic)n(es,)f Fr(\013)c Ft(=)g(\()p Fr(\013)917 278 y Fo(0)937 271 y Fr(;)8 b(\013)990 278 y Fo(1)1010 271 y Fr(;)g(:)g(:)g(:)15 b(;)8 b(\013)1158 278 y Fq(M)1198 271 y Ft(\))13 b Fn(2)h Fl(N)1312 253 y Fo(3)p Fq(M)t Fo(+3)1416 271 y Fd(,)j(we)h(have)278 335 y Fp(\015)278 365 y(\015)278 394 y(\015)278 424 y(\015)305 407 y Fr(@)334 386 y Fq(\013)357 391 y Fg(0)331 419 y Fq(x)351 424 y Fg(1)384 339 y Fp(Z)434 407 y Ft(\()p Fr(@)482 386 y Fq(\013)505 391 y Fg(1)479 419 y Fq(x)499 425 y Ff(P)518 432 y Fg(1)548 407 y Fn(\001)8 b(\001)g(\001)g Fr(@)643 386 y Fq(\013)666 392 y Ff(M)640 419 y Fq(x)660 425 y Ff(P)679 433 y(M)717 407 y Fn(j)p Fr( )r Fn(j)779 386 y Fo(2)798 407 y Ft(\)\()p Fy(x)p Ft(\))p Fr(\036)p Ft(\()p Fy(x)p Ft(\))g Fr(dx)1043 414 y Fo(2)1070 407 y Fn(\001)g(\001)g(\001)g Fr(dx)1189 414 y Fq(N)1223 335 y Fp(\015)1223 365 y(\015)1223 394 y(\015)1223 424 y(\015)1251 456 y Fq(L)1275 447 y Fg(1)1292 456 y Fo(\()p Fm(fj)p Fq(x)1354 461 y Fg(1)1370 456 y Fm(j)p Fq(>")p Fm(g)p Fo(\))1173 529 y Fn(\024)13 b Fr(C)t(L)1297 508 y Fm(j)p Fq(\013)p Fm(j)1341 529 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)e Ft(+)g(1\))1522 508 y Fm(j)p Fq(\013)p Fm(j)1567 529 y Fr(:)41 b Ft(\(4.9\))263 623 y Fy(ii\))19 b Fd(Ther)n(e)f(exist)g (c)n(onstants)h Fr(C)q(;)8 b(L)14 b(>)g Ft(0)k Fd(\(dep)n(ending)h(on)f Fr(")p Fd(\))f(such)h(that)g(for)f(al)r(l)336 681 y Fr(\013)d Fn(2)h Fl(N)464 663 y Fo(3)502 681 y Fd(we)j(have)262 720 y Fp(\015)262 750 y(\015)262 780 y(\015)262 809 y(\015)289 792 y Fr(@)318 771 y Fq(\013)315 804 y(x)335 809 y Fg(1)362 724 y Fp(Z)421 792 y Fn(j)p Fr( )r Fn(j)483 771 y Fo(2)501 792 y Ft(\()p Fy(x)p Ft(\))p Fr(\036)p Ft(\()p Fy(x)p Ft(\))8 b Fr(dx)727 799 y Fo(2)755 792 y Fn(\001)g(\001)g(\001)g Fr(dx)874 799 y Fq(N)908 720 y Fp(\015)908 750 y(\015)908 780 y(\015)908 809 y(\015)936 841 y Fq(L)960 832 y Fg(1)977 841 y Fo(\()p Fm(fj)p Fq(x)1039 846 y Fg(1)1055 841 y Fm(j)p Fq(>")p Fm(g)p Fo(\))1156 792 y Fn(\024)14 b Fr(C)t(L)1281 771 y Fm(j)p Fq(\013)p Fm(j)1325 792 y Ft(\()p Fn(j)p Fr(\013)p Fn(j)d Ft(+)g(1\))1506 771 y Fm(j)p Fq(\013)p Fm(j)1550 792 y Fr(:)34 b Ft(\(4.10\))228 927 y Fd(Pr)n(o)n(of.)19 b Ft(T)l(o)e(a)g(function)f Fr(\036)h Ft(as)g(giv)o(en)f(in)g(\(4.7\))h (w)o(e)f(will)g(asso)q(ciate)h Fr(P)22 b Ft(=)14 b Fr(P)7 b Ft(\()p Fr(\036)p Ft(\))14 b Fn(\032)228 986 y(f)p Ft(1)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(N)d Fn(g)16 b Ft(satisfying)g(1)f Fn(2)f Fr(P)23 b Ft(and)17 b(suc)o(h)f(that)562 1077 y(\(supp)8 b Fr(\036)p Ft(\))j Fn(\\)h(fj)p Fr(x)860 1084 y Fo(1)879 1077 y Fn(j)h Fr(>)h(")p Fn(g)g(\032)f Fr(U)1105 1084 y Fq(P)5 b Fo(\()p Fq(\036)p Fo(\))1183 1077 y Ft(\()p Fr("=)p Ft(\(4)p Fr(N)g Ft(\)\))p Fr(:)210 b Ft(\(4.11\))228 1170 y(W)l(e)17 b(will)f(no)o(w)i(describ)q(e)e(the)h (map)g Fr(\036)e Fn(7!)h Fr(P)7 b Ft(\()p Fr(\036)p Ft(\).)24 b(W)l(e)17 b(note)h(that)f(the)h(follo)o(wing)228 1229 y(construction)e(is)g(similar)e(to)j(the)f(one)g(from)g([1)o(].)21 b(De\014ne)16 b Fr(I)i Ft(=)13 b Fr(I)t Ft(\()p Fr(\036)p Ft(\))g Fn(\032)h(M)i Ft(b)o(y)365 1320 y(\()p Fr(j;)8 b(k)r Ft(\))14 b Fn(2)g Fr(I)t Ft(\()p Fr(\036)p Ft(\))h(if)h(and)h (only)f(if)f Fr(f)956 1327 y Fq(j;k)1016 1320 y Fn(2)f(f)p Fr(\037)1119 1327 y Fo(1)1138 1320 y Fr(;)8 b(@)1189 1299 y Fq(e)1205 1304 y Fg(1)1224 1320 y Fr(\037)1255 1327 y Fo(2)1274 1320 y Fr(;)g(@)1325 1299 y Fq(e)1341 1304 y Fg(2)1360 1320 y Fr(\037)1391 1327 y Fo(2)1411 1320 y Fr(;)g(@)1462 1299 y Fq(e)1478 1304 y Fg(3)1497 1320 y Fr(\037)1528 1327 y Fo(2)1547 1320 y Fn(g)p Fr(:)228 1410 y Ft(In)18 b(other)h(w)o(ords,)g(\()p Fr(j;)8 b(k)r Ft(\))18 b Fn(2)g Fr(I)t Ft(\()p Fr(\036)p Ft(\))f(means)h(precisely)f (that)i Fr(f)1349 1417 y Fq(j;k)1412 1410 y Fn(6)p Ft(=)e Fr(\037)1498 1417 y Fo(2)1518 1410 y Ft(.)28 b(The)18 b(set)228 1469 y Fr(I)t Ft(\()p Fr(\036)p Ft(\))h(generates)i(an)g (equiv)m(alence)d(relation)i(on)h Fn(f)p Ft(1)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(N)d Fn(g)1383 1451 y Fo(2)1423 1469 y Ft(and)21 b(w)o(e)f(de\014ne)228 1527 y Fr(P)7 b Ft(\()p Fr(\036)p Ft(\))16 b(to)g(b)q(e)g(the)g(equiv)m(alence)e (class)i(of)g(1.)22 b(Less)16 b(abstractly)l(,)g(this)g(means)f(that) 291 1601 y Fn(\017)20 b Ft(1)15 b Fn(2)f Fr(P)7 b Ft(\()p Fr(\036)p Ft(\).)291 1659 y Fn(\017)20 b Ft(F)l(or)27 b Fr(j)33 b Fn(\025)e Ft(2)26 b(w)o(e)g(ha)o(v)o(e)f Fr(j)34 b Fn(2)d Fr(P)7 b Ft(\()p Fr(\036)p Ft(\))26 b(i\013)g(there)f(exists)h Fn(f)p Fr(j)1452 1666 y Fo(1)1472 1659 y Fr(;)8 b(:)g(:)g(:)15 b(;)8 b(j)1609 1666 y Fq(s)1628 1659 y Fn(g)30 b(\022)336 1717 y(f)p Ft(1)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(N)d Fn(g)p Ft(,)16 b Fr(s)e Fn(\024)g Fr(N)5 b Ft(,)16 b(satisfying)420 1776 y(\(1)p Fr(;)8 b(j)505 1783 y Fo(1)525 1776 y Ft(\))14 b Fn(2)g Fr(I)t Ft(\()p Fr(\036)p Ft(\),)420 1834 y(\()p Fr(j)459 1841 y Fq(t)474 1834 y Fr(;)8 b(j)516 1841 y Fq(t)p Fo(+1)576 1834 y Ft(\))14 b Fn(2)g Fr(I)t Ft(\()p Fr(\036)p Ft(\))h(or)i(\()p Fr(j)863 1841 y Fq(t)p Fo(+1)923 1834 y Fr(;)8 b(j)965 1841 y Fq(t)980 1834 y Ft(\))13 b Fn(2)i Fr(I)t Ft(\()p Fr(\036)p Ft(\))g(for)i(1)d Fn(\024)f Fr(t)h(<)g(s)p Ft(,)336 1892 y(and)420 1950 y(\()p Fr(j)459 1957 y Fq(s)477 1950 y Fr(;)8 b(j)s Ft(\))14 b Fn(2)g Fr(I)t Ft(\()p Fr(\036)p Ft(\))i(or)g(\()p Fr(j;)8 b(j)851 1957 y Fq(s)870 1950 y Ft(\))14 b Fn(2)g Fr(I)t Ft(\()p Fr(\036)p Ft(\).)228 2025 y(Notice)d(that,)i(since)e Fr(\037)639 2032 y Fo(1)661 2025 y Ft(+)r Fr(\037)732 2032 y Fo(2)765 2025 y Ft(=)j(1,)f(supp)c Fr(@)1006 2006 y Fq(e)1022 2011 y Ff(j)1040 2025 y Fr(\037)1071 2032 y Fo(2)1104 2025 y Fn(\032)k Ft(supp)c Fr(\037)1296 2032 y Fo(1)1316 2025 y Ft(,)j Fr(j)17 b Ft(=)d(1)p Fr(;)8 b Ft(2)p Fr(;)g Ft(3.)20 b(There-)228 2083 y(fore)e(w)o(e)g(get)h (\(4.11\))g(b)o(y)f(the)g(same)f(elemen)o(tary)f(geometrical)g (considerations)228 2141 y(\(the)g(triangle)g(inequalit)o(y\))e(as)j (in)f([1].)278 2199 y(In)e(the)g(pro)q(of)h(w)o(e)f(shall)g(use)g Fr(P)21 b Ft(=)14 b Fr(P)7 b Ft(\()p Fr(\036)p Ft(\))15 b(in)e(order)i(to)g(replace)e(the)h(deriv)m(ativ)o(e)228 2257 y Fr(@)257 2239 y Fq(\013)280 2244 y Fg(0)254 2269 y Fq(x)274 2274 y Fg(1)314 2257 y Ft(outside)i(the)g(in)o(tegral)f(in)h (the)g(left)f(hand)i(side)e(of)20 b(\(4.9\))d(b)o(y)e(the)h(deriv)m (ativ)o(e)228 2315 y Fr(@)257 2297 y Fq(\013)280 2302 y Fg(0)254 2327 y Fq(x)274 2333 y Ff(P)317 2315 y Ft(inside)g(the)g(in) o(tegral.)k(That)d(will)e(enable)h(us)h(to)f(apply)g(Corollary)h(3.2.) 