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Existence and Born-Oppenheimer asymptotics of the
total scattering cross-section in ion-atom collisions.
Thierry Jecko, Markus Klein, Xue Ping Wang
To cite this version:
Thierry Jecko, Markus Klein, Xue Ping Wang. Existence and Born-Oppenheimer asymptotics of the
total scattering cross-section in ion-atom collisions.. Long Time Behaviour of Classical and Quantum
Systems: Proceedings of the Bologna Aptex International Conference, Bologna, Italy 13-17 September
1999, 2001, Series on Concrete and Applicable Mathematics, �10.1142/4640�. �hal-03213428�
S attering Cross-Se tion in Ion-Atom Collisions
Thierry Je ko
Universitede Rennes I
Departement de Mathematiques
F-35042Rennes Cedex, FRANCE e-mail: je komaths.univ-rennes1.fr Markus Klein Universitat Potsdam Institut fur Mathematik D-14469Potsdam, GERMANY e-mail: mkleinmath.uni-potsdam.de
Xue Ping Wang
Universite de Nantes
Departement de Mathematiques
F-44072Nantes Cedex, FRANCE
e-mail: Xue-Ping.Wangmath.univ-nantes.fr
Abstra t
We prove the niteness of the total s attering ross-se tion for ion-atom ollisions withan initial hannelgivenbyasimpleeigenvalueoftheinternal Hamiltoniandes ribingtheneutral luster,i.e. the atom.Undermorerestri tiveassumptions,weshowthatsomee e tiveintera tioninBorn-Oppeheimer approximation ispre isely oforder O(jxj
4
) inthe distan ebetween themass enters oftwo lusters. Wethenextra ttheleadingtermofthes attering ross-se tionintheBorn-Oppenheimerlimit.
I Introdu tion
The s attering pro ess for multi-parti le Coulomb systems with initial two- luster data has been studied in physi s litterature, both experimentally and theoreti ally. In parti ular, in the ollision of a harged luster withaneutralone(ion-atoms attering),itis believedthatiftheneutralsub-systemhasnostati dipolemoment,thetotal ross-se tionwouldbeis nite. In[ES℄,Enss-Simonputforwardasopenquestions to provethe niteness of total ross-se tions in this ase and to give expli it bounds for them. In [CT℄, Combes-Tipsprovedthe nitenessandanalyti ityofforwards atteringamplitudeinele tron-atom s atter-ing. They indi atedte hni aldiÆ ulties to extendtheirresultsto ion-atom ollisionandsuggestedto use Born-Oppenheimerapproximationtostudytheproblem.
Re allthatitiswell-knownintwo-bodys atteringtheory(see,forexample,[Y ℄)thatifthepotentialV onR
3
hasthede ay
jV(x)jC<x>
; 8x2R 3
with>2,thetotal ross-se tionforthes atteringpro essdes ribedby( , +V(x))is nite, while if V(x)
C jxj 2
as jxj ! 1 for some C 6= 0, the total ross-se tion is in nite. In the s attering theory for multi-parti le Coulomb systems with initial two- luster data, the inter lusterintera tion between the two lusters de ays like O(jxj
1
)in general ase, likeO(jxj 2
) ifoneof the lusters is neutral( ion-atom s attering)andlikeO(jxj
3
)iftheboth lustersareneutral(atom-atoms attering). Herex2R 3
denotes therelativepositionofthemass- entersofthetwo lusters. SeeAppendixAformorepre isestatementsand the al ulus. Forion-atoms attering, theknownresultsin two-body asesuggestthat withoutadditional assumption, the total ross-se tion would bein nite. Inthis paper, we provethe niteness of total ross-se tions under theassumption that the atom isin thefundamental statewhi h implies, bythe symmetry of Coulomb potentials, that there is no stati dipole moment for the atom. The quantitive study of the total ross-se tionsin ion-atom s atteringis interestingand non-trivial,sin e theleadingtermsin various
present ase. Inthispaper,weonlystudytheasymptoti sin theBorn-Oppenheimerapproximation,where thesemi lassi alparameter,h,isproportionaltotheratiooftheele toni tonu learmass. Due totheuse of luster oordinateswhi hisneededtodes ribemany-parti les atteringpro esses,thepotentialsbe ome h-dependent. Theperturbationbytheshiftterml(y)=O(h
2
jyj)issingularandthepi tureofeigenvalues of the ele troni HamiltonianP
e
(x;h) de ned below hanges drasti ally from h=0to h 6=0. Our result in Born-Oppenheimerapproximationis basedonthesemi lassi alresolventestimatesof [KMW2℄whi h is establishedintermsoftheweightinx l(y),therelativepositionbetweenthetwonu leus. Wethenusethe adiabati approximationfortotal ross-se tionsandprovethattheeigenvalueoftheele troni Hamiltonian P
e
(x;h) onvergessuÆ ientlyfastasx!1sothat we anextra ttheleadingtermin thelimith!0. Theplan of this paperis as follows. In Se tion II weintrodu e thebasi notationwhi h will be used throughout the paper and we re all a few basi fa ts from N-body s attering theory. We introdu e the hypotheses whi h are relevant for this paper and we state our main results, i.e. Theorem II.2 on the existen e ofthe totals attering ross-se tionand Theorem II.3, whi h givesthe semi lassi alasymptoti s of this ross-se tion. In Se tion III we prove Theorem II.2. The essential point are ertain weighted L
2
estimateswhi hshowthatuponlo alizationinenergyintherelevantspe tralrangethee e tiveintera tion de aysfaster thanO(jxj
2
),whi histheobviousnormestimateonanion-atomintera tion. InSe tionIV, weestablishtherelevantsemi lassi alestimatesonpotentialsandresolvents,using methodsfrom[KMW2℄ and giveasket h ofthe proofof TheoremII.3. InAppendixA wein lude therelevantexpansionsforthe Coulombintera tioninion-atoms atteringwhi hareusedthroughoutthepaper.
II Notation, assumptions and main results
TheHamiltonianofadiatomi mole ulewithN ele trons anbewrittenin theform
P phys = 2 X k =1 1 2m k x k + N+2 X j=3 1 2 x j + Z 1 Z 2 jx 1 x 2 j (II.1) + 2 X k =1 N+2 X j=3 e j Z k jx j x k j + X 2l<jN+2 e l e j jx l x j j wherex k 2R 3
,k=1;2,denotethepositionofthetwonu leiwithmassm k and hargeZ k >0andx j 2R 3 , j =3;:::;N +2,denote the position of N ele tronswithmass 1and hargee
j
2R (in thephysi al ase harges are equal and negative). Plan k's onstant is taken to be 1 in this formula. The result on the existen eoftotal ross-se tionsremainsvalidforanyCoulombsystem.
