%!PS-Adobe-2.0 %%Creator: dvips(k) 5.996 Copyright 2016 Radical Eye Software %%Title: arXiv:1909.08902v2 [math.AP] 27 Jan 2020 %%CreationDate: Wed Jan 12 14:54:05 2022 %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: EUFM10 dsrom10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z Stab2D_rev1_Jan20.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2022.01.12:1454 %%BeginProcSet: texc.pro 0 0 %! /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 /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}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: Stab2D_rev1_Jan20.out.ps 0 0 %! /pdfmark where{pop} {/globaldict where{pop globaldict}{userdict}ifelse/pdfmark/cleartomark load put} ifelse [ /Title(1. 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b(and)e(a)i(\014xed)f(function) 975 3702 y Fw(w)e Fy(:)d Fv(R)1178 3664 y Fu(2)1249 3702 y Ft(7!)g Fv(R)r Fw(;)113 b(w)r Fy(\()p Fw(x)p Fy(\))26 b(=)f Fw(w)r Fy(\()p Ft(\000)p Fw(x)p Fy(\))p Fw(;)107 b(w)r Fy(\()p Fw(x)p Fy(\))64 b Ft(\000)-16 b(!)2481 3766 y Fs(j)p Fr(x)p Fs(j!1)2726 3702 y Fy(0)p Fw(:)86 3905 y Fy(Mathematically)30 b(this)g(means)g(lo)s(oking)f(at)i(the)g (action)g(of)f(the)h(man)m(y-b)s(o)s(dy)e(Sc)m(hr\177)-45 b(odinger)29 b(op)s(erator)259 4140 y SDict begin H.S end 259 4140 a 259 4140 a SDict begin 13 H.A end 259 4140 a 259 4140 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.2) cvn /DEST pdfmark end 259 4140 a Fw(H)335 4154 y Fr(N)427 4140 y Fy(:=)583 4026 y Fr(N)549 4054 y Fq(X)553 4249 y Fr(j)t Fu(=1)695 4039 y Fq(n)756 4140 y Fy(\()p Ft(\000)p Fy(i)p Ft(r)964 4154 y Fr(j)1020 4140 y Fy(+)19 b FC(A)p Fy(\()p Fw(x)1276 4154 y Fr(j)1313 4140 y Fy(\)\))1384 4098 y Fu(2)1444 4140 y Fy(+)h Fw(V)g Fy(\()p Fw(x)1695 4154 y Fr(j)1732 4140 y Fy(\))1767 4039 y Fq(o)1848 4140 y Fy(+)2046 4078 y(1)p 1949 4119 240 4 v 1949 4202 a Fw(N)30 b Ft(\000)20 b Fy(1)2304 4054 y Fq(X)2213 4251 y Fu(1)p Fp(6)p Fr(i<j)t Fp(6)p Fr(N)2541 4140 y Fw(N)2624 4102 y Fu(2)p Fr(\014)2707 4140 y Fw(w)r Fy(\()p Fw(N)2892 4102 y Fr(\014)2940 4140 y Fy(\()p Fw(x)3027 4154 y Fr(i)3075 4140 y Ft(\000)g Fw(x)3218 4154 y Fr(j)3255 4140 y Fy(\)\))174 b(\(1.2\))86 4385 y(on)30 b(the)h(symmetric)e(space)1257 4583 y Fo(H)1322 4597 y Fr(N)1414 4583 y Fy(=)c Fw(L)1572 4545 y Fu(2)1572 4605 y(sym)1700 4583 y Fy(\()p Fv(R)1795 4545 y Fu(2)q Fr(N)1904 4583 y Fy(\))g(=)2098 4469 y Fr(N)2060 4496 y Fq(O)2067 4687 y Fu(sym)2213 4583 y Fw(L)2275 4545 y Fu(2)2314 4583 y Fy(\()p Fv(R)2409 4545 y Fu(2)2455 4583 y Fy(\))p Fw(:)86 4803 y Fy(F)-8 b(or)40 b(nice)e(data)i Fw(V)5 b(;)15 b(w)r(;)g FC(A)p Fy(,)43 b(this)38 b(op)s(erator)i(is)e(b)s(ounded)f(from)h(b)s(elo)m(w)h(with)e (the)j(core)g(domain)e Fo(H)3532 4817 y Fr(N)3625 4803 y Ft(\\)86 4911 y Fw(C)158 4878 y Fs(1)151 4933 y Fr(c)232 4911 y Fy(\()p Fv(R)327 4878 y Fu(2)p Fr(N)436 4911 y Fy(\),)22 b(and)e(th)m(us)g(can)h(b)s(e)e(extended)i(to)g(b)s(e)e(a)i (self-adjoin)m(t)f(op)s(erator)g(b)m(y)g(F)-8 b(riedric)m(hs')20 b(metho)s(d.)36 b(The)86 5019 y(non-trivial)20 b(question)h(is)g(then)g (that)i(of)f(stabilit)m(y)e(of)i(the)h(second)f(kind)e([)2538 5019 y SDict begin H.S end 2538 5019 a Black Fy(17)p Black 2629 4960 a SDict begin H.R end 2629 4960 a 2629 5019 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieSei-09) cvn H.B /ANN pdfmark end 2629 5019 a Fy(],)k(i.e.)38 b(whether)21 b Fw(H)3273 5033 y Fr(N)3365 5019 y Fn(>)k Ft(\000)p Fw(C)7 b(N)86 5127 y Fy(for)31 b(a)h(constan)m(t)h Fw(C)38 b Fy(indep)s(enden)m(t)30 b(of)i Fw(N)10 b Fy(.)44 b(Note)33 b(that)g(when)d(the)i(negativ)m(e)h(part)e Fw(w)2987 5141 y Fs(\000)3074 5127 y Fy(:=)c(min)o Ft(f)p Fw(w)r(;)15 b Fy(0)p Ft(g)34 b Fy(is)86 5235 y(nonzero,)h(the)f(in)m(teraction)f (energy)h(with)e(the)i(rescaled)g(p)s(oten)m(tial)f(in)f(\()2617 5235 y SDict begin H.S end 2617 5235 a Black Fy(1.1)p Black 2733 5176 a SDict begin H.R end 2733 5176 a 2733 5235 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 2733 5235 a Fy(\))i(ma)m(y)g(b)s(e)f(v)m(ery)h (negativ)m(e,)86 5342 y(and)26 b(w)m(e)h(will)c(see)28 b(that)f(the)f(stabilit)m(y)f(question)h(cannot)h(b)s(e)f(answ)m(ered)g (using)f(merely)h(the)h(uncertain)m(t)m(y)86 5450 y(principle.)p Black 86 5534 499 4 v 186 5626 a Fm(Date)6 b FB(:)26 b(Septem)n(b)r(er,)e(2019.)p Black Black 1868 5725 a Fu(1)p Black eop end end %%Page: 2 2 TeXDict begin HPSdict begin 2 1 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.2) cvn /DEST pdfmark end -222 785 a Black 86 559 a Fu(2)1257 b(P)-6 b(.T.)23 b(NAM)g(AND)g(N.)g(R)n(OUGERIE)p Black 186 759 a Fy(Out)29 b(of)i Fw(H)552 773 y Fr(N)649 759 y Fy(one)g(constructs)f(a)h(corresp) s(onding)d(non-linear)h(Sc)m(hr\177)-45 b(odinger)29 b(\(NLS\))h(functional)747 977 y SDict begin H.S end 747 977 a 747 977 a SDict begin 13 H.A end 747 977 a 747 977 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.3) cvn /DEST pdfmark end 747 977 a Ft(E)803 939 y Fu(nls)894 977 y Fy([)p Fw(u)p Fy(])c(=)64 b(lim)1118 1038 y Fr(N)7 b Fs(!1)1366 916 y Fy(1)p 1347 956 83 4 v 1347 1039 a Fw(N)1440 977 y Ft(h)p Fw(u)1527 939 y Fs(\012)p Fr(N)1650 977 y Ft(j)p Fw(H)1751 991 y Fr(N)1818 977 y Ft(j)p Fw(u)1895 939 y Fs(\012)p Fr(N)2017 977 y Ft(i)26 b Fy(=)f Ft(h)p Fw(u)p Ft(j)p Fw(h)p Ft(j)p Fw(u)p Ft(i)2450 997 y Fr(L)2498 978 y Fl(2)2559 977 y Fy(+)2663 916 y Fw(b)p 2659 956 46 4 v 2659 1039 a Fy(2)2730 853 y Fq(Z)2781 1059 y Fk(R)2828 1041 y Fl(2)2882 977 y Ft(j)p Fw(u)p Ft(j)2984 939 y Fu(4)3499 977 y Fy(\(1.3\))86 1194 y(where)1696 1350 y Fw(b)g Fy(=)1856 1227 y Fq(Z)1907 1433 y Fk(R)1954 1414 y Fl(2)2008 1350 y Fw(w)86 1548 y Fy(and)1410 1681 y SDict begin H.S end 1410 1681 a 1410 1681 a SDict begin 13 H.A end 1410 1681 a 1410 1681 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.4) cvn /DEST pdfmark end 1410 1681 a Fw(h)g Fy(=)g(\()q Ft(\000)p Fy(i)p Ft(r)18 b Fy(+)i FC(A)p Fy(\()p Fw(x)p Fy(\)\))2138 1639 y Fu(2)2198 1681 y Fy(+)g Fw(V)1157 b Fy(\(1.4\))86 1835 y(is)39 b(the)i(one-b)s(o)s(dy)e(Hamiltonian.)68 b(Clearly)-8 b(,)42 b Ft(E)1749 1802 y Fu(nls)1880 1835 y Fy(is)e(b)s(ounded)e(from) h(b)s(elo)m(w)h(under)e(the)j(constrain)m(t)86 1943 y Ft(k)p Fw(u)p Ft(k)228 1962 y Fr(L)276 1943 y Fl(2)311 1962 y Fu(\()p Fk(R)385 1943 y Fl(2)421 1962 y Fu(\))478 1943 y Fy(=)25 b(1)31 b(if)e(and)h(only)g(if)1727 2081 y SDict begin H.S end 1727 2081 a 1727 2081 a SDict begin 13 H.A end 1727 2081 a 1727 2081 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.5) cvn /DEST pdfmark end 1727 2081 a Fw(b)25 b Fn(>)g Ft(\000)p Fw(a)2006 2043 y Fs(\003)3499 2081 y Fy(\(1.5\))86 2232 y(where)30 b Fw(a)397 2199 y Fs(\003)467 2232 y Fy(is)f(the)i(optimal)e(constan)m(t)j(for)e(the)g (Gagiardo-Niren)m(b)s(erg)g(inequalit)m(y)809 2324 y Fq(\022)876 2329 y(Z)926 2535 y Fk(R)973 2516 y Fl(2)1028 2453 y Ft(j)p Fw(u)p Ft(j)1130 2415 y Fu(2)1170 2324 y Fq(\023)15 b(\022)1319 2329 y(Z)1370 2535 y Fk(R)1417 2516 y Fl(2)1472 2453 y Ft(jr)p Fw(u)p Ft(j)1650 2415 y Fu(2)1689 2324 y Fq(\023)1782 2453 y Fn(>)1887 2391 y Fw(a)1935 2358 y Fs(\003)p 1887 2432 88 4 v 1908 2515 a Fy(2)2000 2329 y Fq(Z)2051 2535 y Fk(R)2098 2516 y Fl(2)2152 2453 y Ft(j)p Fw(u)p Ft(j)2254 2415 y Fu(4)2294 2453 y Fw(;)107 b Ft(8)p Fw(u)25 b Ft(2)f Fw(H)2722 2415 y Fu(1)2762 2453 y Fy(\()p Fv(R)2857 2415 y Fu(2)2902 2453 y Fy(\))p Fw(:)3499 2453 y SDict begin H.S end 3499 2453 a 3499 2453 a SDict begin 16.25 H.A end 3499 2453 a 3499 2453 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.6) cvn /DEST pdfmark end 3499 2453 a Fy(\(1.6\))86 2678 y(See)31 b([)273 2678 y SDict begin H.S end 273 2678 a Black Fy(29)p Black 364 2620 a SDict begin H.R end 364 2620 a 364 2678 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Weinstein-83) cvn H.B /ANN pdfmark end 364 2678 a Fy(,)419 2678 y SDict begin H.S end 419 2678 a Black Fy(8)p Black 465 2620 a SDict begin H.R end 465 2620 a 465 2678 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GuoSei-13) cvn H.B /ANN pdfmark end 465 2678 a Fy(,)520 2678 y SDict begin H.S end 520 2678 a Black Fy(20)p Black 611 2620 a SDict begin H.R end 611 2620 a 611 2678 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Maeda-10) cvn H.B /ANN pdfmark end 611 2678 a Fy(,)667 2678 y SDict begin H.S end 667 2678 a Black Fy(11)p Black 758 2620 a SDict begin H.R end 758 2620 a 758 2678 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Kwong-89) cvn H.B /ANN pdfmark end 758 2678 a Fy(,)813 2678 y SDict begin H.S end 813 2678 a Black Fy(5)p Black 859 2620 a SDict begin H.R end 859 2620 a 859 2678 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Frank-14) cvn H.B /ANN pdfmark end 859 2678 a Fy(])f(for)g(references.)186 2786 y(Giv)m(en)c(the)g(v)-5 b(ariational)26 b(construction)g(\()1598 2786 y SDict begin H.S end 1598 2786 a Black Fy(1.3)p Black 1714 2727 a SDict begin H.R end 1714 2727 a 1714 2786 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 1714 2786 a Fy(\))q(,)h(Condition)d(\()2257 2786 y SDict begin H.S end 2257 2786 a Black Fy(1.5)p Black 2374 2727 a SDict begin H.R end 2374 2727 a 2374 2786 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.5) cvn H.B /ANN pdfmark end 2374 2786 a Fy(\))j(is)e(necessary)i(for)f Fw(H)3133 2800 y Fr(N)3226 2786 y Fy(to)h(b)s(e)f(stable)86 2894 y(of)33 b(the)f(second)h(kind.)45 b(Su\016cien)m(t)31 b(conditions)g(are)i(in)e(fact)j(more)e(stringen)m(t.)47 b(Simple)30 b(considerations)86 3002 y(sho)m(w)g(that)h(w)m(e)g(m)m (ust)f(demand)g(more)g(than)g(\()1685 3002 y SDict begin H.S end 1685 3002 a Black Fy(1.5)p Black 1802 2943 a SDict begin H.R end 1802 2943 a 1802 3002 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.5) cvn H.B /ANN pdfmark end 1802 3002 a Fy(\).)41 b(Here)31 b(w)m(e)g(w)m(ork)f(with)f(the)i (condition)1610 3219 y SDict begin H.S end 1610 3219 a 1610 3219 a SDict begin 13 H.A end 1610 3219 a 1610 3219 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.7) cvn /DEST pdfmark end 1610 3219 a 1626 3096 a Fq(Z)1676 3302 y Fk(R)1723 3283 y Fl(2)1778 3219 y Ft(j)p Fw(w)1868 3233 y Fs(\000)1927 3219 y Ft(j)26 b Fw(<)f(a)2122 3182 y Fs(\003)3499 3219 y Fy(\(1.7\))86 3441 y(and)30 b(refer)g(to)h([)609 3441 y SDict begin H.S end 609 3441 a Black Fy(13)p Black 700 3382 a SDict begin H.R end 700 3382 a 700 3441 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-14c) cvn H.B /ANN pdfmark end 700 3441 a Fy(])g(for)f(a)h(re\014nemen)m(t)f (\(Hartree)h(stabilit)m(y\).)186 3549 y(In)41 b(this)h(note)h(w)m(e)h (are)f(in)m(terested)f(in)g(the)h(range)g(of)f(diluteness)f(parameter)i Fw(\014)51 b(>)46 b Fy(0)d(for)f(whic)m(h)86 3657 y(the)c(stabilit)m(y) e(of)h(the)h(second)f(kind)f(can)i(b)s(e)f(sho)m(wn)f(to)i(hold.)61 b(Indeed,)39 b Fw(\014)j Fy(measures)37 b(ho)m(w)h(fast)g(the)86 3765 y(in)m(teractions)31 b(con)m(v)m(erge)j(to)e(p)s(oin)m(t-lik)m(e)e (ones.)44 b(Implicit)29 b(in)i(the)g(ab)s(o)m(v)m(e)i(is)d(that)j(the)e (reference)h(length-)86 3873 y(scale)h(of)h(the)f(system)g(is)f (\014xed,)i(set)f(b)m(y)g(that)h(of)f(the)h(one-b)s(o)s(dy)e (Hamiltonian)f Fw(h)j Fy(\(think)e(of)h(particles)86 3983 y(in)e(a)i(\014xed)e(b)s(o)m(x)h(if)f(y)m(ou)i(wish\).)45 b(Hence)33 b Fw(N)1567 3950 y Fs(\000)p Fr(\014)1701 3983 y Fy(measures)f(the)h(range)f(of)h(the)f(in)m(teraction)g(p)s (oten)m(tial)g(in)86 4097 y(units)i(of)i(the)g(reference)g(length)f (scale.)57 b(F)-8 b(rom)37 b(the)f(p)s(oin)m(t)e(of)i(view)f(of)h(in)m (teractions,)h Fw(N)3219 4064 y Fr(\014)s Fs(\000)p Fu(1)p Fr(=)p Fu(2)3463 4097 y Fy(is)e(the)86 4205 y(a)m(v)m(erage)i(n)m(um)m (b)s(er)c(of)h(particles)g(a)g(tagged)i(one)f(in)m(teracts)g(with)e(at) i(a)g(time,)g(for)f Fw(N)3019 4172 y Fs(\000)p Fu(1)p Fr(=)p Fu(2)3218 4205 y Fy(is)f(the)i(mean)86 4313 y(in)m(ter-particle) 29 b(distance.)186 4421 y(It)e(is)g(w)m(ell-kno)m(wn)f(that)i(large)f (quan)m(tum)g(in)m(teracting)h(systems)f(are)h(harder)e(to)i(deal)f (with)g(for)g(large)86 4529 y(v)-5 b(alues)25 b(of)h Fw(\014)5 b Fy(,)27 b(in)d(particular,)h(in)f(our)i(2D)g(case,)i(for)d Fw(\014)31 b(>)25 b Fy(1)p Fw(=)p Fy(2)i(whic)m(h)d(is)h(the)g (threshold)f(to)j(ha)m(v)m(e)g(a)f(dilute)86 4637 y(system)35 b(\(few)h(but)f(strong)g(in)m(ter-particle)g(collisions\).)53 b(In)35 b(this)f(note)i(w)m(e)g(mak)m(e)h(the)e(remark)g(that)h(a)86 4745 y(com)m(bination)d(of)h(the)g(to)s(ols)g(in)e([)1232 4745 y SDict begin H.S end 1232 4745 a Black Fy(14)p Black 1323 4686 a SDict begin H.R