\magnification=\magstep1 \tolerance=10000 \hsize=16.0true cm \hoffset=.0true cm \vsize=22.true cm \voffset=.4true cm \baselineskip=15pt \hfuzz=2pt \parindent=0truept \def\vpush#1{\vphantom{ {#1} _{\lower5pt\hbox{\ }} ^{\lower-5pt\hbox{\ }} } #1 } \parskip=0pt \def\acci{\'\i} \def\Bigskip{\vskip4em} \def\lemma#1{\underbar{\bf{Lemma.}}\ {#1}\vskip 3em} \def\reals#1{{\bf R}^{#1}} \def\rationals{{\bf Q}} \def\norm#1{|\!|{#1}|\!|} \def\lnorm#1{\left|\!\left|{#1}\right|\!\right|} \def\parv{\par\vskip 1em} \def\lapl{\Delta} \def\repulsion{\oh\sum_{i\ne j}{1\over {|x_i-x_j|}}} \def\dzdr{\>{{dz\,dR}\over {R^5}}} \def\dr{{{dR}\over {R^5}}} \def\2int{\int\!\!\!\int} \font\BBigrm=cmr10 scaled \magstep4 \font\Bbigrm=cmr10 scaled \magstep3 \font\Bigrm=cmr10 scaled \magstep2 \def\sss{\scriptscriptstyle} \def\scapro#1,#2{\left<#1,#2\right>} \def\littlescapro#1,#2{\left<\right. #1,#2\left. \right>} \def\smallc{\scriptstyle c} \def\Fra#1,#2{{#1\over #2}} \def\frac#1,#2{{}^{#1}\!\!\kern1pt \lower-1.0pt\hbox{$\scriptscriptstyle /$} \!{}_{#2}} \def\fra#1,#2{\hbox{$ #1\over #2$}} \def\ffra#1,#2{\hbox{$ \scriptscriptstyle{{ {\lower-0.5pt\hbox{$\scriptscriptstyle #1$}}\over #2}}$}} \def\BBracket#1{\Biggl(#1\Biggr)} \def\Bracket#1{\Bigl(#1\Bigr)} \def\bracket#1{{\bigl(#1\bigr)}} \def\bbracket#1{\biggl(#1\biggr)} \def\bra#1{\left(#1\right)} \def\braces#1{\left\{#1\right\}} \def\vs#1,#2{#1_1,\ldots,#1_{#2}} \def\vsz#1,#2{#1_0,\ldots,#1_{#2}} \def\abs#1{\left|#1\right|} \def\muchbigger{>\!\!\!>} \def\muchsmaller{<\!\!\!<} \def\bover#1,#2{{\displaystyle{#1}\over{\displaystyle{#2}}}} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def\Vol#1{{\rm\ Vol\,}\bra{#1}} \def\sign{{\rm sign\ }} \def\for{\qquad{\rm\ for\ }\quad} \def\and{\qquad{\rm\ and\ }\qquad} \def\bigo#1{{\cal O}\!\bra{#1}} \def\littleo#1{{\it o}\!\bra{#1}} \def\eqbydef{\quad{\buildrel {\rm def} \over \equiv}\quad} \def\oh{\hbox{$ 1\over 2$}} \def\dxz{\,dx_1\,\cdots\,dx_Z} \def\brac#1{\left\{{#1}\right\}} \def\dist{{\rm dist}\,} \def\restr{\>\lceil\>} \def\domain{{\cal D}} \def\lipnorm#1{\lnorm{#1}_{\rm Lip}} \def\feffsec#1{ \centerline{\Bigrmb #1} \vskip3em \centerline{\bigrmb Charles L. Fefferman\footnote{*} {\rm Partially supported by a NSF grant at Princeton University}} \vskip1em \centerline{\it Department of Mathematics, Princeton University} \vskip2em \centerline{\bigrmb Luis A. Seco} \vskip1em \centerline{\it Department of Mathematics, California Institute of Technology} \vskip10em } \def\Gutzwiller{[G]} \def\BBGutzwillerBB{ \item{\Gutzwiller} Gutzwiller, M. ``{\sl Chaos in Classical and Quantum Mechanics \/}'' Springer Verlag. \goodbreak\vskip1em} \def\feffsectwo#1#2{ \centerline{\Bigrmb #1} \vskip1em \centerline{\Bigrmb #2} \vskip3em \centerline{\bigrmb Charles L. Fefferman\footnote{$ ^\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\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} \long\def\boxit#1{ \setbox0=\vbox{\kern3pt\strut #1\par\strut \kern3pt} %\wd0=0pt \ht0=0pt \setbox1=\hbox{\vrule\kern3pt\box0 \kern3pt\vrule} %\wd1=0pt \ht1=0pt \vbox{\parskip=0pt\hrule\box1\hrule\par} } \def\boxitt#1{ \setbox0=\hbox{\vrule\kern0pt \strut #1\kern0pt\vrule} \setbox1=\vbox{\hrule\box0\hrule} \box1 } \def\ttm#1,#2,#3,#4{\hbox{$\left(\matrix{#1\cr#3\cr}\right)$}} \def\sltr{{$SL(2,\reals{})$}} \def\sltz{\hbox{$SL(2,{\bf Z})$}} \def\trace{{\rm Tr\ }} \def\rightleftarrow{\leftrightarrow} \def\ep{\varepsilon} \def\torus{\hbox{\bf T}} \def\ttower#1#2{{\scriptstyle #1}\atop{\scriptstyle #2}} \def\thmstyleb{\vskip2em\goodbreak\noindent\bf } \def\thmstylea{\it} \def\intsend{\hskip1pt} \def\intsmid{\hskip1pt{,}\hskip1pt} \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} \def\itemize{\parindent=40pt\parskip=8pt} \def\smallitemize{\parindent=15pt\parskip=8pt} \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} \def\medtype{ \let\rm=\eightrm \let\bf=\eightbf \let \mus=\eightmus \let\tt=\eighttt \let\it=\eightit \let\sl\eightsl \baselineskip=3pt minus 0pt\rm} \font\BBigrm=cmr10 scaled \magstep4 \font\Bbigrm=cmr10 scaled \magstep3 \font\Bigrm=cmr10 scaled \magstep2 \font\bigrm=cmr10 scaled \magstep1 \font\smarm=cmr10 scaled \magstep0 \font\smfpt=cmr5 scaled \magstep0 \font\bfpt=cmr5 scaled \magstep1 \font\Bfpt=cmr5 scaled \magstep2 \font\bbfpt=cmr5 scaled \magstep3 \font\smept=cmr8 scaled \magstep0 \font\bept=cmr8 scaled \magstep1 \font\Bept=cmr8 scaled \magstep2 \font\bbept=cmr8 scaled \magstep3 \font\BBigrmb=cmbx10 scaled \magstep4 \font\Bbigrmb=cmbx10 scaled \magstep3 \font\Bigrmb=cmbx10 scaled \magstep2 \font\bigrmb=cmbx10 scaled \magstep1 \font\sigrmb=cmbx10 scaled \magstep0 \font\eightit=cmti8 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\eightsl=cmsl8 \font\eighttt=cmtt8 \def\refno{\item} \def\Lanford{[La]} \def\BBLanfordBB{ \item{\Lanford} Lanford O. ``{\sl A Computer--Assisted Proof of the Feigenbaum Conjecture \/}'' Bull. AMS 6, 427--34 (1986). \vskip1em} \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, 603--641. \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. ``{\sl On the Leading Energy Correction for the Statistical Model of the Atom: Interacting Case}'' Communications in Mathematical Physics {\bf 112} 471-490 (1987). \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 Phenomena 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\Lieb{[Li]} \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\FeffermanSeconII{[FS9]} \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\SecoSigalSolovej{[SSS]} \def\BBSecoSigalSolovejBB{ \item{\SecoSigalSolovej} Seco, L., Sigal, I. M., Solovej, J. P., ``{\sl Bound on the Ionization Energy of Large Atoms }'' Comm. Math. Phys., 131, 307 -- 315 (1990). \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 }'' Adv. Math. Vol 95 no. 2, 145 -- 305 (1992). \vskip1em} \def\FeffSecc{[FS3]} \def\BBFeffSeccBB{ \item{\FeffSecc} Fefferman, C. and Seco, L. ``{\sl The Eigenvalue Sum for a One--Dimensional Potential }'' Advances in Math, Vol. 108 no. 2, Oct 1994; 263 -- 335. \vskip1em} \def\FeffSecd{[FS4]} \def\BBFeffSecdBB{ \item{\FeffSecd} Fefferman, C. and Seco, L. ``{\sl