278 2374 y(Let)22 b Fr(P)32 b Ft(=)25 b Fr(P)7 b Ft(\()p Fr(\036)p Ft(\))23 b(according)g(to)g(our)g(construction.)41 b(W)l(e)22 b(will)g(pro)o(v)o(e)f(the)228 2433 y(lemm)o(a)13 b(recursiv)o(ely)f(in)i Fn(j)p Fr(P)7 b Fn(j)p Ft(.)21 b(In)15 b(the)f(pro)q(of)i(b)q(elo)o(w)f(w)o(e)f(will)g(freely)f(in)o (terc)o(hange)228 2491 y(the)f(order)g(of)h(di\013eren)o(tiation)f (\(in)g(the)g(distributional)g(sense\))g(and)h(in)o(tegration.)228 2549 y(This)i(is)g(p)q(ermitted,)e(due)i(to)h(Corollary)f(3.2,)h(whic)o (h)e(ensures)i(that)f(the)g(deriv)m(a-)228 2607 y(tiv)o(es)20 b(of)i(the)f(functions)g(in)g(question)g(b)q(elong)h(to)f Fr(L)1231 2589 y Fo(1)1260 2567 y Fp(\000)1282 2607 y Fn(fj)p Fr(x)1349 2614 y Fo(1)1369 2607 y Fn(j)13 b Fr(>)h(\017)p Fn(g)d(\002)f Fl(R)1592 2589 y Fo(3)p Fq(N)t Fm(\000)p Fo(3)1686 2567 y Fp(\001)1709 2607 y Ft(.)p eop %%Page: 11 11 11 10 bop 300 118 a Fj(ANAL)m(YTICITY)14 b(OF)j(THE)f(DENSITY)h(OF)f (ELECTR)o(ONIC)g(W)l(A)l(VEFUNCTIONS)33 b(11)228 213 y Ft(W)l(e)19 b(will)f(only)h(pro)o(v)o(e)f(part)i Fy(i)p Ft(\))f(of)g(Lemma)e(4.3.)30 b(The)20 b(c)o(hanges)f(necessary)g(for) 228 271 y(the)d(case)g Fy(ii)p Ft(\))f(are)i(ob)o(vious)f(and)h (therefore)f(omitted.)278 329 y Fy(Step)25 b(1,)i Fn(j)p Fr(P)7 b Fn(j)24 b Ft(=)g Fr(N)5 b Fy(.)40 b Ft(In)22 b(the)h(case)f(where)g Fr(P)32 b Ft(=)24 b Fn(f)p Ft(1)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(N)d Fn(g)22 b Ft(w)o(e)g(mak)o(e)228 387 y(the)d(c)o(hange)h(of)g(v)m(ariables)f Fr(y)768 394 y Fq(j)806 387 y Ft(=)g Fr(x)891 394 y Fq(j)923 387 y Fn(\000)13 b Fr(x)1003 394 y Fo(1)1042 387 y Ft(for)20 b Fr(j)i Ft(=)e(2)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(N)d Ft(.)31 b(Then)20 b(w)o(e)f(get)228 445 y Fr(x)256 452 y Fq(j)287 445 y Fn(\000)13 b Fr(x)367 452 y Fq(k)407 445 y Ft(=)19 b Fr(y)488 452 y Fq(j)519 445 y Fn(\000)13 b Fr(y)595 452 y Fq(k)635 445 y Ft(for)20 b Fr(j;)8 b(k)21 b Fn(6)p Ft(=)e(1.)30 b(The)20 b(p)q(oin)o(t)f(is)g(that)h Fr(\036)f Ft(only)g(dep)q(ends)h(on)228 503 y(the)14 b(di\013erences)f Fr(x)573 510 y Fq(j)599 503 y Fn(\000)7 b Fr(x)673 510 y Fq(k)694 503 y Ft(,)14 b(and)h(therefore,)f(after)g (the)g(c)o(hange)h(of)f(v)m(ariables,)g(the)228 561 y(only)19 b(dep)q(endence)f(on)h Fr(x)700 568 y Fo(1)738 561 y Ft(will)f(b)q(e)h(in)g Fn(j)p Fr( )r Fn(j)1024 543 y Fo(2)1043 561 y Ft(,)g(where)g(w)o(e)f(can)h(apply)g(Corollary)228 619 y(3.2.)i(Let)c(us)f(carry)g(this)g(out.)278 678 y(Denote)g Fy(y)e Ft(=)f(\()p Fr(y)583 685 y Fo(2)603 678 y Fr(;)8 b(:)g(:)g(:)f(;)h(y)736 685 y Fq(N)770 678 y Ft(\).)21 b(Then)16 b(w)o(e)f(see)g(that)i(after)e(c)o(hange)h(of)g(v)m(ariables) 228 736 y(w)o(e)g(ha)o(v)o(e)839 815 y Fr(\036)p Ft(\()p Fy(x)p Ft(\))d(=)1007 802 y(~)1001 815 y Fr(\036)p Ft(\()p Fy(y)q Ft(\))p Fr(;)228 909 y Ft(for)j(some)g(function)622 896 y(~)616 909 y Fr(\036)o Ft(.)22 b(Explicitly)l(,)13 b(w)o(e)j(see)g(from)f(\(4.7\))h(that)479 1044 y(~)472 1057 y Fr(\036)p Ft(\()p Fr(y)544 1064 y Fo(2)564 1057 y Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(y)706 1064 y Fq(N)739 1057 y Ft(\))14 b(=)846 1010 y Fp(Y)823 1116 y Fo(1)p Fq()g(K)23 b Ft(for)c(some)e(0)i Fn(\024)f Fr(K)k(<)c(N)5 b Ft(.)29 b(W)l(e)18 b(will)g(pro)o(v)o(e)f (the)228 2375 y(statemen)o(t)d(for)j Fn(j)p Fr(P)7 b Fn(j)14 b Ft(=)f Fr(K)t Ft(.)278 2433 y(De\014ne)22 b Fr(Q)h Ft(=)h Fn(f)p Ft(1)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(N)d Fn(g)15 b(n)g Fr(P)7 b Ft(.)39 b(Since)21 b Fn(j)p Fr(P)7 b Fn(j)24 b Fr(<)g(N)5 b Ft(,)23 b Fr(Q)h Fn(6)p Ft(=)f Fn(;)p Ft(.)39 b(Note)22 b(that)228 2492 y(if)d Fr(j)k Fn(2)d Fr(P)7 b Ft(,)21 b Fr(k)h Fn(2)e Fr(Q)g Ft(then,)g(b)o(y)g(de\014nition)f(of)h Fr(I)t Ft(\()p Fr(\036)p Ft(\))f(and)i Fr(P)27 b Ft(=)20 b Fr(P)7 b Ft(\()p Fr(\036)p Ft(\),)21 b(w)o(e)e(ha)o(v)o(e)228 2550 y(\()p Fr(j;)8 b(k)r Ft(\))23 b Fr(=)-30 b Fn(2)17 b Fr(I)t Ft(\()p Fr(\036)p Ft(\))g(and)i(\()p Fr(k)r(;)8 b(j)s Ft(\))23 b Fr(=)-30 b Fn(2)17 b Fr(I)t Ft(\()p Fr(\036)p Ft(\).)26 b(Therefore,)18 b(if)g Fr(j)i(<)c(k)21 b Ft(w)o(e)c(ha)o(v)o(e)h Fr(f)1555 2557 y Fq(j;k)1617 2550 y Ft(=)f Fr(\037)1703 2557 y Fo(2)228 2608 y Ft(and)j(if)e Fr(k)j(<)e(j)j Ft(w)o(e)d(ha)o(v)o(e)f Fr(f)732 2615 y Fq(k)q(;j)798 2608 y Ft(=)h Fr(\037)886 2615 y Fo(2)905 2608 y Ft(.)30 b(So)20 b Fr(\036)f Ft(con)o(tains)g(the)g(factor)h (\(remem)n(b)q(er)p eop %%Page: 12 12 12 11 bop 228 120 a Fj(12)51 b(S.)17 b(F)o(OURNAIS,)f(M.)g(AND)h(T.)f (HOFFMANN-OSTENHOF,)f(T.)h(\037.)k(S\037RENSEN)228 213 y Ft(that)c Fr(\037)364 220 y Fo(2)400 213 y Ft(is)g(rotationally)g (symmetric)o(,)d(in)j(particular)g(ev)o(en\))692 304 y Fr(\036)721 311 y Fq(P)q(;Q)797 304 y Ft(=)889 257 y Fp(Y)849 363 y Fq(j)r Fm(2)p Fq(P)q(;k)q Fm(2)p Fq(Q)1001 304 y Fr(\037)1032 311 y Fo(2)1051 304 y Ft(\()p Fr(x)1098 311 y Fq(j)1127 304 y Fn(\000)11 b Fr(x)1205 311 y Fq(k)1226 304 y Ft(\))p Fr(;)228 436 y Ft(and)17 b(can)f(b)q(e)g(written)g(as)781 517 y Fr(\036)d Ft(=)h Fr(\036)904 524 y Fq(P)944 517 y Fn(\001)d Fr(\036)998 524 y Fq(Q)1039 517 y Fn(\001)g Fr(\036)1093 524 y Fq(P)q(;Q)1156 517 y Fr(;)228 598 y Ft(where)305 678 y Fr(\036)334 685 y Fq(P)377 678 y Ft(=)479 631 y Fp(Y)429 737 y Fq(j;k)q Fm(2)p Fq(P)q(;j)r()1218 1314 y(<)1218 1404 y(>)1218 1419 y(:)1262 1303 y Fr(x)1290 1310 y Fo(1)1460 1303 y Fr(j)g Ft(=)d(1)p Fr(;)1262 1373 y(x)1290 1380 y Fo(1)1321 1373 y Ft(+)d Fr(y)1394 1380 y Fq(j)1460 1373 y Fr(j)17 b Fn(2)d Fr(P)k Fn(n)11 b(f)p Ft(1)p Fn(g)p Fr(;)1262 1443 y(x)1290 1450 y Fq(j)1460 1443 y Fr(j)17 b Fn(2)d Fr(Q:)228 1503 y Ft(Then)289 1530 y Fp(Z)339 1597 y Ft(\()p Fr(@)387 1577 y Fq(\013)410 1582 y Fg(1)384 1610 y Fq(x)404 1616 y Ff(P)423 1623 y Fg(1)453 1597 y Fn(\001)8 b(\001)g(\001)g Fr(@)548 1577 y Fq(\013)571 1583 y Ff(M)545 1610 y Fq(x)565 1616 y Ff(P)584 1624 y(M)622 1597 y Fn(j)p Fr( )r Fn(j)684 1577 y Fo(2)703 1597 y Ft(\)\()p Fy(x)p Ft(\))p Fr(\036)p Ft(\()p Fy(x)p Ft(\))g Fr(dx)948 1604 y Fo(2)975 1597 y Fn(\001)g(\001)g(\001)h Fr(dx)1095 1604 y Fq(N)1598 1597 y Ft(\(4.14\))243 1746 y(=)289 1678 y Fp(Z)339 1746 y Ft(\()p Fr(@)387 1725 y Fq(\013)410 1730 y Fg(1)384 1758 y Fq(x)404 1764 y Ff(P)423 1771 y Fg(1)453 1746 y Fn(\001)f(\001)g(\001)g Fr(@)548 1725 y Fq(\013)571 1731 y Ff(M)545 1758 y Fq(x)565 1764 y Ff(P)584 1772 y(M)622 1746 y Fn(j)p Fr( )r Fn(j)684 1725 y Fo(2)703 1746 y Ft(\)\()p Fy(z)p Ft(\))791 1733 y(~)785 1746 y Fr(\036)814 1753 y Fq(P)843 1746 y Ft(\()p Fy(y)q Ft(\))920 1661 y Fp( )999 1699 y(Y)959 1805 y Fq(j)r Fm(2)p Fq(P)q(;k)q Fm(2)p Fq(Q)1111 1746 y Fr(\037)1142 1753 y Fo(2)1161 1746 y Ft(\()p Fr(x)1208 1753 y Fo(1)1239 1746 y Ft(+)j Fr(y)1312 1753 y Fq(j)1341 1746 y Fn(\000)g Fr(y)1415 1753 y Fq(k)1436 1746 y Ft(\))1455 1661 y Fp(!)1509 1733 y Ft(~)1502 1746 y Fr(\036)1531 1753 y Fq(Q)1561 1746 y Ft(\()p Fy(y)q Ft(\))d Fr(d)p Fy(y)q Fr(:)228 1878 y Ft(Di\013eren)o(tiation)15 b(under)h(the)g(in)o(tegral)g(sign)g (yields)279 1982 y Fn(r)321 1989 y Fq(x)341 1994 y Fg(1)368 1914 y Fp(Z)417 1982 y Ft(\()p Fr(@)465 1962 y Fq(\013)488 1967 y Fg(1)462 1994 y Fq(x)482 2000 y Ff(P)501 2007 y Fg(1)531 1982 y Fn(\001)8 b(\001)g(\001)h Fr(@)627 1962 y Fq(\013)650 1968 y Ff(M)624 1994 y Fq(x)644 2000 y Ff(P)663 2008 y(M)701 1982 y Fn(j)p Fr( )r Fn(j)763 1962 y Fo(2)781 1982 y Ft(\)\()p Fy(x)p Ft(\))p Fr(\036)p Ft(\()p Fy(x)p Ft(\))f Fr(dx)1026 1989 y Fo(2)1054 1982 y Fn(\001)g(\001)g(\001)g Fr(dx)1173 1989 y Fq(N)1598 