end 1323 4686 a 1323 4745 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 1323 4745 a Fy(,)1382 4745 y SDict begin H.S end 1382 4745 a Black Fy(25)p Black 1473 4686 a SDict begin H.R end 1473 4686 a 1473 4745 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-19) cvn H.B /ANN pdfmark end 1473 4745 a Fy(])i(allo)m(ws)f(to)i(pro)m(v)m(e)f(stabilit)m(y)f (of)h(the)g(second)f(kind)f(under)h(the)86 4853 y(condition)d Fw(\014)j(<)27 b Fy(1)p Fw(:)32 b Fy(This)e(extends)i(the)g(range)g(of) g(v)-5 b(alidit)m(y)30 b(of)i(metho)s(ds)f([)2649 4853 y SDict begin H.S end 2649 4853 a Black Fy(13)p Black 2740 4794 a SDict begin H.R end 2740 4794 a 2740 4853 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-14c) cvn H.B /ANN pdfmark end 2740 4853 a Fy(,)2797 4853 y SDict begin H.S end 2797 4853 a Black Fy(14)p Black 2888 4794 a SDict begin H.R end 2888 4794 a 2888 4853 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2888 4853 a Fy(,)2945 4853 y SDict begin H.S end 2945 4853 a Black Fy(3)p Black 2990 4794 a SDict begin H.R end 2990 4794 a 2990 4853 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.CheHol-15) cvn H.B /ANN pdfmark end 2990 4853 a Fy(,)3047 4853 y SDict begin H.S end 3047 4853 a Black Fy(10)p Black 3138 4794 a SDict begin H.R end 3138 4794 a 3138 4853 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.JebPic-17) cvn H.B /ANN pdfmark end 3138 4853 a Fy(])h(dealing)e(with)86 4961 y(the)g(large-)p Fw(N)41 b Fy(limit)28 b(of)j Fw(H)975 4975 y Fr(N)1042 4961 y Fy(.)86 5194 y FC(Ac)m(kno)m(wledgemen)m(ts.)40 b Fy(W)-8 b(e)29 b(thank)e(Mathieu)h(Lewin)e(and)i(F)-8 b(ernando)28 b(G.S.L.)g(Brand~)-45 b(ao)28 b(for)g(helpful)86 5302 y(discussions.)50 b(W)-8 b(e)36 b(receiv)m(ed)e(fundings)e(from)i (the)h(Europ)s(ean)e(Researc)m(h)i(Council)e(\(ER)m(C\))i(under)e(the) 86 5410 y(Europ)s(ean)39 b(Union's)g(Horizon)i(2020)h(Researc)m(h)f (and)f(Inno)m(v)-5 b(ation)40 b(Programme)h(\(Gran)m(t)g(agreemen)m(t) 86 5518 y(CORFR)m(ONMA)-8 b(T)31 b(No)g(758620\),)k(and)30 b(from)g(the)h(Deutsc)m(he)h(F)-8 b(orsc)m(h)m(ungsgemeinsc)m(haft)32 b(\(DF)m(G,)h(Ger-)86 5626 y(man)d(Researc)m(h)h(F)-8 b(oundation\))31 b(under)e(German)m(y's)h(Excellence)h(Strategy)g(\(EX) m(C-2111-39081486)q(8\).)p Black Black eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.3) cvn /DEST pdfmark end -222 785 a Black 830 559 a Fu(IMPR)n(O)n(VED)30 b(ST)-6 b(ABILITY)32 b(F)n(OR)e(2D)h(A)-6 b(TTRA)n(CTIVE)30 b(BOSE)h(GASES)710 b(3)p Black 86 676 a SDict begin H.S end 86 676 a 86 676 a SDict begin 13 H.A end 86 676 a 86 676 a SDict begin [/View [/XYZ H.V]/Dest (section.2) cvn /DEST pdfmark end 86 676 a 1548 759 a Fy(2.)46 b Fx(Main)34 b(resul)-6 b(t)186 921 y Fy(F)e(or)21 b(simplicit)m(y)e(w)m(e)i(assume)g(that)h Fw(V)5 b(;)15 b FC(A)p Fw(;)g(w)24 b Fy(are)e(smo)s(oth.)37 b(The)21 b(actual)g(condition)f(on)h(the)g(in)m(teraction)86 1029 y(p)s(oten)m(tial)30 b(w)m(e)h(need)f(is)1456 1139 y SDict begin H.S end 1456 1139 a 1456 1139 a SDict begin 13 H.A end 1456 1139 a 1456 1139 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.1) cvn /DEST pdfmark end 1456 1139 a Fw(w)e Ft(2)d Fw(L)1697 1102 y Fu(1)1736 1139 y Fy(\()p Fv(R)1831 1102 y Fu(2)1877 1139 y Fy(\))20 b Ft(\\)g Fw(L)2075 1102 y Fu(2)2114 1139 y Fy(\()p Fv(R)2209 1102 y Fu(2)2255 1139 y Fy(\))p Fw(:)1184 b Fy(\(2.1\))86 1269 y(W)-8 b(e)31 b(also)g(assume)f(that)h(there)f(exist)h(p)s(ositiv) m(e)e(constan)m(ts)j Fw(s)24 b(>)h Fy(0)31 b(and)f Fw(c)25 b(>)g Fy(0)31 b(suc)m(h)f(that)1516 1417 y SDict begin H.S end 1516 1417 a 1516 1417 a SDict begin 13 H.A end 1516 1417 a 1516 1417 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.2) cvn /DEST pdfmark end 1516 1417 a Fw(V)20 b Fy(\()p Fw(x)p Fy(\))26 b Fn(>)f Fw(c)1872 1379 y Fs(\000)p Fu(1)1966 1417 y Ft(j)p Fw(x)p Ft(j)2068 1379 y Fr(s)2126 1417 y Ft(\000)20 b Fw(c)1243 b Fy(\(2.2\))86 1565 y(and)1589 1676 y SDict begin H.S end 1589 1676 a 1589 1676 a SDict begin 13 H.A end 1589 1676 a 1589 1676 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.3) cvn /DEST pdfmark end 1589 1676 a Ft(j)p FC(A)p Fy(\()p Fw(x)p Fy(\))p Ft(j)26 b Fn(6)f Fw(ce)2043 1638 y Fr(c)p Fs(j)p Fr(x)p Fs(j)2158 1676 y Fw(:)1316 b Fy(\(2.3\))186 1805 y(Our)29 b(main)g(result)g(is)86 1968 y SDict begin H.S end 86 1968 a 86 1968 a SDict begin 13 H.A end 86 1968 a 86 1968 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.1) cvn /DEST pdfmark end 86 1968 a Black FC(Theorem)34 b(2.1)h Fy(\()p FC(Stabilit)m(y)g(of)g(attractiv) m(e)f(2D)i(Bose)f(gases)p Fy(\))p FC(.)p Black 86 2076 a Fj(Under)f(the)g(ab)-5 b(ove)35 b(assumptions)43 b Fy(\()1324 2076 y SDict begin H.S end 1324 2076 a Black Fy(1.7)p Black 1441 2018 a SDict begin H.R end 1441 2018 a 1441 2076 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.7) cvn H.B /ANN pdfmark end 1441 2076 a Fy(\))p Fj(-)p Fy(\()1544 2076 y SDict begin H.S end 1544 2076 a Black Fy(2.1)p Black 1660 2018 a SDict begin H.R end 1660 2018 a 1660 2076 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.1) cvn H.B /ANN pdfmark end 1660 2076 a Fy(\))q Fj(-)p Fy(\()1764 2076 y SDict begin H.S end 1764 2076 a Black Fy(2.2)p Black 1880 2018 a SDict begin H.R end 1880 2018 a 1880 2076 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 1880 2076 a Fy(\))p Fj(-)p Fy(\()1983 2076 y SDict begin H.S end 1983 2076 a Black Fy(2.3)p Black 2099 2018 a SDict begin H.R end 2099 2018 a 2099 2076 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 2099 2076 a Fy(\))p Fj(,)34 b(for)h(any)f(\014xe)-5 b(d)35 b Fy(0)28 b Fw(<)f(\014)33 b(<)27 b Fy(1)34 b Fj(ther)-5 b(e)35 b(exists)f(a)86 2184 y(c)-5 b(onstant)34 b Fw(C)e(>)25 b Fy(0)33 b Fj(such)g(that)1641 2295 y SDict begin H.S end 1641 2295 a 1641 2295 a SDict begin 13 H.A end 1641 2295 a 1641 2295 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.4) cvn /DEST pdfmark end 1641 2295 a Fw(H)1717 2309 y Fr(N)1809 2295 y Fn(>)25 b Ft(\000)p Fw(C)7 b(N)1378 b Fy(\(2.4\))86 2426 y Fj(as)33 b(an)g(op)-5 b(er)g(ator)36 b(acting)d(on)g Fo(H)1157 2440 y Fr(N)1250 2426 y Fy(=)25 b Fw(L)1408 2393 y Fu(2)1408 2448 y(sym)1536 2426 y Fy(\()p Fv(R)1631 2393 y Fu(2)p Fr(N)1739 2426 y Fy(\))p Fj(.)p Black 86 2589 a(R)-5 b(emarks.)p Black 86 2697 a FC(1.)41 b Fy(The)30 b(b)s(est)f(result)h (preceding)f(the)i(ab)s(o)m(v)m(e)g(is)f(that)h(from)f([)2195 2697 y SDict begin H.S end 2195 2697 a Black Fy(14)p Black 2286 2639 a SDict begin H.R end 2286 2639 a 2286 2697 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2286 2697 a Fy(])g(whic)m(h)f(co)m(v)m(ers)1688 2889 y Fw(\014)h(<)1875 2827 y(s)20 b Fy(+)g(1)p 1875 2868 200 4 v 1875 2951 a Fw(s)g Fy(+)g(2)86 3075 y(with)27 b Fw(s)h Fy(the)h(exp)s(onen)m(t)f(in)f(\()1045 3075 y SDict begin H.S end 1045 3075 a Black Fy(2.2)p Black 1162 3017 a SDict begin H.R end 1162 3017 a 1162 3075 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 1162 3075 a Fy(\).)40 b(This)27 b(had)h(the)g(merit)g(of) h(allo)m(wing)e(a)i(dilute)d(gas,)k Fw(\014)h(>)25 b Fy(1)p Fw(=)p Fy(2,)30 b(whereas)86 3183 y(the)23 b(previous)e(results) h([)896 3183 y SDict begin H.S end 896 3183 a Black Fy(13)p Black 987 3125 a SDict begin H.R end 987 3125 a 987 3183 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-14c) cvn H.B /ANN pdfmark end 987 3183 a Fy(])h(w)m(ere)h(limited)c(to)k Fw(\014)31 b(<)25 b Fy(1)p Fw(=)p Fy(2.)39 b(It)23 b(is)f(not)h(our)g (aim)f(to)i(pretend)e(that)h(increasing)86 3291 y Fw(\014)31 b(>)26 b Fy(0)31 b(is)e(an)i(undertaking)e(that)j(should)c(go)k(on)e (forev)m(er,)i(but)e(there)h(are)g(reasons)g(that)g(mak)m(e)h(us)e (feel)86 3399 y(the)39 b(ab)s(o)m(v)m(e)g(is)f(notew)m(orth)m(y)-8 b(.)67 b(First,)40 b(the)f(anno)m(ying)f(dep)s(endence)f(on)i(the)g (trapping)e(p)s(oten)m(tial)h(gets)86 3507 y(disp)s(ensed)c(with)h (\(pro)m(vided)g(the)i(gro)m(wth)g(is)e(still)f(p)s(olynomial\).)57 b(Second,)38 b(the)e(pro)s(of)g(is)f(somewhat)86 3615 y(cleaner.)48 b(Third,)32 b(a)h(natural)f(barrier)f(seems)i(to)h(ha)m (v)m(e)g(b)s(een)f(reac)m(hed:)46 b(the)33 b(condition)f Fw(\014)j(<)29 b Fy(1)k(is)f(that)86 3723 y(needed)e(to)h(obtain)f(a)h (second-momen)m(t)g(estimate)g(follo)m(wing)e(the)i(tec)m(hniques)f(of) g([)2981 3723 y SDict begin H.S end 2981 3723 a Black Fy(4)p Black 3027 3664 a SDict begin H.R end 3027 3664 a 3027 3723 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ErdYau-01) cvn H.B /ANN pdfmark end 3027 3723 a Fy(,)3082 3723 y SDict begin H.S end 3082 3723 a Black Fy(22)p Black 3173 3664 a SDict begin H.R end 3173 3664 a 3173 3723 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.NamRouSei-15) cvn H.B /ANN pdfmark end 3173 3723 a Fy(,)3229 3723 y SDict begin H.S end 3229 3723 a Black Fy(14)p Black 3320 3664 a SDict begin H.R end 3320 3664 a 3320 3723 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 3320 3723 a Fy(].)86 3871 y FC(2.)69 b Fy(It)39 b(is)g(not)h(ob)m (vious)f(to)i(us)e(whether)g(there)h(should)d(exist)j(a)g(ph)m (ysically)d(natural)i(upp)s(er)f(b)s(ound)86 3979 y(on)k Fw(\014)48 b Fy(in)42 b(the)g(attractiv)m(e)j(case.)78 b(F)-8 b(or)44 b(3D)f(repulsiv)m(e)e(gases)j(this)d(w)m(ould)h(b)s(e)g (giv)m(en)g(b)m(y)h(the)f(Gross-)86 4087 y(Pitaevskii)26 b([)529 4087 y SDict begin H.S end 529 4087 a Black Fy(18)p Black 620 4029 a SDict begin H.R end 620 4029 a 620 4087 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieSeiSolYng-05) cvn H.B /ANN pdfmark end 620 4087 a Fy(,)673 4087 y SDict begin H.S end 673 4087 a Black Fy(26)p Black 763 4029 a SDict begin H.R end 763 4029 a 763 4087 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-EMS) cvn H.B /ANN pdfmark end 763 4087 a Fy(])i(limit)d Fw(\014)31 b Fy(=)25 b(1.)40 b(F)-8 b(or)28 b(2D)g(repulsiv)m(e)d(gases,)30 b(an)m(y)d Fw(\014)k(>)25 b Fy(0)i(should)f(b)s(e)g(allo)m(w)m(ed,)j (for)e(the)86 4195 y(GP)j(limit)f([)492 4195 y SDict begin H.S end 492 4195 a Black Fy(19)p Black 583 4136 a SDict begin H.R end 583 4136 a 583 4195 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieSeiYng-01) cvn H.B /ANN pdfmark end 583 4195 a Fy(,)638 4195 y SDict begin H.S end 638 4195 a Black Fy(9)p Black 684 4136 a SDict begin H.R end 684 4136 a 684 4195 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.JebLeoPic-16) cvn H.B /ANN pdfmark end 684 4195 a Fy(])h(corresp)s (onds)f(to)i(an)g(in)m(teraction)f(scaled)g(exp)s(onen)m(tially)f(with) g Fw(N)10 b Fy(.)86 4343 y FC(3.)37 b Fy(The)20 b(pro)s(of)g(pro)s (ceeds)f(b)m(y)i(com)m(bining)d(the)j(second)f(momen)m(t)h(estimate)g (of)g([)2757 4343 y SDict begin H.S end 2757 4343 a Black Fy(14)p Black 2848 4285 a SDict begin H.R end 2848 4285 a 2848 4343 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2848 4343 a Fy(])f(with)f(the)i (information-)86 4451 y(theoretic)28 b(quan)m(tum)f(de)h(Finetti)f (theorem)h([)1637 4451 y SDict begin H.S end 1637 4451 a Black Fy(2)p Black 1683 4393 a SDict begin H.R end 1683 4393 a 1683 4451 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BraHar-12) cvn H.B /ANN pdfmark end 1683 4451 a Fy(,)1735 4451 y SDict begin H.S end 1735 4451 a Black Fy(15)p Black 1826 4393 a SDict begin H.R end 1826 4393 a 1826 4451 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LiSmi-15) cvn H.B /ANN pdfmark end 1826 4451 a Fy(])g(of)g(Brand~)-45 b(ao-Harro)m(w.)41 b(The)27 b(in)m(terest)h(of)g(the)f(latter)86 4559 y(for)38 b(large)f(b)s(osonic)g(systems)h([)1166 4559 y SDict begin H.S end 1166 4559 a Black Fy(25)p Black 1257 4501 a SDict begin H.R end 1257 4501 a 1257 4559 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-19) cvn H.B /ANN pdfmark end 1257 4559 a Fy(])g(is)f(that)i(it)e(\(almost\))i (giv)m(es)f(a)g(quan)m(titativ)m(e)h(de)e(Finetti)g(theorem)86 4667 y(when)26 b(the)h(one-b)s(o)s(dy)g(state-space)i(is)d(in\014nite)f (dimensional)f(\(more)k(precisely)e(the)h(dimension)e(should)86 4775 y(b)s(e)39 b(still)e(\014nite)i(but)f(only)h(its)g(logarithm)f(en) m(ters)i(relev)-5 b(an)m(t)40 b(estimates\),)j(th)m(us)c(b)m(ypassing)f (the)i(main)86 4883 y(tec)m(hnical)30 b(limitation)e(of)j(the)g(to)s (ols)f(used)f(in)g([)1705 4883 y SDict begin H.S end 1705 4883 a Black Fy(14)p Black 1797 4824 a SDict begin H.R end 1797 4824 a 1797 4883 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 1797 4883 a Fy(].)86 5031 y FC(4.)66 b Fy(F)-8 b(or)40 b(the)f(ab)s(o)m(v)m(e)h (stabilit)m(y)e(result)g(w)m(e)h(ma)m(y)h(assume)e FC(A)i