The Density in a One-Dimensional Potential }'' Advances in Math., Vol 107 no. 2, Sep 1994, 187 -- 364. \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 }'' Advances in Math., Vol 107 no. 1, Aug 1994, 1 -- 185. \vskip1em} \def\fs7{[FS8]} \def\BBfs7BB{ \item{\fs7} Fefferman, C. and Seco, L. ``{\sl Aperiodicity of the Hamiltonian Flow in the Thomas--Fermi Potential}'' Revista Matem\'atica Iberoamericana, Vol 9 no. 3, 409 -- 551 (1993). \vskip1em} \def\Feffcpam{[F3]} \def\BBFeffcpamBB{ \item{\Feffcpam} Fefferman, C. ``{\sl The $N$--Body Problem in Quantum Mechanics. }'' Comm. Pure and App. Math., Vol. 39 no. S, S67--S110 (1986). \vskip1em} \def\Feffatom{[F2]} \def\BBFeffatomBB{ \item{\Feffatom} Fefferman, C. ``{\sl The Atomic and Molecular Structure of Matter }'' Revista Matem\'atica Iberoamericana, Vol. 1 no. 1, 1--44 (1985). \vskip1em} \def\Feffnt{[F1]} \def\BBFeffntBB{ \item{\Feffnt} Fefferman, C. ``{\sl Atoms and Analytic Number Theory }'' A.M.S. Centennial Publication Vol. II (1992), 27 -- 36. \vskip1em} \def\Arnold{[Ar]} \def\BBArnoldBB{ \item{\Arnold} Arnold, V. ``{\sl Mathematical Methods of Classical Mechanics }'' Graduate Texts in Math. no. 60, Springer (1978). \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\FalcoliniFeffermanLlave{[FFL]} \def\BBFalcoliniFeffermanLlaveBB{ \item{\FalcoliniFeffermanLlave} Falcolini, C., Fefferman, C. and Llave, R., {\it In preparation.} \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\/}'', 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 \/}'' Annals of Math. Vol. 138 no. 2, 243 -- 335 (1993) \vskip1em} \def\Sigaln{[Si]} \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\Ruskai{[Ru]} \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 {C}oulomb Singularity. \/}'' Comm. P.D.E., Vol 17, no. 3\&4, 615--639 (1992). \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\cfsa{[CFS1]} \def\BBcfsaBB{ \item{\cfsa} C\'ordoba, A., Fefferman, C., Seco, L., ``{\sl A Trigonometric Sum relevant to the Non--relativistic Theory of Atoms \/}'' Proc. Nat. Acad. Sci. USA, Vol. 91, 5776 -- 5778, June 1994. \goodbreak\vskip1em} \def\cfsb{[CFS2]} \def\BBcfsbBB{ \item{\cfsb} C\'ordoba, A., Fefferman, C., Seco, L., ``{\sl Weyl Sums and Atomic Energy Oscillations \/}'' Revista Matem\'atica Iberoamericana, Vol 11, no. 1. (1995); pp 167 -- 228. \goodbreak\vskip1em} \def\cfsc{[CFS3]} \def\BBcfscBB{ \item{\cfsc} C\'ordoba, A., Fefferman, C., Seco, L., ``{\sl A Number--Theoretic Estimate for the Thomas--Fermi Density \/}'' To appear. \goodbreak\vskip1em} \def\Gutzwiller{[G]} \def\BBGutzwillerBB{ \item{\Gutzwiller} Gutzwiller, M. ``{\sl Chaos in Classical and Quantum Mechanics \/}'' Springer Verlag, 1990. \goodbreak\vskip1em} \def\Liebna{[Li1]} \def\BBLiebnaBB{ \item{\Liebna} Lieb, E. H. ``{\sl Atomic and Molecular Negative Ions \/}'' Phys. Rev. Lett. 52, 315. \goodbreak\vskip1em} \def\Liebnb{[Li2]} \def\BBLiebnbBB{ \item{\Liebnb} Lieb, E. H. ``{\sl Bound on the Maximum Negative Ionization of Atoms and Molecules \/}'' Phys. Rev. A29, 