1982 y Ft(\(4.15\))241 2111 y(=)287 2044 y Fp(Z)337 2111 y Ft(\()p Fn(r)398 2118 y Fq(x)418 2124 y Ff(P)445 2111 y Fr(@)474 2091 y Fq(\013)497 2096 y Fg(1)471 2124 y Fq(x)491 2130 y Ff(P)510 2137 y Fg(1)539 2111 y Fn(\001)g(\001)g(\001)h Fr(@)635 2091 y Fq(\013)658 2097 y Ff(M)632 2124 y Fq(x)652 2130 y Ff(P)671 2138 y(M)709 2111 y Fn(j)p Fr( )r Fn(j)771 2091 y Fo(2)790 2111 y Ft(\)\()p Fy(z)p Ft(\))878 2098 y(~)872 2111 y Fr(\036)901 2118 y Fq(P)930 2111 y Ft(\()p Fy(y)q Ft(\))1007 2071 y Fp(\010)1034 2064 y(Y)995 2170 y Fq(j)r Fm(2)p Fq(P)q(;k)q Fm(2)p Fq(Q)1146 2111 y Fr(\037)1177 2118 y Fo(2)1196 2111 y Ft(\()p Fr(x)1243 2118 y Fo(1)1273 2111 y Ft(+)i Fr(y)1346 2118 y Fq(j)1375 2111 y Fn(\000)g Fr(y)1449 2118 y Fq(k)1470 2111 y Ft(\))1489 2071 y Fp(\011)1525 2098 y Ft(~)1518 2111 y Fr(\036)1547 2118 y Fq(Q)1577 2111 y Ft(\()p Fy(y)q Ft(\))d Fr(d)p Fy(y)290 2268 y Ft(+)374 2221 y Fp(X)339 2327 y Fq(j)r Fm(2)p Fq(P)q(;k)q Fm(2)p Fq(Q)490 2200 y Fp(Z)540 2268 y Ft(\()p Fr(@)588 2248 y Fq(\013)611 2253 y Fg(1)585 2280 y Fq(x)605 2286 y Ff(P)624 2293 y Fg(1)654 2268 y Fn(\001)g(\001)g(\001)g Fr(@)749 2248 y Fq(\013)772 2254 y Ff(M)746 2280 y Fq(x)766 2286 y Ff(P)785 2294 y(M)823 2268 y Fn(j)p Fr( )r Fn(j)885 2248 y Fo(2)904 2268 y Ft(\)\()p Fy(z)p Ft(\))992 2255 y(~)986 2268 y Fr(\036)1015 2275 y Fq(P)1044 2268 y Ft(\()p Fy(y)q Ft(\))1119 2255 y(~)1113 2268 y Fr(\036)1142 2275 y Fq(Q)1171 2268 y Ft(\()p Fy(y)q Ft(\))290 2414 y Fn(\002)339 2359 y Fp(\020)530 2367 y(Y)378 2475 y Fq(j)394 2465 y Fe(0)405 2475 y Fm(2)p Fq(P)q(;k)481 2465 y Fe(0)492 2475 y Fm(2)p Fq(Q;)p Fo(\()p Fq(j)584 2465 y Fe(0)594 2475 y Fq(;k)623 2465 y Fe(0)634 2475 y Fo(\))p Fm(6)p Fo(=\()p Fq(j;k)q Fo(\))755 2414 y Fr(\037)786 2421 y Fo(2)805 2414 y Ft(\()p Fr(x)852 2421 y Fo(1)882 2414 y Ft(+)j Fr(y)955 2421 y Fq(j)971 2412 y Fe(0)996 2414 y Fn(\000)f Fr(y)1069 2421 y Fq(k)1088 2412 y Fe(0)1102 2414 y Ft(\))1121 2359 y Fp(\021)1151 2414 y Ft(\()p Fn(r)p Fr(\037)1243 2421 y Fo(2)1261 2414 y Ft(\)\()p Fr(x)1327 2421 y Fo(1)1358 2414 y Ft(+)h Fr(y)1431 2421 y Fq(j)1460 2414 y Fn(\000)g Fr(y)1534 2421 y Fq(k)1555 2414 y Ft(\))d Fr(d)p Fy(y)q Fr(:)228 2550 y Ft(Let)15 b(us)h(explain)f(roughly)g(ho)o(w)h(w)o(e)f(will)f(pro)q(ceed)i(for)f (higher)g(deriv)m(ativ)o(es)f(with)228 2608 y(resp)q(ect)h(to)h Fr(x)481 2615 y Fo(1)500 2608 y Ft(.)21 b(F)l(or)16 b(eac)o(h)f (consecutiv)o(e)e(di\013eren)o(tiation)i(w)o(e)g(will)f(get)i(terms)e (as)p eop %%Page: 13 13 13 12 bop 300 118 a Fj(ANAL)m(YTICITY)14 b(OF)j(THE)f(DENSITY)h(OF)f (ELECTR)o(ONIC)g(W)l(A)l(VEFUNCTIONS)33 b(13)228 213 y Ft(in)15 b(\(4.15\).)21 b(The)15 b(term)f(where)h(all)f(di\013eren)o (tiations)h(fall)f(on)i Fr( )h Ft(can)e(b)q(e)g(di\013eren-)228 271 y(tiated)e(again)g(in)g(a)h(manner)e(similar)f(to)i(\(4.15\))q(.)20 b(If)13 b(one)g(di\013eren)o(tiation)f(falls)h(on)228 329 y Fr(\037)259 336 y Fo(2)298 329 y Ft(w)o(e)19 b(stop)i(di\013eren) o(tiating)e(that)h(term)e(under)i(the)f(in)o(tegral)g(sign|lea)o(ving) 228 387 y(the)f(rest)f(of)i(the)f(di\013eren)o(tiations)f(outside)h (the)g(in)o(tegral.)25 b(The)18 b(result)g(of)g(this)228 445 y(pro)q(cedure)h(is)h(\(4.16\),)g(the)g(notation)g(of)g(whic)o(h)f (w)o(e)g(will)g(de\014ne)g(b)q(elo)o(w.)31 b(The)228 503 y(imp)q(ortan)o(t)19 b(p)q(oin)o(t)i(is)f(that)h(when)g(all)e (deriv)m(ativ)o(es)g(fall)h(on)h Fr( )r Ft(,)g(w)o(e)f(can)g(apply)228 561 y(Corollary)e(3.2)f(to)h(obtain)g(our)h(conclusion.)25 b(On)17 b(the)g(other)h(hand)g(a)g(di\013eren-)228 619 y(tiation)f(of)g Fr(\037)473 626 y Fo(2)509 619 y Ft(will)e(lead)i(to)g (a)g(situation)g(with)g(a)g(larger)g Fn(j)p Fr(P)7 b Fn(j)p Ft(|so)17 b(these)g(terms)228 678 y(can)f(b)q(e)h(handled)f(b)o (y)g(the)g(induction)f(h)o(yp)q(othesis.)278 736 y(Let)h Fr(\021)389 743 y Fq(s)421 736 y Fn(2)e Fl(N)503 718 y Fo(3)524 736 y Ft(,)i Fn(j)p Fr(\021)592 743 y Fq(s)610 736 y Fn(j)e Ft(=)g(1,)i Fr(s)e Ft(=)f(1)p Fr(;)8 b Ft(2)p Fr(;)g(:)g(:)g(:)17 b(;)8 b(S)s Ft(.)21 b(De\014ne)16 b(for)g Fr(t)e Fn(2)g(f)p Ft(1)p Fr(;)8 b(:)g(:)g(:)16 b(;)8 b(S)s Fn(g)354 865 y Fr(A)391 872 y Fo(0)424 865 y Ft(=)14 b(0)p Fr(;)126 b(A)677 872 y Fq(t)706 865 y Ft(=)757 818 y Fp(X)765 923 y Fq(s)p Fm(\024)p Fq(t)838 865 y Fr(\021)862 872 y Fq(s)880 865 y Fr(;)g(B)1057 872 y Fq(t)1086 865 y Ft(=)1177 803 y Fq(S)1152 818 y Fp(X)1138 922 y Fq(s)p Fo(=)p Fq(t)p Fo(+1)1247 865 y Fr(\021)1271 872 y Fq(s)1290 865 y Fr(;)g(B)1467 872 y Fq(S)1506 865 y Ft(=)14 b(0)p Fr(:)228 998 y Ft(Notice)f(that)i(the)g (de\014nition)f(of)h Fr(B)872 1005 y Fq(t)902 998 y Ft(dep)q(ends)g(on) g Fr(S)s Ft(,)f(i.e.)20 b Fr(B)1337 1005 y Fq(t)1366 998 y Ft(=)13 b Fr(B)1454 1005 y Fq(t)1469 998 y Ft(\()p Fr(S)s Ft(\).)21 b(W)l(e)14 b(get)228 1056 y(the)f(follo)o(wing)h(form) o(ula)e(\(4.16\))i(from)f(\(4.15\))q(,)g(using)i(the)e(pro)q(cedure)h (describ)q(ed)228 1114 y(ab)q(o)o(v)o(e,)j(b)o(y)g(induction)g(with)g (resp)q(ect)g(to)g Fr(S)s Ft(.)25 b(F)l(or)17 b Fr(S)i Ft(=)c(1)j(the)f(equation)h(\(4.16\))228 1172 y(reduces)d(to)i (\(4.15\))q(.)291 1279 y Fr(@)320 1259 y Fq(A)346 1265 y Ff(S)317 1291 y Fq(x)337 1296 y Fg(1)371 1224 y Fp(n)412 1211 y(Z)462 1279 y Ft(\()p Fr(@)510 1259 y Fq(\013)533 1264 y Fg(1)507 1291 y Fq(x)527 1297 y Ff(P)546 1304 y Fg(1)576 1279 y Fn(\001)8 b(\001)g(\001)g Fr(@)671 1259 y Fq(\013)694 1265 y Ff(M)668 1291 y Fq(x)688 1297 y Ff(P)707 1305 y(M)745 1279 y Fn(j)p Fr( )r Fn(j)807 1259 y Fo(2)826 1279 y Ft(\)\()p Fy(x)p Ft(\))p Fr(\036)p Ft(\()p Fy(x)p Ft(\))g Fr(dx)1071 1286 y Fo(2)1098 1279 y Fn(\001)g(\001)g(\001)g Fr(dx)1217 1286 y Fq(N)1251 1224 y Fp(o)1598 1279 y Ft(\(4.16\))281 1409 y(=)333 1341 y Fp(Z)383 1409 y Ft(\()p Fr(@)431 1388 y Fq(A)457 1394 y Ff(S)428 1421 y Fq(x)448 1427 y Ff(P)481 1409 y Fr(@)510 1388 y Fq(\013)533 1393 y Fg(1)507 1421 y Fq(x)527 1427 y Ff(P)546 1434 y Fg(1)576 1409 y Fn(\001)g(\001)g(\001)g Fr(@)671 1388 y Fq(\013)694 1394 y Ff(M)668 1421 y Fq(x)688 1427 y Ff(P)707 1435 y(M)745 1409 y Fn(j)p Fr( )r Fn(j)807 1388 y Fo(2)826 1409 y Ft(\)\()p Fy(x)p Ft(\))p Fr(\036)p Ft(\()p Fy(x)p Ft(\))g Fr(dx)1071 1416 y Fo(2)1098 1409 y Fn(\001)g(\001)g(\001)g Fr(dx)1217 1416 y Fq(N)337 1560 y Ft(+)407 1498 y Fq(S)383 1513 y Fp(X)390 1617 y Fq(t)p Fo(=1)463 1560 y Fr(@)492 1539 y Fq(B)519 1543 y Ff(t)489 1572 y Fq(x)509 1577 y Fg(1)534 1505 y Fp(n)611 1513 y(X)576 1619 y Fq(j)r Fm(2)p Fq(P)q(;k)q Fm(2)p Fq(Q)727 1492 y Fp(Z)777 1560 y Ft(\()p Fr(@)825 1539 y Fq(A)851 1544 y Ff(t)p Fe(\000)p Fg(1)822 1572 y Fq(x)842 1578 y Ff(P)906 1560 y Fr(@)935 1539 y Fq(\013)958 1544 y Fg(1)932 1572 y Fq(x)952 1578 y Ff(P)971 1585 y Fg(1)1001 1560 y Fn(\001)g(\001)g(\001)g Fr(@)1096 1539 y Fq(\013)1119 1545 y Ff(M)1093 1572 y Fq(x)1113 1578 y Ff(P)1132 1586 y(M)1170 1560 y Fn(j)p Fr( )r Fn(j)1232 1539 y Fo(2)1251 1560 y Ft(\)\()p Fy(z)p Ft(\))1339 1547 y(~)1333 1560 y Fr(\036)1362 1567 y Fq(P)1391 1560 y Ft(\()p Fy(y)q Ft(\))1466 1547 