Fy(=)f(0,)j(thanks)c(to)i(the)f(diamagnetic)g(in-)86 5139 y(equalit)m(y)32 b([)459 5139 y SDict begin H.S end 459 5139 a Black Fy(16)p Black 550 5081 a SDict begin H.R end 550 5081 a 550 5139 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieLos-01) cvn H.B /ANN pdfmark end 550 5139 a Fy(,)h(Theorem)f(7.21],)j(as)d(w)m(ell)f(as)i(ignore)f(the)g(b)s (osonic)f(symmetry)h(\(i.e.)47 b(the)32 b(lo)m(w)m(er)h(b)s(ound)86 5249 y(holds)28 b(true)g(on)h(the)h(full)d(space)i Fw(L)1255 5216 y Fu(2)1295 5249 y Fy(\()p Fv(R)1390 5216 y Fu(2)p Fr(N)1498 5249 y Fy(\)\).)41 b(Ho)m(w)m(ev)m(er,)32 b(the)d(presence)g (of)g(the)h(magnetic)f(\014eld)f(and)h(the)86 5357 y(Bose-Einstein)h (statistics)g(are)h(meaningful)d(for)i(the)h(Corollary)2355 5357 y SDict begin H.S end 2355 5357 a Black Fy(2.2)p Black 2471 5298 a SDict begin H.R end 2471 5298 a 2471 5357 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 2471 5357 a 30 w Fy(b)s(elo)m(w.)861 b Fn(\003)186 5518 y Fy(The)25 b(stabilit)m(y)g(result)f(has)i(t)m(w)m (o)h(corollaries)e(regarding)g(the)h(large)g Fw(N)36 b Fy(limit)24 b(of)i(the)g(system)g(at)h(hand.)86 5626 y(First)j(for)g(ground)f(states:)p Black Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end -222 785 a Black 86 559 a Fu(4)1257 b(P)-6 b(.T.)23 b(NAM)g(AND)g(N.)g(R)n(OUGERIE)p Black 86 759 a SDict begin H.S end 86 759 a 86 759 a SDict begin 13 H.A end 86 759 a 86 759 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.2) cvn /DEST pdfmark end 86 759 a Black FC(Corollary)36 b(2.2)g Fy(\()p FC(NLS)f(limit)f(for)h(ground)g(states)p Fy(\))p FC(.)p Black 86 867 a Fj(Under)e(the)g(ab)-5 b(ove)33 b(assumptions,)i(the)e(gr)-5 b(ound)34 b(state)f(ener)-5 b(gy)33 b(p)-5 b(er)33 b(p)-5 b(article)728 1003 y Fw(E)5 b Fy(\()p Fw(N)10 b Fy(\))p 728 1043 227 4 v 799 1127 a Fw(N)989 1064 y Fy(=)1114 1003 y(1)p 1095 1043 83 4 v 1095 1127 a Fw(N)1203 1064 y Fy(inf)21 b Fw(\033)1381 1079 y Fi(H)1429 1090 y Fh(N)1491 1064 y Fy(\()p Fw(H)1602 1078 y Fr(N)1669 1064 y Fy(\))k(=)1854 1003 y(1)p 1835 1043 V 1835 1127 a Fw(N)1943 1064 y Fy(inf)6 b Ft(fh)p Fy(\011)p Fw(;)15 b(H)2321 1078 y Fr(N)2389 1064 y Fy(\011)p Ft(i)25 b Fy(:)g Ft(k)p Fy(\011)p Ft(k)2731 1079 y Fi(H)2779 1090 y Fh(N)2867 1064 y Fy(=)g(1)p Ft(g)86 1245 y Fj(c)-5 b(onver)g(ges)33 b(when)h Fw(N)h Ft(!)25 b(1)32 b Fj(to)i(the)f(gr)-5 b(ound)34 b(state)f(ener)-5 b(gy)33 b(of)g(the)g(NLS)f(functional)h(in) 40 b Fy(\()3185 1245 y SDict begin H.S end 3185 1245 a Black Fy(1.3)p Black 3301 1187 a SDict begin H.R end 3301 1187 a 3301 1245 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 3301 1245 a Fy(\))q Fj(,)32 b(i.e.)1191 1402 y Fw(E)1263 1365 y Fu(nls)1379 1402 y Fy(=)25 b(inf)6 b Ft(fE)1687 1365 y Fu(nls)1778 1402 y Fy(\()p Fw(u)p Fy(\))26 b(:)g Ft(k)p Fw(u)p Ft(k)2119 1422 y Fr(L)2167 1403 y Fl(2)2202 1422 y Fu(\()p Fk(R)2276 1403 y Fl(2)2312 1422 y Fu(\))2369 1402 y Fy(=)f(1)p Ft(g)p Fw(:)86 1587 y Fj(Mor)-5 b(e)g(over,)28 b(the)f(r)-5 b(e)g(duc)g(e)g(d)28 b(density)e(matric)-5 b(es)28 b Ft(f)p Fw(\015)1727 1539 y Fu(\()p Fr(k)r Fu(\))1722 1615 y(\011)1777 1626 y Fh(N)1839 1587 y Ft(g)1884 1601 y Fr(N)1978 1587 y Fj(of)e(gr)-5 b(ound)27 b(states)g Fy(\011)2692 1601 y Fr(N)2785 1587 y Fj(of)g Fw(H)2962 1601 y Fr(N)3054 1587 y Fj(c)-5 b(onver)g(ge)27 b(to)g(c)-5 b(on-)86 1698 y(vex)30 b(c)-5 b(ombinations)32 b(of)f(pr)-5 b(oje)g(ctions)33 b(on)d(NLS)g(minimizers,)i(namely)f(ther)-5 b(e)32 b(exists)e(a)h(Bor)-5 b(el)32 b(pr)-5 b(ob)g(ability)86 1806 y(me)g(asur)g(e)34 b Fw(\026)e Fj(supp)-5 b(orte)g(d)36 b(on)d(the)g(minimizers)g(of)g Fw(E)1861 1773 y Fu(nls)1985 1806 y Fj(such)f(that,)i(along)g(a)f(subse)-5 b(quenc)g(e)32 b Fw(N)j Ft(!)25 b(1)p Fj(,)916 2006 y Fy(lim)877 2066 y Fr(N)7 b Fs(!1)1096 2006 y Fy(T)-8 b(r)1205 1874 y Fq(\014)1205 1928 y(\014)1205 1983 y(\014)1205 2037 y(\014)1236 2006 y Fw(\015)1288 1958 y Fu(\()p Fr(k)r Fu(\))1283 2034 y(\011)1338 2045 y Fh(N)1420 2006 y Ft(\000)1510 1882 y Fq(Z)1617 2006 y Ft(j)p Fw(u)1694 1968 y Fs(\012)p Fr(k)1791 2006 y Ft(ih)p Fw(u)1913 1968 y Fs(\012)p Fr(k)2012 2006 y Ft(j)p Fw(d\026)p Fy(\()p Fw(u)p Fy(\))2261 1874 y Fq(\014)2261 1928 y(\014)2261 1983 y(\014)2261 2037 y(\014)2317 2006 y Fy(=)25 b(0)p Fw(;)109 b Ft(8)p Fw(k)28 b Ft(2)d Fv(N)7 b Fw(:)3499 2006 y SDict begin H.S end 3499 2006 a 3499 2006 a SDict begin 16.25 H.A end 3499 2006 a 3499 2006 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.5) cvn /DEST pdfmark end 3499 2006 a Fy(\(2.5\))186 2262 y(Next,)31 b(for)f(dynamics:)86 2426 y SDict begin H.S end 86 2426 a 86 2426 a SDict begin 13 H.A end 86 2426 a 86 2426 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.3) cvn /DEST pdfmark end 86 2426 a Black FC(Corollary)36 b(2.3)g Fy(\()p FC(NLS)f(limit)f(for)h(dynamics)p Fy(\))p FC(.)p Black 86 2534 a Fj(Under)e(the)g(ab)-5 b(ove)33 b(assumptions)i(and)e(with)h FC(A)25 b Ft(\021)g Fy(0)p Fj(,)33 b(the)g(many-b)-5 b(o)g(dy)35 b(Schr\177)-46 b(odinger)34 b(dynamics)1458 2684 y Fw(i@)1537 2698 y Fr(t)1567 2684 y Fy(\011)1638 2698 y Fr(N)1705 2684 y Fy(\()p Fw(t)p Fy(\))26 b(=)f Fw(H)2006 2698 y Fr(N)2073 2684 y Fy(\011)2144 2698 y Fr(N)2210 2684 y Fy(\()p Fw(t)p Fy(\))86 2836 y Fj(starting)38 b(fr)-5 b(om)38 b(a)f(wel)5 b(l-pr)-5 b(ep)g(ar)g(e)g(d)40 b(initial)d(datum,)h(e.g.)54 b Fy(\011)2125 2850 y Fr(N)2192 2836 y Fy(\(0\))33 b(=)g Fw(u)p Fy(\(0\))2611 2803 y Fs(\012)p Fr(N)2771 2836 y Fj(with)k Fw(u)p Fy(\(0\))h Fj(smo)-5 b(oth,)40 b(c)-5 b(on-)86 2944 y(ver)g(ges)33 b(when)g Fw(N)i Ft(!)25 b(1)33 b Fj(to)g(the)g(NLS)f(dynamics)1313 3094 y Fw(i@)1392 3108 y Fr(t)1423 3094 y Fw(u)p Fy(\()p Fw(t)p Fy(\))25 b(=)g Fw(hu)p Fy(\()p Fw(t)p Fy(\))c(+)f Fw(b)p Ft(j)p Fw(u)p Fy(\()p Fw(t)p Fy(\))p Ft(j)2262 3056 y Fu(2)2303 3094 y Fw(u)p Fy(\()p Fw(t)p Fy(\))86 3244 y Fj(in)32 b(the)h(sense)g(of)g(r)-5 b(e)g(duc)g(e)g(d)34 b(density)g(matric)-5 b(es)741 3416 y Fy(lim)702 3477 y Fr(N)7 b Fs(!1)921 3416 y Fy(T)-8 b(r)1030 3312 y Fq(\014)1030 3366 y(\014)1030 3421 y(\014)1061 3416 y Fw(\015)1113 3368 y Fu(\()p Fr(k)r Fu(\))1108 3449 y(\011)1163 3460 y Fh(N)1220 3449 y Fu(\()p Fr(t)p Fu(\))1325 3416 y Ft(\000)20 b(j)p Fw(u)p Fy(\()p Fw(t)p Fy(\))1596 3379 y Fs(\012)p Fr(k)1694 3416 y Ft(ih)p Fw(u)p Fy(\()p Fw(t)p Fy(\))1919 3379 y Fs(\012)p Fr(k)2018 3416 y Ft(j)2043 3312 y Fq(\014)2043 3366 y(\014)2043 3421 y(\014)2099 3416 y Fy(=)25 b(0)p Fw(;)109 b Ft(8)p Fw(k)28 b Ft(2)c Fv(N)7 b Fw(;)114 b Ft(8)p Fw(t)25 b Ft(2)g Fv(R)s Fw(:)3499 3416 y SDict begin H.S end 3499 3416 a 3499 3416 a SDict begin 16.25 H.A end 3499 3416 a 3499 3416 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 3499 3416 a Fy(\(2.6\))p Black 86 3614 a Fj(R)-5 b(emarks.)p Black 86 3722 a FC(1.)41 b Fy(Recall)30 b(that)h(the)f Fw(k)s Fy(-particles)g(reduced)g(densit)m (y)f(matrix)h(of)h(an)m(y)f(\011)25 b Ft(2)g Fo(H)2750 3736 y Fr(N)2848 3722 y Fy(is)30 b(de\014ned)f(as)1406 3887 y Fw(\015)1458 3840 y Fu(\()p Fr(k)r Fu(\))1453 3916 y(\011)1581 3887 y Fy(:=)c(T)-8 b(r)1796 3902 y Fr(k)r Fu(+1)p Fs(!)p Fr(N)2078 3887 y Ft(j)p Fy(\011)p Ft(ih)p Fy(\011)p Ft(j)p Fw(:)86 4054 y Fy(Here)45 b(T)-8 b(r)408 4069 y Fr(k)r Fu(+1)p Fs(!)p Fr(N)719 4054 y Fy(means)44 b(the)h(partial)e(trace)i(o)m(v)m(er)h Fw(N)39 b Ft(\000)30 b Fw(k)47 b Fy(factors)e(of)g Fo(H)2739 4068 y Fr(N)2855 4054 y Fy(=)2974 3986 y Fq(N)3075 4012 y Fr(N)3075 4081 y Fu(sym)3218 4054 y Fw(L)3280 4021 y Fu(2)3319 4054 y Fy(\()p Fv(R)3414 4021 y Fu(2)3460 4054 y Fy(\).)83 b(In)86 4200 y(particular,)30 b(for)g(the)h(pro)s (duct)f(state)i(w)m(e)g(ha)m(v)m(e)g Fw(\015)1790 4152 y Fu(\()p Fr(k)r Fu(\))1785 4236 y Fr(u)1826 4217 y Fg(\012)p Fh(N)1962 4200 y Fy(=)26 b Ft(j)p Fw(u)2136 4167 y Fs(\012)p Fr(k)2234 4200 y Ft(ih)p Fw(u)2356 4167 y Fs(\012)p Fr(k)2454 4200 y Ft(j)p Fy(.)43 b(Th)m(us)29 b(the)j(con)m(v)m(ergences)h(\()3509 4200 y SDict begin H.S end 3509 4200 a Black Fy(2.5)p Black 3625 4141 a SDict begin H.R end 3625 4141 a 3625 4200 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.5) cvn H.B /ANN pdfmark end 3625 4200 a Fy(\))q(,)86 4312 y(\()121 4312 y SDict begin H.S end 121 4312 a Black Fy(2.6)p Black 237 4254 a SDict begin H.R end 237 4254 a 237 4312 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.6) cvn H.B /ANN pdfmark end 237 4312 a Fy(\))e(tells)e(us)g(that)h(the)h(corresp) s(onding)c(man)m(y-b)s(o)s(dy)i(w)m(a)m(v)m(e)j(functions)d(are)h (close)g(to)h(pro)s(duct)e(states)86 4420 y(in)g(a)i(w)m(eak)g(sense)f (\(it)h(is)e(w)m(ell)g(kno)m(wn)h(that)h(they)g(are)g(not)f(close)h(in) e Fw(L)2519 4387 y Fu(2)2558 4420 y Fy(-norm\).)86 4570 y FC(2.)51 b Fy(Although)33 b(the)h(statemen)m(ts)i(co)m(v)m(er)g(also) e(repulsiv)m(e)e(gases)j(\()p Fw(w)f Fn(>)c Fy(0\),)36 b(the)e(no)m(v)m(elt)m(y)i(lies)c(mainly)g(in)86 4678 y(the)38 b(attractiv)m(e)h(case.)64 b(If)37 b Fw(w)j Fn(>)e Fy(0)g(the)f(pro)s(of)g(is)g(m)m(uc)m(h)h(simpler)d(indeed)h (and)h(one)h(can)g(reac)m(h)h(m)m(uc)m(h)86 4786 y(higher)29 b(v)-5 b(alues)30 b(of)g Fw(\014)5 b Fy(;)31 b(see)g([)1022 4786 y SDict begin H.S end 1022 4786 a Black Fy(19)p Black 1113 4727 a SDict begin H.R end 1113 4727 a 1113 4786 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieSeiYng-01) cvn H.B /ANN pdfmark end 1113 4786 a Fy(])g(for)f(ground)f(states)j(and)d([) 2076 4786 y SDict begin H.S end 2076 4786 a Black Fy(9)p Black 2122 4727 a SDict begin H.R end 2122 4727 a 2122 4786 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.JebLeoPic-16) cvn H.B /ANN pdfmark end 2122 4786 a Fy(])i(for)f(dynamics.)86 4936 y FC(3.)39 b Fy(W)-8 b(e)27 b(do)e(not)h(ev)m(en)h(sk)m(etc)m(h)g (the)e(pro)s(of)g(of)h(Corollary)1990 4936 y SDict begin H.S end 1990 4936 a Black Fy(2.2)p Black 2106 4877 a SDict begin H.R end 2106 4877 a 2106 4936 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 2106 4936 a Fy(.)40 b(The)25 b(reader)g(should)f(ha)m(v)m(e)j(no)e (di\016cult)m(y)f(in)86 5044 y(\014guring)g(out)h(that,)i(giv)m(en)f (Theorem)1401 5044 y SDict begin H.S end 1401 5044 a Black Fy(2.1)p Black 1517 4985 a SDict begin H.R end 1517 4985 a 1517 5044 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 1517 5044 a Fy(,)h(the)e(pro)s(of)g (of)g([)2076 5044 y SDict begin H.S end 2076 5044 a Black Fy(14)p Black 2168 4985 a SDict begin H.R end 2168 4985 a 2168 5044 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2168 5044 a Fy(,)h(Theorem)f(1])h (applies)e(m)m(utatis)h(m)m(utandis)86 5152 y(for)38 b Fw(\014)43 b(<)38 b Fy(1,)j(at)e(least)g(under)d(the)j(additional)d (condition)h Ft(j)p FC(A)p Fy(\()p Fw(x)p Fy(\))p Ft(j)2412 5119 y Fu(2)2491 5152 y Fn(6)h Fw(V)20 b Fy(\()p Fw(x)p Fy(\).)65 b(The)37 b(latter)i(tec)m(hnical)86 5260 y(condition)29 b(can)i(b)s(e)e(replaced)h(b)m(y)i(\()1297 5260 y SDict begin H.S end 1297 5260 a Black Fy(2.3)p Black 1413 5201 a SDict begin H.R end 1413 5201 a 1413 5260 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 1413 5260 a Fy(\))f(follo)m(wing)e(argumen)m(ts)h(from)g([)2545 5260 y SDict begin H.S end 2545 5260 a Black Fy(22)p Black 2636 5201 a SDict begin H.R end 2636 5201 a 2636 5260 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.NamRouSei-15) cvn H.B /ANN pdfmark end 2636 5260 a Fy(,)h(Section)f(4.2].)86 5410 y FC(4.)58 b Fy(F)-8 b(or)36 b(the)h(dynamical)d(statemen)m(t,)40 b(the)c(metho)s(d)g(of)g([)2086 5410 y SDict begin H.S end 2086 5410 a Black Fy(10)p Black 2177 5351 a SDict begin H.R end 2177 5351 a 2177 5410 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.JebPic-17) cvn H.B /ANN pdfmark end 2177 5410 a Fy(])g(allo)m(ws)f(to)i(deriv)m(e)f(NLS)f(dynamics)g(pro-)86 5518 y(vided)28 b(\()365 5518 y SDict begin H.S end 365 5518 a Black Fy(2.4)p Black 482 5459 a SDict begin H.R end 482 5459 a 482 5518 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 482 5518 a Fy(\))i(holds.)39 b(Hence,)31 b(Theorem)1498 5518 y SDict begin H.S end 1498 5518 a Black Fy(2.1)p Black 1614 5459 a SDict begin H.R end 1614 5459 a 1614 5518 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 1614 5518 a 30 w Fy(extends)f(the)g(range)g(of)g(v)-5 b(alidit)m(y)28 b(of)i(their)f(main)f(result)h(to)86 5626 y(all)i Fw(\014)j(<)28 b Fy(1.)47 b(Suc)m(h)32 b(a)g(range)h(w)m(as)g(obtained)f(previously)e (in)h([)2187 5626 y SDict begin H.S end 2187 5626 a Black Fy(21)p Black 2278 5567 a SDict begin H.R end 2278 5567 a 2278 5626 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.NamNap-17) cvn