3018--3028. \goodbreak\vskip1em} \def\Zhislin{[Zh]} \def\BBZhislinBB{ \item{\Zhislin} Zhislin, G. ``{\sl Discussion of the Spectrum of Schr\"odinger Operators for System of many Particles \/}'' Tr. Mosk. Mat. Obs. 9, 81--128. \goodbreak\vskip1em} \def\AizenmannLieb{[AL]} \def\BBAizenmannLiebBB{ \item{\AizenmannLieb} Aizenmann, M. and Lieb, E. ``{\sl Magnetic Properties of some Itinerant--Electron Systems at $T>0$ \/}'' Phys. Rev. Lett. 65, 1470 (1990). \goodbreak\vskip1em} \def\LiebMattis{[LM]} \def\BBLiebMattisBB{ \item{\LiebMattis} Lieb, E. and Mattis, D. ``{\sl Theory of Ferromagnetism and the Ordering of Electronic Energy Levels\/}'' Phys. Rev. 125, 164 (1962). \goodbreak\vskip1em} \def\Bach{[Ba]} \def\BBBachBB{ \item{\Bach} Bach, V. ``{\sl Accuracy of Mean Field Approximations for Atoms and Molecules \/}'' Comm. Math. Phys. 155 no. 2, 295 -- 310 (1993). \goodbreak\vskip1em} \def\GrafSolovej{[GS]} \def\BBGrafSolovejBB{ \item{\GrafSolovej} Graf, G. M. and Solovej, J. P. ``{\sl A Correlation Estimate with Applications to Quantum Systems with Coulomb Interactions \/}'' Reviews in Math. Phys. Vol 6, No 5a (1994), 977--997. \goodbreak\vskip1em} \def\CyconFroeseHinzSimon{[CFHS]} \def\BBCyconFroeseHinzSimonBB{ \item{\CyconFroeseHinzSimon} ? ``{\sl \/}'' \goodbreak\vskip1em} \def\Huxley{[Hx]} \def\BBHuxleyBB{ \item{\Huxley} M. N. Huxley ``{\sl Exponential Sums and Lattice Points \/}'' Proc. London Math. Soc. Vol 60 (1990), pp 471 -- 502. \goodbreak\vskip1em} \def\ChamizoIwaniec{[CI]} \def\BBChamizoIwaniecBB{ \item{\ChamizoIwaniec} F. Chamizo, H. Iwaniec ``{\sl On the Sphere Problem. \/}'' Rev. Mat. Iberoamericana, To appear. \goodbreak\vskip1em} \def\IwaniecMozzochi{[IM]} \def\BBIwaniecMozzochiBB{ \item{\IwaniecMozzochi} H. Iwaniec, C. J. Mozzochi ``{\sl On the Divisor and Circle Problems. \/}'' J. Number Theory, Vol 29 (1988) pp 60 -- 93. \goodbreak\vskip1em} \def\HeathBrown{[HB]} \def\BBHeathBrownBB{ \item{\HeathBrown} D. R. Heath--Brown ``{\sl The Distribution and Moments of the Error Term in the Dirichlet Divisor Problem \/}'' Acta Arith. Vol 60 (1992) pp 389 -- 415. \goodbreak\vskip1em} \def\Blehera{[B1]} \def\BBBleheraBB{ \item{\Blehera} Bleher, P. ``{\sl Distribution of Energy Levels of a Quantum Free Particle on a Surface of Revolution \/}'' To appear in Duke Math. Journal. \goodbreak\vskip1em} \def\Bleherb{[B2]} \def\BBBleherbBB{ \item{\Bleherb} P. Bleher ``{\sl On the Distribution of the Number of the Number of Lattice Points Inside a Family of Convex Ovals \/}'' Duke Math. Journal, Vol. 67, no 3, (1992), pp 461 -- 481. \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} \magnification=\magstep1 \headline={\ifnum\pageno>1 \hfil{\tenrm \titlerunning} \tenrm\ Page \folio\hfill\today \else\hfil\fi} \def\titlerunning{Weyl Sums and Atomic Energy Oscillations} \def\lema{lema} \def\lemb{lemb} \def\lemc{lemc} \def\lemd{lemd} \def\leme{leme} \def\lemf{lemf} \def\lemg{lemg} \def\lemh{lemh} \def\lemi{lemi} \def\lemj{lemj} \def\lemk{lemk} \def\leml{leml} \def\lemm{lemm} \def\lemn{lemn} \def\lemp{lemp} \def\lemq{lemq} \def\lemr{lemr} \def\lems{lems} \def\corola{corola} \def\corolb{corolb} \def\corolc{corolc} \def\thma{thma} \def\thmb{thmb} \def\thmc{thmc} \def\thmd{thmd} \def\thme{thme} \def\thmfs{thmfs} \def\eqa{eqa} \def\eqaa{\hbox{(\eqa