y(~)1460 1560 y Fr(\036)1489 1567 y Fq(Q)1518 1560 y Ft(\()p Fy(y)q Ft(\))578 1706 y Fn(\002)628 1651 y Fp(\020)819 1659 y(Y)666 1767 y Fq(j)682 1757 y Fe(0)694 1767 y Fm(2)p Fq(P)q(;k)770 1757 y Fe(0)781 1767 y Fm(2)p Fq(Q;)p Fo(\()p Fq(j)873 1757 y Fe(0)883 1767 y Fq(;k)912 1757 y Fe(0)923 1767 y Fo(\))p Fm(6)p Fo(=\()p Fq(j;k)q Fo(\))1043 1706 y Fr(\037)1074 1713 y Fo(2)1094 1706 y Ft(\()p Fr(x)1141 1713 y Fo(1)1171 1706 y Ft(+)j Fr(y)1244 1713 y Fq(j)1260 1704 y Fe(0)1284 1706 y Fn(\000)g Fr(y)1358 1713 y Fq(k)1377 1704 y Fe(0)1390 1706 y Ft(\))1409 1651 y Fp(\021)894 1856 y Fn(\002)g Ft(\()p Fr(@)992 1835 y Fq(\021)1010 1839 y Ff(t)1025 1856 y Fr(\037)1056 1863 y Fo(2)1075 1856 y Ft(\)\()p Fr(x)1141 1863 y Fo(1)1172 1856 y Ft(+)g Fr(y)1245 1863 y Fq(j)1274 1856 y Fn(\000)f Fr(y)1347 1863 y Fq(k)1369 1856 y Ft(\))e Fr(dy)1445 1863 y Fo(2)1473 1856 y Fn(\001)g(\001)g(\001)g Fr(dy)1588 1863 y Fq(N)1622 1801 y Fp(o)1655 1856 y Fr(:)228 1953 y Ft(W)l(e)16 b(will)f(use)h(\(4.16\))h(with)f Fr(S)h Ft(=)d Fn(j)p Fr(\013)884 1960 y Fo(0)903 1953 y Fn(j)p Ft(,)i Fr(A)984 1960 y Fq(S)1023 1953 y Ft(=)d Fr(\013)1105 1960 y Fo(0)1125 1953 y Ft(.)21 b(Consider)c(the)f(function)306 2044 y Fr(\036)335 2051 y Fq(j;k)394 2044 y Ft(=)e Fr(\036)475 2051 y Fq(P)515 2044 y Fn(\001)d Fr(\036)569 2051 y Fq(Q)610 2044 y Fn(\001)g Ft(\()p Fr(@)683 2024 y Fq(\021)701 2028 y Ff(t)716 2044 y Fr(\037)747 2051 y Fo(2)766 2044 y Ft(\)\()p Fr(x)832 2051 y Fq(j)861 2044 y Fn(\000)g Fr(x)939 2051 y Fq(k)960 2044 y Ft(\))1140 1997 y Fp(Y)987 2105 y Fq(j)1003 2095 y Fe(0)1014 2105 y Fm(2)p Fq(P)q(;k)1090 2095 y Fe(0)1102 2105 y Fm(2)p Fq(Q;)p Fo(\()p Fq(j)1194 2095 y Fe(0)1204 2105 y Fq(;k)1233 2095 y Fe(0)1244 2105 y Fo(\))p Fm(6)p Fo(=\()p Fq(j;k)q Fo(\))1364 2044 y Fr(\037)1395 2051 y Fo(2)1414 2044 y Ft(\()p Fr(x)1461 2051 y Fq(j)1477 2042 y Fe(0)1502 2044 y Fn(\000)f Fr(x)1579 2051 y Fq(k)1598 2042 y Fe(0)1612 2044 y Ft(\))p Fr(:)228 2182 y Ft(By)17 b(construction)g(w)o(e)h(ha)o(v)o(e)e Fn(j)p Fr(P)7 b Ft(\()p Fr(\036)873 2189 y Fq(j;k)919 2182 y Ft(\))p Fn(j)16 b Fr(>)g Fn(j)p Fr(P)7 b Ft(\()p Fr(\036)p Ft(\))p Fn(j)p Ft(.)25 b(Therefore,)17 b(w)o(e)g(get)g(b)o(y) g(the)228 2241 y(induction)f(h)o(yp)q(othesis)g(on)h Fn(j)p Fr(P)7 b Fn(j)16 b Ft(that)307 2280 y Fp(\015)307 2310 y(\015)307 2340 y(\015)307 2370 y(\015)335 2352 y Fr(@)364 2332 y Fq(B)391 2336 y Ff(t)361 2364 y Fq(x)381 2369 y Fg(1)414 2284 y Fp(Z)464 2352 y Ft(\()p Fr(@)512 2332 y Fq(A)538 2337 y Ff(t)p Fe(\000)p Fg(1)509 2364 y Fq(x)529 2370 y Ff(P)593 2352 y Fr(@)622 2332 y Fq(\013)645 2337 y Fg(1)619 2364 y Fq(x)639 2370 y Ff(P)658 2377 y Fg(1)688 2352 y Fn(\001)8 b(\001)g(\001)g Fr(@)783 2332 y Fq(\013)806 2338 y Ff(M)780 2364 y Fq(x)800 2370 y Ff(P)819 2379 y(M)857 2352 y Fn(j)p Fr( )r Fn(j)919 2332 y Fo(2)938 2352 y Ft(\)\()p Fy(x)p Ft(\))p Fr(\036)1054 2359 y Fq(j;k)1099 2352 y Ft(\()p Fy(x)p Ft(\))g Fr(dx)1228 2359 y Fo(2)1256 2352 y Fn(\001)g(\001)g(\001)g Fr(dx)1375 2359 y Fq(N)1409 2280 y Fp(\015)1409 2310 y(\015)1409 2340 y(\015)1409 2370 y(\015)1436 2402 y Fq(L)1460 2392 y Fg(1)1478 2402 y Fo(\()p Fm(fj)p Fq(x)1540 2407 y Fg(1)1556 2402 y Fm(j)p Fq(>")p Fm(g)p Fo(\))313 2466 y Fn(\024)13 b Fr(C)t(L)437 2445 y Fq(p)457 2466 y Ft(\()p Fr(p)e Ft(+)g(1\))603 2445 y Fq(p)623 2466 y Fr(;)228 2550 y Ft(where)19 b Fr(p)h Ft(=)g Fn(j)p Fr(B)525 2557 y Fq(t)540 2550 y Fn(j)13 b Ft(+)g Fn(j)p Fr(A)669 2557 y Fq(t)p Fm(\000)p Fo(1)729 2550 y Fn(j)g Ft(+)g Fn(j)p Fr(\013)852 2557 y Fo(1)872 2550 y Fn(j)g Ft(+)g Fr(:)8 b(:)g(:)13 b Ft(+)g Fn(j)p Fr(\013)1117 2557 y Fq(M)1157 2550 y Fn(j)19 b Ft(=)h Fn(j)p Fr(\013)p Fn(j)13 b(\000)g Ft(1.)32 b(F)l(urthermore,)228 2608 y(using)18 b(Corollary)g(3.2)g(on)g(the)g (\014rst)g(term)e(on)i(the)f(righ)o(t)h(hand)g(side)g(in)f(\(4.16\))q (,)p eop %%Page: 14 14 14 13 bop 228 120 a Fj(14)51 b(S.)17 b(F)o(OURNAIS,)f(M.)g(AND)h(T.)f (HOFFMANN-OSTENHOF,)f(T.)h(\037.)k(S\037RENSEN)228 213 y Ft(w)o(e)c(obtain)228 251 y Fp(\015)228 281 y(\015)228 311 y(\015)228 341 y(\015)255 256 y(Z)305 323 y Ft(\()p Fr(@)353 303 y Fq(A)379 309 y Ff(S)350 336 y Fq(x)370 342 y Ff(P)403 323 y Fr(@)432 303 y Fq(\013)455 308 y Fg(1)429 336 y Fq(x)449 342 y Ff(P)468 349 y Fg(1)498 323 y Fn(\001)8 b(\001)g(\001)h Fr(@)594 303 y Fq(\013)617 309 y Ff(M)591 336 y Fq(x)611 342 y Ff(P)630 350 y(M)668 323 y Fn(j)p Fr( )r Fn(j)730 303 y Fo(2)748 323 y Ft(\)\()p Fy(x)p Ft(\))p Fr(\036)p Ft(\()p Fy(x)p Ft(\))f Fr(dx)993 330 y Fo(2)1021 323 y Fn(\001)g(\001)g(\001)g Fr(dx)1140 330 y Fq(N)1174 251 y Fp(\015)1174 281 y(\015)1174 311 y(\015)1174 341 y(\015)1201 373 y Fq(L)1225 363 y Fg(1)1243 373 y Fo(\()p Fm(fj)p Fq(x)1305 378 y Fg(1)1321 373 y Fm(j)p Fq(>")p Fm(g)p Fo(\))1422 323 y Fn(\024)13 b Fr(C)t(L)1546 303 y Fq(p)1566 323 y Ft(\()p Fr(p)p Ft(+1\))1690 303 y Fq(p)1711 323 y Fr(;)228 457 y Ft(with)j Fr(p)e Ft(=)g Fn(j)p Fr(A)480 464 y Fq(S)505 457 y Fn(j)d Ft(+)g Fn(j)p Fr(\013)624 464 y Fo(1)643 457 y Fn(j)g Ft(+)g Fr(:)d(:)g(:)j Ft(+)g Fn(j)p Fr(\013)880 464 y Fq(M)919 457 y Fn(j)j Ft(=)f Fn(j)p Fr(\013)p Fn(j)p Ft(.)278 515 y(Th)o(us)h(the)f(desired)g (estimate)f(holds)i(for)g(the)f(individual)g(terms)f(on)i(the)g(righ)o (t)228 573 y(hand)i(side)e(in)h(\(4.16\))q(.)21 b(Since)14 b(the)h(n)o(um)o(b)q(er)e(of)j(terms)d(is)i(b)q(ounded)i(b)o(y)d Fr(c)p Fn(j)p Fr(\013)p Fn(j)h Ft(this)228 631 y(\014nishes)h(the)g (pro)q(of)h(of)j(\(4.9\))q(.)p 1687 631 2 33 v 1689 600 30 2 v 1689 631 V 1718 631 2 33 v 557 770 a Fs(Appendix)d Ft(A.)27 b Fs(Pr)o(oof)17 b(of)h(Lemma)h(3.1)278 857 y Ft(In)f(this)h(app)q(endix)g(w)o(e)f(will)g(pro)o(v)o(e)g(Lemma)f (3.1.)29 b(F)l(or)19 b(con)o(v)o(enience)e(de\014ne)228 916 y Fr(H)268 923 y Fq(E)312 916 y Ft(=)d Fr(H)f Fn(\000)c Fr(E)s Ft(,)15 b(with)g Fr(E)k Ft(b)q(eing)c(the)g(eigen)o(v)m(alue)f (corresp)q(onding)j(to)e(the)h(eigen-)228 974 y(function)21 b Fr( )r Ft(,)g(i.e.)34 b Fr( )23 b Ft(satis\014es)e Fr(H)870 981 y Fq(E)900 974 y Fr( )j Ft(=)e(0.)36 b(Recall)20 b(the)g(notations)i(giv)o(en)f(in)228 1032 y(\(3.3\).)278 1090 y(Let)13 b(us)h(start)g(the)f(pro)q(of)i(b)o(y)e(stating)h(a)g(w)o (ell)e(kno)o(wn)i(result)f(explicitly)l(.)k(Since)228 1148 y(the)f(domain)f(of)i Fr(H)581 1155 y Fq(E)627 1148 y Ft(is)f(kno)o(wn)h(to)f(b)q(e)h Fr(W)1010 1130 y Fo(2)p Fq(;)p Fo(2)1057 1148 y Ft(\()p Fl(R)1115 1130 y Fo(3)o Fq(N)1163 1148 y Ft(\),)f(w)o(e)g(get)228 1243 y Fy(Lemm)o(a)g(A.1.)24 b Fd(L)n(et)d Fr(v)g Fn(2)f Fr(W)791 1225 y Fo(1)p Fq(;)p Fo(2)838 1243 y Ft(\()p Fl(R)896 1225 y Fo(3)p Fq(N)944 1243 y Ft(\))p Fd(.)33 b(Then)21 b Fr(v)g Fn(2)f Fr(W)1292 1225 y Fo(2)p Fq(;)p Fo(2)1339 1243 y Ft(\()p Fl(R)1397 1225 y Fo(3)p Fq(N)1446 1243 y Ft(\))g Fd(if)h(and)g(only)228 1301 y(if)e Fr(H)317 1308 y Fq(E)347 1301 y Fr(v)f Fn(2)e Fr(L)471 1283 y Fo(2)491 1301 y Ft(\()p Fl(R)549 1283 y Fo(3)p Fq(N)597 1301 y Ft(\))p Fd(.)27 b(F)l(urthermor)n(e,)18 b(ther)n(e)h(exists)h(a)f(c)n(onstant)g