H.B /ANN pdfmark end 2278 5626 a Fy(])h(\(without)g(using)g (\()2987 5626 y SDict begin H.S end 2987 5626 a Black Fy(2.4)p Black 3104 5567 a SDict begin H.R end 3104 5567 a 3104 5626 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 3104 5626 a Fy(\)\),)h(and)f(a)h(more)p Black Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end -222 785 a Black 830 559 a Fu(IMPR)n(O)n(VED)30 b(ST)-6 b(ABILITY)32 b(F)n(OR)e(2D)h(A)-6 b(TTRA)n(CTIVE)30 b(BOSE)h(GASES)710 b(5)p Black 86 759 a Fy(restricted)29 b(one)g(in)f([)785 759 y SDict begin H.S end 785 759 a Black Fy(3)p Black 831 700 a SDict begin H.R end 831 700 a 831 759 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.CheHol-15) cvn H.B /ANN pdfmark end 831 759 a Fy(])h(\(in)f(the)i(sp)s(ecial)d(case)j Fw(s)25 b Fy(=)g(2\).)41 b(W)-8 b(e)31 b(also)e(refer)g(to)h(these)f (references)h(for)e(precise)86 867 y(descriptions)g(of)j(the)f(w)m (ell-prepared)f(initial)f(data)j(that)g(can)f(b)s(e)g(co)m(v)m(ered.) 3615 975 y Fn(\003)86 1137 y SDict begin H.S end 86 1137 a 86 1137 a SDict begin 13 H.A end 86 1137 a 86 1137 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.4) cvn /DEST pdfmark end 86 1137 a Black Fj(R)-5 b(emark)44 b Fy(2.4)33 b(\(Classically)c(stable)h(case)h(in)e(3D\))p Fj(.)p Black 86 1245 a Fy(Stabilit)m(y)g(of)h(the)h(second)f(kind)f(is)g(also) h(an)h(issue)e(for)h(3D)h(Bose)h(gases)f(with)e(p)s(oten)m(tials)1598 1394 y Fw(N)1681 1356 y Fu(3)p Fr(\014)s Fs(\000)p Fu(1)1854 1394 y Fw(w)r Fy(\()p Fw(N)2039 1356 y Fr(\014)2087 1394 y Fw(x)p Fy(\))86 1540 y(ha)m(ving)i(an)g(attractiv)m(e)i(part,)f(in)e (the)h(dilute)f(regime)h Fw(\014)h(>)26 b Fy(1)p Fw(=)p Fy(3.)45 b(Because)33 b(of)e(the)h(resp)s(ectiv)m(e)f(scalings)86 1648 y(of)39 b(the)h(mean-\014eld)e(in)m(teraction)i(and)e(kinetic)h (energy)g(one)h(m)m(ust)f(then)g(assume)h([)3043 1648 y SDict begin H.S end 3043 1648 a Black Fy(13)p Black 3134 1590 a SDict begin H.R end 3134 1590 a 3134 1648 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-14c) cvn H.B /ANN pdfmark end 3134 1648 a Fy(,)3198 1648 y SDict begin H.S end 3198 1648 a Black Fy(28)p Black 3289 1590 a SDict begin H.R end 3289 1590 a 3289 1648 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Triay-17) cvn H.B /ANN pdfmark end 3289 1648 a Fy(])g(that)g(the)86 1756 y(unscaled)29 b(p)s(oten)m(tial)h Fw(w)j Fy(is)c(classically)g(stable)1431 1830 y Fq(X)1340 2027 y Fu(1)p Fp(6)p Fr(i<j)t Fp(6)p Fr(N)1668 1917 y Fw(w)r Fy(\()p Fw(x)1822 1931 y Fr(i)1871 1917 y Ft(\000)20 b Fw(x)2014 1931 y Fr(j)2050 1917 y Fy(\))26 b Fn(>)f Ft(\000)p Fw(C)7 b(N)86 2176 y Fy(for)30 b(all)f Fw(x)403 2190 y Fu(1)443 2176 y Fw(;)15 b(:)g(:)g(:)h(;)f(x)696 2190 y Fr(N)789 2176 y Ft(2)25 b Fv(R)935 2143 y Fr(d)981 2176 y Fy(.)41 b(This)28 b(holds)h(for)h(examp)s(e)h(if)45 b Fq(b)-67 b Fw(w)28 b Fn(>)d Fy(0.)186 2283 y(Our)k(metho)s(d)h(also)h (co)m(v)m(ers)i(this)c(case,)k(but)d(w)m(e)h(refrain)e(from)i(stating)g (details,)f(for)h(the)g(range)g(of)g Fw(\014)86 2391 y Fy(w)m(e)g(are)f(allo)m(w)m(ed)g(to)h(reac)m(h)h(is)d(1)p Fw(=)p Fy(3)d Fw(<)f(\014)31 b(<)25 b Fy(9)p Fw(=)p Fy(26.)42 b(T)-8 b(ria)m(y)g(,)31 b(in)e(his)g(study)g(of)i(the)f(dip)s(olar)e (Bose)j(gas)g([)3544 2391 y SDict begin H.S end 3544 2391 a Black Fy(28)p Black 3635 2333 a SDict begin H.R end 3635 2333 a 3635 2391 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Triay-17) cvn H.B /ANN pdfmark end 3635 2391 a Fy(],)86 2499 y(has)f(already)h(obtained)f(1)p Fw(=)p Fy(3)d Fw(<)f(\014)31 b(<)25 b Fy(1)p Fw(=)p Fy(3)d(+)e Fw(s=)p Fy(\(45)i(+)e(42)p Fw(s)p Fy(\))32 b(where)e Fw(s)g Fy(is)g(the)h(exp)s(onen)m(t)f(in)g(\()3273 2499 y SDict begin H.S end 3273 2499 a Black Fy(2.2)p Black 3389 2441 a SDict begin H.R end 3389 2441 a 3389 2499 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 3389 2499 a Fy(\))q(.)41 b(Only)86 2607 y(for)30 b(rather)h(small)e(v)-5 b(alues)30 b(of)h Fw(s)g Fy(do)s(es)f(our)g(estimate)i(impro)m(v)m(e)f(on)f(his.)41 b(The)30 b(3D)i(case)f(for)g(a)g(restricted)86 2715 y(class)25 b(of)h(p)s(oten)m(tials)f(with)f(an)h(attractiv)m(e)j(part)d(has)g (also)h(b)s(een)e(considered)h([)2743 2715 y SDict begin H.S end 2743 2715 a Black Fy(12)p Black 2834 2657 a SDict begin H.R end 2834 2657 a 2834 2715 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Lee-09) cvn H.B /ANN pdfmark end 2834 2715 a Fy(,)2884 2715 y SDict begin H.S end 2884 2715 a Black Fy(30)p Black 2975 2657 a SDict begin H.R end 2975 2657 a 2975 2715 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Yin-10b) cvn H.B /ANN pdfmark end 2975 2715 a Fy(])h(for)f(homogeneous)86 2823 y(gases)31 b(in)e(the)i(thermo)s (dynamic)e(limit.)2192 b Ft(\005)86 2920 y SDict begin H.S end 86 2920 a 86 2920 a SDict begin 13 H.A end 86 2920 a 86 2920 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 86 2920 a 1686 3010 a Fy(3.)46 b Fx(Pr)n(oof)186 3172 y Fy(This)31 b(b)s(eing)g(for)i(a)g(large)g (part)g(an)g(impro)m(v)m(emen)m(t)g(on)g([)2119 3172 y SDict begin H.S end 2119 3172 a Black Fy(14)p Black 2210 3113 a SDict begin H.R end 2210 3113 a 2210 3172 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2210 3172 a Fy(,)2268 3172 y SDict begin H.S end 2268 3172 a Black Fy(25)p Black 2359 3113 a SDict begin H.R end 2359 3113 a 2359 3172 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-19) cvn H.B /ANN pdfmark end 2359 3172 a Fy(],)h(w)m(e)f(shall)e(b)s(e)i(brief.)46 b(See)33 b([)3362 3172 y SDict begin H.S end 3362 3172 a Black Fy(1)p Black 3408 3113 a SDict begin H.R end 3408 3113 a 3408 3172 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BenPorSch-15) cvn H.B /ANN pdfmark end 3408 3172 a Fy(,)3466 3172 y SDict begin H.S end 3466 3172 a Black Fy(7)p Black 3512 3113 a SDict begin H.R end 3512 3113 a 3512 3172 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Golse-13) cvn H.B /ANN pdfmark end 3512 3172 a Fy(,)3570 3172 y SDict begin H.S end 3570 3172 a Black Fy(18)p Black 3661 3113 a SDict begin H.R end 3661 3113 a 3661 3172 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieSeiSolYng-05) cvn H.B /ANN pdfmark end 3661 3172 a Fy(,)86 3280 y SDict begin H.S end 86 3280 a Black Fy(26)p Black 177 3221 a SDict begin H.R end 177 3221 a 177 3280 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-EMS) cvn H.B /ANN pdfmark end 177 3280 a Fy(,)232 3280 y SDict begin H.S end 232 3280 a Black Fy(23)p Black 323 3221 a SDict begin H.R end 323 3221 a 323 3280 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-LMU) cvn H.B /ANN pdfmark end 323 3280 a Fy(,)379 3280 y SDict begin H.S end 379 3280 a Black Fy(24)p Black 470 3221 a SDict begin H.R end 470 3221 a 470 3280 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-spartacus) cvn H.B /ANN pdfmark end 470 3280 a Fy(,)525 3280 y SDict begin H.S end 525 3280 a Black Fy(27)p Black 616 3221 a SDict begin H.R end 616 3221 a 616 3280 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Schlein-08) cvn H.B /ANN pdfmark end 616 3280 a Fy(])e(for)f(general) g(bac)m(kground)g(on)h(large)f Fw(N)40 b Fy(limits)28 b(of)j(b)s(osonic)e(quan)m(tum)h(systems.)186 3388 y(Recall)g(that)h(w) m(e)g(write)f(the)g(pro)s(of)g(for)h FC(A)25 b Ft(\021)h Fy(0,)31 b(whic)m(h)e(implies)f(the)j(general)f(statemen)m(t)j(using)c (the)86 3495 y(diamagnetic)h(inequalit)m(y)f([)1043 3495 y SDict begin H.S end 1043 3495 a Black Fy(16)p Black 1134 3437 a SDict begin H.R end 1134 3437 a 1134 3495 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieLos-01) cvn H.B /ANN pdfmark end 1134 3495 a Fy(,)h(Theorem)h(7.21].)86 3574 y SDict begin H.S end 86 3574 a 86 3574 a SDict begin 13 H.A end 86 3574 a 86 3574 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.1) cvn /DEST pdfmark end 86 3574 a 86 x Fy(3.1.)47 b FC(Old)34 b(argumen)m(ts.)45 b Fy(Consider)29 b(the)h(sp)s(ectral)g(pro)5 b(jectors)1408 3807 y SDict begin H.S end 1408 3807 a 1408 3807 a SDict begin 13 H.A end 1408 3807 a 1408 3807 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.1) cvn /DEST pdfmark end 1408 3807 a Fw(P)39 b Fy(=)25 b Ff(1)1657 3822 y Fr(h)p Fp(6)p Fu(\003)1801 3807 y Fw(;)107 b(Q)25 b Fy(=)g Ff(1)20 b Ft(\000)g Fw(P)1148 b Fy(\(3.1\))86 3953 y(with)29 b(\003)c Fw(>)g Fy(0)31 b(a)g(\(large)g(enough\))f(one-b)s(o)s(dy)g (energy)g(cut-o\013.)42 b(Let)1153 4107 y Fw(H)1229 4121 y Fu(2)1293 4107 y Fy(:=)26 b Fw(h)1467 4121 y Fu(1)1527 4107 y Fy(+)19 b Fw(h)1669 4121 y Fu(2)1729 4107 y Fy(+)h Fw(N)1903 4069 y Fu(2)p Fr(\014)1986 4107 y Fw(w)r Fy(\()p Fw(N)2171 4069 y Fr(\014)2219 4107 y Fy(\()p Fw(x)2306 4121 y Fu(1)2366 4107 y Ft(\000)g Fw(x)2509 4121 y Fu(2)2548 4107 y Fy(\)\))86 4254 y(b)s(e)31 b(the)g(t)m(w)m(o-b)s(o)s(dy)h (Hamiltonian)e(asso)s(ciated)i(with)e(\()1972 4254 y SDict begin H.S end 1972 4254 a Black Fy(1.2)p Black 2088 4195 a SDict begin H.R end 2088 4195 a 2088 4254 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.2) cvn H.B /ANN pdfmark end 2088 4254 a Fy(\))q(.)44 b(Let)31 b(\011)2427 4268 y Fr(N)2526 4254 y Fy(b)s(e)g(a)g(ground)g(state)i(of)e Fw(H)3441 4268 y Fr(N)3539 4254 y Fy(and)86 4383 y(let)f Fw(\015)269 4335 y Fu(\()p Fr(k)r Fu(\))264 4411 y Fr(N)397 4383 y Fy(b)s(e)g(its)g Fw(k)s Fy(-b)s(o)s(dy)f(densit)m(y)h(matrix.)40 b(Then)1008 4517 y(1)p 989 4558 83 4 v 989 4641 a Fw(N)1082 4578 y(E)5 b Fy(\()p Fw(N)10 b Fy(\))26 b(=)1458 4517 y(1)p 1439 4558 V 1439 4641 a Fw(N)1532 4578 y Ft(h)p Fy(\011)1638 4592 y Fr(N)1705 4578 y Ft(j)p Fw(H)1806 4592 y Fr(N)1873 4578 y Ft(j)p Fy(\011)1969 4592 y Fr(N)2037 4578 y Ft(i)f Fy(=)2203 4517 y(1)p 2203 4558 46 4 v 2203 4641 a(2)2274 4578 y(T)-8 b(r)2382 4478 y Fq(\020)2437 4578 y Fw(H)2513 4592 y Fu(2)2552 4578 y Fw(\015)2604 4531 y Fu(\(2\))2599 4607 y Fr(N)2698 4478 y Fq(\021)2768 4578 y Fw(:)3499 4578 y SDict begin H.S end 3499 4578 a 3499 4578 a SDict begin 16.25 H.A end 3499 4578 a 3499 4578 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 3499 4578 a Fy(\(3.2\))186 4756 y(W)g(e)31 b(start)g(with)e(a)i(lo)s(calization)e(lemma,)h(essen)m(tially)g(a)g (restatemen)m(t)j(of)d([)2781 4756 y SDict begin H.S end 2781 4756 a Black Fy(14)p Black 2872 4698 a SDict begin H.R end 2872 4698 a 2872 4756 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2872 4756 a Fy(,)h(Equation)f(\(46\)]:)86 4919 y SDict begin H.S end 86 4919 a 86 4919 a SDict begin 13 H.A end 86 4919 a 86 4919 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.1) cvn /DEST pdfmark end 86 4919 a Black FC(Lemma)j(3.1)i Fy(\()p FC(Lo)s(calization)p Fy(\))p FC(.)p Black 86 5027 a Fj(With)e(the)g(ab)-5 b(ove)33 b(notation,)h(for)g(any)f Fw(\016)c(>)c Fy(1)p Fw(=)p Fy(2)34 b Fj(ther)-5 b(e)33 b(exists)g(a)g Fw(C)2374 5042 y Fr(\016)2437 5027 y Fw(>)25 b Fy(0)33 b Fj(such)g(that)184 5225 y Fy(T)-8 b(r)293 5124 y Fq(\020)347 5151 y(\000)389 5225 y Fw(H)465 5239 y Fu(2)524 5225 y Ft(\000)20 b Fw(P)686 5187 y Fs(\012)p Fu(2)781 5225 y Fw(H)857 5239 y Fu(2)896 5225 y Fw(P)967 5187 y Fs(\012)p Fu(2)1061 5151 y Fq(\001)1118 5225 y Fw(\015)1170 5177 y Fu(\(2\))1165 5253 y(\011)1220 5264 y Fh(N)1282 5124 y Fq(\021)1361 5225 y Fn(>)25 b Ft(\000)p Fw(C)1593 5240 y Fr(\016)1631 5225 y Fy(\003)1694 5187 y Fu(\()p Fr(\016)r Fs(\000)p Fu(1\))p Fr(=)p Fu(2)1962 5124 y Fq(\020)2017 5225 y Fy(T)-8 b(r)2126 5124 y Fq(\020)2180 5225 y Fw(h)15 b(\015)2299 5177 y Fu(\(1\))2294 5253 y(\011)2349 5264 y Fh(N)2411 5124 y Fq(\021\021)2520 5146 y Fu(\(1)p Fs(\000)p Fr(\016)r Fu(\))p Fr(=)p Fu(2)2788 5124 y Fq(\020)2843 5225 y Fy(T)-8 b(r)2951 5124 y Fq(\020)3006 5225 y Fw(h)20 b Ft(\012)g Fw(h)15 b(\015)3288 5177 y Fu(\(2\))3283 5253 y(\011)3338 5264 y Fh(N)3401 5124 y Fq(\021\021)3509 5146 y Fr(\016)3562 5225 y Fw(:)186 5410 y Fy(T)-8 b(o)31 b(put)f(the)i(ab)s(o)m(v)m(e)g(to)g(go)s(o)s(d)f (use)g(w)m(e)g(need)g(a)h(priori)c(b)s(ounds)h(on)i(the)g(\014rst)g (and)f(second)h(momen)m(ts)86 5518 y(of)d(the)h(one-b)s(o)s(dy)e (Hamiltonian.)39 b(The)28 b(follo)m(wing)f(is)g([)1975 5518 y SDict begin H.S end 1975 5518 a Black Fy(14)p Black 2066 5459 a SDict begin H.R end 2066 5459 a 2066 5518 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2066 5518 a Fy(,)i(Lemma)g(5],)g(and)f(this)f(is)h (where)f(w)m(e)i(use)f(our)86 5626 y(main)h(assumption)g Fw(\014)h(<)25 b Fy(1)31 b(and)f(also)g(the)h(requiremen)m(t)e Fw(w)f Ft(2)d Fw(L)2309 5593 y Fu(2)2348 5626 y Fy(\()p Fv(R)2443 5593 y Fu(2)2489 5626 y Fy(\).)p Black Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end -222 785 a Black 86 559 a Fu(6)1257 b(P)-6 b(.T.)23 b(NAM)g(AND)g(N.)g(R)n(OUGERIE)p Black 86 759 a SDict begin H.S end 86 759 a 86 759 a SDict begin 13 H.A end 86 759 a 86 759 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.2) cvn /DEST pdfmark end 86 759 a Black FC(Lemma)33 b(3.2)i Fy(\()p FC(Momen)m(ts)p Fy(\))p FC(.)p Black 86 867 a Fj(L)-5 b(et)33 b Fy(0)26 b Fw(<)f(\014)30 b(<)25 b Fy(1)p Fj(.)42 b(F)-7 b(or)34 b(al)5 b(l)33 b Fw(")26 b Ft(2)e Fy(\(0)p Fw(;)15 b Fy(1\))35 b Fj(we)e(have)504 1085 y SDict begin H.S end 504 1085 a 504 1085 a SDict begin 13 H.A end 504 1085 a 504 1085 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.3) cvn /DEST pdfmark end 504 1085 a 15 w Fy(T)-8 b(r)628 984 y Fq(\020)682 1085 y Fw(h)15 b(\015)801 1037 y Fu(\(1\))796 1113 y(\011)851 1124 y Fh(N)913 984 y Fq(\021)993 1085 y Fn(6)25 b Fw(C)1170 1022 y Fy(1)c(+)f Ft(j)p Fw(e)1394 1036 y Fr(N)s(;")1510 1022 y Ft(j)p 1170 1064 365 4 v 1331 1147 a Fw(")1638 1085 y Fj(and)109 b Fy(T)-8 b(r)1999 984 y Fq(\020)2053 1085 y Fw(h)21 b Ft(\012)f Fw(h)15 b(\015)2336 1037 y Fu(\(2\))2331 1113 y(\011)2386 1124 y Fh(N)2448 984 y Fq(\021)2528 1085 y Fn(6)25 b Fw(C)2710 956 y Fq(\022)2787 1022 y Fy(1)c(+)f Ft(j)p Fw(e)3011 1036 y Fr(N)s(;")3127 1022 y Ft(j)p 2787 1064 V 2948 1147 a Fw(")3162 956 y Fq(\023)3229 979 y Fu(2)3499 1085 y Fy(\(3.3\))86 1281 y Fj(wher)-5 b(e)926 1477 y SDict begin H.S end 926 1477 a 926 1477 a SDict begin 13 H.A end 926 1477 a 926 1477 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.4) cvn /DEST pdfmark end 926 1477 a Fw(e)968 1491 y Fr(N)s(;")1109 1477 y Fy(:=)25 b Fw(N)1313 1439 y Fs(\000)p Fu(1)1588 1477 y Fy(inf)1422 1542 y Fu(\011)p Fs(2)p Fi(H)1572 1553 y Fh(N)1630 1542 y Fr(;)p Fs(k)p Fu(\011)p Fs(k)p Fu(=1)1880 1321 y Fq(*)1953 1477 y Fy(\011)2039 1290 y Fq(\014)2039 1345 y(\014)2039 1399 y(\014)2039 1454 y(\014)2039 1508 y(\014)2039 1563 y(\014)2070 1477 y Fw(H)2146 1491 y Fr(N)2233 1477 y Ft(\000)20 b Fw(")2415 1363 y Fr(N)2381 1390 y Fq(X)2385 1586 y Fr(j)t Fu(=1)2528 1477 y Fw(h)2580 1491 y Fr(j)2616 1290 y Fq(\014)2616 1345 y(\014)2616 1399 y(\014)2616 1454 y(\014)2616 1508 y(\014)2616 1563 y(\014)2662 1477 y Fy(\011)2733 1321 y Fq(+)2821 1477 y Fw(:)653 b Fy(\(3.4\))86 1672 y SDict begin H.S end 86 1672 a 86 1672 a SDict begin 13 H.A end 86 1672 a 86 1672 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.2) cvn /DEST pdfmark end 86 1672 a 76 x Fy(3.2.)47 b FC(New)27 b(argumen)m(t.)44 b Fy(W)-8 b(e)26 b(rely)d(on)h(a)h(v)m (ersion)f(of)g(the)h(quan)m(tum)e(de)h(Finetti)g(theorem)h(from)e([) 3449 1748 y SDict begin H.S end 3449 1748 a Black Fy(2)p Black 3495 1690 a SDict begin H.R end 3495 1690 a 3495 1748 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BraHar-12) cvn H.B /ANN pdfmark end 3495 1748 a Fy(,)3545 1748 y SDict begin H.S end 3545 1748 a Black Fy(15)p Black 3635 1690 a SDict begin H.R end 3635 1690 a 3635 1748 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LiSmi-15) cvn H.B /ANN pdfmark end 3635 1748 a Fy(]:)86 1911 y SDict begin H.S end 86 1911 a 86 1911 a SDict begin 13 H.A end 86 1911 a 86 1911 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.3) cvn /DEST pdfmark end 86 1911 a Black FC(Lemma)33 b(3.3)i Fy(\()p FC(de)g(Finetti)p Fy(\))p FC(.)p Black 86 2020 a Fj(L)-5 b(et)38 b Fo(H)h Fj(b)-5 b(e)38 b(a)g(c)-5 b(omplex)40 b(sep)-5 b(ar)g(able)40 b(Hilb)-5 b(ert)38 b(sp)-5 b(ac)g(e,)41 b(and)e Fo(H)2118 2034 y Fr(N)2220 2020 y Fy(=)c Fo(H)2391 1987 y Fs(\012)2446 1995 y Fl(sym)2557 1987 y Fr(N)2662 2020 y Fj(the)k(c)-5 b(orr)g(esp)g(onding)41 b(b)-5 b(osonic)86 2144 y(sp)g(ac)g(e.)43 b(L)-5 b(et)33 b Fw(\015)569 2096 y Fu(\(2\))564 2172 y Fr(N)696 2144 y Fj(b)-5 b(e)32 b(the)h Fy(2)p Fj(-b)-5 b(o)g(dy)34 b(r)-5 b(e)g(duc)g(e)g(d)34 b(density)f(matrix)h(of)f(a)g Fw(N)10 b Fj(-b)-5 b(o)g(dy)33 b(state)h(ve)-5 b(ctor)33 b Fy(\011)3221 2158 y Fr(N)3313 2144 y Ft(2)25 b Fo(H)3464 2158 y Fr(N)3564 2144 y Fj(\(or)86 2252 y(a)33 b(gener)-5 b(al)33 b(mixe)-5 b(d)34 b(state\).)186 2381 y(L)-5 b(et)40 b Fw(P)53 b Fj(b)-5 b(e)40 b(a)h(\014nite)f(dimensional)i(ortho)-5 b(gonal)43 b(pr)-5 b(oje)g(ctor.)67 b(Ther)-5 b(e)41 b(exists)g(a)f(Bor)-5 b(el)41 b(me)-5 b(asur)g(e)42 b Fw(\026)3592 2333 y Fu(\(2\))3592 2409 y Fr(N)86 2488 y Fj(with)33 b(total)h(mass)g Fn(6)25 b Fy(1)33 b Fj(on)g(the)g(set)g (of)g(one-b)-5 b(o)g(dy)34 b(mixe)-5 b(d)33 b(states)780 2635 y SDict begin H.S end 780 2635 a 780 2635 a SDict begin 13 H.A end 780 2635 a 780 2635 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.5) cvn /DEST pdfmark end 780 2635 a Ft(S)835 2649 y Fr(P)919 2635 y Fy(:=)26 b Ft(f)p Fw(\015)38 b Fj(p)-5 b(ositive)33 b(tr)-5 b(ac)g(e-class)34 b(op)-5 b(er)g(ator)36 b(on)d Fw(P)13 b Fo(H)p Fw(;)31 b Fy(T)-8 b(r)15 b Fw(\015)31 b Fy(=)25 b(1)p Ft(g)508 b Fy(\(3.5\))86 2782 y Fj(such)33 b(that)464 2983 y SDict begin H.S end 464 2983 a 464 2983 a SDict begin 13 H.A end 464 2983 a 464 2983 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.6) cvn /DEST pdfmark end 464 2983 a 15 w Fy(T)-8 b(r)588 2852 y Fq(\014)588 2906 y(\014)588 2961 y(\014)588 3015 y(\014)618 2983 y Fw(A)20 b Ft(\012)g Fw(B)886 2855 y Fq(\022)953 2983 y Fw(P)1024 2946 y Fs(\012)p Fu(2)1118 2983 y Fw(\015)1170 2935 y Fu(\(2\))1165 3012 y Fr(N)1265 2983 y Fw(P)1336 2946 y Fs(\012)p Fu(2)1450 2983 y Ft(\000)1541 2860 y Fq(Z)1647 2983 y Fw(\015)1699 2946 y Fs(\012)p Fu(2)1793 2983 y Fw(d\026)1895 2935 y Fu(\(2\))1895 3012 y Fr(N)1990 2983 y Fy(\()p Fw(\015)5 b Fy(\))2112 2855 y Fq(\023)2180 2852 y(\014)2180 2906 y(\014)2180 2961 y(\014)2180 3015 y(\014)2235 2983 y Fn(6)25 b Fw(C)2403 2836 y Fq(r)p 2494 2836 502 4 v 2504 2922 a Fy(log)q(\(dim)o(\()p Fw(P)13 b Fy(\)\))p 2504 2962 482 4 v 2703 3046 a Fw(N)3010 2983 y Ft(j)-15 b(j)p Fw(A)p Ft(j)h(j)15 b(j)-15 b(j)p Fw(B)5 b Ft(j)-15 b(j)191 b Fy(\(3.6\))86 3180 y Fj(for)34 b(al)5 b(l)34 b Fw(A;)15 b(B)39 b Fj(self-adjoint)34 b(op)-5 b(er)g(ators)37 b(on)d Fw(P)13 b Fo(H)p Fj(.)45 b(The)34 b(norm)h(in)e(the)h(right-hand)h (side)f(is)g(the)g(op)-5 b(er)g(ator)86 3288 y(norm.)p Black 86 3450 a(Pr)g(o)g(of.)p Black 43 w Fy(The)29 b(pro)s(of)g(of)g ([)917 3450 y SDict begin H.S end 917 3450 a Black Fy(2)p Black 963 3392 a SDict begin H.R end 963 3392 a 963 3450 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BraHar-12) cvn H.B /ANN pdfmark end 963 3450 a Fy(])h(giv)m(es)g(the)g(statemen)m (t)h(with)d Fw(A;)15 b(B)35 b Fy(replaced)29 b(b)m(y)g(quan)m(tum)g (measuremen)m(ts.)86 3558 y(In)42 b([)237 3558 y SDict begin H.S end 237 3558 a Black Fy(25)p Black 328 3499 a SDict begin H.R end 328 3499 a 328 3558 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-19) cvn H.B /ANN pdfmark end 328 3558 a Fy(,)47 b(Prop)s(osition)41 b(3.2])j(it)f(is)f (explained)f(ho)m(w)j(the)f(statemen)m(t)i(with)d Fw(A;)15 b(B)48 b Fy(p)s(ositiv)m(e)42 b(op)s(erators)86 3666 y(follo)m(ws.)e(The)30 b(full)e(result)h(is)g(obtained)h(b)m(y)g (decomp)s(osing)g(self-adjoin)m(t)f(op)s(erators)i(in)e(the)i(manner) 1466 3812 y Fw(A)26 b Fy(=)f Ff(1)1712 3826 y Fr(A<)p Fu(0)1858 3812 y Fw(A)c Fy(+)f Ff(1)2094 3826 y Fr(A)p Fp(>)p Fu(0)2237 3812 y Fw(A)86 3959 y Fy(as)31 b(used)e(already)h(in)f ([)857 3959 y SDict begin H.S end 857 3959 a Black Fy(6)p Black 903 3900 a SDict begin H.R end 903 3900 a 903 3959 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Girardot-19) cvn H.B /ANN pdfmark end 903 3959 a Fy(].)2662 b Fn(\003)186 4118 y Fy(T)-8 b(o)38 b(apply)f(the)h(ab)s(o)m(v)m(e)i(w)m(e)f(shall,)f (as)h(in)e([)1663 4118 y SDict begin H.S end 1663 4118 a Black Fy(25)p Black 1754 4059 a SDict begin H.R end 1754 4059 a 1754 4118 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rougerie-19) cvn H.B /ANN pdfmark end 1754 4118 a Fy(],)j(decomp)s(ose)f (the)f(in)m(teraction)g(op)s(erator)h(using)e(the)86 4226 y(F)-8 b(ourier)30 b(transform)f(in)h(the)g(manner)205 4396 y Fw(N)288 4359 y Fu(2)p Fr(\014)370 4396 y Fw(w)r Fy(\()p Fw(N)555 4359 y Fr(\014)603 4396 y Fy(\()p Fw(x)21 b Ft(\000)f Fw(y)s Fy(\)\))25 b(=)1041 4273 y Fq(Z)1092 4479 y Fk(R)1139 4460 y Fl(2)1194 4396 y Fw(N)1277 4359 y Fu(2)p Fr(\014)1400 4366 y Fv(\\)1359 4396 y Fw(w)r Fy(\()p Fw(N)1544 4370 y Fr(\014)1592 4396 y Ft(\001)p Fy(\))q(\()p Fw(k)s Fy(\))p Fw(e)1815 4359 y Fr(ik)r Fs(\001)p Fr(x)1942 4396 y Fw(e)1984 4359 y Fs(\000)p Fr(ik)r Fs(\001)p Fr(y)2163 4396 y Fw(dk)945 4638 y Fy(=)1041 4514 y Fq(Z)1092 4720 y Fk(R)1139 4701 y Fl(2)1210 4638 y Fq(b)-67 b Fw(w)r Fy(\()p Fw(N)1379 4600 y Fs(\000)p Fr(\014)1482 4638 y Fw(k)s Fy(\))15 b(\()q(cos)h(\()p Fw(k)23 b Ft(\001)e Fw(x)p Fy(\))15 b(cos)h(\()p Fw(k)24 b Ft(\001)c Fw(y)s Fy(\))h(+)f(sin)13 b(\()q Fw(k)23 b Ft(\001)e Fw(x)p Fy(\))15 b(sin)f(\()p Fw(k)24 b Ft(\001)c Fw(y)s Fy(\)\))c Fw(dk)3499 4638 y SDict begin H.S end 3499 4638 a 3499 4638 a SDict begin 16.25 H.A end 3499 4638 a 3499 4638 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.7) cvn /DEST pdfmark end 3499 4638 a Fy(\(3.7\))86 4830 y(and)22 b(apply)g(Lemma)815 4830 y SDict begin H.S end 815 4830 a Black Fy(3.3)p Black 931 4772 a SDict begin H.R end 931 4772 a 931 4830 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.3) cvn H.B /ANN pdfmark end 931 4830 a 23 w Fy(for)h(eac)m(h)h Fw(k)s Fy(.)38 b(W)-8 b(e)24 b(in)m(terject)g(a)f(simple)e(con)m(trol)i(of)g(the)g(in)m(v)m(olv)m (ed)g(m)m(ultiplication)86 4938 y(op)s(erators:)86 5101 y SDict begin H.S end 86 5101 a 86 5101 a SDict begin 13 H.A end 86 5101 a 86 5101 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.4) cvn /DEST pdfmark end 86 5101 a Black FC(Lemma)33 b(3.4)i Fy(\()p FC(Multiplication)h(b)m(y)f(plane)g(w)m(a)m (v)m(es)p Fy(\))p FC(.)p Black 86 5209 a Fj(L)-5 b(et)37 b Fw(k)g Ft(2)c Fv(R)485 5176 y Fu(2)530 5209 y Fw(;)15 b(k)37 b Ft(6)p Fy(=)c(0)38 b Fj(and)g FC(e)1070 5224 y Fr(k)1150 5209 y Fj(b)-5 b(e)37 b(the)g(multiplic)-5 b(ation)40 b(op)-5 b(er)g(ator)40 b(on)e Fw(L)2559 5176 y Fu(2)2598 5209 y Fy(\()p Fv(R)2693 5176 y Fu(2)2738 5209 y Fy(\))g Fj(by)f(either)g Fy(cos)q(\()p Fw(k)27 b Ft(\001)d Fw(x)p Fy(\))37 b Fj(or)86 5317 y Fy(sin)o(\()p Fw(k)23 b Ft(\001)e Fw(x)p Fy(\))p Fj(.)42 b(L)-5 b(et)33 b Fw(P)45 b Fj(b)-5 b(e)33 b(the)g(sp)-5 b(e)g(ctr)g(al)35 b(pr)-5 b(oje)g(ctor)35 b(in)k Fy(\()1887 5317 y SDict begin H.S end 1887 5317 a Black Fy(3.1)p Black 2004 5258 a SDict begin H.R end 2004 5258 a 2004 5317 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 2004 5317 a Fy(\))p Fj(.)j(As)32 b(op)-5 b(er)g(ators)1302 5545 y SDict begin H.S end 1302 5545 a 1302 5545 a SDict begin 13 H.A end 1302 5545 a 1302 5545 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.8) cvn /DEST pdfmark end 1302 5545 a 20 w Ft(\006)20 b Fw(P)13 b FC(e)1532 5560 y Fr(k)1574 5545 y Fw(P)38 b Fn(6)25 b Fy(min)1933 5390 y Fq(\()2006 5545 y Fy(1)p Fw(;)15 b(C)2174 5484 y Fy(\003)2237 5451 y Fu(1)p Fr(=)p Fu(2)p 2174 5524 174 4 v 2210 5608 a Ft(j)p Fw(k)s Ft(j)2357 5390 y Fq(\))2445 5545 y Fw(:)1029 b Fy(\(3.8\))p Black Black eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end -222 785 a Black 830 559 a Fu(IMPR)n(O)n(VED)30 b(ST)-6 b(ABILITY)32 b(F)n(OR)e(2D)h(A)-6 b(TTRA)n(CTIVE)30 b(BOSE)h(GASES)710 b(7)p Black Black 86 759 a Fj(Pr)-5 b(o)g(of.)p Black 43 w Fy(The)35 b(\014rst)h(upp)s(er)e(b)s(ound)g(1)i(is)f(ob)m(vious)h (as)g Ft(j)p FC(e)2008 774 y Fr(k)2051 759 y Ft(j)f Fn(6)g Fy(1.)58 b(W)-8 b(e)37 b(write)f(the)g(pro)s(of)f(of)h(the)h(second)86 867 y(b)s(ound)g(for)i FC(e)571 882 y Fr(k)653 867 y Fy(the)h(m)m(ultiplication)c(b)m(y)k(cos)q(\()p Fw(k)29 b Ft(\001)e Fw(x)p Fy(\))40 b(\(the)f(case)i(of)f(sin)n(\()p Fw(k)30 b Ft(\001)c Fw(x)p Fy(\))40 b(is)f(similar\).)65 b(F)-8 b(or)41 b(an)m(y)86 975 y(smo)s(oth)30 b(compactly)h(supp)s (orted)e(function)g Fw(f)10 b Fy(,)29 b(in)m(tegrating)i(b)m(y)f (parts,)1088 1189 y Ft(h)p Fw(f)10 b Ft(j)p FC(e)1251 1204 y Fr(k)1294 1189 y Ft(j)p Fw(f)g Ft(i)25 b Fy(=)g Ft(\000)1616 1065 y Fq(Z)1666 1271 y Fk(R)1713 1252 y Fl(2)1778 1127 y Fy(sin)o(\()p Fw(k)f Ft(\001)c Fw(x)p Fy(\))p 1778 1168 351 4 v 1903 1251 a Ft(j)p Fw(k)s Ft(j)2173 1127 y Fw(k)p 2148 1168 101 4 v 2148 1251 a Ft(j)p Fw(k)s Ft(j)2279 1189 y(\001)h(r)2416 1115 y Fq(\000)2457 1189 y Ft(j)p Fw(f)10 b Ft(j)2562 1151 y Fu(2)2601 1115 y Fq(\001)2658 1189 y Fw(;)86 1404 y Fy(whence)619 1626 y Ft(jh)p Fw(f)g Ft(j)p FC(e)807 1641 y Fr(k)849 1626 y Ft(j)p Fw(f)g