We are interested in s attering pro esses where the in oming s attering hannel is a two- luster one, whiletheout-goings attering hannel anbearbitrary. Leta=(a
1 ;a 2 )beatwo- lusterde omposition of f1;:::;N+2g, i.e. a partition (a 1 ;a 2
) of the parti le labels f1;:::;N +2g, where j 2 a j
, for j = 1;2. Adaptedtothis lusterde omposition,we hooseso alled lustered atomi oordinates(x;y)2R
3 R 3N : h = 1 2M 1 + 1 2M 2 1=2 ; M k =m k +ja 0 k j;a 0 k =a k nfkg;k=1;2; (II.2) R k = 1 M k m k x k + X j2a 0 k x j ;k=1;2; x = R 1 R 2 ; (II.3) y j = x j x k ;j2a 0 k ;k=1;2; (II.4) l(y) = 1 M 1 X j2a 0 1 y j 1 M 2 X j2a 0 2 y j : (II.5) Noti ethatR k
isthe enterofmassofthe lustera k
,fork=1;2,andthatxistherelativepositionofthese entersofmass. These oordinatesarewelladaptedtodes ribetwo- lusters atteringofdiatomi mole ules
phys bewritteninthissystemof oordinatesas
P = h 2 x +P e (x;h); P e (x;h) = P a (h)+I a (x;h); (II.6)
wherethesub-HamiltonianP a (h)isgivenby P a (h)=P a1 (h)+P a2 (h); (II.7) with P a k (h)= X j2a 0 k 1 2 yj + Z k e j jy j j 1 2m k X j2a 0 k yj 2 + X l;j2a 0 k l<j e l e j jy l y j j ;
andtheinter- lusterintera tionI a (x;h) by I a (x;h)= Z 1 Z 2 jx l(y)j + X k 2a 0 1 j2a 0 2 e k e j jy k y j +x l(y)j + X j2a 0 1 Z 2 e j jy j +x l(y)j + X j2a 0 2 Z 1 e j jx l(y) y j j : (II.8) Finally,weset P a (h) = h 2 x +P a (h): (II.9)
P is onsidered asa self-adjoint operator in L 2
(R 3(N+1)
;dxdy ). Note that l(y) = O(h 2
jyj) and that the studyofthedependen e onhofthespe traofP
e
(x;h)iste hni al. Infa t,eventoprovetheterme Z1Z2 jx l(y)j isuniformly(w.r.t. h)
y
-bounded,theauthorsof[KMW2℄usedthefa tsthatx2R 3 andZ 1 Z 2 >0. Foranarbitrary lusterde omposition =(
1 ;:::; k )off1;:::;N+2g,i.e. 1 [[ k =f1;:::;N+2g and j \ k
=;,forj6=k,we analso hooseadapted oordinates(x ;y ). We allP thesub-Hamiltonian, x 2 R 3 (k 1)
theinter- luster oordinates, y
the intra- luster oordinates, and I (x ;y )the inter- luster intera tion. By D x (resp. D y ) and by x (resp. y
), we denote i times the gradient and the Lapla ianintheinter- luster(resp. intra- luster) oordinates. Itiswellknown(seee.g. [DG ℄)that,forthis S hrodingeroperatorP, themodi edwaveoperators
; = s lim t!1 e itP e it x + R t 0 I (sDx ;0)ds+E J (II.10)
exist for any s attering hannel = ( ;E
;
), where is an arbitrary luster de omposition, is an eigenfun tion of P with eigenvalueE : P = E , and where J
denotes theidenti ation operator, whi hisde nedforanyL
2
-fun tionf ofthevariablex by (J f)(x ;y )=f(x ) (y ): (II.11)
Furthermore,thefamilyof waveoperatorsf ;
;8 gisasymptoti ally omplete. It isequallywellknown (see[Ra℄)that,ifa=(a
1 ;a
2
)isatwo- lusterde ompositionwithoneneutral luster(anatom),saya 1 ,i.e. X j2a 0 1 e j = Z 1 ; (II.12)
then,forany hannel =(a;E
;
)withE
outsidethethresholdsofP a
,one ande nethewaveoperators withoutmodi er,namelyby
0 ; = s lim t!1 e itP e it x a +E J : (II.13) Inthis ase, ; = 0 ; e i (D xa )
,where isarealfun tion. Thereforetheresultonasymptoti omplete-ness remains trueif we repla e
; by
0 ;
when the latterexists. So wejust set ;
= 0 ;
to hannelÆ by S = +; ; ; T Æ = S Æ ; (II.14) whereÆ =1if = and0otherwise.
Let us now de ne the total s attering ross-se tions in many-parti le s attering. Sin e few is known about the s attering amplitudes in many-body s attering theory (see [V℄ for results in this subje t), we de ne thetotals attering ross-se tionsa ordingtothephilosophy of[ES℄. ForE
(h), weintrodu e themagnitudeofthemomentumasso iatedwiththekineti energyoftherelativemotionofthetwo lusters in thes attering hannel via
n (;h):= 1=2 (h); (h):= E (h): (II.15) Forg2C 1 0 (I ;C), I =℄E (h);+1[,and!2S 2
,we onsiderthewavepa ket
R 3 3x 7! g ! (x) =~g(!x) (II.16) where ~ g()= 1 2 p h Z R e ih 1 n (;h) g() n (;h) 1=2 d:
Thenormalizationis hosensu hthat
kgk L 2 (R) =k~gk L 2 (R) :
DenotingbyCthesetofall hannels,wewanttoapply,forÆ2C,T Æ
tog !
(x)
(y;h). Sin ethisfun tion doesnotbelong toL
2 (R
3(N+1)
)-it de aysrapidlyonly inthe dire tionde ned by ! -weregularizeitby multipli ation witha fun tion h
R;! 2L
1 (R
3
), depending onlyon thevariable x (!x)! transversalto thedire tion! ofthein identwavepa ketg
!
(x),su hthatpointwisely
lim R!1
h R;!
= 1: (II.17)
Forthepurposeofthispaperweshallspe ifythis ut-o fun tion tobeaGaussian,i.e. wetake
h R;! (x)=e (x (!x)!) 2 =R (II.18) De nition. For2I and! 2S 2
, weshallsay thatthe total ross-se tion
(;!)with thein oming hannel existsattheenergywiththein identdire tion!,ifthefollowinglimitis niteandwellde ned:
(;!) := lim n!1 lim R!1 X Æ2C kT Æ h R;! g n;! k 2 ; (II.19) whereg n;!
isde ned asin(II.16)withg repla edbyg n : g n ()=n 1=2 h(( )=n) andhisanyC 1 0 (R )-fun ti on normalizedby R R jh()j 2 d=1.
Re allthatin[ES℄and[W℄,thetotal ross-se tionisde ned asdistributionin 2I by Z +1 E (h) (;!)jg()j 2 d = lim R!1 X Æ2C kT Æ h R;! g ! k 2 ; (II.20) forall g2C 1 0 (I ;C). Sin e jg n ()j 2 onvergesto Æ
(), theDira measure at, asn!1,thede nitions (II.19)and(II.20) oin ide ifthedistributionde nedin(II.20) anbeidenti edwitha ontinuousfun tion in a neighbourhood of ,whi h is truein the ase whenone knowsto prove theexisten e in the sense of
oneintwo-body ase,see[ES℄,[RW℄,[W℄,[Je ℄. Forsome hannels ;Æandsomein identdire tion!,total s attering ross-se tionsmaynotexistonanyintervalI(see[W ℄). Usuallyitisrequiredthattheintera tions de ayquite rapidlyto ensuretheirexisten e. Inthepresent situationwith Coulomb intera tions, whi h a prioridonotde aysuÆ ientlyfast,weshallshowtheexisten e,i.e. niteness,of
onlyforsomespe ial hannel des ribingion-atoms attering,forallin identdire tions!2S
2
. The onditionson are olle ted in thefollowinghypothesis.
Hypothesis 1. Let =(a;E ; )bea hannelwithE 2 dis (P a
)and lusterde ompositiona=(a 1
;a 2
) su hthat ea h luster ontainsanu leusandsu hthat a
1
isneutral(an atom),that is X j2a 0 1 e j +Z 1 =0: (II.21) Assumethat E = E ;1 +E ;2 with E ;j 2 dis (P a j );j =1;2; (II.22) whereP aj
standsfor theinternal Hamiltonianof luster a j
andE ;1
(theeigenvalueofthe neutral luster) isnon-degenerate.
RemarkII.1. Writey=(y 0
;y 00
)for the ele troni oordinates inthe lusters a 1 ;a 2 andput (y) = ;1 y 0 ;2 y 00 ; 8y2R 3N ; (II.23) with P aj ;j = E ;j ;j :
By the spheri al symmetry of Coulomb potential and the non-degenera y of E ;1
, it an be dedu ed that j ;1 y 0 j=j ;1 y 0 j. Therefore, Z R 3ja 0 1 j y j j ;1 y 0 j 2 dy 0 =0; 8j2a 0 1 : Sin ea 1
isneutral,an elementary al ulus usingthe Taylor expansionof I a iny showsthat <I a (x;h) ; > y =O(jxj 3 ): <;> y
denotes the s alar produ tinL 2
(R 3N y
;dy).
WedenotebyR(z;h)theresolventofP(h)andre allthatitsboundaryvalueR(i0;h):L 2;s
!L 2; s
is well de ned outsidetheset T of thethresholdsand theeigenvaluesof P(h) asanoperator betweenthe weightedL
2
spa es,foranys>1=2.
Our rstmain result on ernstheexisten eof
andgivesausefulformulaforit.
TheoremII.2. Let =(a;E
(h);
(h))be as attering hannelsatisfyingHypothesis 1. Weset
F(z;!;h)= D R(z;h)I a e ;I a e E ; Im z6=0; (II.24) where e (x;y)=e ih 1 n (;h)!x (y;h):
Let T be the set of thresholds and eigenvalues of P. Then, for any energy 2 I
nT and any in ident dire tion !2S 2 ,the limit F(+i0;!;h)= lim !0+ F(+i;!;h) (II.25)
exists and de nes a ontinuous fun tion in . The total s attering ross-se tion
(;!) exists for any energy2I
nT andanyin ident dire tion !2S 2
andone hasthe opti al formula
(;!)= 1 hn (;h)
Sin eI a
e
doesnotbelongtoL ,forsomes>1=2,thisresultisnottrivial. Itsproof-giveninSe tion III - depends ru ially on the de ay of some appropriate e e tive potentials, ombinedwith phase spa e analysis,i.e. anappropriatelo alizationintherelativekineti energyofthetwo lusters.
NextweareinterestedintheBorn-Oppenheimerapproximation(h!0)of
. Werestri tourselvesto thegroundstateenergyofP
a
and demandsomestabilitypropertyw.r.t. x andh.
Hypothesis 2. Leth 0
>0besmallenough. LetE
(h),satisfyingHypthesis1,bethebottomofthespe trum ofP a (h),0hh 0 . Let 0 >E
(0). From(II.7),weseethat,forsomeÆ>0, 0 Æ>E (h),0<hh 0 . Let 1
(x;h) bethe bottom of the spe trumof P e
(x;h). We assumethat for x inaneighborhoodO 0 of the non- ompa t set fx2R 3 ; 1 (x;0) 0 g; 1
(x;h) isasimpleeigenvalueandistheuniqueeigenvalueofP e
(x;h) thattendstoE
(h)asjxj!1,and the uniqueeigenvalueof P
e
(x;h) that tendsto 1
(x;0) ash!0. Furthermore, wedemand that
1 (x;h) ! E (h) as jxj !1;uniformlyw.r.t.hh 0 ; (II.27) 1 (x;h) ! 1 (x;0) as h !0;uniformlyw.r.t. x2O 0 : (II.28)
Notethat thereexistsÆ 0
>0, su hthat, forh 0
smallenough and0hh 0 , fx2R 3 ; 1 (x;h) 0 +Æ 0 g O 0 :
Wealso impose thatfor 0hh 0 , inf x2O 0 P e (x;h) nf 1 (x;h)g > 0 +2Æ 0 ; (II.29) where(P e
(x;h))denotes the spe trumofP e (x;h). Forx2O 0 ,let e (x;h)beanormalizedeigenfun tionofP e (x;h)asso iatedto 1 (x;h). Asin[KMW2℄, we anextendittoasmooth,normalizedfun tion
e
(x;h)ofx su hthat,forsomeÆ 1 >0, P e (x;h) e (x;h); e (x;h) 0 +Æ 1 ; (II.30) forall0hh 0
andforallxin some ompa tneighborhoodK ofthe omplementofO 0 ,satisfying K fx2R 3 ; 1 (x;h) > 0 ;0hh 0 g:
We denote the orthogonal proje tion on the one-dimensional spa e generated by e (x;h) in L 2 (R 3N y ) by (x;h). It indu esaproje tion(h) onL 2 (R 3(N+1)
). Theorthogonalproje tion 0
(h)onto
(h) (intro-du edin Hypothesis1)alsoindu es aproje tiononL
2 (R
3(N+1)
),whi hwestill denoteby 0
(h). Wethen de netheadiabati operatorasso iatedwiththespe tralproje tion(h) by
P AD
(h):=(h)P(h):
WedenotebyR AD
(z;h)itsresolventandset ^ (h)=1 (h) and ^ 0 (h)=1 0 (h). We onsideranenergyrangeJ ℄E
(0);
0 [. Let
t
betheHamiltonian owofthee e tiveHamiltonian fun tion H e (x;)=jj 2 + 1 (x;0) E (0): (II.31)
Anenergy2Risnon-trappingforH e
if,forall(x;)belongingtotheenergysurfa eofH e
ofenergy, thepoint
t
(x;)goestoin nityastand tgoto +1.
Hypothesis 3. Let J an open interval of R su h that J is non-trapping for the e e tive Hamiltonian fun tion H
e
,i.e. isanon-trapping energyfor H e
Inour ontextweneedsu hahypothesistoobtainasemi lassi alestimateontheresolvent.
Undertheprevioushypotheses,weshallderiveinPropositionIV.1semi lassi alestimatesonR(i0) and R
AD
(i0), for 2 J, using arguments developed in [KMW2℄. Finally we introdu e the e e tive potentialswhi hgoverntheleadingtermsof
. DenotingbyC 2
theele troni hargeofa 2 ,thatis C 2 = X j2a 0 2 e j ; (II.32)
wede nethefun tion
C(^x ;y) = C 2 +Z 2 X l2a 0 1 e l ^ xy l ; (II.33)
where x^=x=jxjandwheredenotes thestandards alarprodu tin R 3
. Physi ally,thisfun tion des ribes the intera tion ofthe dipoles formed by theele trons in luster a
1
with the e e tive hargeof luster a 2 . De ne ^ R a (h)=(P a (h) ^ 0 (h) E (h)) 1 ^ 0 (h): (II.34)
Thee e tivepotentialinthe ontextoftheBorn-Oppenheimerapproximationisgivenby
I e (x):= 1 (x;0) E (0): (II.35)
While the intuitive e e tiveterm < I a (x;0) (0); a (0) > y
may de ay exponentially, weshall provethat I
e
(x) isexa tlyoftheorderO(jxj 4
). Infa t,weprovein LemmaIV.2that
jI e (x) ^ I e (x)j=O(jxj 5 ); asjxj!1 (II.36) where ^ I e (x):= 2 ^ R a (0) ^ 0 (0)C(^x ;y) (0); ^ 0 (0)C(^x ;y) (0) L 2 (R 3N y ) jxj 4 ; (II.37) iseverywherenegativeifC 2 +Z 2
6=0. Itisessentiallythisfa twhi hallowstoextra t theleadingorderof thetotals attering ross-se tioninequation(II.39)below. Nowwe anstateourse ondmainresult,whi h givesthesemi lassi alasymptoti sof
.
TheoremII.3. Let =(a;E
(h);
(h))beas attering hannelsatisfyingHypothesis1andHypothesis2. LetJ beareal interval satisfyingHypothesis 3. Then wehave
(;!) = O h 2=3 ; (II.38)
lo ally uniformly w.r.t. 2 J and ! 2 S 2 . We set n (;0) = ( E (0)) 1=2 and we denote by H ! the hyperplane orthogonal to!. Thenthereexistssome
0
>0su hthat,for either hoi e ofe e tivepotential, i.e. for I=I e andI= ^ I e ,wehave (;!) = 4 Z H! sin 2 1 4hn (;0) Z R I(u+s!)ds du + O h 2=3+0 ; (II.39)
lo ally uniformlyw.r.t. 2J and !2S 2
. If a 2
isnot neutral (i.e. the ele troni harge C 2 of a 2 satis es C 2 6= Z 2
),the leadingterm (II.39) withI= ^ I e isexa tly of orderh 2=3 andthusis .
Theorem II.3 shows that the Born-Oppenheimer approximation orre tly des ribesthe asymptoti s of thetotals attering ross-se tionin thesituation onsideredinthis paper,asexpe tedin [CT℄.
In thisse tion weshallprove theexisten eof thetotal s attering ross-se tionasstated in Theorem II.2. Theparameterhplaysnoroleinthisse tionandwillbesetto1. Weshallassumethroughoutthisse tion thattheinitial hannel isasso iatedtoatwo- lusterde ompositiona=(a
1 ;a 2 )witha 1 aneutral luster, that is,(II.21)holds fora
1
. Asa rststep,weestablishthefollowingrepresentationformula. Here weuse thefun tionu R;! =g ! h R;! ,whereg ! ;h R;!
arede ned in(II.16)and(II.18).
LemmaIII.1. Forg2C 1 0 (I ;C), I :=℄E ;+1[,one has X 2C kT u R;! k 2 = 4 Z I ImhR (+i0)I a u R;! ();I a u R;! ()id (III.1) where u R;! (;x)= R 8 n () 3=2 Z S 2 + e in ()x R 4 ( 2 2 + 2 3 ) p 1 g( 2 1 +E )d; (III.2) where 1 =!; the omponents 2 ; 3
denote the dire tionsorthogonal to! 2S 2
and S 2 +
denotes the half sphere
1
>0;2S 2
.
Theproof isthesameasin [RW,W℄ andisomittedhere. Remarkthat theasymptoti ompletenessof waveoperatorsplaysanessentialrolein theproof.
Writing 0 =( 2 ; 3 ),setting B ;R =f 0 2R 2 ;j 0 jR (1 )=2 gandusingd=(1 0 2 ) 1=2 d 0 onS 2 + , wenotethat equation(III.2)implies
u R;! (;x)= R 8 n () 3=2 Z B ;R e in ()(x 1 p 1 02 +x 0 0 ) e R 4 02 (1 0 2 ) 1=4 g( 0 2 )d 0 +O (jR j 1 ); (III.3) uniformlyinx2R 3 . Forj 0 jR (1 )=2
we hangevariablesvia = p
R 0
and, onsideringseparatelythe regionsjxj>R
=2
andjxj<R =2
,weobservethat, forsuÆ ientlysmall,
hxi je in ()(x 1 p 1 2 =R+ p R x 0 ) e in ()x1 jCn ()(1+ 2 )R =2 :
Taylorexpansionoftheintegrandinequation(III.3) ombinedwiththeevaluationoftheGaussianintegral Z R 2 e R 4 02 d 0 = 4 R gives
LemmaIII.2. Forany >0;N 2N thereexistsC>0su hthat
ju R;! (x;) 1 2 1 n () 1=2 g()e in ()x! jChxi R =2 jn ()j N (III.4) uniformlyin x2R 3 ;R1andn () >0:
WeshallnowderiveTheorem II.2asaneasy onsequen eof
TheoremIII.3. Let 2C 1 0
(R) beequalto1on[ Æ=2;Æ=2℄with supp ( Æ;Æ):AssumingHypothesis 1, thereexistsÆ>0su hthatfor any 2I
nT andfor u;v2L 1 (R 3 x )with ( x )u=u; ( x )v=v (III.5) one has jhR (+i)I a u;I a vijC s jjhxi s ujj L 1jjhxi s vjj L 1 (III.6)
s hR (+i0)I a u;I a vi:= lim !0+ hR (+i)I a u;I a vi (III.7)
existsandde nes a ontinuousfun tion of inI
nT.
Proof of TheoremII.2: Itiswellknownthatthemap
(I nT)37!h(x;y)i s R (+i0)h(x;y)i s
is ontinuousforanys>1=2:FromTheoremIII.3, weseethatthefun tion
F(+i0;!):=hR (+i0)I a e in ()x! ;I a e in ()x! i (III.8)
is wellde ned and ontinuous for 2I
nT: Letu R;!
() bethe fun tion de ned in Lemma III.1. Then u R;! ()ande in ()x! areL 1
fun tions satisfyingthe ondition(III.5)in TheoremIII.3. Therefore, om-biningLemmaIII.2withthede nition off
andu
R;!
(),we ndthat forsome0<s<1=2
hR (+i0)I a u R;! ();I a u R;! ()i jg()j 2 4n () F(;!) C hxi s u R;! () g() 2(n ()) 1=2 e in ()x! L 1 C M R s=2 jn ()j M ; (III.9) forallM;jn
()j >0. Thisestimateprovesthatforanyg2C 1 0 (I nT);thelimit lim R!1 X 2C kT h R;! g ! k 2
existsandis equalto
Z Im F(+i0;!) jg()j 2 n () d Nowwerepla eg byg n
intheaboveformulaandtakethelimitn!1. Sin eF(+i0;!)is ontinuousin 2I
nT,weobtainfromthede nitionoftotal rossse tionthat (;!)existsand (;!)= 1 n ()
ImF(+i0;!) (III.10)
for2I
nT and!2S 2
.
Theremainingpartofthisse tionisdevotedtoprovingTheoremIII.3. Thisisdividedintoseveralsteps whi hshallbestatedasdistin tLemmata. HereweareinspiredbytheweightedL
2
estimatesandthephase spa ede ompositionin [CT℄. LemmaIII.4. If u2L 1 (R 3 x )satis es ( x
)u=u,with asin Theorem III.3,then
( 1 ( x ))I a u2L 2;s (R 3(N+1) )
for any s<3=2and
jj(1 ( x ))I a ujj L 2;s (R 3(N+1) ) C s;s 0 jjhxi s 0 ujj L 1 (III.11) for any s;s 0 withs+s 0 <3=2.
(x;y). We hoosea ut-o fun tion ~2C 1 0
(R 3(N+1)
)with0 ~1,whi h isequalto1in asmall oni neighborhoodof andvanishesoutsideaslightlybigger oni neighborhood. Then
~ I a u2L 2;s (R 3(N+1) ) and (1 ( x )) I~ a u2L 2;s (R 3(N+1) ); 8s>0
Onthesupportof1 ;~ theintera tionpotentialI a
issmooth,andsin ethe lustera 1 isneutral,wehave for ~ I a =(1 )I~ a ~ I a (x;y) =O(jxj 2 ); x ~ I a (x;y) =O(jxj 3 ) in L 2;s (R 3N y ); 8s>0: Nextwerewrite ( 1 ( x ))( ~ I a u)= [ ( x ); ~ I a ℄( u)
Thekernelofthe ommutator [ ( x ); ~ I a ℄isgivenby K(x;x 0 ) = 1 (2) 3 Z ~ I a (x;y) ~ I a (x 0 ;y) e i(x x 0 ) ( 2 )d = i (2) 3 Z e ih 1 (x x 0 ) Z 1 0 2 x ~ I a (x 0 +t(x x 0 );y)dt 0 ( 2 )d (III.12)
An easyanalysis showsthat
[ ( x ); ~ I a ℄( u)=O(jxj 3 ) in L 2;s (R 3N y );8s>0:
This implies the rst statement of the Lemma. The asserted norm estimate (III.11) is evident from the aboveproof.
LemmaIII.5. Let beanormalizedeigenfun tionofP a :P a =E b witheigenvalueE E :Then hI a ; i y 2L 2;s (R 3 x ) 8s<1=2; (III.13)
andin the aseE
=E
we havethe improvedestimate
hI a ; i y 2L 2;s (R 3 x ) 8s<3=2: (III.14)
Proof: Weusean expli it omputation to he kthe aseE
=E
: Inthis ase,Hypothesis 1implies that (y)= ;1 (y 1 ) ;2 (y 2 ) where P a2 ;2 =E ;2 ;2 ; jj ;2 jj=1 Setting x^= x jxj
,wehavemodulo aterminL 2;s
(R 3 x
);foranys<3=2andforjxj>1;
hI a ; i y = 1 jxj 2 ((C 1 +Z 1 ) 2; (^x) (C 2 +Z 2 ) 1; (^x )) (III.15) where C j = X k 2a 0 j e k ; j=1;2 and j; (^x ) = X k 2a 0 j e k Z ^ xy k (y) (y)dy = X k 2a 0 j e k Z ^ xy k j ;1 (y 1 )j 2 ;2 (y 2 ) ;2 (y 2 )dy; y=(y 1 ;y 2 )
Sin eC 1
= Z 1
,onehas,moduloaterminL (R x );foranys<3=2, hI a ; i y = 1 jxj 2 (C 2 +Z 2 ) 1; (^x )
UsingHypothesis1(seetheremarkfollowingit),wesee that
y 0 7! X k 2a 0 1 e k ^ xy k j ;1 (y 0 )j 2 is anoddfun tion of y 0 , where y 0 =(y k ;k2a 0 1
). Thus itsintegralvanishes and 1;
(^x ) =0; 8x;^ whi h proves(III.14). Theproofof(III.13) issimilar.
Weshall now lo alizein energy usingthe spe tral proje tionsfor P a . Weset 2Æ :=dist (E ;(P a )n fE g)>0and denoteby 1
thespe tralproje tionof P a asso iatedwithE and by 2 ; 3 the spe tral proje tionsasso iatedwiththeintervals℄ 1;E
[and℄E
;1[. Theproje tions j
areregardedasoperators in L
2 (R
3(N+1)
):Itisthenpossibleto estimateontherangeofthespe tralproje tions 2 ; 3 theresolvent R a (z)=(P a z) 1 oftheHamiltonianP a
des ribing thefreemotionof the lusters,whi hwasde ned in (II.9). One nds
LemmaIII.6. Let 2 C 1 0 (℄ Æ;Æ[) and u 2 L 1 (R 3 x ): For j = 2;3, R a () j ( x a ) are bounded operatorsandwehave theweightedestimate
jjhyi s hxi s 0 R a () j ( x a )(I a u)jjCjjhxi s 00 ujj L 1
for alls>0andfor alls 0 ;s 00 satisfying s 0 +s 0 0 <1=2: Proof: Setting =I a a u,wehave 2 = P E <E h ; i L 2 (R 3N y ) ;wheref gisanorthonormalset ofeigenfun tionsofP a witheigenvalueE <E . Byde nitionofÆ,ifj 2 j<Æ;then 2 +E = 2 +E E
isinvertible. Thus,using thesupportpropertiesof , thefun tiong (;)= ( 2 )( 2 +E ) 1 is boundedandsmooth,forE
asabove. Furthermore R a () ( x ) 2 = X E <E g (D x ;)h ; i L 2 (R 3N y ) : (III.16) Using de ayof
in thevariable y one anapply Lemma III.5 withs=s 0
+s 00
<1=2to gettheasserted estimateforj=2:Forj =3,wehaveP
a 3 := 3 P a 3 (E +2Æ) 3
:ApplyingtheFouriertransformation withrespe tto thex-variable, weseeasabovethat
R a () ( x ) 3 =(P a 3 x ) 1 ( x ) 3 (III.17)
iswellde nedasaboundedoperatoronL 2
(R 3(N+1)
):Applyingthemethodof ommutators,one anverify byindu tionthat hyi s hxi s 0 (P a 3 x ) 1 ( x ) 3 hyi s hxi s 0 L(L 2 ) C foranys;s 0
2R: Grantedthis,theestimateforj=3followsfrom thefollowingweightedestimateon hyi s hxi s 0 I a u Ckuk L 1; foranys>0;s 0
<1=2;whi hisan easy onsequen eofde ayof
in y and fall-o proportionalto jxj 2 ofkI a k y .
a into4pie esvia
= 3 X j=0 j ; 0 =(1 (D 2 x )) ; j = j ( x ) ; j=1;2;3: (III.18) Similarly,forv2L 1 (R 3 x
), withu;vsatisfyingequation(III.5),wede ompose :=I a v:= P 3 j=0 j :This gives hR (+i) ; i= 3 X j;k =0 hR (+i) j ; k i: (III.19)
Forj =0;1; we get from Lemma III.4 and III.5 that j ; j 2 L 2;s (R 3(N+1)
); 8s < 3=2: This gives for j;k=0;1,usingtheweightedestimatefortheresolvent,
jhR (+i) j ; k ij Ckh(x;y)i s j kkh(x;y)i s k k C 1 khxi s 0 uk L 1 khxi s 0 vk L 1; (III.20) foranys>1=2; 0<s 0
<3=2 s:Thisestimateandthosebelowarealluniformin2[ 1;0[[℄0;1℄. Inthe asej=0;1,butk=2;3,wede omposefurtherusingtheresolventequation
R (+i)=R a (+i) R a (+i)I a R (+i): Thisgives jhR (+i) j ; k ij C khxi s j kkhxi s R a ( i) k k+kh(x;y)i s j kkhxi 1+s R a ( i) k k C 1 khxi s 0 uk L 1 khxi s 0 vk L 1 ; (III.21) foranys>1=2; 0<s 0
<3=2 s. HerewehaveusedtheweightedestimateontheresolventR (i)and on
j
;for j =0;1,- as explainedafter equation (III.19) -to estimate the ontribution of j
and wehave used Lemma III.6to estimate the ontribution of
k
. Inter hangingj;k weobtainthesameestimates for theother rosstermsj =2;3andk=0;1.
Finally,to treatthe asej;k=2;3,weiterate theresolventequationon emore:
R (+i)=R a (+i) R a (+i)I a R a (+i)+R a (+i)I a R (+i)I a R a (+i): (III.22)
The rst2termsontherhsofthis equationareeasily handledbyLemmaIII.6andgive
jhR a (+i) j ; k ij Ckhxi s 0 uk L 1 khxi s 0 vk L 1; 8s 0 <1=2 jhR a (+i)I a R a (+i) j ; k ij Ckhxi s 0 uk L 1 khxi s 0 vk L 1; 8s 0 <1: (III.23)
ForthethirdtermontherhsofequationIII.22weobtain,againviaLemma III.6,
j hR a (+i)I a R (+i)I a R a (+i) j ; k ij Ckhxi 1+s R a (+i) j kkhxi 1+s R a ( i) k k Ckhxi s 0 uk L 1khxi s 0 vk L 1 ; (III.24) foranys>1=2; 0<s 0
<3=2 s:Choosingsarbitrarily loseto1=2andaddingequations(III.20),(III.21), (III.23) and (III.24) provesthe uniform boundedness in Theorem III.3. Tosee theexisten e of the weak limit,weusethesamede ompositions. The desiredresultfollowsfromLemma III.5,Lemma III.6 andthe existen eoftheboudaryvaluesR (i0)asoperatorfrom L
2;s toL
2; s
Thisse tionisdevotedtoasket hoftheproofofTheoremII.3. Withinthisse tion,weshallalwaysassume Hypotheses1,2and3andwereinserttheparameterhwhi hisde nedin (II.2).
Tostudy
givenby(II.26),weneed variousapproximationsand estimatesfor theboundary valueof theresolventandforthefun tion I
a e
,whi hare olle tedin thefollowing
PropositionIV.1. Let be the h-dependent setof all possible ollisions (de nedin A.5) and let be an h-dependentsmooth fun tion on R
3(N+1)
,equal toone on some oni neighborhoodof ,andequal to zero onsome bigger oni neighborhood. Forany s0,wehave, uniformly w.r.t. h,
I a 2 L 2 s R 3(N+1) ; (IV.1) k(1 )I a k y = O hxi 2 ; (IV.2) wherekk y
denotes thenormon L 2 (R 3N y ). Forjxj>1, uniformlyw.r.t. h, k(x;h)I a (x;;h) k y = O jxj 4 ; (IV.3) I e (x) := 1 (x;0) E (0) = O jxj 4 ; (IV.4) k(x;h) I a (x;;h) I e (x) k y = O h 2 jxj 4 +O jxj 5 ; (IV.5) k(x;h) I a (x;;h) ^ I e (x) k y = O h 2 jxj 4 +O jxj 5 ; (IV.6)
Furthermore, the smooth fun tion R 3
nf0g3x7!(x;h) has the following properties. Thereexists >0 su hthat,uniformlyw.r.t. x andh,
X j j2 hxi 4+j j e hyi (x;h) x (x;h) 0 (h) 0 (h) L(L 2 (R 3N y )) = O(1): (IV.7)
Notethat (IV.7) remainstrue ifthe rstproje tor (x;h) isrepla edby 0
(h). The resolvents satisfy,for alls>1=2andlo ally uniformlyfor 2J,
hx l(y)i s R(i0)hx l(y)i s + hxi s R AD (i0;h)hxi s = O(h 1 ); (IV.8) hx l(y)i s R(i0;h) ^ = O(1); (IV.9) hxi s R(i0) R AD (i0;h) hxi s = O(1): (IV.10) uniformlyin h2℄0;h 0 ℄.
Proof: (IV.1)followsfromtheexponentialde ayoftheeigenfun tions
(h),whi hisuniformw.r.t. h. A ordingto AppendixA,equation(IV.2)holdsforjxj>1anduniformly w.r.t. h,and
0 (h)+ 0 (0) I a (x;h)+I a (x;0) 0 (h)+ 0 (0) y = A(^x ;h) jxj 5 +O jxj 6 ; (IV.11)
where x^=x=jxjandA(^x ;h)is uniformlybounded ash!0. Furthermore,theoperator hxi 2 I a (x;h) 0 (h) isuniformlybounded. Usingthisfa t,we anshow,asin[KMW2℄, that
X j j2 hxi 2+j j e hyi x (x;h) 0 (h) y = O(1): (IV.12)
Using(IV.12) and(IV.11),weobtain
(x;h)I a (x;h) 0 (h) = (x;h) 0 (h) I a (x;h) 0 (h) + 0 (h)I a (x;h) 0 (h) = O jxj 4 : (IV.13)
Next,weshowthat
1 (x;h) E (h) = ^ I e (x) + O h 2 jxj 4 +O jxj 5 : (IV.14)
k(x;h) (h)k 2 y = 1+O jxj 2 :
Thus, a ording to (IV.11), for jxjlarge enough and writingh;i y
for the s alarprodu t in L 2 (R 3 N y ), we have 1 (x;h) E (h) = (x;h) (h);I a (x;h)(x;h) (h) y =k(x;h) (h)k 2 y = (x;h) (h);I a (x;h)(x;h) (h) y +O jxj 6 = 2< (x;h) 0 (h) (h);I a (x;h)(x;h) (h) y +O jxj 5 = 2< (x;h) 0 (h) (h);I a (x;h) (h) y + O jxj 5 (IV.15)
Next,weusethefollowinglemma,whi h willbeprovedafterthepresentproof.