Fr(K)1494 1308 y Fo(0)1531 1301 y Fr(>)e Ft(0)i Fd(such)228 1359 y(that)e(for)g(al)r (l)i Fr(v)c Fn(2)f Fr(W)614 1341 y Fo(2)p Fq(;)p Fo(2)661 1359 y Ft(\()p Fl(R)719 1341 y Fo(3)p Fq(N)767 1359 y Ft(\))468 1453 y Fn(k)p Fr(v)r Fn(k)544 1462 y Fq(W)582 1453 y Fg(2)p Ff(;)p Fg(2)623 1462 y Fo(\()p Fh(R)664 1453 y Fg(3)p Ff(N)705 1462 y Fo(\))734 1453 y Fn(\024)g Fr(K)828 1460 y Fo(0)856 1412 y Fp(\000)879 1453 y Fn(k)p Fr(H)944 1460 y Fq(E)974 1453 y Fr(v)r Fn(k)1025 1462 y Fq(L)1049 1453 y Fg(2)1066 1462 y Fo(\()p Fh(R)1107 1453 y Fg(3)p Ff(N)1148 1462 y Fo(\))1175 1453 y Ft(+)d Fn(k)p Fr(v)r Fn(k)1300 1462 y Fq(L)1324 1453 y Fg(2)1340 1462 y Fo(\()p Fh(R)1381 1453 y Fg(3)p Ff(N)1422 1462 y Fo(\))1438 1412 y Fp(\001)1469 1453 y Fr(:)278 1549 y Ft(This)23 b(follo)o(ws)h(from)e(the)h(fact)h(that)g Fr(V)35 b Ft(is)23 b(in\014nitesimally)d(small)i(\(in)h(the)228 1607 y(op)q(erator)17 b(sense\))f(with)h(resp)q(ect)e(to)i Fn(\000)p Ft(\001.)278 1665 y(W)l(e)f(no)o(w)g(state)h(and)g(pro)o(v)o (e)e(an)i Fd(a)g(priori)e Ft(estimate.)228 1760 y Fy(Lemm)o(a)h(A.2)j Ft(\(A)d(priori)g(estimate\))p Fy(.)22 b Fd(Ther)n(e)12 b(exists)i(a)f(c)n(onstant)h Fr(C)1501 1767 y Fo(0)1534 1760 y Fd(such)f(that)228 1818 y(for)18 b(al)r(l)i Fr(\021)r(;)8 b(\021)450 1825 y Fo(1)487 1818 y Fn(2)16 b Ft(\(0)p Fr(;)8 b Ft(1\))p Fd(,)20 b(al)r(l)h Fr(\013)782 1825 y Fo(0)818 1818 y Fn(2)c Fl(N)903 1800 y Fo(3)p Fq(N)955 1818 y Fd(,)i Fr(\013)1020 1800 y Fm(0)1049 1818 y Fn(2)e Fl(N)1134 1800 y Fo(3)p Fq(M)1211 1818 y Fd(with)i Fn(j)p Fr(\013)1363 1825 y Fo(0)1383 1818 y Fn(j)12 b Ft(+)g Fn(j)p Fr(\013)1504 1800 y Fm(0)1516 1818 y Fn(j)k(\024)h Ft(2)i Fd(and)228 1877 y(al)r(l)g Fr(v)c Fn(2)f Fr(W)437 1859 y Fo(2)p Fq(;)p Fo(2)484 1877 y Ft(\()p Fr(U)536 1884 y Fb(P)566 1877 y Ft(\()p Fr("=)p Ft(2)d(+)g Fr(\021)740 1884 y Fo(1)760 1877 y Ft(\)\))17 b Fd(we)i(have)f(the)g(estimate)250 1974 y Fr(\021)276 1953 y Fm(j)p Fq(\013)309 1958 y Fg(0)326 1953 y Fm(j)p Fo(+)p Fm(j)p Fq(\013)396 1941 y Fe(0)407 1953 y Fm(j)419 1974 y Fn(k)p Fr(@)473 1953 y Fq(\013)496 1958 y Fg(0)514 1974 y Fr(@)543 1953 y Fq(\013)566 1941 y Fe(0)540 1986 y Fq(x)560 1992 y Fc(P)587 1974 y Fr(v)r Fn(k)638 1982 y Fq(L)662 1973 y Fg(2)679 1982 y Fo(\()p Fq(U)717 1988 y Fc(P)742 1982 y Fo(\()p Fq("=)p Fo(2+)p Fq(\021)q Fo(+)p Fq(\021)899 1987 y Fg(1)916 1982 y Fo(\)\))1610 1974 y Ft(\(A.1\))380 2069 y Fn(\024)p Fr(C)454 2076 y Fo(0)473 2014 y Fp(n)507 2069 y Fr(\021)533 2049 y Fo(2)552 2069 y Fn(k)p Fr(H)617 2076 y Fq(E)647 2069 y Fr(v)r Fn(k)698 2078 y Fq(L)722 2068 y Fg(2)739 2078 y Fo(\()p Fq(U)777 2084 y Fc(P)802 2078 y Fo(\()p Fq("=)p Fo(2+)p Fq(\021)913 2083 y Fg(1)930 2078 y Fo(\)\))970 2069 y Ft(+)1046 2022 y Fp(X)978 2131 y Fq(\014)r Fm(2)p Fh(N)1050 2122 y Fg(3)p Ff(N)1091 2131 y Fq(;)p Fm(j)p Fq(\014)r Fm(j)p Fq(<)p Fo(2)1195 2069 y Fr(\021)1221 2049 y Fm(j)p Fq(\014)r Fm(j)1264 2069 y Fn(k)p Fr(@)1318 2049 y Fq(\014)1341 2069 y Fr(v)r Fn(k)1392 2078 y Fq(L)1416 2068 y Fg(2)1433 2078 y Fo(\()p Fq(U)1471 2084 y Fc(P)1496 2078 y Fo(\()p Fq("=)p Fo(2+)p Fq(\021)1607 2083 y Fg(1)1623 2078 y Fo(\)\))1653 2014 y Fp(o)1686 2069 y Fr(:)228 2221 y Fd(F)l(urthermor)n(e,)h(if)g(the)g(right)h(hand)f(side)h(of)29 b Ft(\(A.1\))19 b Fd(is)g(\014nite)i(for)d(al)r(l)j Fr(\021)r(;)8 b(\021)1606 2228 y Fo(1)1643 2221 y Fr(>)17 b Ft(0)228 2279 y Fd(then)h Fr(v)d Fn(2)f Fr(W)475 2261 y Fo(2)p Fq(;)p Fo(2)522 2279 y Ft(\()p Fr(U)574 2286 y Fb(P)604 2279 y Ft(\()p Fr("=)p Ft(2)e(+)f Fr(\021)779 2286 y Fo(1)799 2279 y Ft(\)\))17 b Fd(for)g(al)r(l)i Fr(\021)1026 2286 y Fo(1)1059 2279 y Fr(>)14 b Ft(0)p Fd(.)228 2374 y(Pr)n(o)n(of.)19 b Ft(Since)13 b Fr(U)535 2381 y Fb(P)565 2374 y Ft(\()p Fr("=)p Ft(2)6 b(+)g Fr(\021)i Ft(+)e Fr(\021)805 2381 y Fo(1)823 2374 y Ft(\))14 b Fn(\032)g Fr(U)942 2381 y Fb(P)972 2374 y Ft(\()p Fr("=)p Ft(2)6 b(+)g Fr(\021)1136 2381 y Fo(1)1156 2374 y Ft(\),)13 b(the)h(estimate)d(is)j(ob)o(viously)228 2432 y(true)j(for)h Fn(j)p Fr(\013)453 2439 y Fo(0)472 2432 y Fn(j)12 b Ft(+)g Fn(j)p Fr(\013)593 2414 y Fm(0)604 2432 y Fn(j)k Fr(<)g Ft(2.)25 b(Let)17 b Fr(\013)870 2439 y Fo(0)906 2432 y Fn(2)f Fl(N)990 2414 y Fo(3)p Fq(N)1042 2432 y Ft(,)h Fr(\013)1104 2414 y Fm(0)1132 2432 y Fn(2)f Fl(N)1216 2414 y Fo(3)p Fq(M)1292 2432 y Ft(with)h Fn(j)p Fr(\013)1449 2439 y Fo(0)1469 2432 y Fn(j)11 b Ft(+)h Fn(j)p Fr(\013)1589 2414 y Fm(0)1601 2432 y Fn(j)k Ft(=)f(2.)228 2490 y(Cho)q(ose)21 b Fr(\036)f Fn(2)h Fr(C)543 2472 y Fm(1)580 2490 y Ft(\()p Fl(R)638 2472 y Fo(3)o Fq(N)686 2490 y Ft(\),)g(0)f Fn(\024)g Fr(\036)g Fn(\024)g Ft(1,)h(with)f Fr(\036)h Fn(\021)f Ft(1)g(on)h Fr(U)1383 2497 y Fb(P)1413 2490 y Ft(\()p Fr("=)p Ft(2)14 b(+)f Fr(\021)j Ft(+)d Fr(\021)1683 2497 y Fo(1)1703 2490 y Ft(\))228 2550 y(and)23 b(supp)9 b Fr(\036)25 b Fn(\032)f Fr(U)588 2557 y Fb(P)618 2550 y Ft(\()p Fr("=)p Ft(2)16 b(+)g Fr(\021)802 2557 y Fo(1)822 2550 y Ft(\),)23 b(satisfying)g Fn(k)p Fr(@)1156 2532 y Fq(\015)1178 2550 y Fr(\036)p Fn(k)1232 2557 y Fm(1)1294 2550 y Fn(\024)h Fr(C)1392 2557 y Fq(\015)1414 2550 y Fr(\021)1440 2532 y Fm(\000j)p Fq(\015)r Fm(j)1510 2550 y Ft(,)g(with)e Fr(C)1700 2557 y Fq(\015)228 2608 y Ft(indep)q(enden)o (t)15 b(of)i Fr(\021)r(;)8 b(\021)632 2615 y Fo(1)651 2608 y Ft(.)p eop %%Page: 15 15 15 14 bop 300 118 a Fj(ANAL)m(YTICITY)14 b(OF)j(THE)f(DENSITY)h(OF)f (ELECTR)o(ONIC)g(W)l(A)l(VEFUNCTIONS)33 b(15)278 213 y Ft(W)l(e)12 b(can)g(no)o(w)h(estimate,)d(using)j(Lemma)d(A.1)i(in)g (the)g(third)g(inequalit)o(y)e(b)q(elo)o(w)327 293 y Fn(k)p Fr(@)381 272 y Fq(\013)404 277 y Fg(0)422 293 y Fr(@)451 272 y Fq(\013)474 261 y Fe(0)448 305 y Fq(x)468 311 y Fc(P)495 293 y Fr(v)r Fn(k)546 301 y Fq(L)570 292 y Fg(2)587 301 y Fo(\()p Fq(U)625 307 y Fc(P)650 301 y Fo(\()p Fq("=)p Fo(2+)p Fq(\021)q Fo(+)p Fq(\021)807 306 y Fg(1)824 301 y Fo(\)\))867 293 y Fn(\024)k(k)p Fr(@)974 272 y Fq(\013)997 277 y Fg(0)1015 293 y Fr(@)1044 272 y Fq(\013)1067 261 y Fe(0)1041 305 y Fq(x)1061 311 y Fc(P)1089 293 y Ft(\()p Fr(\036v)r Ft(\))p Fn(k)1207 302 y Fq(L)1231 293 y Fg(2)1247 302 y Fo(\()p Fh(R)1288 293 y Fg(3)p Ff(N)1329 302 y Fo(\))384 373 y Fn(\024)8 b Fr(c)p Fn(k)p Ft(\()p Fr(\036v)r Ft(\))p Fn(k)595 383 y Fq(W)633 373 y Fg(2)p Ff(;)p Fg(2)674 383 y Fo(\()p Fh(R)715 373 y Fg(3)p Ff(N)756 383 y Fo(\))785 373 y Fn(\024)14 b Fr(C)885 333 y Fp(\000)907 373 y Fn(k)p Fr(H)972 380 y Fq(E)1003 373 y Ft(\()p Fr(\036v)r Ft(\))p Fn(k)1121 383 y Fq(L)1145 373 y Fg(2)1161 383 y Fo(\()p Fh(R)1202 373 y Fg(3)p Ff(N)1243 383 y Fo(\))1270 373 y Ft(+)d Fn(k)p Fr(\036v)r Fn(k)1424 383 y Fq(L)1448 373 y Fg(2)1464 383 y Fo(\()p Fh(R)1505 373 y Fg(3)p Ff(N)1546 383 y Fo(\))1561 333 y Fp(\001)384 469 y Fn(\024)d Fr(C)470 414 y Fp(n)502 469 y Fn(k)p Fr(\036H)596 476 y Fq(E)626 469 y Fr(v)r Fn(k)677 478 y Fq(L)701 469 y Fg(2)718 478 y Fo(\()p Fh(R)759 469 y Fg(3)p Ff(N)800 478 y Fo(\))827 469 y Ft(+)j Fn(k)p Ft(\(\001)p Fr(\036)p Ft(\))p Fr(v)r Fn(k)1060 478 y Fq(L)1084 469 y Fg(2)1099 478 y Fo(\()p Fh(R)1140 469 y Fg(3)p Ff(N)1181 478 y Fo(\))525 577 y Ft(+)g(2)p Fn(kr)p Fr(\036)g Fn(\001)f(r)p Fr(v)r Fn(k)822 587 y Fq(L)846 577 y Fg(2)863 587 y Fo(\()p Fh(R)903 577 y Fg(3)p Ff(N)944 587 y Fo(\))971 577 y Ft(+)h Fn(k)p Fr(\036v)r Fn(k)1125 587 y Fq(L)1149 577 y Fg(2)1165 587 y Fo(\()p Fh(R)1206 577 y Fg(3)p Ff(N)1247 587 y Fo(\))1263 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g(b)o(y)g(m)o(ultiplying)228 946 y(with)f Fr(\021)365 928 y Fo(2)384 946 y Ft(.)278 1004 y(The)d(last)g(statemen)o(t)f(of)h (the)g(lemma)e(follo)o(ws)i(easily)f(from)g(Lemma)f(A.1.)p 1687 1004 2 33 v 1689 973 30 2 v 1689 1004 V 1718 1004 2 33 v 278 1087 a(Finally)l(,)k(w)o(e)i(state)g(and)h(pro)o(v)o(e)e (the)h(prop)q(erties)g(of)g Fr(V)29 b Ft(that)17 b(w)o(e)g(need)g(in)f (the)228 1146 y(pro)q(of)h(of)g(Lemma)d(3.1.)228 1232 y Fy(Lemm)o(a)i(A.3)j Ft(\(Prop)q(erties)d(of)h Fr(V)11 b Ft(\))p Fy(.)24 b Fd(L)n(et)f Fr(V)34 b Fd(b)n(e)24 b(the)g(Coulomb)f(p)n(otential)h(de-)228 1290 y(\014ne)n(d)18 b(in)j Ft(\(1.7\))q Fd(.)278 1358 y Ft(1.)f Fd(Ther)n(e)d(exists)i Fr(C)645 1365 y Fq(V)689 1358 y Fr(>)14 b Ft(0)j Fd(such)h(that)g(for)f (al)r(l)h Fr(v)e Fn(2)e Fr(W)1280 1340 y Fo(1)p Fq(;)p Fo(2)1327 1358 y Ft(\()p Fl(R)1385 1340 y Fo(3)o Fq(N)1433 1358 y Ft(\))j Fd(we)i(have)580 1437 y Fn(k)p Ft(\()p Fr(V)k Fn(\000)10 b Fr(E)s Ft(\))p Fr(v)r Fn(k)833 1446 y Fq(L)857 1437 y Fg(2)874 1446 y 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y(<)718 2327 y(:)762 2245 y Ft(0)492 b(for)16 b Fr(j)23 b(=)-30 b Fn(2)14 b Fr(P)1467 2252 y Fq(s)762 2323 y Fn(j)p Fr(P)807 2330 y Fq(s)826 2323 y Fn(j)840 2305 y Fm(\000j)p Fq(\013)p Fm(j)p Fq(=)p Fo(2)955 2323 y Fr(@)984 2305 y Fq(\013)981 2336 y(x)1008 2323 y Fn(j)p Fr(x)p Fn(j)1064 2305 y Fm(\000)p Fo(1)1111 2266 y Fp(\014)1111 2296 y(\014)1111 2326 y(\014)1127 2358 y Fq(x)p Fo(=)p Fq(x)1194 2363 y Ff(j)1278 2323 y Ft(for)i Fr(j)h Fn(2)d Fr(P)1467 2330 y Fq(s)1486 2323 y Fr(;)228 2421 y Ft(and)228 2547 y Fr(@)257 2527 y Fq(\013)254 2559 y(x)274 2565 y Ff(P)293 2569 y(s)314 2547 y Fn(j)p Fr(x)356 2554 y Fq(j)378 2547 y Fn(\000)5 b Fr(x)450 2554 y Fq(k)470 2547 y Fn(j)484 2527 y Fm(\000)p Fo(1)545 2547 y Ft(=)597 2445 y Fp(8)597 2490 y(<)597 2580 y(:)641 2498 y Ft(0)542 b(for)17 b Fr(j;)8 b(k)21 b(=)-29 b Fn(2)14 b Fr(P)1443 2505 y Fq(s)1478 2498 y Ft(or)i Fr(j;)8 b(k)16 b Fn(2)e Fr(P)1698 2505 y Fq(s)641 2576 y Fn(j)p Fr(P)686 2583 y Fq(s)705 2576 y Fn(j)719 2558 y Fm(\000j)p Fq(\013)p Fm(j)p Fq(=)p Fo(2)834 2576 y Fr(@)863 2558 y Fq(\013)860 2588 y(x)887 2576 y Fn(j)p Fr(x)p Fn(j)943 2558 y Fm(\000)p Fo(1)990 2519 y Fp(\014)990 2549 y(\014)990 2579 y(\014)1006 2611 y Fq(x)p Fo(=)p Fq(x)1073 2616 y Ff(j)1090 2611 y Fm(\000)p Fq(x)1137 2617 y Ff(k)1207 2576 y Ft(for)j Fr(j)f Fn(2)e Fr(P)1396 2583 y Fq(s)1415 2576 y Fr(;)8 b(k)22 b(=)-30 b Fn(2)14 b Fr(P)1556 2583 y Fq(s)1575 2576 y Fr(:)p eop %%Page: 16 16 16 15 bop 228 120 a Fj(16)51 b(S.)17 b(F)o(OURNAIS,)f(M.)g(AND)h(T.)f (HOFFMANN-OSTENHOF,)f(T.)h(\037.)k(S\037RENSEN)228 213 y Ft(Therefore,)14 b(\(A.3\))f(follo)o(ws)h(from)f(the)h(structure)g (of)g Fr(V)e Ft(,)i(the)g(real)f(analyticit)o(y)g(of)228 271 y Fr(x)g Fn(7!)h(j)p Fr(x)p Fn(j)389 253 y Fm(\000)p Fo(1)452 271 y Ft(a)o(w)o(a)o(y)i(from)f(0)i(and)f(the)g(de\014nitions) g(of)h Fr(U)1233 278 y Fb(P)1263 271 y Ft(\()p Fr("=)p Ft(2\))g(and)g Fr(@)1513 253 y Fq(\013)1510 283 y(x)1530 289 y Fc(P)1557 271 y Ft(.)p 1687 271 2 33 v 1689 239 30 2 v 1689 271 V 1718 271 2 33 v 228 402 a Fd(Pr)n(o)n(of)f(of)h(L)n (emma)f(3.1.)21 b Ft(Notice)14 b(that)j(\(A.3\))e(trivially)e(implies)g (that)j(for)g Fr(j;)8 b(\021)16 b(>)228 460 y Ft(0,)g Fr(j)s(\021)g(<)d Ft(1,)k Fn(j)p Fr(\013)p Fn(j)c(\025)h Ft(1)j(w)o(e)f(ha)o(v)o(e)535 585 y Fr(\021)561 564 y Fm(j)p Fq(\013)p Fm(j)606 585 y Fn(k)p Fr(@)660 564 y Fq(\013)657 597 y(x)677 603 y Fc(P)704 585 y Fr(V)11 b Fn(k)768 593 y Fq(L)792 583 y Fe(1)824 593 y Fo(\()p Fq(U)862 599 y Fc(P)887 593 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)r(\021)q Fo(\)\))1058 585 y Fn(\024)j Fr(L)1144 559 y Fm(j)p Fq(\013)p Fm(j)p Fo(+1)1144 598 y Fq(V)1234 585 y Fn(j)p Fr(\013)p Fn(j)p Ft(!)p Fr(j)1330 564 y Fm(\000j)p Fq(\013)p Fm(j)1401 585 y Fr(:)195 b Ft(\(A.4\))278 704 y(W)l(e)18 b(will)f(pro)o(v)o(e)h(that)h(there)e(exists)h Fr(L)996 711 y Fq( )1040 704 y Fr(>)g Ft(0,)h(suc)o(h)f(that)h(for)g (all)e Fr(\021)j Fn(2)e Ft(\(0)p Fr(;)8 b Ft(1\))228 763 y(and)13 b(all)f Fr(j)17 b Fn(2)d Fl(N)g Ft(with)e Fr(j)s(\021)k(<)e Ft(1)f(w)o(e)f(ha)o(v)o(e,)g(for)h(all)f Fr(\013)j Fn(2)f Fl(N)1229 745 y Fo(3)p Fq(M)1287 763 y Ft(,)f Fr(\013)1345 770 y Fo(0)1378 763 y Fn(2)h Fl(N)1460 745 y Fo(3)p Fq(N)1512 763 y Ft(,)f Fn(j)p Fr(\013)1584 770 y Fo(0)1604 763 y Fn(j)h(\024)f Ft(2,)228 821 y Fn(j)p Fr(\013)p Fn(j)e Ft(+)g Fn(j)p Fr(\013)392 828 y Fo(0)411 821 y Fn(j)j Fr(<)g Ft(2)d(+)g Fr(j)s Ft(,)507 946 y Fr(\021)533 926 y Fm(j)p Fq(\013)p Fm(j)p Fo(+)p Fm(j)p Fq(\013)636 931 y Fg(0)653 926 y Fm(j)665 946 y Fn(k)p Fr(@)719 926 y Fq(\013)742 931 y Fg(0)760 946 y Fr(@)789 926 y Fq(\013)786 959 y(x)806 965 y Fc(P)833 946 y Fr( )r Fn(k)892 955 y Fq(L)916 945 y Fg(2)933 955 y Fo(\()p Fq(U)971 961 y Fc(P)996 955 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)r(\021)q Fo(\)\))1167 946 y Fn(\024)i Fr(L)1252 921 y Fm(j)p Fq(\013)p Fm(j)p Fo(+)p Fm(j)p Fq(\013)1355 926 y Fg(0)1372 921 y Fm(j)p Fo(+1)1252 960 y Fq( )1429 946 y Fr(:)167 b Ft(\(A.5\))228 1066 y(Before)18 b(pro)o(ving)h (\(A.5\),)g(let)f(us)h(note)g(that)h(Lemma)c(3.1)j(follo)o(ws)g(easily) f(from)228 1125 y(it.)30 b(In)20 b(fact,)f(let)g Fr(\013)h Fn(2)f Fl(N)695 1107 y Fo(3)p Fq(M)754 1125 y Ft(,)g Fn(j)p Fr(\013)p Fn(j)h(\025)f Ft(1)h(and)g(c)o(ho)q(ose)g Fn(j)p Fr(\013)1269 1132 y Fo(0)1289 1125 y Fn(j)f Ft(=)g(0,)i Fr(\021)g Ft(=)e Fr("=)p Ft(\(2)p Fn(j)p Fr(\013)p Fn(j)p Ft(\),)228 1183 y Fr(j)e Ft(=)c Fn(j)p Fr(\013)p Fn(j)p Ft(.)22 b(Then)16 b(\(A.5\))g(b)q(ecomes)585 1308 y Fn(k)p Fr(@)639 1287 y Fq(\013)636 1320 y(x)656 1326 y Fc(P)683 1308 y Fr( )r Fn(k)742 1316 y Fq(L)766 1307 y Fg(2)783 1316 y Fo(\()p Fq(U)821 1322 y Fc(P)846 1316 y Fo(\()p Fq(")p Fo(\)\))919 1308 y Fn(\024)e Fr(L)1005 1283 y Fm(j)p Fq(\013)p Fm(j)p Fo(+1)1005 1322 y Fq( )1094 1308 y Ft(\(2)p Fr(=")p Ft(\))1203 1287 y Fm(j)p Fq(\013)p Fm(j)1248 1308 y Fn(j)p Fr(\013)p Fn(j)1307 1287 y Fm(j)p Fq(\013)p Fm(j)1351 1308 y Fr(;)228 1425 y Ft(whic)o(h)h(is)h(the)g (statemen)o(t)f(of)h(Lemma)f(3.1.)278 1483 y(W)l(e)f(no)o(w)i(pro)o(v)o (e)e(\(A.5\))g(b)o(y)h(induction)g(in)f Fr(j)s Ft(.)21 b(F)l(or)15 b Fr(j)i Ft(=)d(0)p Fr(;)8 b Ft(1,)15 b(there)g(is)g (nothing)228 1542 y(to)21 b(pro)o(v)o(e)f(since)f(w)o(e)h(kno)o(w)h (that)g Fr( )i Fn(2)e Fr(W)1031 1523 y Fo(2)p Fq(;)p Fo(2)1078 1542 y Ft(\()p Fl(R)1136 1523 y Fo(3)p Fq(N)1185 1542 y Ft(\).)34 b(Let)20 b Fr(L)1376 1549 y Fq( )1424 1542 y Ft(b)q(e)g(su\016cien)o(tly)228 1600 y(large)c(for)h(\(A.5\))e (to)i(b)q(e)f(true)g(for)h Fr(j)g Ft(=)c(0)p Fr(;)8 b Ft(1)17 b(and)g(satisfying)f(furthermore,)620 1732 y Fr(L)653 1739 y Fq( )693 1732 y Fn(\025)e Ft(max)845 1677 y Fp(n)878 1732 y Ft(2)p Fr(L)935 1739 y Fq(V)966 1732 y Fr(;)8 b(C)1023 1739 y Fo(0)1042 1732 y Ft(\(1)k(+)1153 1684 y Fp(X)1146 1792 y Fm(j)p Fq(\014)r Fm(j)p Fq(<)p Fo(2)1240 1732 y Ft(1\))1283 1677 y Fp(o)1317 1732 y Fr(:)279 b Ft(\(A.6\))228 1909 y(Here)14 b(the)h(sum)f(is)h(o)o(v)o(er) g(all)g Fr(\014)h Fn(2)e Fl(N)872 1891 y Fo(3)p Fq(N)939 1909 y Ft(with)h Fn(j)p Fr(\014)s Fn(j)e Fr(<)h Ft(2,)h Fr(C)1261 1916 y Fo(0)1296 1909 y Ft(is)g(the)h(constan)o(t)f(from)228 1967 y(Lemma)f(A.2,)h(and)i Fr(L)634 1974 y Fq(V)681 1967 y Ft(is)f(the)g(constan)o(t)h(from)e(\(A.3\))o(.)278 2025 y(Supp)q(ose)k(that)f(w)o(e)g(ha)o(v)o(e)f(pro)o(v)o(ed)g(\(A.5\)) g(for)i(all)e Fr(j)j Fn(\024)c Fr(j)1317 2032 y Fo(0)1355 2025 y Ft(and)j(all)e Fr(\021)i Fn(2)e Ft(\(0)p Fr(;)8 b Ft(1\))228 2083 y(with)20 b Fr(\021)r(j)k(<)d Ft(1.)34 b(W)l(e)21 b(will)e(pro)o(v)o(e)h(that)g(\(A.5\))g(holds)h(for)g Fr(j)j Ft(=)d Fr(j)1441 2090 y Fo(0)1475 2083 y Ft(+)14 b(1)21 b(and)g(all)228 2141 y Fr(\021)15 b Fn(2)g Ft(\(0)p Fr(;)8 b Ft(1\))16 b(with)g(\()p Fr(j)589 2148 y Fo(0)620 2141 y Ft(+)11 b(1\))p Fr(\021)16 b(<)e Ft(1.)228 2199 y(Let)e Fn(j)p Fr(\013)p Fn(j)s Ft(+)s Fn(j)p Fr(\013)459 2206 y Fo(0)478 2199 y Fn(j)i Fr(<)f Ft(2)s(+)s Fr(j)645 2206 y Fo(0)665 2199 y Ft(.)20 b(Then)12 b(clearly)f Fr(U)1007 2206 y Fb(P)1037 2199 y Ft(\()p Fr("=)p Ft(2)s(+)s(\()p Fr(j)1210 2206 y Fo(0)1232 2199 y Ft(+)s(1\))p Fr(\021)r Ft(\))j Fn(\032)f Fr(U)1460 2206 y Fb(P)1490 2199 y Ft(\()p Fr("=)p Ft(2)s(+)s Fr(j)1644 2206 y Fo(0)1664 2199 y Fr(\021)r Ft(\).)228 2258 y(Therefore,)238 2375 y Fr(\021)264 2355 y Fm(j)p Fq(\013)p Fm(j)p Fo(+)p Fm(j)p Fq(\013)367 2360 y Fg(0)383 2355 y Fm(j)395 2375 y Fn(k)p Fr(@)449 2355 y Fq(\013)472 2360 y Fg(0)490 2375 y Fr(@)519 2355 y Fq(\013)516 2387 y(x)536 2393 y Fc(P)563 2375 y Fr( )r Fn(k)622 2384 y Fq(L)646 2374 y Fg(2)663 2384 y Fo(\()p Fq(U)701 2390 y Fc(P)726 2384 y Fo(\()p Fq("=)p Fo(2+\()p Fq(j)847 2389 y Fg(0)864 2384 y Fo(+1\))p Fq(\021)q Fo(\)\))985 2375 y Fn(\024)h Fr(\021)1064 2355 y Fm(j)p Fq(\013)p Fm(j)p Fo(+)p Fm(j)p Fq(\013)1167 2360 y Fg(0)1183 2355 y Fm(j)1195 2375 y Fn(k)p Fr(@)1249 2355 y Fq(\013)1272 2360 y Fg(0)1290 2375 y Fr(@)1319 2355 y Fq(\013)1316 2387 y(x)1336 2393 y Fc(P)1363 2375 y Fr( )r Fn(k)1422 2384 y Fq(L)1446 2374 y Fg(2)1463 2384 y Fo(\()p Fq(U)1501 2390 y Fc(P)1526 2384 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)1633 2389 y Fg(0)1650 2384 y Fq(\021)q Fo(\)\))1699 2375 y Fr(;)228 2492 y Ft(and)23 b(the)f(result)g(holds)h(b)o(y)f(the)g (induction)g(h)o(yp)q(othesis.)41 b(So)23 b(w)o(e)f(only)g(ha)o(v)o(e) 228 2550 y(to)i(consider)g(the)g(case)g Fn(j)p Fr(\013)p Fn(j)17 b Ft(+)f Fn(j)p Fr(\013)871 2557 y Fo(0)891 2550 y Fn(j)27 b Ft(=)g(2)17 b(+)f Fr(j)1112 2557 y Fo(0)1132 2550 y Ft(.)45 b(Cho)q(ose)26 b(a)e(decomp)q(osition)228 2608 y Fr(\013)18 b Ft(=)g Fr(\013)364 2590 y Fm(0)389 2608 y Ft(+)13 b Fr(\013)471 2590 y Fm(0)o(0)492 2608 y Ft(,)19 b(with)g Fn(j)p Fr(\013)684 2590 y Fm(0)695 2608 y Fn(j)f Ft(=)g(2)13 b Fn(\000)g(j)p Fr(\013)917 2615 y Fo(0)936 2608 y Fn(j)p Ft(,)19 b(i.e.)28 b(with)18 b Fn(j)p Fr(\013)1233 2590 y Fm(00)1254 2608 y Fn(j)g Ft(=)g Fr(j)1362 2615 y Fo(0)1382 2608 y Ft(.)29 b(Using)18 b(Lemma)p eop %%Page: 17 17 17 16 bop 300 118 a Fj(ANAL)m(YTICITY)14 b(OF)j(THE)f(DENSITY)h(OF)f (ELECTR)o(ONIC)g(W)l(A)l(VEFUNCTIONS)33 b(17)228 215 y Ft(A.2)16 b(with)g Fr(\021)454 222 y Fo(1)487 215 y Ft(=)e Fr(j)559 222 y Fo(0)579 215 y Fr(\021)r Ft(,)i Fr(v)f Ft(=)f Fr(@)755 197 y Fq(\013)778 185 y Fe(0)o(0)752 227 y Fq(x)772 233 y Fc(P)799 215 y Fr( )k Ft(w)o(e)e(\014nd)269 317 y Fr(\021)295 296 y Fo(2+)p Fq(j)354 301 y Fg(0)374 317 y Fn(k)p Fr(@)428 296 y Fq(\013)451 301 y Fg(0)469 317 y Fr(@)498 296 y Fq(\013)495 329 y(x)515 335 y Fc(P)542 317 y Fr( )r Fn(k)601 325 y Fq(L)625 316 y Fg(2)642 325 y Fo(\()p Fq(U)680 331 y Fc(P)705 325 y Fo(\()p Fq("=)p Fo(2+\()p Fq(j)826 330 y Fg(0)843 325 y Fo(+1\))p Fq(\021)q Fo(\)\))607 412 y Fn(\024)e Fr(C)695 419 y Fo(0)715 357 y Fp(n)748 412 y Fr(\021)774 392 y Fo(2+)p Fq(j)833 397 y Fg(0)852 412 y Fn(k)p Fr(H)917 419 y Fq(E)948 412 y Fr(@)977 392 y Fq(\013)1000 380 y Fe(0)o(0)974 425 y Fq(x)994 431 y Fc(P)1021 412 y Fr( )r Fn(k)1080 421 y Fq(L)1104 411 y Fg(2)1121 421 y Fo(\()p Fq(U)1159 427 y Fc(P)1184 421 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)1291 426 y Fg(0)1308 421 y Fq(\021)q Fo(\)\))764 521 y Ft(+)820 473 y Fp(X)813 581 y Fm(j)p Fq(\014)r Fm(j)p Fq(<)p Fo(2)908 521 y Fr(\021)934 500 y Fm(j)p Fq(\014)r Fm(j)p Fo(+)p Fq(j)1017 505 y Fg(0)1036 521 y Fn(k)p Fr(@)1090 500 y Fq(\014)1113 521 y Fr(@)1142 500 y Fq(\013)1165 489 y Fe(0)o(0)1139 533 y Fq(x)1159 539 y Fc(P)1186 521 y Fr( )r Fn(k)1245 529 y Fq(L)1269 520 y Fg(2)1286 529 y Fo(\()p Fq(U)1324 535 y Fc(P)1349 529 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)1456 534 y Fg(0)1473 529 y Fq(\021)q Fo(\)\))1521 466 y Fp(o)1555 521 y Fr(:)41 b Ft(\(A.7\))228 657 y(Since)15 b Fr(H)395 664 y Fq(E)425 657 y Fr( )h Ft(=)d(0,)k(w)o(e)e(get)267 740 y Fr(\021)293 719 y Fo(2+)p Fq(j)352 724 y Fg(0)371 740 y Fn(k)p Fr(H)436 747 y Fq(E)467 740 y Fr(@)496 719 y Fq(\013)519 707 y Fe(0)o(0)493 752 y Fq(x)513 758 y Fc(P)540 740 y Fr( )r Fn(k)599 748 y Fq(L)623 739 y Fg(2)640 748 y Fo(\()p Fq(U)678 754 y Fc(P)703 748 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)810 753 y Fg(0)827 748 y Fq(\021)q Fo(\)\))281 850 y Ft(=)e Fr(\021)358 829 y Fo(2+)p Fq(j)417 834 y Fg(0)437 793 y Fp(\015)437 823 y(\015)437 853 y(\015)557 803 y(X)473 909 y Fq(\015)r(<\013)543 900 y Fe(00)564 909 y Fq(;\014)r Fo(+)p Fq(\015)r Fo(=)p Fq(\013)693 900 y Fe(00)721 780 y Fp(\022)758 816 y Fr(\013)789 798 y Fm(00)770 884 y Fr(\015)810 780 y Fp(\023)855 810 y(\000)878 850 y Fr(@)907 829 y Fq(\014)904 862 y(x)924 868 y Fc(P)951 850 y Ft(\()p Fr(V)23 b Fn(\000)10 b Fr(E)s Ft(\))1128 810 y Fp(\001)1159 850 y Fr(@)1188 829 y Fq(\015)1185 862 y(x)1205 868 y Fc(P)1233 850 y Fr( )1267 793 y Fp(\015)1267 823 y(\015)1267 853 y(\015)1294 884 y Fq(L)1318 875 y Fg(2)1335 884 y Fo(\()p Fq(U)1373 890 y Fc(P)1398 884 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)1505 889 y Fg(0)1522 884 y Fq(\021)q Fo(\)\))1610 850 y Ft(\(A.8\))228 985 y(W)l(e)17 b(no)o(w)i(use)f(\(A.4\))o(,)g(the)g(com)o(binatorical)e(result)h(from) g(Prop)q(osition)i(2.1)g(to-)228 1043 y(gether)d(with)g(the)g (induction)g(h)o(yp)q(othesis,)g(to)g(estimate)f(\(A.8\))g(as)500 1129 y Fr(\021)526 1108 y Fo(2+)p Fq(j)585 1113 y Fg(0)605 1129 y Fn(k)p Fr(H)670 1136 y Fq(E)700 1129 y Fr(@)729 1108 y Fq(\013)752 1097 y Fe(0)o(0)726 1141 y Fq(x)746 1147 y Fc(P)773 1129 y Fr( )r Fn(k)832 1137 y Fq(L)856 1128 y Fg(2)873 