Ft(ij)26 b Fn(6)1148 1565 y Fy(2)p 1121 1605 V 1121 1689 a Ft(j)p Fw(k)s Ft(j)1246 1503 y Fq(Z)1297 1709 y Fk(R)1344 1690 y Fl(2)1399 1626 y Ft(j)p Fw(f)10 b Ft(jjr)p Fw(f)g Ft(j)24 b Fn(6)1843 1565 y Fy(2)p 1815 1605 V 1815 1689 a Ft(j)p Fw(k)s Ft(j)1941 1498 y Fq(\022)2008 1503 y(Z)2059 1709 y Fk(R)2106 1690 y Fl(2)2160 1626 y Ft(j)p Fw(f)10 b Ft(j)2265 1589 y Fu(2)2305 1498 y Fq(\023)2372 1521 y Fu(1)p Fr(=)p Fu(2)2497 1498 y Fq(\022)2564 1503 y(Z)2614 1709 y Fk(R)2661 1690 y Fl(2)2716 1626 y Ft(jr)p Fw(f)g Ft(j)2897 1589 y Fu(2)2936 1498 y Fq(\023)3003 1521 y Fu(1)p Fr(=)p Fu(2)3128 1626 y Fw(:)86 1859 y Fy(Applying)31 b(the)k(ab)s(o)m(v)m(e)g(with)d Fw(f)41 b Fy(=)31 b Fw(P)13 b(g)s Fy(,)36 b Fw(g)f Ft(2)30 b Fw(L)1712 1826 y Fu(2)1752 1859 y Fy(\()p Fv(R)1847 1826 y Fr(d)1893 1859 y Fy(\))k(pro)m(v)m(es)h(the)f(claim)f(\(recall)h(w)m (e)g(w)m(ork)g(without)f(a)86 1967 y(magnetic)e(\014eld\).)2912 b Fn(\003)186 2152 y Fy(W)-8 b(e)31 b(can)g(\014nally)d(giv)m(e)j(the)p Black 86 2337 a Fj(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)814 2337 y SDict begin H.S end 814 2337 a Black Fj(2.1)p Black 935 2279 a SDict begin H.R end 935 2279 a 935 2337 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 935 2337 a Fj(.)p Black 42 w Fy(According)30 b(to)h(Lemma)1872 2337 y SDict begin H.S end 1872 2337 a Black Fy(3.1)p Black 1988 2279 a SDict begin H.R end 1988 2279 a 1988 2337 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 1988 2337 a 31 w Fy(and)e(Lemma)2520 2337 y SDict begin H.S end 2520 2337 a Black Fy(3.2)p Black 2636 2279 a SDict begin H.R end 2636 2279 a 2636 2337 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.2) cvn H.B /ANN pdfmark end 2636 2337 a 30 w Fy(w)m(e)i(ha)m(v)m(e)505 2526 y(T)-8 b(r)613 2426 y Fq(\020)668 2526 y Fw(H)744 2540 y Fu(2)783 2526 y Fw(\015)835 2479 y Fu(\(2\))830 2555 y Fr(N)929 2426 y Fq(\021)1009 2526 y Fn(>)25 b Fy(T)-8 b(r)1213 2426 y Fq(\020)1268 2526 y Fw(P)1339 2489 y Fs(\012)p Fu(2)1433 2526 y Fw(H)1509 2540 y Fu(2)1548 2526 y Fw(P)1619 2489 y Fs(\012)p Fu(2)1713 2526 y Fw(\015)1765 2479 y Fu(\(2\))1760 2555 y Fr(N)1860 2426 y Fq(\021)1185 2749 y Ft(\000)20 b Fw(C)1341 2764 y Fr(\016)1379 2749 y Fy(\003)1442 2711 y Fu(\()p Fr(\016)r Fs(\000)p Fu(1\))p Fr(=)p Fu(2)1711 2648 y Fq(\020)1765 2749 y Fy(T)-8 b(r)1874 2648 y Fq(\020)1928 2749 y Fw(h\015)2032 2701 y Fu(\(1\))2027 2778 y Fr(N)2127 2648 y Fq(\021\021)2236 2671 y Fu(\(1)p Fs(\000)p Fr(\016)r Fu(\))p Fr(=)p Fu(2)2504 2648 y Fq(\020)2559 2749 y Fy(T)g(r)2668 2648 y Fq(\020)2722 2749 y Fw(h)20 b Ft(\012)g Fw(h\015)2989 2701 y Fu(\(2\))2984 2778 y Fr(N)3084 2648 y Fq(\021\021)3193 2671 y Fr(\016)1009 2999 y Fn(>)25 b Fy(T)-8 b(r)1213 2898 y Fq(\020)1268 2999 y Fw(P)1339 2961 y Fs(\012)p Fu(2)1433 2999 y Fw(H)1509 3013 y Fu(2)1548 2999 y Fw(P)1619 2961 y Fs(\012)p Fu(2)1713 2999 y Fw(\015)1765 2951 y Fu(\(2\))1760 3027 y Fr(N)1860 2898 y Fq(\021)1934 2999 y Ft(\000)20 b Fw(C)2090 3014 y Fr(\016)2128 2999 y Fy(\003)2191 2961 y Fu(\()p Fr(\016)r Fs(\000)p Fu(1\))p Fr(=)p Fu(2)2460 2870 y Fq(\022)2537 2936 y Fy(1)g(+)g Ft(j)p Fw(e)2760 2950 y Fr(N)s(;")2876 2936 y Ft(j)p 2537 2978 365 4 v 2698 3061 a Fw(")2912 2870 y Fq(\023)2979 2893 y Fu(\(1+3)p Fr(\016)r Fu(\))p Fr(=)p Fu(2)3499 2999 y SDict begin H.S end 3499 2999 a 3499 2999 a SDict begin 16.25 H.A end 3499 2999 a 3499 2999 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.9) cvn /DEST pdfmark end 3499 2999 a Fy(\(3.9\))86 3220 y(for)30 b(an)m(y)h(1)p Fw(=)p Fy(2)26 b Fw(<)f(\016)k Fn(6)c Fy(1)31 b(and)f(0)c Fw(<)f(")g(<)g Fy(1.)41 b(On)30 b(the)g(other)h(hand)e(w)m(e)i(ma)m(y)g(use)f(Lemma)3099 3220 y SDict begin H.S end 3099 3220 a Black Fy(3.3)p Black 3215 3162 a SDict begin H.R end 3215 3162 a 3215 3220 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.3) cvn H.B /ANN pdfmark end 3215 3220 a Fy(:)40 b(denoting)1434 3434 y Fq(f)-102 b Fw(\015)1470 3448 y Fr(N)1537 3392 y Fu(\(2\))1656 3434 y Fy(:=)1778 3310 y Fq(Z)1884 3434 y Fw(\015)1936 3396 y Fs(\012)p Fu(2)2030 3434 y Fw(d\026)2132 3386 y Fu(\(2\))2132 3462 y Fr(N)2227 3434 y Fy(\()p Fw(\015)5 b Fy(\))86 3642 y(w)m(e)31 b(ha)m(v)m(e)184 3840 y(T)-8 b(r)293 3739 y Fq(\020)347 3840 y Fw(P)418 3803 y Fs(\012)p Fu(2)513 3840 y Fw(H)589 3854 y Fu(2)628 3840 y Fw(P)699 3803 y Fs(\012)p Fu(2)793 3840 y Fw(\015)845 3792 y Fu(\(2\))840 3869 y Fr(N)939 3739 y Fq(\021)1019 3840 y Fn(>)25 b Fy(T)-8 b(r)1224 3739 y Fq(\020)1278 3840 y Fw(P)1349 3803 y Fs(\012)p Fu(2)1443 3840 y Fw(H)1519 3854 y Fu(2)1558 3840 y Fw(P)1629 3803 y Fs(\012)p Fu(2)1735 3840 y Fq(f)-102 b Fw(\015)1771 3854 y Fr(N)1838 3799 y Fu(\(2\))1932 3739 y Fq(\021)711 4121 y Ft(\000)20 b Fw(C)874 3974 y Fq(r)p 964 3974 272 4 v 974 4060 a Fy(log)r(\(\003\))p 974 4100 252 4 v 1059 4184 a Fw(N)1251 3938 y Fq(0)1251 4102 y(@)1330 4121 y Fy(\003)h(+)1789 4035 y Fq(X)1505 4236 y Fe(e)1542 4248 y Fh(k)1580 4236 y Fs(2f)p Fu(cos)q(\()p Fr(k)r Fs(\001)p Fr(x)p Fu(\))p Fr(;)p Fu(sin\()p Fr(k)r Fs(\001)p Fr(x)p Fu(\))p Fs(g)2219 3997 y Fq(Z)2269 4204 y Fk(R)2316 4185 y Fl(2)2371 4121 y Ft(j)-15 b(j)q Fw(P)13 b FC(e)2526 4136 y Fr(k)2568 4121 y Fw(P)g Ft(j)-15 b(j)2675 4079 y Fu(2)2729 4121 y Ft(j)17 b Fq(b)-68 b Fw(w)s Fy(\()p Fw(N)2940 4084 y Fs(\000)p Fr(\014)3042 4121 y Fw(k)s Fy(\))p Ft(j)p Fw(dk)3249 3938 y Fq(1)3249 4102 y(A)3345 4121 y Fw(:)3454 4121 y SDict begin H.S end 3454 4121 a 3454 4121 a SDict begin 16.25 H.A end 3454 4121 a 3454 4121 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.10) cvn /DEST pdfmark end 3454 4121 a Fy(\(3.10\))86 4397 y(Here)42 b(w)m(e)h(ha)m(v)m(e)g (decomp)s(osed)f(the)g(in)m(teraction)g(term)f(as)i(in)d(\()2340 4397 y SDict begin H.S end 2340 4397 a Black Fy(3.7)p Black 2457 4339 a SDict begin H.R end 2457 4339 a 2457 4397 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.7) cvn H.B /ANN pdfmark end 2457 4397 a Fy(\),)45 b(used)c(the)h(triangle)g (inequalit)m(y)86 4505 y(with)36 b(\()335 4505 y SDict begin H.S end 335 4505 a Black Fy(3.6)p Black 451 4447 a SDict begin H.R end 451 4447 a 451 4505 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.6) cvn H.B /ANN pdfmark end 451 4505 a Fy(\))h(and)g(recalled)f(that)h(dim)o(\()p Fw(P)13 b Fy(\))38 b(dep)s(ends)d(at)i(w)m(orst)h(p)s(olynomially)33 b(on)k(\003)g(as)g(recalled)g(e.g.)86 4613 y(in)c([)221 4613 y SDict begin H.S end 221 4613 a Black Fy(13)p Black 313 4554 a SDict begin H.R end 313 4554 a 313 4613 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-14c) cvn H.B /ANN pdfmark end 313 4613 a Fy(,)j(Lemma)e(3.3])i(\(this)e(is)g(the)h (place)g(where)f(Condition)f(\()2320 4613 y SDict begin H.S end 2320 4613 a Black Fy(2.2)p Black 2436 4554 a SDict begin H.R end 2436 4554 a 2436 4613 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 2436 4613 a Fy(\))i(is)f(used\).)54 b(The)34 b(error)g(term)h(also)86 4721 y(con)m(tains)30 b(the)h(op)s(erator)g(norm)e(2\003)i(of)g Fw(P)1517 4688 y Fs(\012)p Fu(2)1611 4721 y Fy(\()p Fw(h)1698 4735 y Fu(1)1758 4721 y Fy(+)20 b Fw(h)1901 4735 y Fu(2)1941 4721 y Fy(\))p Fw(P)2047 4688 y Fs(\012)p Fu(2)2142 4721 y Fy(.)186 4829 y(The)42 b(second)g(term)h(on)g(the)f(righ)m(t)h(side)e (of)50 b(\()1812 4829 y SDict begin H.S end 1812 4829 a Black Fy(3.10)p Black 1974 4770 a SDict begin H.R end 1974 4770 a 1974 4829 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 1974 4829 a Fy(\))43 b(can)g(b)s(e)f(estimated)h(using)e(Lemma)3381 4829 y SDict begin H.S end 3381 4829 a Black Fy(3.4)p Black 3497 4770 a SDict begin H.R end 3497 4770 a 3497 4829 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.4) cvn H.B /ANN pdfmark end 3497 4829 a 42 w Fy(and)86 4937 y(Condition)28 b(\()545 4937 y SDict begin H.S end 545 4937 a Black Fy(2.1)p Black 662 4878 a SDict begin H.R end 662 4878 a 662 4937 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.1) cvn H.B /ANN pdfmark end 662 4937 a Fy(\))i(\(the)h(latter)g(ensures)e (that)47 b Fq(b)-67 b Fw(w)28 b Ft(2)d Fw(L)1924 4904 y Fu(2)1984 4937 y Ft(\\)19 b Fw(L)2126 4904 y Fs(1)2201 4937 y Fy(\):)283 5027 y Fq(Z)334 5233 y Fk(R)381 5214 y Fl(2)435 5150 y Ft(j)-14 b(j)p Fw(P)13 b FC(e)590 5165 y Fr(k)632 5150 y Fw(P)g Ft(j)-15 b(j)739 5108 y Fu(2)793 5150 y Ft(j)17 b Fq(b)-68 b Fw(w)s Fy(\()p Fw(N)1004 5113 y Fs(\000)p Fr(\014)1106 5150 y Fw(k)s Fy(\))p Ft(j)p Fw(dk)30 b Fn(6)1436 5027 y Fq(Z)1486 5233 y Fk(R)1533 5214 y Fl(2)1588 5150 y Fy(min)o Ft(f)p Fy(1)p Fw(;)15 b(C)7 b Fy(\003)p Ft(j)p Fw(k)s Ft(j)2105 5113 y Fs(\000)p Fu(2)2201 5150 y Ft(gj)17 b Fq(b)-68 b Fw(w)s Fy(\()p Fw(N)2457 5113 y Fs(\000)p Fr(\014)2559 5150 y Fw(k)s Fy(\))p Ft(j)p Fw(dk)293 5392 y Fn(6)389 5268 y Fq(Z)440 5474 y Fs(j)p Fr(k)r Fs(j)p Fp(6)p Fu(1)624 5392 y Ft(k)16 b Fq(b)-67 b Fw(w)s Ft(k)782 5406 y Fr(L)830 5387 y Fg(1)899 5392 y Fw(dk)24 b Fy(+)c Fw(C)1194 5268 y Fq(Z)1245 5474 y Fu(1)p Fr(<)p Fs(j)p Fr(k)r Fs(j)p Fp(6)p Fr(N)1529 5455 y Fh(\014)1588 5392 y Fy(\003)p Ft(j)p Fw(k)s Ft(j)1751 5354 y Fs(\000)p Fu(2)1847 5392 y Ft(k)c Fq(b)-67 b Fw(w)r Ft(k)2004 5406 y Fr(L)2052 5387 y Fg(1)2122 5392 y Fw(dk)24 b Fy(+)c Fw(C)2417 5268 y Fq(Z)2468 5474 y Fs(j)p Fr(k)r Fs(j)p Fr(>N)2665 5455 y Fh(\014)2724 5392 y Fy(\003)p Ft(j)p Fw(k)s Ft(j)2887 5354 y Fs(\000)p Fu(2)2983 5392 y Ft(j)c Fq(b)-67 b Fw(w)r Fy(\()p Fw(N)3193 5354 y Fs(\000)p Fr(\014)3296 5392 y Fw(k)s Fy(\))p Ft(j)p Fw(dk)293 5589 y Fn(6)25 b Fw(C)i Fy(+)20 b Fw(C)7 b Fy(\003)15 b(log)h Fw(N)30 b Fy(+)20 b Fw(C)7 b Fy(\003)p Fw(:)p Black Black eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end -222 785 a Black 86 559 a Fu(8)1257 b(P)-6 b(.T.)23 b(NAM)g(AND)g(N.)g(R)n(OUGERIE)p Black 186 759 a Fy(As)30 b(regards)g(the)h(\014rst)e(term)i(on)f(the)h(righ)m(t-hand)e(side)g (of)38 b(\()2259 759 y SDict begin H.S end 2259 759 a Black Fy(3.10)p Black 2421 700 a SDict begin H.R end 2421 700 a 2421 759 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 2421 759 a Fy(\))30 b(w)m(e)h(write)1069 975 y(T)-8 b(r)1178 874 y Fq(\020)1232 975 y Fw(P)1303 937 y Fs(\012)p Fu(2)1397 975 y Fw(H)1473 989 y Fu(2)1512 975 y Fw(P)1583 937 y Fs(\012)p Fu(2)1689 975 y Fq(f)-102 b Fw(\015)1725 989 y Fr(N)1792 933 y Fu(\(2\))1886 874 y Fq(\021)1966 975 y Fy(=)2062 851 y Fq(Z)2168 975 y Ft(E)2224 937 y Fu(H)2281 975 y Fy([)p Fw(\015)5 b Fy(])p Fw(d\026)2485 927 y Fu(\(2\))2485 1003 y Fr(N)2580 975 y Fy(\()p Fw(\015)g Fy(\))86 1186 y(where)607 1341 y Ft(E)663 1304 y Fu(H)720 1341 y Fy([)p Fw(\015)g Fy(])26 b(=)f(T)-8 b(r)15 b(\()p Fw(h\015)5 b Fy(\))21 b(+)1349 1280 y(1)p 1349 1320 46 4 v 1349 1404 a(2)1420 1217 y Fq(Z)9 b(Z)1531 1424 y Fk(R)1578 1405 y Fl(2)1613 1424 y Fs(\002)p Fk(R)1715 1405 y Fl(2)1770 1341 y Fw(\032)1817 1355 y Fr(\015)1862 1341 y Fy(\()p Fw(x)p Fy(\))p Fw(N)2067 1304 y Fu(2)p Fr(\014)2150 1341 y Fw(w)r Fy(\()p Fw(N)2335 1304 y Fr(\014)2383 1341 y Fy(\(\()p Fw(x)21 b Ft(\000)e Fw(y)s Fy(\)\))p Fw(\032)2781 1355 y Fr(\015)2826 1341 y Fy(\()p Fw(y)s Fy(\))p Fw(dxdy)s(:)86 1549 y Fy(Here)31 b(the)f(densit)m(y)g Fw(\032)818 1563 y Fr(\015)862 1549 y Fy(\()p Fw(x)p Fy(\))c(=)f Fw(\015)5 b Fy(\()p Fw(x;)15 b(x)p Fy(\))32 b(\(de\014ned)d(prop)s(erly)f(b)m(y)j (the)f(sp)s(ectral)g(decomp)s(osition\))f(satis\014es)1527 1646 y Fq(Z)1577 1852 y Fk(R)1624 1833 y Fl(2)1679 1769 y Fw(\032)1726 1783 y Fr(\015)1796 1769 y Fy(=)c(T)-8 b(r)15 b Fw(\015)30 b Fy(=)25 b(1)p Fw(:)86 1989 y Fy(Using)35 b(the)g(diamagnetic)h(inequalit)m(y)e([)1476 1989 y SDict begin H.S end 1476 1989 a Black Fy(16)p Black 1567 1931 a SDict begin H.R end 1567 1931 a 1567 1989 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieLos-01) cvn H.B /ANN pdfmark end 1567 1989 a Fy(,)j(Theorem)e(7.21])j(and)d(the)g(con)m(v)m(exit) m(y)j(of)d(the)h(kinetic)f(en-)86 2097 y(ergy)c([)311 2097 y SDict begin H.S end 311 2097 a Black Fy(16)p Black 402 2039 a SDict begin H.R end 402 2039 a 402 2097 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LieLos-01) cvn H.B /ANN pdfmark end 402 2097 a Fy(,)f(Theorem)g(7.8])i(w)m(e)f(ha)m(v)m (e)h(the)e(Ho\013mann-Ostenhof-t)m(yp)s(e)h(inequalit)m(y)1428 2313 y(T)-8 b(r)15 b(\()p Fw(h\015)5 b Fy(\))26 b Fn(>)1833 2190 y Fq(Z)1884 2396 y Fk(R)1931 2377 y Fl(2)1986 2313 y Ft(jr)2087 2257 y(p)p 2162 2257 92 4 v 2162 2313 a Fw(\032)2209 2327 y Fr(\015)2254 2313 y Ft(j)2279 2276 y Fu(2)2319 2313 y Fw(:)86 2529 y Fy(Then)j(inserting)g(the)h(Cauc)m(h) m(y-Sc)m(h)m(w)m(arz)i(inequalit)m(y)d(in)g(the)i(in)m(teraction)f (term,)347 2754 y Fw(N)430 2716 y Fu(2)p Fr(\014)512 2754 y Fw(w)r Fy(\()p Fw(N)697 2716 y Fr(\014)745 2754 y Fy(\(\()p Fw(x)21 b Ft(\000)f Fw(y)s Fy(\)\))p Fw(\032)1144 2768 y Fr(\015)1189 2754 y Fy(\()p Fw(x)p Fy(\))p Fw(\032)1358 2768 y Fr(\015)1403 2754 y Fy(\()p Fw(y)s Fy(\))p Fw(dxdy)29 b Fn(>)c Ft(\000)p Fw(N)1991 2716 y Fu(2)p Fr(\014)2073 2754 y Ft(j)p Fw(w)2163 2768 y Fs(\000)2223 2754 y Fy(\()p Fw(N)2341 2716 y Fr(\014)2388 2754 y Fy(\(\()p Fw(x)c Ft(\000)f Fw(y)s Fy(\)\))p Ft(j)2775 2692 y Fw(\032)2822 2706 y Fr(\015)2867 2692 y Fy(\()p Fw(x)p Fy(\))2989 2659 y Fu(2)3049 2692 y Fy(+)g Fw(\032)3187 2706 y Fr(\015)3232 2692 y Fy(\()p Fw(y)s Fy(\))3350 2659 y Fu(2)p 2775 2733 615 4 v 3060 2816 a Fy(2)3400 2754 y Fw(;)86 2954 y Fy(and)29 b(com)m(bining)e(with)h(the)i(Gagiardo-Niren)m(b)s(erg)f(inequalit)m(y) f(\()2331 2954 y SDict begin H.S end 2331 2954 a Black Fy(1.6)p Black 2448 2895 a SDict begin H.R end 2448 2895 a 2448 2954 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.6) cvn H.B /ANN pdfmark end 2448 2954 a Fy(\))h(and)g(Assumption)e (\()3225 2954 y SDict begin H.S end 3225 2954 a Black Fy(1.7)p Black 3342 2895 a SDict begin H.R end 3342 2895 a 3342 2954 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.7) cvn H.B /ANN pdfmark end 3342 2954 a Fy(\),)j(w)m(e)g(get)946 3178 y SDict begin H.S end 946 3178 a 946 3178 a SDict begin 13 H.A end 946 3178 a 946 3178 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.11) cvn /DEST pdfmark end 946 3178 a Ft(E)1002 3140 y Fu(H)1059 3178 y Fy([)p Fw(\015)5 b Fy(])26 b Fn(>)1283 3054 y Fq(Z)1334 3260 y Fk(R)1381 3241 y Fl(2)1436 3178 y Ft(jr)1537 3121 y(p)p 1612 3121 92 4 v 1612 3178 a Fw(\032)1659 3192 y Fr(\015)1704 3178 y Ft(j)1729 3140 y Fu(2)1789 3178 y Ft(\000)1890 3116 y Fy(1)p 1890 3157 46 4 v 1890 3240 a(2)1960 3050 y Fq(\022)2027 3054 y(Z)2133 3178 y Ft(j)p Fw(w)2223 3192 y Fs(\000)2283 3178 y Ft(j)2308 3050 y Fq(\023)2390 3054 y(Z)2440 3260 y Fk(R)2487 3241 y Fl(2)2542 3178 y Fw(\032)2589 3140 y Fu(2)2589 3200 y Fr(\015)2659 3178 y Fn(>)f Fy(0)p Fw(:)629 b Fy(\(3.11\))86 3402 y(Th)m(us)29 b(the)i(\014rst)e(term)i (on)f(the)h(righ)m(t)f(side)f(of)38 b(\()1709 3402 y SDict begin H.S end 1709 3402 a Black Fy(3.10)p Black 1871 3343 a SDict begin H.R end 1871 3343 a 1871 3402 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 1871 3402 a Fy(\))30 b(is)g(nonnegativ)m(e.)41 b(Therefore,)30 b(\()3040 3402 y SDict begin H.S end 3040 3402 a Black Fy(3.9)p Black 3157 3343 a SDict begin H.R end 3157 3343 a 3157 3402 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 3157 3402 a Fy(\))h(reduces)f(to)375 3649 y Fw(e)417 3663 y Fr(N)509 3649 y Fy(=)25 b(T)-8 b(r)714 3549 y Fq(\020)769 3649 y Fw(H)845 3663 y Fu(2)884 3649 y Fw(\015)936 3602 y Fu(\(2\))931 3678 y Fr(N)1030 3549 y Fq(\021)1110 3649 y Fn(>)25 b Ft(\000)p Fw(C)1349 3502 y Fq(r)p 1439 3502 272 4 v 1449 3588 a Fy(log)q(\(\003\))p 1449 3629 252 4 v 1533 3712 a Fw(N)1710 3649 y Fy(\003)15 b(log)i Fw(N)30 b Ft(\000)20 b Fw(C)2180 3664 y Fr(\016)2218 3649 y Fy(\003)2281 3612 y Fu(\()p Fr(\016)r Fs(\000)p Fu(1\))p Fr(=)p Fu(2)2549 3521 y Fq(\022)2626 3587 y Fy(1)h(+)f Ft(j)p Fw(e)2850 3601 y Fr(N)s(;")2966 3587 y Ft(j)p 2626 3629 365 4 v 2787 3712 a Fw(")3001 3521 y Fq(\023)3068 3544 y Fu(\(1+3)p Fr(\016)r Fu(\))p Fr(=)p Fu(2)3372 3649 y Fw(:)86 3868 y Fy(Moreo)m(v)m(er)32 b(b)m(y)f(a)f(simple)f(trial)g(state)j(argumen)m (t)f(w)m(e)f(kno)m(w)h(that)g Fw(e)2408 3882 y Fr(N)2501 3868 y Fn(6)24 b Fw(C)7 b Fy(.)41 b(In)29 b(summary)-8 b(,)30 b(w)m(e)h(ha)m(v)m(e)614 4111 y Ft(j)p Fw(e)681 4125 y Fr(N)749 4111 y Ft(j)25 b Fn(6)g Fw(C)i Fy(+)20 b Fw(C)1150 3963 y Fq(r)p 1240 3963 272 4 v 1250 4050 a Fy(log)q(\(\003\))p 1250 4090 252 4 v 1334 4173 a Fw(N)1511 4111 y Fy(\003)15 b(log)i Fw(N)30 b Fy(+)20 b Fw(C)1981 4126 y Fr(\016)2019 4111 y Fy(\003)2082 4074 y Fu(\()p Fr(\016)r Fs(\000)p Fu(1\))p Fr(=)p Fu(2)2351 3983 y Fq(\022)2428 4049 y Fy(1)g(+)g Ft(j)p Fw(e)2651 4063 y Fr(N)s(;")2767 4049 y Ft(j)p 2428 4090 365 4 v 2589 4173 a Fw(")2802 3983 y Fq(\023)2869 4006 y Fu(\(1+3)p Fr(\016)r Fu(\))p Fr(=)p Fu(2)3454 4111 y SDict begin H.S end 3454 4111 a 3454 4111 a SDict begin 16.25 H.A end 3454 4111 a 3454 4111 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.12) cvn /DEST pdfmark end 3454 4111 a Fy(\(3.12\))86 4335 y(for)30 b(all)f(1)p Fw(=)p Fy(2)e Fw(<)e(\016)k Fn(6)c Fy(1)31 b(and)e(0)d Fw(<)f(")h(<)f Fy(1.)186 4443 y(No)m(w)31 b(w)m(e)f(b)s(o)s(otstrap)g(the)h(ab)s(o)m(v)m(e)g(argumen) m(ts.)42 b(Let)30 b Fw(")2038 4457 y Fu(0)2103 4443 y Ft(2)25 b Fy(\(0)p Fw(;)15 b Fy(1\))33 b(b)s(e)c(a)i(\014xed)f(constan) m(t)i(suc)m(h)e(that)1441 4535 y Fq(Z)1491 4742 y Fk(R)1538 4723 y Fl(2)1593 4659 y Fw(w)1658 4673 y Fs(\000)1743 4659 y Fw(>)24 b Ft(\000)p Fw(a)1957 4622 y Fs(\003)1997 4659 y Fy(\(1)d Ft(\000)f Fw(")2231 4673 y Fu(0)2270 4659 y Fy(\))p Fw(:)86 4874 y Fy(Assume)30 b(w)m(e)h(kno)m(w)f(that)h (there)g(is)e(a)i Fw(\013)25 b(>)g Fy(0)31 b(suc)m(h)f(that)1205 5043 y SDict begin H.S end 1205 5043 a 1205 5043 a SDict begin 13 H.A end 1205 5043 a 1205 5043 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.13) cvn /DEST pdfmark end 1205 5043 a Ft(j)p Fw(e)1272 5057 y Fr(N)s(;")1388 5043 y Ft(j)c Fn(6)f Fw(C)1600 5057 y Fr(";\013)1701 5043 y Fw(N)1784 5006 y Fr(\013)1864 5043 y Fy(for)30 b(all)f(0)d Fw(<)f(")h(<)f(")2502 5057 y Fu(0)2542 5043 y Fw(:)887 b Fy(\(3.13\))86 5218 y(W)-8 b(e)31 b(can)g(start)g(the)g(b) s(o)s(otstrap)f(from)f Fw(\013)d Fy(=)f(2)p Fw(\014)5 b Fy(,)31 b(using)e(the)i(simple)d(one-b)s(o)s(dy)i(inequalit)m(y)3218 5218 y SDict begin H.S end 3218 5218 a -35 x Fu(1)3251 5218 y SDict begin 13 H.L end 3251 5218 a 3251 5218 a SDict begin [/Subtype /Link/Dest (Hfootnote.1) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 3251 5218 a 1278 5389 a Ft(\000)p Fy(\001)1425 5403 y Fr(x)1488 5389 y Ft(\000)20 b Fw(N)1662 5351 y Fu(2)p Fr(\014)1745 5389 y Fw(w)r Fy(\()p Fw(N)1930 5351 y Fr(\014)1978 5389 y Fw(x)p Fy(\))25 b Fn(>)g Ft(\000)p Fw(C)7 b(N)2412 5351 y Fu(2)p Fr(\014)p Black 86 5526 499 4 v 186 5597 a Fu(1)221 5626 y SDict begin H.S end 221 5626 a 221 5626 a SDict begin H.R end 221 5626 a 221 5626 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.1) cvn /DEST pdfmark end 221 5626 a FB(Here)25 b(w)n(e)h(do)g(not)g (need)f(the)g(impro)n(v)n(ed)f(b)r(ound)h Fz(O)r FB(\()p Fz(N)1801 5594 y Fl(2)p Fh(\014)s Fg(\000)p Fl(1)1952 5626 y FB(\))h(in)f([)2118 5626 y SDict begin H.S end 2118 5626 a Black FB(14)p Black 2195 5578 a SDict begin H.R end 2195 5578 a 2195 5626 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LewNamRou-15) cvn H.B /ANN pdfmark end 2195 5626 a FB(,)h(Lemma)f(2])h(whic)n(h)g(requires)39 b Fd(b)-57 b Fz(w)24 b Fc(2)d Fz(L)3313 5594 y Fl(1)3348 5626 y FB(.)p Black Black Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end -222 785 a Black 830 559 a Fu(IMPR)n(O)n(VED)30 b(ST)-6 b(ABILITY)32 b(F)n(OR)e(2D)h(A)-6 b(TTRA)n(CTIVE)30 b(BOSE)h(GASES)710 b(9)p Black 86 759 a Fy(Then,)27 b(w)m(e)h(can)f(apply)f(the)i(ab)s(o)m (v)m(e)g(argumen)m(ts)g(to)g(the)g Fw(")p Fy(-p)s(erturb)s(ed)d (Hamiltonian)h(in)g(\()3149 759 y SDict begin H.S end 3149 759 a Black Fy(3.4)p Black 3265 700 a SDict begin H.R end 3265 700 a 3265 759 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 3265 759 a Fy(\))i(to)g(b)s (ound)86 867 y Fw(e)128 881 y Fr(N)s(;")244 867 y Fy(.)80 b(Note)45 b(that)g(the)e(lo)m(w)m(er)h(b)s(ound)e(\()1542 867 y SDict begin H.S end 1542 867 a Black Fy(3.11)p Black 1704 808 a SDict begin H.R end 1704 808 a 1704 867 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.11) cvn H.B /ANN pdfmark end 1704 867 a Fy(\))i(remains)e(v)-5 b(alid)42 b(with)h Fw(w)j Fy(replaced)d(b)m(y)g(\(1)30 b Ft(\000)f Fw(")p Fy(\))3498 834 y Fs(\000)p Fu(1)3593 867 y Fw(w)r Fy(,)86 975 y(pro)m(vided)g(that)i(1)26 b Fw(<)f(")g(<)g(")1029 989 y Fu(0)1069 975 y Fy(.)41 b(Com)m(bining)28 b(with)h(\()1839 975 y SDict begin H.S end 1839 975 a Black Fy(3.13)p Black 2001 916 a SDict begin H.R end 2001 916 a 2001 975 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.13) cvn H.B /ANN pdfmark end 2001 975 a Fy(\))h(w)m(e)h(obtain)428 1212 y Ft(j)p Fw(e)495 1226 y Fr(N)s(;")611 1212 y Ft(j)26 b Fn(6)f Fw(C)823 1226 y Fr(")880 1212 y Fy(+)20 b Fw(C)1036 1226 y Fr(")1072 1064 y Fq(r)p 1163 1064 272 4 v 1173 1150 a Fy(log)r(\(\003\))p 1173 1191 252 4 v 1257 1274 a Fw(N)1435 1212 y Fy(\003)15 b(log)h Fw(N)31 b Fy(+)19 b Fw(C)1904 1227 y Fr(\016)o(;")1991 1212 y Fy(\003)2054 1174 y Fu(\()p Fr(\016)r Fs(\000)p Fu(1\))p Fr(=)p Fu(2)2307 1212 y Fw(N)2390 1174 y Fr(\013)p Fu(\(1+3)p Fr(\016)r Fu(\))p Fr(=)p Fu(2)2724 1212 y Fw(;)107 b Ft(8)p Fy(0)25 b Fw(<)g(")g(<)g(")3278 1226 y Fu(0)3318 1212 y Fw(:)86 1411 y Fy(If)30 b(w)m(e)h(c)m(ho)s(ose)g(\003)25 b(=)g Fw(N)866 1378 y Fr(\013)p Fu(+)p Fr(a)1038 1411 y Fy(and)30 b Fw(\016)f Fy(=)c(1)p Fw(=)p Fy(2)d(+)e Fw(b)25 b Ft(2)g Fy(\(1)p Fw(=)p Fy(2)p Fw(;)15 b Fy(1\))33 b(in)c(the)i(ab)s(o)m(v)m(e) g(w)m(e)g(obtain)766 1598 y Ft(j)p Fw(e)833 1612 y Fr(N)s(;")949 1598 y Ft(j)25 b Fn(6)g Fw(C)1160 1613 y Fr(";a;b)1319 1497 y Fq(\020)1373 1598 y Fy(1)c(+)f Fw(N)1613 1560 y Fr(\013)p Fu(+)p Fr(a)p Fs(\000)p Fu(1)p Fr(=)p Fu(2)1931 1598 y Fy(log)c Fw(N)30 b Fy(+)20 b Fw(N)2340 1560 y Fr(\013)p Fu(+2)p Fr(b\013)p Fu(+)p Fr(ab=)p Fu(2)p Fs(\000)p Fr(a=)p Fu(4)2911 1497 y Fq(\021)2981 1598 y Fw(:)86 1785 y Fy(Clearly)29 b(w)m(e)i(can)f(pic)m(k)g(0)c Fw(<)f(a)g(<)g Fy(1)p Fw(=)p Fy(2)32 b(and)e Fw(b)25 b(>)g Fy(0)31 b(small)d(enough)j (so)f(that)h(this)e(implies)1434 1944 y Ft(j)p Fw(e)1501 1958 y Fr(N)s(;")1617 1944 y Ft(j)c Fn(6)g Fw(C)1828 1958 y Fr(")1880 1871 y Fq(\000)1922 1944 y Fy(1)c(+)e Fw(N)2161 1907 y Fr(\013)p Fs(\000)p Fr(c)2296 1871 y Fq(\001)86 2104 y Fy(for)26 b(a)h(\014xed)e(constan)m(t)j Fw(c)d(>)g Fy(0)i(\(indep)s(enden)m(t)d(of)j Fw(N)5 b(;)15 b(";)g(\013)p Fy(\).)41 b(Th)m(us)25 b(w)m(e)i(ha)m(v)m(e)g(sho)m(wn)f (that)h(if)32 b(\()3255 2104 y SDict begin H.S end 3255 2104 a Black Fy(3.13)p Black 3417 2045 a SDict begin H.R end 3417 2045 a 3417 2104 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.13) cvn H.B /ANN pdfmark end 3417 2104 a Fy(\))26 b(holds)86 2212 y(with)h Fw(\013)f(>)f Fy(0,)k(it)f(also)g (holds)f(with)g Fw(\013)h Fy(replaced)g(b)m(y)g(max\()p Fw(\013)17 b Ft(\000)f Fw(c;)f Fy(0\))p Fw(:)30 b Fy(After)f (\014nitely)d(man)m(y)i(steps)h(of)f(this)86 2320 y(pro)s(cedure,)e(w)m (e)h(deduce)e(that)i(actually)f Ft(j)p Fw(e)1560 2334 y Fr(N)s(;")1676 2320 y Ft(j)g Fy(is)f(b)s(ounded)f(indep)s(enden)m (tly)f(of)j Fw(N)10 b Fy(,)28 b(whic)m(h)c(implies)g(the)86 2428 y(theorem.)3181 b Fn(\003)1621 2649 y SDict begin H.S end 1621 2649 a 1621 2649 a SDict begin 13 H.A end 1621 2649 a 1621 2649 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end 1621 2649 a Fx(References)86 2703 y SDict begin H.S end 86 2703 a 86 2703 a SDict begin 11 H.A end 86 2703 a 86 2703 a SDict begin [/View [/XYZ H.V]/Dest (cite.BenPorSch-15) cvn /DEST pdfmark end 86 2703 a Black 124 2795 a FB([1])p Black 40 w FA(N.)22 b(Benedikter,)g(M.)h (Por)-5 b(t)g(a,)25 b(and)d(B.)g(Schlein)p FB(,)f Fm(E\013e)l(ctive)i (Evolution)h(Equations)g(fr)l(om)e(Quantum)h(Dynamics)p FB(,)244 2886 y(Springer)i(Briefs)j(in)d(Mathematical)i(Ph)n(ysics,)f (Springer,)g(2016.)86 2901 y SDict begin H.S end 86 2901 a 86 2901 a SDict begin 11 H.A end 86 2901 a 86 2901 a SDict begin [/View [/XYZ H.V]/Dest (cite.BraHar-12) cvn /DEST pdfmark end 86 2901 a Black 124 2977 a FB([2])p Black 40 w FA(F.)j(Brand)594 2971 y(~)592 2977 y(ao)f(and)h(A.)f(Harr)n(o)n (w)p FB(,)h Fm(Quantum)g(de)f(Finetti)h(The)l(or)l(ems)h(under)f(L)l(o) l(c)l(al)g(Me)l(asur)l(ements)i(with)d(Appli-)244 3069 y(c)l(ations)p FB(,)f(Comm)n(un.)d(Math.)i(Ph)n(ys.,)g(353)h(\(2017\),) g(pp.)e(469{506.)86 3087 y SDict begin H.S end 86 3087 a 86 3087 a SDict begin 11 H.A end 86 3087 a 86 3087 a SDict begin [/View [/XYZ H.V]/Dest (cite.CheHol-15) cvn /DEST pdfmark end 86 3087 a Black 124 3160 a FB([3])p Black 40 w FA(X.)h(Chen)g(and)f(J.)i(Holmer)p FB(,)e Fm(The)h(rigor)l(ous)i (derivation)f(of)e(the)i(2D)f(cubic)h(fo)l(cusing)f(NLS)g(fr)l(om)g (quantum)h(many-)244 3251 y(b)l(o)l(dy)h(evolution)p FB(,)f(In)n(t)e(Math)h(Res)f(Notices,)i(\(2016\).)86 3270 y SDict begin H.S end 86 3270 a 86 3270 a SDict begin 11 H.A end 86 3270 a 86 3270 a SDict begin [/View [/XYZ H.V]/Dest (cite.ErdYau-01) cvn /DEST pdfmark end 86 3270 a Black 124 3343 a FB([4])p Black 40 w FA(L.)21 b(Erd)490 3337 y(\177)487 3343 y(os)h(and)f(H.