a)}} \def\eqab{\hbox{(\eqa b)}} \def\eqac{\hbox{(\eqa c)}} \def\eqad{\hbox{(\eqa d)}} \def\eqb{(eqb)} \def\eqc{(eqc)} \def\eqd{(eqd)} \def\eqeaa{(eqeaa)} \def\eqea{(eqea)} \def\eqeb{(eqeb)} \def\eqf{(eqf)} \def\eqg{(eqg)} \def\eqh{eqh} \def\eqha{(\eqh a)} \def\eqhb{(\eqh b)} \def\eqhc{(\eqh c)} \def\eqi{(eqi)} \def\eqj{(eqj)} \def\eqk{(eqk)} \def\eql{eql} \def\eqla{(\eql a)} \def\eqlb{(\eql b)} \def\eqlc{(\eql c)} \def\eqm{eqm} \def\eqma{(\eqm a)} \def\eqmb{(\eqm b)} \def\eqn{(eqn)} \def\eqp{(eqp)} \def\eqq{(eqq)} \def\eqr{(eqr)} \def\eqs{eqs} \def\eqt{eqt} \def\eqta{(\eqt a)} \def\eqtb{(\eqt b)} \def\eqsa{(\eqs a)} \def\eqsb{(\eqs b)} \def\equ{(equ)} \def\eqv{(eqv)} \def\eqw{(eqw)} \def\eqx{(eqx)} \def\eqxa{(eqxa)} \def\eqxb{(eqxb)} \def\eqy{(eqy)} \def\eqz{(eqz)} \def\sda{(sda)} \def\sdb{(sdb)} \def\sdc{(sdc)} \def\sdd{(sdd)} \def\sde{(sde)} \def\one{(one)} \def\eqt{1} \def\eqxb{(2)} \def\lema{1} \def\eqa{3} \def\lemb{2} \def\eqh{4} \def\eqi{(5)} \def\lemr{3} \def\leme{4} \def\eqx{(6)} \def\eqf{(7)} \def\lemd{5} \def\thmb{6} \def\eqy{(8)} \def\eqg{(9)} \def\thmc{7} \def\thmfs{8} \def\lemg{9} \def\eqz{(10)} \def\lemc{10} \def\eqeaa{(11a)} \def\eqea{(11b)} \def\eqeb{(11c)} \def\lemh{11} \def\lemi{12} \def\eqd{(12)} \def\lemf{13} \def\eqj{(13)} \def\eqk{(14)} \def\eqc{(15)} \def\lemj{14} \def\sdd{(16)} \def\sdb{(17)} \def\sde{(18)} \def\lemk{15} \def\corola{16} \def\leml{17} \def\equ{(19)} \def\eqxa{(20)} \def\eqv{(21)} \def\thmd{18} \def\eqr{(22)} \def\eqs{23} \def\eqq{(24)} \def\eqp{(25)} \def\eqm{26} \def\eqn{(27)} \def\lemn{19} \def\lemm{20} \def\lems{21} \def\corolb{22} \def\corolc{23} \def\eqw{(28)} \def\thme{24} \def\lemq{25} \def\lemp{26} \def\one{(29)} \centerline{\bigrmb Weyl Sums} \vskip1em \centerline{\bigrmb and Atomic Energy Oscillations} \vskip3em \centerline{\bf Antonio C\'ordoba} \vskip1em \centerline{\it Departamento de Matem\'aticas} \centerline{\it Universidad Aut\'onoma de Madrid} \centerline{\it Cantoblanco, 28049 Madrid} \centerline{\it SPAIN} \vskip2em \centerline{\bf Charles L. Fefferman} \vskip1em \centerline{\it Department of Mathematics} \centerline{\it Princeton University} \centerline{\it Princeton NJ 08544} \centerline{\it USA} \vskip2em \centerline{\bf Luis A. Seco} \vskip1em \centerline{\it Department of Mathematics} \centerline{\it University of Toronto} \centerline{\it 100 St. George St} \centerline{\it Toronto ONTARIO M5S 1A1} \centerline{\it CANADA} \hfill \vtop{\hsize 200pt \obeylines \medtype \it \hfill``...cuando hay vino beben vino \hfill cuando no hay vino, agua fresca.'' \vskip10pt \hfill\bf A. Machado.} \vskip3em \centerline{\vbox{\hsize=10true cm {\bf Abstract:} We extend Van der Corput's method for exponential sums to study an oscillating term appearing in the quantum theory of large atoms. We obtain an interpretation in terms of classical dynamics and we produce sharp asymptotic upper and lower bounds for the oscillations. }} \vskip2em \vfill\eject \Title{Introduction.