1137 y Fo(\()p Fq(U)911 1143 y Fc(P)936 1137 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)1043 1142 y Fg(0)1060 1137 y Fq(\021)q Fo(\)\))420 1262 y Fn(\024)505 1196 y Fm(j)p Fq(\013)538 1184 y Fe(00)558 1196 y Fm(j)500 1215 y Fp(X)504 1321 y Fq(k)q Fo(=1)581 1192 y Fp(\022)617 1229 y Fn(j)p Fr(\013)662 1211 y Fm(00)684 1229 y Fn(j)644 1297 y Fr(k)697 1192 y Fp(\023)734 1262 y Fr(L)767 1242 y Fq(k)q Fo(+1)767 1276 y Fq(V)834 1262 y Fr(k)r Ft(!)p Fr(j)898 1242 y Fm(\000)p Fq(k)895 1275 y Fo(0)946 1262 y Fr(L)979 1237 y Fm(j)p Fq(\013)1012 1225 y Fe(00)1032 1237 y Fm(j\000)p Fq(k)q Fo(+1)979 1276 y Fq( )1149 1262 y Fn(\024)1207 1196 y Fm(j)p Fq(\013)1240 1184 y Fe(0)o(0)1260 1196 y Fm(j)1202 1215 y Fp(X)1206 1321 y Fq(k)q Fo(=1)1282 1262 y Fr(L)1315 1242 y Fq(k)q Fo(+1)1315 1276 y Fq(V)1382 1262 y Fr(L)1415 1237 y Fm(j)p Fq(\013)1448 1225 y Fe(00)1468 1237 y Fm(j\000)p Fq(k)q Fo(+1)1415 1276 y Fq( )420 1441 y Fn(\024)41 b Fr(L)533 1416 y Fm(j)p Fq(\013)566 1404 y Fe(00)586 1416 y Fm(j)p Fo(+2)533 1455 y Fq( )656 1375 y Fm(j)p Fq(\013)689 1364 y Fe(00)709 1375 y Fm(j)651 1394 y Fp(X)655 1500 y Fq(k)q Fo(=1)723 1441 y Ft(\()p Fr(L)775 1448 y Fq(V)806 1441 y Fr(=L)863 1448 y Fq( )890 1441 y Ft(\))909 1421 y Fq(k)q Fo(+1)989 1441 y Fn(\024)14 b Fr(L)1075 1416 y Fm(j)p Fq(\013)1108 1404 y Fe(0)o(0)1128 1416 y Fm(j)p Fo(+2)1075 1455 y Fq( )1185 1441 y Fr(:)411 b Ft(\(A.9\))228 1570 y(Here)11 b(w)o(e)g(used)i(the)e(assumption)h Fr(L)880 1577 y Fq( )921 1570 y Fn(\025)h Ft(2)p Fr(L)1030 1577 y Fq(V)1073 1570 y Ft(from)e(\(A.6\))h(in)f(the)h(last)h (estimate.)278 1628 y(Due)h(to)g(the)g(induction)f(h)o(yp)q(othesis)i (w)o(e)e(can)h(also)h(estimate)d(the)i(other)g(term)228 1686 y(in)i(\(A.7\))407 1731 y Fp(X)400 1839 y Fm(j)p Fq(\014)r Fm(j)p Fq(<)p Fo(2)495 1778 y Fr(\021)521 1757 y Fm(j)p Fq(\014)r Fm(j)p Fo(+)p Fq(j)604 1762 y Fg(0)623 1778 y Fn(k)p Fr(@)677 1757 y Fq(\014)700 1778 y Fr(@)729 1757 y Fq(\013)752 1746 y Fe(0)o(0)726 1790 y Fq(x)746 1796 y Fc(P)773 1778 y Fr( )r Fn(k)832 1786 y Fq(L)856 1777 y Fg(2)873 1786 y Fo(\()p Fq(U)911 1792 y Fc(P)936 1786 y Fo(\()p Fq("=)p Fo(2+)p Fq(j)1043 1791 y Fg(0)1060 1786 y Fq(\021)q Fo(\)\))1150 1778 y Fn(\024)1237 1731 y Fp(X)1230 1839 y Fm(j)p Fq(\014)r Fm(j)p Fq(<)p Fo(2)1325 1778 y Fr(L)1358 1753 y Fm(j)p Fq(\013)1391 1741 y Fe(00)1411 1753 y Fm(j)p Fo(+)p Fm(j)p Fq(\014)r Fm(j)p Fo(+1)1358 1792 y Fq( )1536 1778 y Fr(:)35 b Ft(\(A.10\))228 1917 y(So)17 b(using)g(\(A.9\))f(and)g(\(A.10\),)g(w)o(e)g(can)g(estimate)f (\(A.7\))g(as)262 2043 y Fr(\021)288 2023 y Fo(2+)p Fq(j)347 2028 y Fg(0)366 2043 y Fn(k)p Fr(@)420 2023 y Fq(\013)443 2028 y Fg(0)462 2043 y Fr(@)491 2023 y Fq(\013)488 2056 y(x)508 2062 y Fc(P)535 2043 y Fr( )r Fn(k)594 2052 y Fq(L)618 2042 y Fg(2)634 2052 y Fo(\()p Fq(U)672 2058 y Fc(P)697 2052 y Fo(\()p Fq("=)p Fo(2+\()p Fq(j)818 2057 y Fg(0)835 2052 y Fo(+1\))p Fq(\021)q Fo(\)\))957 2043 y Fn(\024)e Fr(L)1042 2018 y Fm(j)p Fq(\013)p Fm(j)p Fo(+)p Fm(j)p Fq(\013)1145 2023 y Fg(0)1162 2018 y Fm(j)p Fo(+1)1042 2057 y Fq( )1227 1958 y Fp( )1271 2001 y Fr(C)1306 2008 y Fo(0)1326 2001 y Ft(\(1)e(+)1430 1964 y Fp(P)1482 2016 y Fm(j)p Fq(\014)r Fm(j)p Fq(<)p Fo(2)1579 2001 y Ft(1\))p 1271 2032 351 2 v 1417 2078 a Fr(L)1450 2085 y Fq( )1627 1958 y Fp(!)1675 2043 y Fr(:)228 2170 y Ft(The)18 b(last)h(factor)g(is)f Fn(\024)f Ft(1,)i(b)o(y)f(the)g(c)o(hoice)f(of)i Fr(L)1125 2177 y Fq( )1170 2170 y Ft(\(see)f(\(A.6\))o(\).)28 b(That)19 b(\014nishes)228 2228 y(the)d(pro)q(of)h(of)j(\(A.5\))c(and)h (therefore)e(of)i(Lemma)d(3.1.)p 1687 2228 2 33 v 1689 2197 30 2 v 1689 2228 V 1718 2228 2 33 v 228 2317 a Fy(Ac)n(kno)n (wledgemen)n(t.)23 b Ft(All)11 b(four)i(authors)h(thank)g(the)e (organizers)i(of)f(the)f(pro-)228 2375 y(gram)h Fd(Partial)i(Di\013er)n (ential)g(Equations)h(and)f(Sp)n(e)n(ctr)n(al)g(The)n(ory)d Ft(for)i(in)o(vitations)228 2433 y(to)k(the)f(Mittag)h(Le\017er)f (institute)g(where)g(part)h(of)g(the)g(w)o(ork)f(w)o(as)h(done.)26 b(F)l(ur-)228 2492 y(thermore,)10 b(parts)i(of)f(this)g(w)o(ork)g(has)h (b)q(een)f(carried)f(out)i(at)f(v)m(arious)h(institutions,)228 2550 y(whose)17 b(hospitalit)o(y)e(is)i(gratefully)e(ac)o(kno)o (wledged:)21 b(Aalb)q(org)c(Univ)o(ersit)o(y)d(\(SF,)228 2608 y(MHO,)i(THO\),)h(The)g(Erwin)h(Sc)o(hr\177)-24 b(o)q(dinger)17 b(Institute)g(\(T\037S\))g(and)i(Univ)o(ersit)o(\023) -23 b(e)p eop %%Page: 18 18 18 17 bop 228 120 a Fj(18)51 b(S.)17 b(F)o(OURNAIS,)f(M.)g(AND)h(T.)f (HOFFMANN-OSTENHOF,)f(T.)h(\037.)k(S\037RENSEN)228 213 y Ft(P)o(aris-Sud)e(\(T\037S\).)g(Financial)f(supp)q(ort)h(from)f(the)h (Danish)g(Natural)g(Science)228 271 y(Researc)o(h)g(Council,)i(Europ)q (ean)g(Science)e(F)l(oundation)i(Programme)d Fd(Sp)n(e)n(ctr)n(al)228 329 y(The)n(ory)d(and)j(Partial)e(Di\013er)n(ential)i(Equations)e Ft(\(SPECT\),)g(and)g(EU)f(IHP)g(net-)228 387 y(w)o(ork)e Fd(Postdo)n(ctor)n(al)h(T)l(r)n(aining)h(Pr)n(o)n(gr)n(am)e(in)i (Mathematic)n(al)h(A)o(nalysis)f(of)f(L)n(ar)n(ge)228 445 y(Quantum)i(Systems)p Ft(,)f(con)o(tract)f(no.)g(HPRN-CT-2002-0027) q(7)j(is)d(gratefully)g(ac-)228 503 y(kno)o(wledged.)228 561 y(SF)j(w)o(as)h(supp)q(orted)g(b)o(y)f(a)g(gran)o(t)h(from)e(the)h (Carlsb)q(erg)h(F)l(oundation.)228 619 y(Finally)l(,)d(SF)i(wishes)f (to)h(thank)g(B.)f(Hel\013er)g(and)h(L.)g(T'Jo)o(\177)-23 b(en)16 b(for)g(useful)f(discus-)228 678 y(sions)i(and)f(encouragemen)o (t.)830 853 y Fs(References)249 932 y Fx([1])j(S.)28 b(F)m(ournais,)i(M.)d(Ho\013mann-Ostenhof,)j(T.)d(Ho\013mann-Ostenhof,) j(and)e(T.)f(\037.)313 981 y(S\034rensen.)11 b(The)g(Electron)g(Densit) o(y)f(is)g(Smo)q(oth)f(a)o(w)o(a)o(y)g(from)f(the)j(Nuclei.)f Fk(Comm.)h(Math.)313 1031 y(Phys.)p Fx(,)j(228\(3\):401{415,)c(2002.) 249 1081 y([2])19 b(S.)28 b(F)m(ournais,)i(M.)d(Ho\013mann-Ostenhof,)j (T.)d(Ho\013mann-Ostenhof,)j(and)e(T.)f(\037.)313 1131 y(S\034rensen.)18 b(On)f(the)h(regularit)o(y)e(of)h(the)g(densit)o(y)h (of)e(electronic)i(w)o(a)o(v)o(efunctions.)e Fk(Con-)313 1181 y(temp.)f(Math.,)g Fx(in)e(prin)o(t.,)g(307,)f(2002.)249 1231 y([3])19 b(M.)i(Ho\013mann-Ostenhof,)h(T.)e(Ho\013mann-Ostenhof,)i (and)f(T.)g(\037.)g(S\034rensen.)h(Elec-)313 1280 y(tron)11 b(w)o(a)o(v)o(efunctions)g(and)g(densities)h(for)f(atoms.)e Fk(A)o(nn.)k(Henri)e(Poinc)n(ar)o(\023)-20 b(e)p Fx(,)12 b(2\(1\):77{100,)313 1330 y(2001.)249 1380 y([4])19 b(M.)i (Ho\013mann-Ostenhof,)h(T.)f(Ho\013mann-Ostenhof,)i(and)e(H.)g (Stremnitzer.)h(Lo)q(cal)313 1430 y(prop)q(erties)14 b(of)f(Coulom)o(bic)d(w)o(a)o(v)o(e)j(functions.)f Fk(Comm.)i(Math.)g (Phys.)p Fx(,)f(163\(1\):185{215,)313 1480 y(1994.)249 1529 y([5])19 b(L.)f(H\177)-21 b(ormander.)17 b Fk(Line)n(ar)i(p)n (artial)f(di\013er)n(ential)g(op)n(er)n(ators)p Fx(.)f(Springer)i(V)m (erlag,)f(Berlin,)313 1579 y(1976.)249 1629 y([6])h(T.)12 b(Kato.)g(On)h(the)g(eigenfunctions)g(of)f(man)o(y-particle)f(systems)i 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