-T.)g(Y)-7 b(a)n(u)p FB(,)21 b Fm(Derivation)h(of)f(the)i(nonline)l(ar)f(Schr\177)-39 b(odinger)23 b(e)l(quation)g(fr)l(om)e(a)h(many)f(b)l(o)l(dy)i(Coulomb) 244 3434 y(system)p FB(,)k(Adv.)e(Theor.)i(Math.)f(Ph)n(ys.,)g(5)g (\(2001\),)h(pp.)e(1169{1205.)86 3453 y SDict begin H.S end 86 3453 a 86 3453 a SDict begin 11 H.A end 86 3453 a 86 3453 a SDict begin [/View [/XYZ H.V]/Dest (cite.Frank-14) cvn /DEST pdfmark end 86 3453 a Black 124 3525 a FB([5])p Black 40 w FA(R.)j(L.)g(Frank)p FB(,)e Fm(Gr)l(ound)j(states)h (of)d(semi-line)l(ar)h(PDEs)p FB(.)e(Lecture)g(notes,)g(2014.)86 3544 y SDict begin H.S end 86 3544 a 86 3544 a SDict begin 11 H.A end 86 3544 a 86 3544 a SDict begin [/View [/XYZ H.V]/Dest (cite.Girardot-19) cvn /DEST pdfmark end 86 3544 a Black 124 3617 a FB([6])p Black 40 w FA(T.)i(Girardot)p FB(,)d Fm(A)n(ver)l(age)k(\014eld)f(appr)l(oximation)h(for)e(almost)h (b)l(osonic)h(anyons)f(in)g(a)f(magnetic)i(\014eld)p FB(,)d(2019.)86 3635 y SDict begin H.S end 86 3635 a 86 3635 a SDict begin 11 H.A end 86 3635 a 86 3635 a SDict begin [/View [/XYZ H.V]/Dest (cite.Golse-13) cvn /DEST pdfmark end 86 3635 a Black 124 3708 a FB([7])p Black 40 w FA(F.)44 b(Golse)p FB(,)h Fm(On)c(the)h(Dynamics)g(of)f(L)l(ar)l(ge)i (Particle)f(Systems)h(in)e(the)h(Me)l(an)g(Field)f(Limit)p FB(,)k(ArXiv)39 b(e-prin)n(ts)244 3799 y(1301.5494,)j(\(2013\).)37 b(Lecture)f(notes)g(for)h(a)f(course)g(at)g(the)g(NDNS+)e(Applied)h (Dynamical)g(Systems)f(Summer)244 3891 y(Sc)n(ho)r(ol)26 b("Macroscopic)i(and)e(large)g(scale)h(phenomena",)e(Univ)n(ersiteit)h (Tw)n(en)n(te,)h(Ensc)n(hede)e(\(The)h(Netherlands\).)86 3909 y SDict begin H.S end 86 3909 a 86 3909 a SDict begin 11 H.A end 86 3909 a 86 3909 a SDict begin [/View [/XYZ H.V]/Dest (cite.GuoSei-13) cvn /DEST pdfmark end 86 3909 a Black 124 3982 a FB([8])p Black 40 w FA(Y.)f(Guo)h(and)e(R.)i (Seiringer)p FB(,)21 b Fm(Symmetry)27 b(br)l(e)l(aking)f(and)g(c)l(ol)t (lapse)g(in)e(Bose-Einstein)j(c)l(ondensates)h(with)d(attr)l(ac-)244 4073 y(tive)j(inter)l(actions)p FB(,)g(Lett.)e(Math.)g(Ph)n(ys.,)g(104) g(\(2014\),)i(pp.)d(141{156.)86 4092 y SDict begin H.S end 86 4092 a 86 4092 a SDict begin 11 H.A end 86 4092 a 86 4092 a SDict begin [/View [/XYZ H.V]/Dest (cite.JebLeoPic-16) cvn /DEST pdfmark end 86 4092 a Black 124 4165 a FB([9])p Black 40 w FA(M.)30 b(Jeblick,)i(N.)f(Leopold,)g(and)f(P.)h(Pickl)p FB(,)f Fm(Derivation)h(of)e(the)i(time)e(dep)l(endent)j(gr)l (oss-pitaevskii)h(e)l(quation)244 4256 y(in)27 b(two)h(dimensions)p FB(.)e(arXiv:1608.05326,)k(2016.)86 4271 y SDict begin H.S end 86 4271 a 86 4271 a SDict begin 11 H.A end 86 4271 a 86 4271 a SDict begin [/View [/XYZ H.V]/Dest (cite.JebPic-17) cvn /DEST pdfmark end 86 4271 a Black 76 x FB([10])p Black 40 w FA(M.)e(Jeblick)i(and)d(P.)i(Pickl)p FB(,)f Fm(Derivation)g(of)g(the)g(time)g(dep)l(endent)i(two)e(dimensional)g (fo)l(cusing)g(NLS)g(e)l(quation)p FB(.)244 4438 y(arXiv:1707.06523,)i (2017.)86 4453 y SDict begin H.S end 86 4453 a 86 4453 a SDict begin 11 H.A end 86 4453 a 86 4453 a SDict begin [/View [/XYZ H.V]/Dest (cite.Kwong-89) cvn /DEST pdfmark end 86 4453 a Black 77 x FB([11])p Black 40 w FA(M.)24 b(K.)g(Kw)n(ong)p FB(,)g Fm(Uniqueness)h(of)f(p)l(ositive)h(solutions)f (of)g FB(\001)p Fz(u)9 b Fc(\000)g Fz(u)g FB(+)g Fz(u)2302 4498 y Fh(p)2361 4530 y FB(=)21 b(0)j Fm(in)g Fb(R)2661 4498 y Fh(n)2703 4530 y FB(,)f(Arc)n(h.)e(Rational)i(Mec)n(h.)f(Anal.,) 244 4621 y(105)27 b(\(1989\),)g(pp.)e(243{266.)86 4640 y SDict begin H.S end 86 4640 a 86 4640 a SDict begin 11 H.A end 86 4640 a 86 4640 a SDict begin [/View [/XYZ H.V]/Dest (cite.Lee-09) cvn /DEST pdfmark end 86 4640 a Black 72 x FB([12])p Black 40 w FA(J.)i(O.)g(Lee)p FB(,)d Fm(Gr)l(ound)k (state)h(ener)l(gy)f(of)e(dilute)g(Bose)i(gases)g(in)e(smal)t(l)f(ne)l (gative)j(p)l(otential)g(c)l(ase)p FB(,)e(Journal)g(of)f(Statis-)244 4804 y(tical)h(Ph)n(ysics,)h(134)g(\(2009\),)g(pp.)e(1{18.)86 4822 y SDict begin H.S end 86 4822 a 86 4822 a SDict begin 11 H.A end 86 4822 a 86 4822 a SDict begin [/View [/XYZ H.V]/Dest (cite.LewNamRou-14c) cvn /DEST pdfmark end 86 4822 a Black 73 x FB([13])p Black 40 w FA(M.)g(Lewin,)h(P.)f(Nam,)h(and)f (N.)g(R)n(ougerie)p FB(,)d Fm(The)k(me)l(an-\014eld)g(appr)l(oximation) g(and)g(the)g(non-line)l(ar)g(Schr\177)-39 b(odinger)244 4986 y(functional)27 b(for)h(tr)l(app)l(e)l(d)i(Bose)e(gases)p FB(,)f(T)-6 b(rans.)27 b(Amer.)d(Math.)j(So)r(c,)f(368)h(\(2016\),)g (pp.)e(6131{6157.)86 5005 y SDict begin H.S end 86 5005 a 86 5005 a SDict begin 11 H.A end 86 5005 a 86 5005 a SDict begin [/View [/XYZ H.V]/Dest (cite.LewNamRou-15) cvn /DEST pdfmark end 86 5005 a Black 73 x FB([14])p Black 244 5065 191 4 v 231 w(,)f Fm(A)i(note)h(on)f(2D)g(fo)l(cusing)g(many-b)l (oson)i(systems)p FB(,)e(Pro)r(c.)f(Ame.)e(Math.)i(So)r(c.,)g(145)g (\(2017\),)h(pp.)d(2441{2454.)86 5096 y SDict begin H.S end 86 5096 a 86 5096 a SDict begin 11 H.A end 86 5096 a 86 5096 a SDict begin [/View [/XYZ H.V]/Dest (cite.LiSmi-15) cvn /DEST pdfmark end 86 5096 a Black 73 x FB([15])p Black 40 w FA(K.)36 b(Li)f(and)g(G.)h(Smith)p FB(,)e Fm(Quantum)h(de)g(Finetti)g(The)l(or)l(ems)h(under)f(ful)t(ly-one-way)h (adaptative)g(me)l(asur)l(ements)p FB(,)244 5260 y(Ph)n(ys.)25 b(Rev.)g(Lett.)h(114,)h(114)g(\(2015\),)g(p.)f(160503.)86 5279 y SDict begin H.S end 86 5279 a 86 5279 a SDict begin 11 H.A end 86 5279 a 86 5279 a SDict begin [/View [/XYZ H.V]/Dest (cite.LieLos-01) cvn /DEST pdfmark end 86 5279 a Black 73 x FB([16])p Black 40 w FA(E.)g(H.)f(Lieb)g(and)g(M.)h(Loss)p FB(,)f Fm(A)n(nalysis)p FB(,)f(v)n(ol.)g(14)f(of)h(Graduate)g(Studies)f (in)g(Mathematics,)h(American)f(Mathemat-)244 5443 y(ical)k(So)r(ciet)n (y)-6 b(,)25 b(Pro)n(vidence,)h(RI,)f(2nd)h(ed.,)g(2001.)86 5458 y SDict begin H.S end 86 5458 a 86 5458 a SDict begin 11 H.A end 86 5458 a 86 5458 a SDict begin [/View [/XYZ H.V]/Dest (cite.LieSei-09) cvn /DEST pdfmark end 86 5458 a Black 76 x FB([17])p Black 40 w FA(E.)d(H.)f(Lieb)h(and)f(R.)h (Seiringer)p FB(,)c Fm(The)k(Stability)g(of)g(Matter)h(in)e(Quantum)i (Me)l(chanics)p FB(,)e(Cam)n(bridge)f(Univ.)f(Press,)244 5626 y(2010.)p Black Black eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop -300 300 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -300 300 a -222 785 a SDict begin H.S end -222 785 a -222 785 a SDict begin H.R end -222 785 a -222 785 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end -222 785 a Black 86 559 a Fu(10)1222 b(P)-6 b(.T.)23 b(NAM)g(AND)g(N.)g(R)n(OUGERIE)p Black 86 676 a SDict begin H.S end 86 676 a 86 676 a 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86 1051 a 86 1051 a SDict begin [/View [/XYZ H.V]/Dest (cite.Maeda-10) cvn /DEST pdfmark end 86 1051 a Black 73 x FB([20])p Black 40 w FA(M.)32 b(Maed)n(a)p FB(,)e Fm(On)h(the)h (symmetry)g(of)f(the)h(gr)l(ound)h(states)g(of)e(nonline)l(ar)g (Schr\177)-39 b(odinger)33 b(e)l(quations)g(with)e(p)l(otential)p FB(,)244 1215 y(Adv.)24 b(Nonlinear)j(Stud.,)e(10)h(\(2010\),)h(pp.)f (895{925.)86 1234 y SDict begin H.S end 86 1234 a 86 1234 a SDict begin 11 H.A end 86 1234 a 86 1234 a SDict begin [/View [/XYZ H.V]/Dest (cite.NamNap-17) cvn /DEST pdfmark end 86 1234 a Black 73 x FB([21])p Black 40 w FA(P.)c(Nam)g(and)g (M.)g(Napi)986 1301 y(\023)983 1307 y(ork)n(o)n(wski)p FB(,)h Fm(Norm)f(appr)l(oximation)i(for)e(many-b)l(o)l(dy)i(quantum)f (dynamics:)33 b(fo)l(cusing)23 b(c)l(ase)244 1398 y(in)k(low)g (dimensions)p FB(.)f(arXiv:1710.09684,)31 b(2017.)86 1413 y SDict begin H.S end 86 1413 a 86 1413 a SDict begin 11 H.A end 86 1413 a 86 1413 a SDict begin [/View [/XYZ H.V]/Dest (cite.NamRouSei-15) cvn /DEST pdfmark end 86 1413 a Black 76 x FB([22])p Black 40 w FA(P.)39 b(T.)f(Nam,)j(N.)d(R)n (ougerie,)i(and)e(R.)h(Seiringer)p FB(,)d Fm(Gr)l(ound)j(states)g(of)d (lar)l(ge)i(Bose)g(systems:)55 b(The)38 b(Gr)l(oss-)244 1581 y(Pitaevskii)28 b(limit)e(r)l(evisite)l(d)p FB(,)h(Analysis)f(and) g(PDEs,)g(9)g(\(2016\),)h(pp.)f(459{485.)86 1599 y SDict begin H.S end 86 1599 a 86 1599 a SDict begin 11 H.A end 86 1599 a 86 1599 a SDict begin [/View [/XYZ H.V]/Dest (cite.Rougerie-LMU) cvn /DEST pdfmark end 86 1599 a Black 73 x FB([23])p Black 40 w FA(N.)89 b(R)n(ougerie)p FB(,)100 b Fm(De)82 b(Finetti)i(the)l(or)l(ems,)97 b(me)l(an-\014eld)84 b(limits)e(and)h(Bose-Einstein)h(c)l(ondensation)p FB(.)244 1763 y(arXiv:1506.05263,)30 b(2014.)d(LMU)f(lecture)g(notes.)86 1778 y SDict begin H.S end 86 1778 a 86 1778 a SDict begin 11 H.A end 86 1778 a 86 1778 a SDict begin [/View [/XYZ H.V]/Dest (cite.Rougerie-spartacus) cvn /DEST pdfmark end 86 1778 a Black 77 x FB([24])p Black 244 1842 191 4 v 231 w(,)f Fm(Th)n(\023)-37 b(eor)n(\022)g(emes)29 b(de)f(De)f(Finetti,)g(limites) g(de)h(champ)g(moyen)f(et)h(c)l(ondensation)i(de)d(Bose-Einstein)p FB(,)h(Les)d(cours)244 1946 y(P)n(eccot,)i(Spartacus)e(IDH,)g(P)n (aris,)i(2016.)h(Cours)e(P)n(eccot,)h(Coll)n(\022)-36 b(ege)28 b(de)d(F)-6 b(rance)26 b(:)34 b(f)n(\023)-36 b(evrier-mars)26 b(2014.)86 1961 y SDict begin H.S end 86 1961 a 86 1961 a SDict begin 11 H.A end 86 1961 a 86 1961 a SDict begin [/View [/XYZ H.V]/Dest (cite.Rougerie-19) cvn /DEST pdfmark end 86 1961 a Black 76 x FB([25])p Black 244 2024 191 4 v 231 w(,)g Fm(Non)h(line)l(ar)h(schr\177)-39 b(odinger)30 b(limit)c(of)h(b)l(osonic)i(gr)l(ound)g(states,)g(again)p FB(.)d(arXiv:1901.09561,)k(2019.)86 2056 y SDict begin H.S end 86 2056 a 86 2056 a SDict begin 11 H.A end 86 2056 a 86 2056 a SDict begin [/View [/XYZ H.V]/Dest (cite.Rougerie-EMS) cvn /DEST pdfmark end 86 2056 a Black 73 x FB([26])p Black 244 2116 191 4 v 231 w(,)c Fm(Sc)l(aling)h(limits)g(of)g(b)l (osonic)i(gr)l(ound)g(states)p FB(.)f(arXiv,)d(2019.)j(In)d (preparation.)86 2147 y SDict begin H.S end 86 2147 a 86 2147 a SDict begin 11 H.A end 86 2147 a 86 2147 a SDict begin [/View [/XYZ H.V]/Dest (cite.Schlein-08) cvn /DEST pdfmark end 86 2147 a Black 73 x FB([27])p Black 40 w FA(B.)37 b(Schlein)p FB(,)h Fm(Derivation)f(of)f(e\013e)l(ctive)i(evolution)g(e) l(quations)g(fr)l(om)e(micr)l(osc)l(opic)i(quantum)f(dynamics)p FB(,)i(arXiv)244 2311 y(eprin)n(ts,)26 b(\(2008\).)h(Lecture)e(Notes)h (for)h(a)f(course)g(at)g(ETH)g(Zuric)n(h.)86 2330 y SDict begin H.S end 86 2330 a 86 2330 a SDict begin 11 H.A end 86 2330 a 86 2330 a SDict begin [/View [/XYZ H.V]/Dest (cite.Triay-17) cvn /DEST pdfmark end 86 2330 a Black 73 x FB([28])p Black 40 w FA(A.)20 b(Tria)-5 b(y)p FB(,)19 b Fm(Derivation)i(of)g(the)g(dip)l(olar)g(Gr)l(oss{Pitaevskii)i(ener)l (gy)p FB(,)f(SIAM)17 b(J.)i(Math.)g(Anal.,)h(50)f(\(2018\),)i(pp.)d (33{63.)86 2421 y SDict begin H.S end 86 2421 a 86 2421 a SDict begin 11 H.A end 86 2421 a 86 2421 a SDict begin [/View [/XYZ H.V]/Dest (cite.Weinstein-83) cvn /DEST pdfmark end 86 2421 a Black 73 x FB([29])p Black 40 w FA(M.)32 b(I.)h(Weinstein)p FB(,)d Fm(Nonline)l(ar)i(Schr\177)-39 b(odinger)34 b(e)l(quations)f(and)f(sharp)h(interp)l(olation)g (estimates)p FB(,)g(Comm.)c(Math.)244 2585 y(Ph)n(ys.,)d(87)g (\(1983\),)h(pp.)f(567{576.)86 2604 y SDict begin H.S end 86 2604 a 86 2604 a SDict begin 11 H.A end 86 2604 a 86 2604 a SDict begin [/View [/XYZ H.V]/Dest (cite.Yin-10b) cvn /DEST pdfmark end 86 2604 a Black 73 x FB([30])p Black 40 w FA(J.)37 b(Yin)p FB(,)f Fm(The)g(gr)l(ound)h(state)h(ener)l (gy)f(of)e(the)i(dilute)e(Bose)i(gas)f(in)f(p)l(otentials)i(with)f(p)l (ositive)h(sc)l(attering)h(length)p FB(,)244 2768 y(Comm)n(unications) 25 b(in)g(Mathematical)i(Ph)n(ysics,)f(295)h(\(2010\),)g(pp.)f(1{27.) 186 2939 y FA(Dep)-5 b(ar)g(tment)37 b(of)i(Ma)-5 b(thema)g(tics,)41 b(LMU)d(Munich,)i(Theresienstrasse)e(39,)j(80333)g(Munich,)f(and)e(Mu-) 86 3030 y(nich)28 b(Center)f(f)n(or)i(Quantum)f(Science)f(and)g (Technology)i(\(MCQST\),)d(Schellingstr.)37 b(4,)28 b(80799)i(Munich,) 86 3122 y(Germany)186 3213 y Fm(E-mail)d(addr)l(ess)6 b FB(:)36 b Fa(nam@math.lmu.de)186 3368 y FA(Universit)550 3362 y(\023)550 3368 y(e)k(Grenoble-Alpes)h(&)h(CNRS,)e(LPMMC)i(\(UMR)f (5493\),)k(B.P.)d(166,)j(F-38042)f(Grenoble,)86 3460 y(France)186 3551 y Fm(E-mail)27 b(addr)l(ess)6 b FB(:)36 b Fa(nicolas.rougerie@grenoble.cnr)q(s.fr)p Black Black eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF