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\def\feffsec#1{
    \centerline{\Bigrmb #1}
    \vskip3em
    \centerline{\bigrmb Charles L. Fefferman\footnote{*}
    {\rm Partially supported by a NSF grant at Princeton University}}
    \vskip1em
    \centerline{\it Department of Mathematics, Princeton University}
    \vskip2em
    \centerline{\bigrmb Luis A. Seco}
    \vskip1em
    \centerline{\it Department of Mathematics, 
	California Institute of Technology}
    \vskip10em
}
\def\feffsectwo#1#2{
    \centerline{\Bigrmb #1}
    \vskip1em
    \centerline{\Bigrmb #2}
    \vskip3em
    \centerline{\bigrmb Charles L. Fefferman\footnote{$ ^\ast $}
    {\medtype\rm Partially supported by an NSF grant at Princeton University}}
    \vskip1em
    \centerline{\it Department of Mathematics, Princeton University}
    \vskip2em
    \centerline{\bigrmb Luis A. Seco}
    \vskip1em
    \centerline{\it Department of Mathematics, 
         California Institute of Technology}
    \vskip5em
}
\def\myaddress{\bigskip
  \centerline{Luis A. Seco}
  \centerline{	    Department of Mathematics}
	   \centerline{ Princeton University}
	  \centerline{  Princeton NJ 08544}
	  \centerline{  U.S.A}
	  \bigskip}

\def\mycopenhagenaddress{\bigskip
  \centerline{Luis A. Seco}
  \centerline{Matematisk Institut}
	   \centerline{K\o benhavns Universitet}
	   \centerline{ Universitetsparken 5}
	  \centerline{DK--2100 K\o benhavn \O}
	  \centerline{Denmark}
	  \centerline{Tel. (Gerd Grubb) 45--3135 3133 Ext. 443}
	  \centerline{Fax. 45--3135 4254}
	  \centerline{e--mail (P. Solovej): philip@math.ku.dk}
	  \bigskip}

\def\myaarhusaddress{\bigskip
  \centerline{Luis A. Seco}
  \centerline{c/o E. Balslev}
  \centerline{Matematisk Institut}
	   \centerline{ Ny Munkegade}
	   \centerline{ Universitetsparken, 8000 \AA rhus}
	  \centerline{Denmark}
	  \centerline{Fax. 45--86--131769}
	  \centerline{e-mail (E. Balslev): balslev@mi.aau.dk}
	  \bigskip}

\def\mymadridaddress{\bigskip
  \centerline{Luis A. Seco}
  \centerline{c/o Antonio C\'ordoba}
  \centerline{	Division de Matem\'aticas C--XVI}
	   \centerline{ Facultad de Ciencias}
	   \centerline{ Universidad Aut\'onoma de Madrid}
	   \centerline{ Cantoblanco,  28049 Madrid}
	  \centerline{  Spain}
	  \centerline{Tel. (A. C\'ordoba): 34--1--3974986}
	  \centerline{Fax. 34--1--3974889}
	  \centerline{e-mail (A. Sanchez--Calle): sanca@emduam11.bitnet}
	  \bigskip}

\def\ourspainaddress{\bigskip
  \centerline{Luis A. Seco}
  \centerline{Maria T. Quintanilla}
  \centerline{	Chacoli Ategorri 4-6A}
	   \centerline{ 48004 Bilbao}
	  \centerline{  Spain}
	  \centerline{Tel. 34--4--4117088}
	  \bigskip}

\def\luisgeorge{\bigskip
  \centerline{Luis A. Seco}
  \centerline{212 St. George St, Apt 303}
	   \centerline{ Toronto, ONTARIO M5R 2N5}
	  \centerline{ Canada}
	  \centerline{Tel. 416--966 2548}
	  \bigskip}

\def\luiswoodlawn{\bigskip
  \centerline{Luis A. Seco}
  \centerline{37 Woodlawn Av. West}
	   \centerline{ Toronto, ONTARIO M4V 1G6}
	  \centerline{ Canada}
	  \centerline{Tel. 416--978 2033}
	  \bigskip}

\def\maitewoodlawn{\bigskip
  \centerline{Maria T. Quintanilla}
  \centerline{37 Woodlawn Av. West}
	   \centerline{ Toronto, ONTARIO M4V 1G6}
	  \centerline{ Canada}
	  \centerline{Tel. 416--978 2033}
	  \bigskip}

\def\myspainaddress{\bigskip
  \centerline{Luis A. Seco}
  \centerline{	Chacoli Ategorri 4-6A}
	   \centerline{ 48004 Bilbao}
	  \centerline{  Spain}
	  \centerline{Tel. 34--4--4117088}
	  \bigskip}

\def\mycaladdress{\bigskip
  \centerline{Luis A. Seco}
  \centerline{253--37 Caltech}
	  \centerline{ Pasadena CA 91125}
	  \bigskip}
\def\supp{{\rm supp\ }}
\def\center#1{\hfil #1\hfil}
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      %\wd1=0pt \ht1=0pt
   \vbox{\parskip=0pt\hrule\box1\hrule\par}
}
\def\boxitt#1{
   \setbox0=\hbox{\vrule\kern0pt
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 \setbox1=\vbox{\hrule\box0\hrule} 
    \box1 }

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\def\thmstylea{\it}
\def\intsend{\hskip1pt}
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\def\theorem#1{{\thmstyleb Theorem #1:} \thmstylea}
\def\algorithm#1{{\thmstyleb Algorithm #1:} \thmstylea}
\def\endt{\vskip2em\goodbreak\rm}
\def\proof{\endt{\bf{Proof:  }}}
\def\cproof{\endt{\bf{Proof (Computer--Assisted):  }}}
\def\description{\endt{\bf{Description:  }}}
\def\corol#1{{\thmstyleb Corollary #1:} \thmstylea}
\def\lem#1{{\thmstyleb Lemma #1:} \thmstylea}
\def\propo#1{{\thmstyleb Proposition #1:} \thmstylea}
\def\definition{\thmstyleb{\bf Definition: }\sl}
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\def\ttitle#1{\goodbreak\vskip4em\centerline{\BBigrmb{#1}}\vskip4em}
\def\Title#1{\goodbreak\vskip4em{\Bigrmb{#1}}\vskip2em}
\def\title#1{\vskip8pt\goodbreak\noindent{\bf{#1}}\enspace }
\def\chapter#1{\vfill\eject\hbox{\ }\vskip60pt{\Bbigrmb #1}\vskip6em}
\def\senia#1{\hskip200pt\vtop{
  \hfuzz=5pt\baselineskip=4pt\hsize1.9in\raggedright
   \smept{#1}}}
\def\mathendpf{\hbox{ \it
   Q\kern-2.5pt\lower2.5pt\hbox{\rm E}\lower-2pt\hbox{\kern-4.5pt D}}
   }
\def\endpf{\hfill{ \it
   Q\kern-2.5pt\lower2.5pt\hbox{\rm E}\lower-2pt\hbox{\kern-4.5pt D}}
   \goodbreak
   \vskip2em}
\def\coint#1,#2{[\intsend #1\intsmid #2\intsend)}
\def\ocint#1,#2{(\intsend #1\intsmid #2\intsend]}
\def\ooint#1,#2{(\intsend #1\intsmid #2\intsend)}
\def\ccint#1,#2{[\intsend #1\intsmid #2\intsend]}
\def\bibliography{ \vskip3em \Title{References}\vskip1em
  \medtype\parindent13pt}
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   \let\rm=\eightrm \let\bf=\eightbf
   \let \mus=\eightmus \let\tt=\eighttt
   \let\it=\eightit \let\sl\eightsl \baselineskip=3pt minus 0pt\rm}
\def\Calogero{[Ca]}
\def\BBcalogeroBB{\item{\calogero} 
Calogero, F. ``{\sl Variable Phase Approach to the
Potential Scattering\/}''
Academic Press, NY 1967.\vskip1em}


\def\Simon{[Si]}
\def\BBSimonBB{
\item{\Simon}Simon B. (1984) ``{\sl 
Fifteen Problems in Mathematical Physics\/}''
Prespectives in Mathematics, Anniversary of Oberwolfach.
\vskip1em}

\def\Liebtf{[Li]}
\def\BBLiebtfBB{
\item{\Liebtf}Lieb, E. (1981) ``{\sl Thomas--Fermi and Related Theories of
Atoms and Mol\-ecules}'' Reviews of Modern Physics Vol 53 no. 4.
\vskip1em}

\def\LiebSimon{[LS]}
\def\BBLiebSimonBB{
\item{\LiebSimon}Lieb, E. and Simon, B. (1977) ``{\sl Thomas--Fermi Theory of
Atoms, Molecules and Solids}'' Adv. Math. 23, pp 22---116.
\vskip1em}

\def\SiedentopWeikard{[SW1]}
\def\BBSiedentopWeikardBB{
\item{\SiedentopWeikard} Siedentop, H., Weikard, R. (1987) ``{\sl
On the Leading Energy
Correction for the Statistical Model of the Atom: Interacting Case}''
Communications in Mathematical Physics {\bf 112} 471-490
\vskip1em}


\def\SiedentopWeikardLBII{[SW3]}
\def\BBSiedentopWeikardLBIIBB{
\item{\SiedentopWeikardLBII} Siedentop, H., Weikard, R. (1990) ``{\sl
A New Phase Space Localization Technique with Applications to
the Sum of Negative Eigenvalues of Schr\"odinger Operators.
}''
Ann. Scient. Ecole Normale Superieure 24, 215 -- 225, (1991).
\vskip1em}


\def\SiedentopWeikardLB{[SW2]}
\def\BBSiedentopWeikardLBBB{
\item{\SiedentopWeikardLB} Siedentop, H., Weikard, R. ``{\sl
On the Leading 
Correction of the Thomas--Fermi Model: Lower Bound}''
and an appendix by A.M.K. M\"uller.
Inv. Math., Vol., 97, pp 159---193, 1989.
\vskip1em}

\def\EnglertSchwingera{[ES1]}
\def\BBEnglertSchwingeraBB{
\item{\EnglertSchwingera} Englert, B. G. and
Schwinger, J. (1985) ``{\sl
Semiclassical Atom
}\/''
Physical Review A, {\bf 32} no 1, 26 -- 35.
\goodbreak\vskip1em}

\def\Englert{[En]}
\def\BBEnglertBB{
\item{\Englert} Englert, B. G.
``{\sl
Semiclassical Theory of the Atom
}\/''
Springer Verlag Lecture Notes in Physics, vol 301.
\goodbreak\vskip1em}

\def\EnglertSchwingerb{[ES2]}
\def\BBEnglertSchwingerbBB{
\item{\EnglertSchwingerb} Englert, B. G. and
Schwinger, J. (1985) ``{\sl
Atomic--binding-energy Oscillations
}\/''
Physical Review A, {\bf 32} no 1, 47 -- 63.
\goodbreak\vskip1em}

\def\Schwinger{[Sch]}
\def\BBSchwingerBB{
\item{\Schwinger} Schwinger, J. (1981) ``{\sl
Thomas--Fermi Model: The Second Correction}''
Physical Review A24, {\bf 5}, 2353---2361.
\goodbreak\vskip1em}

\def\Dirac{[Di]}
\def\BBDiracBB{
\item{\Dirac} Dirac, P. (1930) ``{\sl
Note on Exchange Fenomena in the Thomas--Fermi Atom}''
Proc. Cambridge Philos. Soc. {\bf 26}, 376---385.
\vskip1em}

\def\Thomas{[Th]}
\def\BBThomasBB{
\item{\Thomas} Thomas, L. H. (1927) ``{\sl
The Calculation of Atomic Fields}''
Proc. Cambridge Philos. Soc. {\bf 23} 542---548.
\vskip1em}

\def\Fermi{[Fe]}
\def\BBFermiBB{
\item{\Fermi} Fermi, E. (1927) ``{\sl
Un Metodo Statistico per la Determinazione di alcune Priorieta dell'Atome}''
Rend. Accad. Naz. Lincei {\bf 6}, 602---607.
\vskip1em}

\def\Scott{[Sc]}
\def\BBScottBB{
\item{\Scott} Scott, J. M. C. (1952)
``{\sl 
The Binding Energy of the Thomas--Fermi Atom 
}''
Phil. Mag. {\bf 43} 859---867.
\vskip1em}

\def\BBLiebBB{
\item{\Lieb} Lieb, E. H. (1979)
``{\sl 
A Lower Bound for Coulomb Energies
}''
Phys. Lett. {\bf 70A} 444---446.
\vskip1em}

\def\GilbarTrudinger{[GT]}
\def\BBGilbarTrudingerBB{
\item{\GilbarTrudinger} Gilbar, and Trudinger, (1979)
``{\sl 
Elliptic Partial Differential Equations
}''
Springer-Verlag
\vskip1em}


\def\thesis{[Se1]}
\def\BBthesisBB{
\item{\thesis}Seco, L.
``{\sl 
Lower Bounds for the Ground State Energy of Atoms
}''
Thesis, Princeton University, 1989.
\vskip1em}


\def\BBFeffermanSeconIBB{
\item{\FeffermanSeconI} Fefferman, C., Seco, L. (1989) ``{\sl
An Upper Bound for the
Number of Electrons in a Large Ion}'' 
Proceedings of the Nat. Acad. Sci., USA Vol. {\bf 86}, no. 10, May 1989,
pp 3464--3465.\vskip1em}

\def\BBFeffermanSeconIIBB{
\item{\FeffermanSeconII} Fefferman, C., Seco, L. (1990) ``{\sl
Asymptotic Neutrality of Large Ions
}'' 
Comm. Math. Phys. Vol. 128, pp 109---130.
\vskip1em}


\def\BBSecoSigalSolovejBB{
\item{\SecoSigalSolovej} Seco, L., Sigal, I. M., Solovej, J. P., ``{\sl
Bound on the Ionization Energy of Large Atoms
}'' To appear in {\it
Comm. Math. Phys.}
\vskip1em}

\def\Hille{[Hi]}
\def\BBHilleBB{
\item{\Hille} Hille, E.,
``{\sl 
On the Thomas--Fermi Equation
}'' 
Proc. Nat. Acad. Sci, USA, 62, 7---10.
\vskip1em}


\def\FeffSeca{[FS1]}
\def\BBFeffSecaBB{
\item{\FeffSeca} Fefferman, C. and Seco, L.
``{\sl 
The Ground--State Energy of a Large Atom
}'' 
Bull. A.M.S., Vol {\bf 23} no. 2, 525---530, 1990
\vskip1em}

\def\FeffSecb{[FS2]}
\def\BBFeffSecbBB{
\item{\FeffSecb} Fefferman, C. and Seco, L.
``{\sl 
Eigenvalues and Eigenfunctions of Ordinary Differential Operators
}'' 
To appear in {\sl Adv. Math.}
\vskip1em}

\def\FeffSecc{[FS3]}
\def\BBFeffSeccBB{
\item{\FeffSecc} Fefferman, C. and Seco, L.
``{\sl 
The Eigenvalue Sum for a One--Dimensional Potential
}'' 
To appear in {\sl Adv. Math.}
\vskip1em}

\def\FeffSecd{[FS4]}
\def\BBFeffSecdBB{
\item{\FeffSecd} Fefferman, C. and Seco, L.
``{\sl 
The Density in a One-Dimensional Potential
}'' 
To appear in {\sl Adv. Math.}
\vskip1em}

\def\FeffSece{[FS5]}
\def\BBFeffSeceBB{
\item{\FeffSece} Fefferman, C. and Seco, L.
``{\sl 
The Eigenvalue Sum for a Three--Dimensional Radial Potential
}'' 
To appear in {\sl Adv. Math.}
\vskip1em}

\def\FeffSecf{[FS6]}
\def\BBFeffSecfBB{
\item{\FeffSecf} Fefferman, C. and Seco, L.
``{\sl 
The Density in a Three--Dimensional Radial Potential
}'' 
To appear in {\sl Adv. Math.}
\vskip1em}

\def\FeffSecg{[FS7]}
\def\BBFeffSecgBB{
\item{\FeffSecg} Fefferman, C. and Seco, L.
``{\sl 
On the Dirac and Schwinger Corrections to the Ground--State Energy
of an Atom
}'' 
To appear in {\sl Adv. Math.}
\vskip1em}

\def\fs7{[FS8]}
\def\BBfs7BB{
\item{\fs7} Fefferman, C. and Seco, L.
``{\sl 
Aperiodicity of the Hamiltonian Flow in the Thomas--Fermi Potential
}'' 
To appear in {\sl Revista Matmem\'atica Iberoamericana}
\vskip1em}


\def\Feffnt{[Feff]}
\def\BBFeffntBB{
\item{\Feffnt} Fefferman, C.
``{\sl 
Atoms and Analytic Number Theory
}'' 
Proceedings of the AMS, 1991.
\vskip1em}



\def\Arnold{[Ar]}
\def\BBArnoldBB{
\item{\Arnold} Arnold, V.
``{\sl 
Mathematical Methods of Classical Mechanics
}'' 
Springer.
\vskip1em}


\def\EKW{[EKW]}
\def\BBEKWBB{
 \item{\EKW}
Eckmann, J. P., Koch, H. and Wittwer, P.
``{\sl A computer Assisted Proof of Universality in Area Preserving
Maps}''
Memoirs, A.M.S., Vol 289 (1984).
\vskip1em}


\def\EckmannWittwer{[EW]}
\def\BBEckmannWittwerBB{
\item{\EckmannWittwer} Eckmann, J. P. and Wittwer, P.
``{\sl Computer Methods and Borel Summability Applied to Feigenbaum's
equation}''
Lecture Notes in Mathematics {\bf 227}, Springer Verlag (1985).
\vskip1em}

\def\FeffermanLlave{[FL]}
\def\BBFeffermanLlaveBB{
\item{\FeffermanLlave}
 Fefferman, C. and Llave, R., ``{\sl Relativistic Stability
of Matter, I\/}'', {Revista Mate{\-}m\'atica Ibero{\-}ameri\-cana} Vol {2}
no.1\&2, pp. 119-213 (1986)
\vskip1em}

\def\KaucherMiranker{[KM]}
\def\BBKaucherMirankerBB{
\item{\KaucherMiranker} Kaucher, E. W. and Miranker, W. L.,
``{\sl Self-validating Numerics for Function Space Problems\/}'',
{ Academic Press, New York (1984)}.
\vskip1em}

\def\LanfordLlave{[LL]}
\def\BBLanfordLlaveBB{
\item{\LanfordLlave}Lanford, O. and Llave, R. ``{\sl Solution of the Functional
Equation for Critical Circle Mappings with Golden Rotation Number}''
{\it In preparation}.
\vskip1em}

\def\Llave{[Ll]}
\def\BBLlaveBB{
\item{\Llave} Llave, R. ``{\sl Computer Assisted Bounds in Stability
of Matter}'' 
Computer Aided Proofs in
Analysis, IMA Series in Math. and Appl. Vol {\bf 28}. Cincinnati (1989).
Springer.
\vskip1em}

\def\LlaveRana{[LR]}
\def\BBLlaveRanaBB{
\item{\LlaveRana} Llave, R. and Rana, D. ``{\sl Algorithms for the
Rigorous Proof of Existence of Special Orbits\/}'', To appear.
\vskip1em}

\def\Moore{[Mo]}
\def\BBMooreBB{
\item{\Moore} Moore, R. E., ``{\sl Methods and Applications
of Interval Analysis\/}'', \break
 S.I.A.M., Philadelphia (1979).
\vskip1em}

\def\Rana{[Ra]}
\def\BBRanaBB{
\item{\Rana} Rana , D., ``{\sl Proof of Accurate Upper and Lower
Bounds for Stability Domains in
Denominator Problems\/}'', Thesis, Princeton University (1987)
\vskip1em}

\def\Seco{[Se2]}
\def\BBSecoBB{
\item{\Seco} Seco, L., ``{\sl Computer Assisted Lower Bounds for
Atomic Energies\/}''
Computer Aided Proofs in
Analysis, IMA Series in Math. and Appl. Vol {\bf 28}, 241---251.
 Cincinnati (1989).
Springer.
\vskip1em}

\def\Lohner{[Lo]}
\def\BBLohnerBB{
\item{\Lohner} Lohner, R., ``{\sl Einschlie\ss ung der
L\"osung gew\"ohnlicher Anfangs-- und Rand\-wert\-auf\-gaben und
Anwendungen
\/}''
Dissertation, Universit\"at Karlsruhe (TH), 1988.
\vskip1em}

\def\BBIvriiBB{\refno{\Ivrii} Ivrii, V. ``{\sl Weyl's Asymptotics for the
Laplace--Beltrami Operator in Riemann Polyhedra}''
Dokl. A. N. SSSR {\bf 38}, 35 -- 38.
\vskip1em}

\def\IvriiSigal{[IS]}
\def\BBIvriiSigalBB{
\item{\IvriiSigal } Ivrii, V. and Sigal, I. M. ``{\sl
Asymptotics of the Ground State Energies of Large Coulomb Systems
\/}''
To appear in Annals of Math.
\vskip1em}

\def\BBSigalnBB{
\refno{\Sigaln .} Sigal, I. M. (1982) ``{\sl
Geometric Methods in the Quantum Many--Body Problem. Nonexistence of Very
Negative Ions
\/}''
Comm. Math. Phys. 85, 309--324.
\vskip1em}

\def\BBRuskaiBB{
\refno{\Ruskai .} Ruskai, M.B. (1982) ``{\sl
Absence of Discrete Spectrum in Highly Negative Ions\/}'' {\bf I \& II}
Comm. Math. Phys. 82, 457---469 and 85, 325---327.
\vskip1em}

\def\BBSolovejhfBB{
\refno{\Solovejhf .} Solovej, J.P. (1991) ``{\sl
Proof of the Ionization Conjecture in a Reduced Hartree--Fock Model\/}''
Inv. Math. 104, 291--311.
\vskip1em}

\def\HKSW{[HKSW]}
\def\BBHKSWBB{
\item{\HKSW} Helffer, B., Knauf, A.,  Siedentop, H., Weikard, R.
``{\sl
On the Absence  of a First--Order Correction for the Number of
Bound States of a Schr\" odinger Operator with Coulomb Singularity
\/}''
To appear in ``{\it Comm. P.D.E.}''
\vskip1em}

\def\MarchPlaskett{[MP]}
\def\BBMarchPlaskettBB{
\item{\MarchPlaskett} March, N. H. and Plaskett, J. S.
Proc. Roy. Soc. A., {\bf 235}, 419.
\vskip1em}

\def\Hughes{[Hu]}
\def\BBHughesBB{
\item{\Hughes} Hughes, W. 
``{\sl
An Atomic Energy Lower Bound that Agrees with Scott's
Correction.
\/}''
Advances in Mathematics, {\bf 79}, 213--270, 1990.
\vskip1em}


\def\HertelLiebThirring{[HLT]}
\def\BBHertelLiebThirringBB{
\item{\HertelLiebThirring} Hertel, Lieb and Thirring
``{\sl
\/}''
\goodbreak\vskip1em}


\def\GrahamKolesnik{[GK]}
\def\BBGrahamKolesnikBB{
\item{\GrahamKolesnik} S. W. Graham and G. Kolesnik
``{\sl
Van der Corput's Method of Exponential Sums
\/}''
Cambridge University Press. London Math. Soc. Lecture Notes Series, 126.
\goodbreak\vskip1em}


\def\Gutzwiller{[Gu]}
\def\BBGutzwillerBB{
\item{\Gutzwiller} Gutzwiller, M.
``{\sl
Chaos in Classical and Quantum Mechanics
\/}''
Springer Verlag, 1990.
\goodbreak\vskip1em}


\def\fsca{[CFS1]}
\def\BBfscaBB{
\item{\fsca} C\'ordoba, A., Fefferman, C., Seco, L.
``{\sl
A Trigonometric Sum relevant to the Non--relativistic
Theory of Atoms
\/}''
To appear, P.N.A.S.
\goodbreak\vskip1em}

\def\fscb{[CFS2]}
\def\BBfscbBB{
\item{\fscb} C\'ordoba, A., Fefferman, C., Seco, L.
``{\sl
Weyl Sums and Atomic Energy Oscillations
\/}''
To appear, Revista Matem\'atica Iberoamericana.
\goodbreak\vskip1em}

\def\fsca{[FSC1]}
\def\fscb{[FSC2]}
\def\thma{1}
\def\thmb{2}
\def\eqa{(1)}
\centerline{\bigrmb
Number Theory, Classical Mechanics}
\vskip1em
\centerline{\bigrmb and the Theory of
Large Atoms}
\vskip2em
\centerline{\bf L. A. Seco}
\vskip2em

\def\PsiQ{
\sum_{l=1}^{l_{\rm TF}}
\>
{2l+1\over
\displaystyle
{1\over\pi}\int\bra{V_{\rm TF}^Z(r)-{l(l+1)\over r^2}}_+^{-\frac 1,2}
\>dr}\>
\mu\bra{{1\over\pi}\int\bra{V_{\rm TF}^Z(r)-{l(l+1)\over r^2}}_+^
{\frac 1,2}\>
dr}}
\def\mux{\dist(x,{\bf Z})^2-\fra 1,{12}}
\def\defVTF{$$V^1_{\rm TF}(r)=
{y(a\cdot r)\over r} \qquad a=\bra{3\pi\over 2}^{\frac 2,3}$$}
\def\TFeqn{
$$
\left.
\eqalign{
y''(r)&={y^{\frac 3,2}(r)\over r^{\frac 1,2}}\cr
y(0)&=1\cr
\lim_{r\to\infty}y(r)&=0\cr}
\right\}
$$         }

\def\HZN{\sum_{i=1}^N\bra{-\lapl_{x_i}-{Z\over |x_i|}}
+\repulsion}
\def\defEZ{$$E(Z)=\inf_{N\ge 0} E(Z,N)\qquad
E(Z,N)=\inf_{{\scriptstyle\phi\in {\cal H}}
\atop
{\scriptstyle\norm\phi=1}}\scapro {H_{Z,N}\phi},\phi$$}
\def\EZseries{-c_{\rm TF}Z^{\frac 7,3}+\fra 1,8 Z^2-c_sZ^{\frac 5,3}
+\bigo{Z^{\ffra 5,3-a}}\qquad a>0}

A non--relativistic atom of nuclear charge $Z$ fixed at the
origin, and $N$ quantized electrons at positions $x_i\in \reals 3$
is described by the Hamiltonian

$$H_{Z,N}=\HZN $$

which acts on the Hilbert space ${\cal H}$ of antisymmetric functions
in $L^2(\reals {3N})$.                     The ground state energy
of such a system is given by

\defEZ

As $Z$ goes to infinity, the energy $E(Z)$ admits an asymptotic
expansion of the form
$$E(Z)=\EZseries$$
The first term above was introduced by Thomas and Fermi in
\Thomas, \Fermi, and proved rigorously in {\LiebSimon}
(See also {\Liebtf} for a review of Thomas--Fermi theory).
The $Z^2$ term was discovered by Scott in {\Scott} and
proved to be true in a series of papers by Hughes--Siedentop--Weikard,
in {\Hughes}, {\SiedentopWeikard}, {\SiedentopWeikardLB} and
{\SiedentopWeikardLBII}. Its 
generalization to molecules was obtained by
Ivrii--Sigal ({\IvriiSigal}).
The $Z^{\frac 5,3}$ term was
obtained by Schwinger in {\Schwinger}, and proved to be
correct in {\FeffSeca}, {\FeffSecb}, {\FeffSecc}, {\FeffSecd},
{\FeffSece}, {\FeffSecf}, {\FeffSecg} and {\fs7}.
\vskip1em
In view of (11 --- 18), it is naturally conjectured
(see {\Feffnt}) that the next term in the energy asymptotics
for $E(Z)$ above is given by the following sum
$$\Psi_Q(Z)=\PsiQ$$
where $\mux$, $V^{Z}_{\rm TF}$ is the Thomas--Fermi potential
for an atom with charge $Z$ (see {\Liebtf}), which satisfies
the perfect scaling condition
$$V^Z_{\rm TF}(r)=Z^{\frac 4,3}V^1_{\rm TF}\bra{Z^{\frac 1,3}\cdot r}$$
where we have
\defVTF
and $y$ is the Thomas--Fermi function, solution of the
Thomas--Fermi equation
\TFeqn
Finally, $l_{\rm TF}$ is the greatest integer such that
$V_{\rm TF}^Z(r)-l(l+1)/r^2$ is positive somewhere.
\vskip1em
The book of Englert ({\Englert}; see also references thereof) contains
a discussion of oscillatory terms in the asymptotics of $E(Z)$.

\vskip1em
It was proved in {\fsca} and {\fscb} that this sum $\Psi_Q$ corresponds to
a sum of classical data of a certain classical hamiltonian, which
would then suggest that the expansion for $E(Z)$ is a trace
formula which one would expect from a path integral picture.

\vskip1em
The purpose of this paper is to describe the analysis
involved in understanding the sum $\Psi_Q(Z)$ as a
function of $Z$, which turns out
to be an adaptation of a well--known method
in analytic number theory developed mostly by
Van der Corput to understand the number of lattice points
in a large circle. We begin with a few remarks about analytic number theory.

\Title{Number Theory}
Consider sums of the form
$$S(\lambda)=\sum_{l=1}^{[\lambda]}f\bracket{{l\over\lambda}}
\cdot \mu\bra{\lambda\cdot\phi\bracket{{l\over\lambda}}}$$
where $\lambda$ is a large number, $\mu$ is a periodic function
with average 0, $f$ is an amplitude function
which can be viewed as constant
and $\phi$ is a smooth function which satisfies
the crucial non--degeneracy condition
$$|\phi''(x)|\ge c_0>0.$$
Particular cases of sums of this kind give rise to two
well--known problems in analytic number theory, namely

{\itemize
\item{1.} $f\equiv 1$, $\mu(x)=e^{2\pi i x}$, $\phi(x)=x^2$.
In this case, $S(\lambda)$, for $\lambda$ integer, corresponds
to the Gauss sums. The value of $S$ is then known explicitly, and
satisfies the estimate
$$S(\lambda)=\bigo{\lambda^{\frac 1,2}}$$

\item{2.} $f\equiv 1$, $\mu(x)=\dist(x,[x])-\oh$,
$\phi(x)=\sqrt{1-x^2}$. In this case, $S$ is related to
the error $E(\lambda)$
in the
lattice point problem for the circle in
$\reals 2$, which can is defined as follows:
take a large circle on $\reals 2$ of radius $\lambda$, and denote
by $N(\lambda)$ the number of lattice points in ${\bf Z}^2$ which
fall inside this circle. Then
$$E(\lambda)=N(\lambda)-\pi\lambda^2$$
and it is an old problem in number theory to prove that
$$E(\lambda)=\bigo{\lambda^\alpha}$$
for the best posssible value of $\alpha$.
It was observed very early, but Gauss and Dirichlet, that
one can take $\alpha=1$ which is an obvious geometric
fact, and is also obviously satisfied by
$S(\lambda)$. Different probabilistic approaches (as the one by
Cramer, for instance) indicate that
$\alpha $ above will not be smaller than $\oh$.
What follows is a {\it brief} historic overview
of the estimates for $\alpha$ (see {\GrahamKolesnik} for details).

\itemitem{$\alpha = $}1, Gauss--Dirichlet, 1849.
\itemitem{.}$\fra 2,3=0.666..$, Voronoi 1904, Hardy, 1917. 
\itemitem{.}$\fra 66,{100}=0.6600$, Van der Corput 1922.
\itemitem{.}$\fra 163,{247}=0.659919..$, Walfisz 1927.
\itemitem{.}$\fra 27,{41}=0.6585..$, Nieland--Van der Corput 1928.
\itemitem{.}$\fra 15,{23}=0.6521..$, Tichmarsh 1935.
\itemitem{.}$\fra 13,{20}=0.6500$, Loo Keng Hua 1942.
\itemitem{.}$\fra 24,{37}=0.6486..$, Kolesnik--Yin Wen Lin 1962.
\itemitem{.}$\fra 35,{54}=0.6481..$, Kolesnik 1971.
\itemitem{.}$\fra 278,{429}=0.648018..$, Kolesnik 1985.
\itemitem{.}$\fra 7,{11}=0.636636..$, Iwaniec--Mozzochi 1988.
Huxley 1992.


}
\vskip1em
Note now that the perfect scaling condition
of the Thomas--Fermi potential shows that our sum $\Psi_Q$
is (almost exactly) of the form $S(\lambda)$ as defined above,
where $\mu(x)=\dist(x,{\bf Z})^2-\fra 1,{12}$, $\lambda=Z^{\frac 1,3}$,
and 
$$\phi(x)=\int\bra{{y(r)\over r}-{x^2\over r^2}}_+^{\frac 1,2}
\>dr\eqno\eqa$$
The proof of the non--degeneracy condition for $\phi$ was
done in {\fs7}, and it has the peculiarity that its 
is a computer assisted proof. 

\vskip1em
A natural question then arises: what is the level of difficulty
in analysing the size of $\Psi_Q$?. Is is as simple as
the analysis of the gauss sums above? Or so hard as
the analysis of the lattice point problem?.

\vskip1em
A method devised by Van der Corput
(or at least, a variant of it), in his attempts to understand
the lattice point problem 
provides the answer: we compute our sum using Poisson summation,
and then we expand each Fourier integral using stationary phase.
In doing this, we end up with a sum in which $\mu$ is replaced
by its Fourier coefficients $\hat \mu(n)$. If they decrease fast
enough
(like $|n|^{-\frac 3,2}$, it so happens), 
the sum is bounded by $\lambda^{\frac 1,2}$.
In our case, $\mu(n)\sim |n|^{-2}$, therefore, after realizing that
the size of our amplitude function is $Z^{\frac 4,3}$, we can
conclude
that $\Psi\sim Z^{\ffra 4,3+\ffra 1,6}$.

\Title{Classical Mechanics}
The Van der Corput method does more than merely tell us the
size of $\Psi$. A mechanical analysis of the result
given by the stationary phase expansion shows that the
function $\Psi$ is (to leading order) a sum of classical data
associated to a certain classical hamiltonian, which displays
the relationship between classical and quantum mechanics
reminiscent of the Feynmann path integral representation
of Schr\"odinger operators.
\vskip1em
In order to explain our main result, consider a classical
particle with mass $\oh$, moving in the negative radial potential
given by $-V^Z_{\rm TF}$. Such motion is planar, and we
consider the closed orbits at energy
0 which arise for angular momentum $M$;
all such orbits for a fixed $M$ can be obtained by rotations
of each other.   On each of those orbits, the particle
``oscillates'' in the sense that the distance to the origin
$r(t)$ varies between $r_{\rm min}(M)$ and $r_{\rm max}(M)$
and closes after passing $n(M)$ times through, say, $r_{\rm max}(M)$.
We also consider the integer $l(M)$, the winding number of the
orbit around $0$, the action $S(M)$, the period $T(M)$, and the following
locally defined quantity: for a closed trajectory with angular momentum
$M$, and $n$ oscillations, and given $\ep$ small, consider
a trajectory (not necessarily closed) with angular momentum $M+\ep$
which, after $n(M)$ oscillations between succesive $r_{\rm max}(M+\ep)$,
misses to close by an angle $2\pi\alpha_M(\ep)$, where we take
$\alpha$ between 0 and 1. Then, we define
$$D(M)=\lim_{\ep \to 0}\ep^{-1}\alpha_M(\ep)$$
In this classical formalism, the
non--degeneracy condition $\phi''(x)\ne 0$
stated amounts
to the fact that $D(M)\ne 0$ for all closed trajectories.
This means that closed trajectories are isolated once we
factor out the trivial symmetry given by the rotation group.
Its role in the proof is similar to the non--vanishing curvature
of the sphere in the circle problem.


With this notation, our main result is as follows:
\theorem{\thma}
$$\Psi_Q(Z)=\Psi_0(Z)+\littleo{Z^{\frac 3,2}}$$
where
$$\Psi_0(Z)=2\pi \cdot Z^{\frac 3,2}
\cdot
\sum_{\ttower{\rm closed\ trajectories}{\rm at\  energy\  0}}
\delta\cdot
{\hat\mu(n)\over n}\cdot
{M\over T}
\cdot\abs{D(M)}^{-\frac 1,2}
\cdot e^{i\Bracket{Z^{\frac 1,3} S-\pi\cdot (l+\fra{\sign n},4)}}
$$
where $\delta=1$ except in the case of a perfect circular trajectory,
when we have $\delta=\oh$.
Furthermore, the sum is absolutely convergent.
\endt
We now state precisely the result mentioned in the previous section
for the size of $\Psi_Q(Z)$
\theorem{\thmb}
There are universal constants $K$ (large) and $\kappa_0$ (small but
strictly positive), such that
$$|\psi_Q(Z)|\le K\cdot Z^{\frac 3,2}
\qquad
\int_{Z_0}^{Z_0+Z}
\abs{\psi_Q(z)}^2\>{dz\over z^3}\ge \kappa_0\cdot Z$$
whenever $Z\ge K\,Z_0^{\frac 2,3}$.
Furthermore, 
$$\liminf_{\ttower{Z\to\infty}{Z=1,2,3,\ldots}}
\abs{Z^{-\frac 3,2}\psi_Q(Z)}\ne 0$$
\endt

The lower bound requires some extra work, and hinges on the fact
that a certain number
is not zero: the relevance of the  non--vanishing of this
number is analog to the non vanishing of the $L(\chi,1)$
in Dirichlet's theorem on
the number of primes  in arithmetic progressions.
\vskip1em
It is interesting to note that the size of the error
term above $\littleo{Z^{\frac 3,2}}$ depends on whether a certain number is
rational or not. More precisely, let $r_{\rm max}$ be the radius of
the circular closed trajectory corresponding to the maximal
angular momentum $M_{\rm max}$ at energy 0. Then, if
$$\sqrt{1-\oh r_{\rm max}\cdot M_{\rm max}}$$
is rational, then the error term can be seen to be
$\bigo{Z^{\ffra 3,2-\ep}}$. Otherwise, specially if this number
is very well approximated by rationals (say, a Louiville number), then
in general one cannot improve the {\it o}--result.
However, if in the sum over closed trajectories above one includes
also
the trajectories with complex period, then the error term
is always of size
$\bigo{Z^{\ffra 3,2-\ep}}$.
\vskip1em
The proofs of theorems {\thma} and {\thmb} can be found
in {\fscb}.
\vskip1em
The book of Gutzwiller {\Gutzwiller} contains a discussion
of the interplay between classical and quantum mechanics
in relation with trace formulas.
\vskip1em


\vskip1em
\title{Acknowledgements.}
This reasearch was supported by a N.A.T.O research grant
no. {CRG921184},
by NSERC grants no. OGP0121848
and {EQPEQ336}, by a CICYT grant and by a Connaught Fellowship.
\bibliography
%\def\Thomas{1}
%\def\Fermi{2}
%\def\LiebSimon{3}
%\def\Liebtf{4}
%\def\Scott{5}
%\def\Hughes{6}
%\def\SiedentopWeikard{7}
%\def\SiedentopWeikardLB{8}
%\def\SiedentopWeikardLBII{9}
%\def\Schwinger{10}
%\def\FeffSeca{11}
%\def\FeffSecb{12}
%\def\FeffSecc{13}
%\def\FeffSecd{14}
%\def\FeffSece{15}
%\def\FeffSecf{16}
%\def\FeffSecg{17}
%\def\fs7{18}
%\def\Feffnt{19}
%\def\Englert{20}
%\def\GrahamKolesnik{21}
%\def\Gutzwiller{22}
\BBEnglertBB
\BBFeffSecaBB 
\BBFeffSecbBB 
\BBFeffSeccBB 
\BBFeffSecdBB
\BBFeffSeceBB 
\BBFeffSecfBB 
\BBFeffSecgBB 
\BBfs7BB
\BBFeffntBB
\BBfscaBB
\BBfscbBB
\BBFermiBB 
\BBGrahamKolesnikBB
\BBGutzwillerBB
\BBHughesBB 
\BBLiebSimonBB 
\BBLiebtfBB 
\BBSchwingerBB 
\BBScottBB
\BBSiedentopWeikardBB  
\BBSiedentopWeikardLBBB
\BBSiedentopWeikardLBIIBB
\BBThomasBB  
\end