LemmaIV.2. Setting ^ R a (z;h) = (P a (h) ^ 0 (h) z) 1 ^ 0 (h) and ^ R a (h) = ^ R a (E (h);h) (as in equation (II.34)), wehave, for jxjlargeenough anduniformlyw.r.t. h,
2< (x;h) 0 (h) (h);I a (x;h) (h) y = 2 ^ R a (h) ^ 0 (h)C(^x;y) (h); ^ 0 (h)C(^x;y) (h) y jxj 4 +O jxj 5 : (IV.16)
In parti ular, thetwoformsof the e e tive potential satisfyequation (II.36),i.e.
jI e (x) ^ I e (x)j=O(jxj 5 ); asjxj!1:
Furthermore, the rstterm onthe rhsof (IV.16)isnegative, forall x6=0, ifthe lustera 2
isnot neutral.
UsingLemmaIV.2,weobtain(IV.14). ByaTaylorexpansionw.r.t. handusingthepreviousestimates,
(x;h)I a (x;h) 0 (h) = (x;h)I a (x;0) 0 (h) +O hjxj 6 = (x;0)I a (x;0) 0 (0) +O h 2 jxj 5 +O jxj 6 = (x;0) (x;0) E (0) 0 (0) + O h 2 jxj 5 +O jxj 6 = (x;h) (x;0) E (0) 0 (h) +O h 2 jxj 5 + O jxj 6 :
Wethenhaveproved(IV.5). Using(IV.15),wederive(IV.6)from(IV.5).
Finally, wefollowthe argumentsin [KMW2℄ to derive(IV.7)from (IV.13). Stillfollowing[KMW2℄,we obtainresolventestimateswiththeweighthx l(y)i. Asalreadyremarkedin[KMW2℄,hxi
s
(x)hx l(y)i s
is uniformly bounded. Thus we may repla ethis weightbyhxi if is present. Wedo thisfor these ond termin(IV.8)andin (IV.10).
Proof of LemmaIV.2: Equation(II.36) simplyfollowsfrom(IV.15) and(IV.16),for h=0. Toprove (IV.16),wewritetheproje tionsas ontourintegrals. Let a omplex ontouren losingE
(h)and 1
(x;h) forhsuÆ ientlysmallandjxjsuÆ ientlylarge. Forbrevity,weshallnownotationallysuppressthe depen-den eonh. Werewritethelhsofequation(IV.16) as
lhs(IV.16) = 2< D (P e (x) z) (P a z) ; 1 2i I (P e (x) z) 1 (P a (x) z) 1 dz E y = 2< 1 2i I dz (E z) (P e (x) z) 1 ; y +(E z) 1 I a (x) ; y = 2< 1 2i I dz(E z) (P e (x) z) 1 ; y + O jxj 5 ; (IV.17)
by (IV.11). So weneed to ompute 0 R e (z) 0 , where R e (z) = (P e (x) z) 1
. To this end, we use the resolventequation R e (z)=R a (z) R a (z)I a R a (z)+R a (z)I a (z)R e (z)I a R a (z) (IV.18)
0 R e (z) 0 =R a (z) 0 +R a (z) 0 I a ^ 0 R e (z) ^ 0 I a 0 R a (z)+O(jxj 5 ) (IV.19)
Inserting theseestimates into (IV.17) andusing Appendix C and(IV.11) again, wearriveat (IV.16) with ^ R a (h)repla edby ^ 0 R e (E (h)) ^ 0 . But k ^ 0 ( R e (E ) R a (E )) ^ 0 C(^x ;y) k y =O(jxj 2 ); (IV.20)
uniformly w.r.t. h. This follows from aNeumann expansion of R e
(z), exponentialde ay of
, uniform boundednessoftheweightedredu edresolventhyi
M ^ R a hyi M
,foranyM0, ombinedwithkI a hyi M k y = O(jxj 2
):Thisprovesequation(IV.20) andthus (IV.16). Sin e ^ R a (E
(h)) b > 0, uniformly w.r.t. h, the rst term on the rhs of (IV.16) is bounded above by bk ^ 0 C(^x ;y) (h)k 2 =jxj 4 . Sin e k 0 C(^x ;) (h)k 2 y =0
bytherotationalinvarian eof ;1
(seeAppendix A),wehave
k ^ 0 C(^x ;y) (h)k 2 y =kC(^x;y) (h)k 2 y > 0: (IV.21)
Re allfrom(II.26)that
(;!;h) = h 1 n (;h) Im R(+i0;h)I a e ; I a e :
Inviewoftheadiabati approximationoftheresolvent,weintrodu e
ad (;!;h):= h 1 n (;h) Im R AD (+i0;h)I a e ; I a e : (IV.22) ad
isalmostthetotal ross-se tionforthes atteringpro essofthepairofoperators(P AD (h);(h)( h 2 x + E
(h))(h)),asshownin[Je ℄. It thusshould beagoodapproximationfor
. Indeed,we laimthat
PropositionIV.3. Forall >0small enough,there existssomeC >
0su hthat, for allh>0suÆ iently smallandlo allyuniformly in(;!)2JS
2 , (;!;h)= ad (;!;h)+O(h 2=3+(1=2 ) ): (IV.23)
WiththeestimatesgiveninPropositionIV.1, theproofofPropositionIV.3followsthesamearguments asin[Je ℄.
Theadiabati operatorP AD (h)isequalto(h)( h 2 x + 1 (x;h))(h) in O 0
whi hlookslikea two-body S hrodinger operator with operator-valued potential. We an use themethods of [RT℄ and [Je ℄ to provethat ad (;!;h)=O(h 2=3 ) andthat ad (;!;h) = 4 Z H! sin 2 1 4n (;0)h Z +1 1 I e (x ! +u!)du dx ! + O h 2=3+ 0 ; (IV.24)
whi h,a ordingtoPropositionIV.3,gives(II.39)forI=I e
. Sin ethepotential ^ I e
hasthesameproperties asI e and ^ I e I e =O(jxj 5
)(seePropositionIV.1),we anshowthat(IV.24)stillholdswithI e repla ed by ^ I e
. Wethus obtaintheformula(IV.24)with ^ I e
, whi his(II.39)forI= ^ I e
. NowweassumethatC
2 +Z 2 6=0. Toshowthat ad (andthus )isexa tlyoforderh 2=3 ,weestimate asin[Je ℄ theintegralin (II.39) forI=
^ I e . Re allthat ^ I e
(x)is oftheformA(^x ;0)jxj 4
2 2 asin[Y ℄that,forsomeh-independent onstantb6=0,
4 Z H! sin 2 1 4n (;0)h Z +1 1 ^ I e (x ! +u!)du dx ! =bh 2=3 Z S 1 ! (!; ) 2=3 d ; (IV.25) whereS 1 !
istheunit spherein H !
andwhereisgivenby
(!; )= Z 0 ^ I e
( os!+sin )sin 2
d (IV.26)
for 2S 1 !
. ByLemma IV.2 we knowthat theintegrand -and thus -is negativeeverywhere. Thus the rhsof(IV.25) isexa tlyoforderh
2=3 .
A Expansion of the potentials
Inthisse tionwe olle ttherelevantexpansionsoftheCoulombintera tionsforatom-ions attering,whi h involvethee e tivedipolemomentsandquadrupolemomentsofthetwo lustersa
1 ;a
2
. Theyare ertainly wellknowninthephysi sliterature. Forthesakeofthereader,westatethem as
LemmaA.1. Let =(a;E
(h);
(h)) beas attering hannelsatisfying Hypothesis1. Then
kI a (x;h) (h)k y = O(hxi 2 ); (A.1) I a C(^x ;) jxj 2 (h) y = O(hxi 3 ); C(^x ;y)= C 2 +Z 2 X l2a 0 1 e l ^ xy l ; (A.2) h (h);I a (x;h) (h)i y = O(hxi 3 ); (A.3) uniformly w.r.t. h, for 0hh 0
. Assumingin addition that satis es Hypothesis 2, we even have the strongerestimate h (h);I a (x;h) (h)i y =O(hxi 5 ); (A.4) uniformlyw.r.t. h, for 0hh 0 .
Proof: Be ause of the Coulomb singularities, weseparate the ontribution of ollisions. Let be the (h-dependent)set ofallpossible ollisions,thatis
:= (x;y)2R 3(N+1) ; 9l2a 0 1 ;9j2a 0 2 ; x= y l +l(y) or x=y j y l +l(y) or x=l(y) or x=y j +l(y) : (A.5) Let 2C 1 (R 3(N+1)
)su hthat 0 1, equals 1on asmall oni neighborhoodof ,and equals0 outsideaslightlybigger oni neighborhood. Wealsodemandthat iseveniny. Thankstotheexponential de ay(uniformlyw.r.t. h) oftheeigenfun tions
(h), wehave k (x;y;h)hyi L I a (x;h) (h)k y =O(hxi M ); 8L;M2N: (A.6)
Thus,weonlyhaveto estimatethe ontributionoftheregularpart
I reg ( ):=h (h); ~ I a (x;h) (h)i y ; ~ I a (x;h):=(1 (x;y;h))I a (x;h): (A.7)
A ordingto (II.8),wewanttoexpandtermsoftheform
jx+ ~ l (y)j 1 = jxj 1 ^ x+ ~ l(y)=jxj 1
for largejxj. To thisend, we useaTaylorexpansionat zeroof thefun tion f :R 3r 7! ju+rvj for non-zero ve torsu;v 2 R
3
. More pre isely, one obtainsby Taylor expansion, that for ea h r 2 R, there existssome2℄0;1[su hthat
f(r) = 3 X k =0 r k k! f (k ) (0)+ r 4 4! f (k ) (r) (A.8)
Weobservethatthe rstorderterm(the terminjxj 1 )oftheexpansionofI reg ( )vanishesbyneutralityof a 1
( f. (II.21)). These ondordertermisgivenby
(h);C(^x ;) (h) y jxj 2 ; C(^x ;y)=(C 2 +Z 2 ) X l2a 0 1 e l ^ xy l ; (A.9) where C 2 = P l2a 0 2 e l
is theele troni hargeof a 2
and where wehaveused anestimate similarto(A.6)to get k (x;y;h) (h)C(^x ;) (h)k y =O(hxi M ); 8M 2N:
Bythe re exion symmetry, wesee that the se ond order termalso vanishes. Therest of the expansionis seentobeO(jxj
3
),uniformlyw.r.t. h. Inviewofequation(A.6),thisproves(A.2)and(A.3). Theproofof (A.1) issimilarandinvolvesanon-vanishingtermofse ondorder. Nextweshallprovetheestimate(A.4). It ru ially depends on thefull rotational symmetryof thewavefun tion
= ;1 ;2 in both lusters (whi hisa onsequen eofHypothesis 2). For onvenien e, we hoosethe ut-o in su h awaythat the new ut-o alsohasthesamesymmetryproperties. Tothisend, we onsider
~
, de nedastheunionofthe orbitsunder the a tiono : y !oy =(oy
1
;;oy N
)of O(3;R) onthe y-variablesof ea h pointin (forh=0). Asbefore,we onstru ta ut-o ~2C
1 (R
3(N+1)
)su h that0 ~1, ~equals1onasmall oni neighborhoodof
~
, and ~equals0outsideaslightlybigger oni neighborhood. Noti e that, onthe support of ,~ thepreviouspropertiesarepreservedsin ejxjand jyjareequivalentthere. Thus(A.6) holds in this asealso,anditagainsuÆ estoestimatetheregularpartde nedin (A.7). Obviously,the rstand se ondorder termoftheexpansionarezero. Expandingfurther, we ndthat thethirdordertermis
(h);F 3 (^x ;y) (h) jxj 3 ; (A.10)
wherewehaveestimatedthe ontributionoftheregion utoutby ~asaboveandwhere
F 3 (^x;y)= (C 2 +Z 2 ) X l2a 0 1 e l jy l j 2 3(y l x )^ 2 +2 X l2a 0 1 ;j2a 0 2 e l e j (y l y j 3(y l x )(y^ j ^x)): (A.11)
BytherotationsymmetryofCoulombpotentials,we anrepla ein(A.10)x^bythe anoni albasisve tors b 1 ;b 2 ;b 3 ofR 3 . Sin e P k F 3 (b k
;y)= 0,it followsthat the third order termalso vanishes. Forthefourth orderterm,weget
(h);F 4 (^x ;y) (h) jxj 4 ; (A.12)
wherethefun tion F 4 satis es F 4 (^x ;y 1 ; y 2 )= F 4 (^x ;y 1 ;y 2 ); y=(y 1 ;y 2 );
sin eitishomogeneousofdegree3iny. ByHypothesis2theeigenvalueE ;2
issimpleand ;2
isinvariant under there e tion y
2 7! y
2
. Thus thefourth ordertermis zero,and astandardappli ation of Taylor's theorem(A.8) showsthattheremainderoftheexpansionisO(jxj
5
),uniformlyw.r.t. h.
A knowledgements. X.P. Wangthanksthe organizors ofthe BolognaAPTEXInternational Conferen e, September 1999,for their invitation andhospitality.
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