} The purpose of this paper is to study a certain sum that plays a crucial role in the asymptotic analysis of non--relativistic atomic energies. The sum is given by the expression $$\Psi_Q(Z) =\sum_{l=1}^{l_{\rm TF}} \> {2l+1\over \displaystyle {1\over\pi}\int\bra{V_{\rm TF}^Z(r)-{l(l+1)\over r^2}}_+^{-\frac 1,2} \>dr}\> \mu\bra{{1\over\pi}\int\bra{V_{\rm TF}^Z(r)-{l(l+1)\over r^2}}_+^ {\frac 1,2}\> dr} $$ where $\mu(x)=\dist(x,{\bf Z})^2-\fra 1,{12}$, $V_{\rm TF}^Z$ is the Thomas--Fermi potential with charge $Z$ (see {\Liebtf}), which satisfies the perfect scaling condition $$V^Z_{\rm TF}(r)=Z^{\frac 4,3}V^1_{\rm TF}\bra{Z^{\frac 1,3}\cdot r} \eqno{\eqta}$$ and we have $$V^1_{\rm TF}(r)={y(a\cdot r)\over r}, \qquad a=\bra{3\pi\over 2}^{\frac 2,3} \eqno\eqtb$$ and $y$ is the Thomas--Fermi function, solution of the Thomas--Fermi equation $$ \left. \eqalign{ y''(r)&={y^{\frac 3,2}(r)\over r^{\frac 1,2}}\cr y(0)&=1\cr \lim_{r\to\infty}y(r)&=0\cr} \right\} $$ and $l_{\rm TF}$ is the greatest integer such that $V^Z_{\rm TF}(r)-l(l+1)/r^2 $ is positive somewhere. Here, and throughout this article, we set $$(x)_+^{-\frac 1,2}=\cases{x^{-\frac 1,2}&if $x>0$\cr \center 0&if $x\le 0$\cr}$$ \vskip1em The role of the function $\Psi_Q(Z)$ in atomic physics is as follows: Consider a non--relativistic atom, consisting of a nucleus of charge $Z$ fixed at the origin, and $N$ quantized electrons at positions $x_i\in\reals{3}$. The hamiltonian of such a system is given by $$H_{Z,N}=\sum_{i=1}^N\bra{-\lapl_{x_i}-{Z\over |x_i|}} +\repulsion$$ acting on $$\psi\in{\cal H}= \bigwedge_{i=1}^NL^2\bra{\reals{3}\otimes {\bf Z}_2}$$ We define the energy of such an atom as $$E(Z)=\inf_{N\ge 0} E(Z,N),\qquad E(Z,N)=\inf_{{\scriptstyle\phi\in {\cal H}} \atop {\scriptstyle\norm\phi=1}}\scapro {H_{Z,N}\phi},\phi$$ The computation of $E(Z)$ can only be done explicitly for $Z=1$, when it equals $-\fra 1,4$. For $Z=2$ good upper and lower bounds are known, but the situation gets more and more complicated as $Z$ grows. It was observed very early in the history of quantum mechanics, in 1927 (see {\Thomas} and {\Fermi}), by Thomas and Fermi, that for $Z$ large, $E(Z)$ must approximately equal $c_{\rm TF}Z^{\frac 7,3}$ for $c_{\rm TF}$ a well known explicit constant. This was made rigorous by Lieb and Simon in 1973 ({\LiebSimon} and ({\Liebtf}), a very beautiful result which also holds for molecules. Comparisons with numerical results showed that the Thomas--Fermi approximation was only good up to a term of size $Z^2$, and Scott ({\Scott}) in 1950 was the first to realize that this $Z^2$ effect was due to electrons very near the nucleus, which behave as if they were in the exactly solvable model without electronic interaction. His argument was make rigorous in a series of papers by Hughes--Siedentop--Weikard ({\Hughes}, {\SiedentopWeikard}, {\SiedentopWeikardLB} and {\SiedentopWeikardLBII}) in 1985---89. This was proved to be true also for molecules by Ivrii--Sigal {\IvriiSigal}. A smaller effect, of size $Z^{\frac 5,3}$ was observed by Dirac, in 1930 ({\Dirac}), which comes from a delicate analysis of electronic correlations. Additional effects were also found by Scott ({\Scott}), corrected by March and Plaskett {\MarchPlaskett}, and then finally established by Schwinger ({\Schwinger)}, who argued that the asymptotic energy expansion should then contain the term $c_{DS}Z^{\frac 5,3}$, for $c_{DS}$ an explicit constant. The proof of Schwinger's result was announced in {\FeffSeca}, and is as follows: $$E(Z)=c_{\rm TF}\,Z^{\frac 7,3}+\fra 1,8\,Z^2+c_{DS}\,Z^{\frac 5,3} +\bigo{Z^{\ffra 5,3-a}},\qquad a>0.$$ Its complete proof appears in {\FeffSecb}, {\FeffSecc}, {\FeffSecd}, {\FeffSece}, {\FeffSecf}, {\FeffSecg} and {\fs7}. It has been known for some time that nice asymptotics for atomic energies in powers of $Z^{\frac 1,3}$ will stop after the Dirac--Schwinger term. This can most easily be conjectured by looking at simpler, exactly solvable models such as the harmonic oscillator (see {\Simon}). Comparisons with numerical results also show that the next correction will be oscillatory in nature. We refer the reader to the book of Englert ({\Englert}; see also {\EnglertSchwingera} and {\EnglertSchwingerb}) for a physical discussion of the energy asymptotics up to including oscillatory terms. The exact form of the function $\Psi_Q$ above originates from the proof of the Dirac--Schwinger's term in {\FeffSeca}, where it is seen that $$E(Z)=c_{\rm TF}\,Z^{\frac 7,3}+\fra 1,8\,Z^2+C_{DS}\,Z^{\frac 5,3} +\Psi_Q(Z) +\bigo{Z^{\ffra 5,3-a}},\qquad a>0\eqno\eqxb$$ although the current estimates for $a$ above do not yet guarantee that $\Psi_Q$ really dominates over the ${\cal O}$--term. \vskip1em Note that in establishing {\eqxb} not only do we need estimates for the error terms with $a$ large enough, but also we need lower bounds for the size of the function $\Psi_Q$, which are not completely obvious. It follows from our results in the present article that one would need $a>\fra 1,6$ in order to show that $\Psi_Q$ dominates over the error terms contained in the ${\cal O}$--term. \vskip1em From the abstract mathematical perspective, sums such as $\Psi_Q$ are quite old, the best known going back to Gauss, which is related to estimating the number of integral lattice points inside a convex curve: most notably, a circle, which gave rise to the {\it circle} problem, and a hyperbola, which comes from the {\it divisor} problem, two of the most elusive problems in analytic number theory (see {\GrahamKolesnik} for a general description; {\IwaniecMozzochi} and {\Huxley} for the latest results). It is worth noting the close similarity between our problem and the circle problem, which comes from a refined analysis of the number of bound states of quantum free particles in a box. A step higher in sophistication, but still within the same realm of problems, is the Selberg trace formula, which, very loosely speaking, expresses spectral information about the laplacian on an abstract manifold in terms of the closed geodesics on that manifold, which can also be seen as the mathematical version of the Feynmann Path integrals for abstract systems. We refer the reader to {\Gutzwiller} and references thereof for a wealth of ideas in the theory of trace formulas, quantum chaos, classical mechanics, and all that. \vskip1em Our work is organized as follows: First, after making some trivial modifications to the well known stationary phase lemma (Section~ 1), we set out (in Section~2) to study sums of the type $$S(\lambda)=\sum_{l=1}^\lambda f\bra{{l\over\lambda}}\mu\bra{{ \lambda\cdot\phi\bbracket{{l\over\lambda}}}}$$ where $\phi''(x)\ge c_0>0$, and $\mu$ is a periodic function of average 0. Examples of such sums are {\itemize \item{1.} If $f\equiv 1$, $\mu(x)=e^{2\pi ix}$, $\phi(x)=x^2$, we have the well--known Gauss sums modulo $\lambda$. \item{2.} If $f\equiv 1$, $\mu(x)=x-[x]-\oh$, then $S$ represents the error term in the lattice point problem for a curve $\phi$ dilated by $\lambda$. } \vskip1em While the first item above is well understood, the second remains very hard. In our analysis, we will have to deal only with functions $\mu$ whose Fourier coefficients decrease rather rapidly ($\hat\mu(n)\sim |n|^{-\frac 3,2+\ep}$), and this allows a complete analysis of the sums via the usual method of Van der Corput (Poisson summation followed by stationary phase; see {\GrahamKolesnik}), since all expressions turn out to be absolutely convergent in this case. A little elementary number theory will be needed here to rule out the possibility of a small denominator problem, which gives rise to an error term whose size depends on whether a certain number is rational or irrational. \vskip1em In Section~3, we apply the results of Section~2 to $\Psi_Q$, obtaining a new sum $\Psi_0$, a leading ``dual'' version of $\Psi_Q$, reminiscent of the Jacobi identity for the modular function. Sharp upper bounds for $\Psi_Q$ are an easy consequence of this. However, obtaining the right regularity properties for the curve and amplitude involved in the formula for $\Psi_Q$ turns out to be rather tedious. \vskip1em In Section~4 we obtain {\it lower} bounds for $\Psi_Q$ in the form of an $\Omega$--result, by understanding how $\Psi_0$ behaves on average. \vskip1em In Section~5 we use the {\it dual} expression $\Psi_0$ to give us a dynamical interpretation of the sum $\Psi_Q$ as a sum of classical data extended over all closed trajectories of a classical hamiltonian. This result appears to have similarities also with a recent result of Bleher {\Blehera}. \vskip1em Section~6 is devoted to side issues. \Title{1. Stationary Phase Estimates} We begin with a review of stationary phase. Consider $f\in C^\infty_0(\reals{})$. Then, if $t>0$, $$\eqalign{ \int_{-\infty}^\infty e^{itx^2}f(x)\,dx&= e^{\ffra \pi i,4}\sqrt{{\pi\over t}}\int _{-\infty}^\infty e^{-\pi^2 i\xi^2/t} \hat f(\xi)\,d\xi\cr}$$ Using the identity $$e^s=1+\int_0^1e^{su}s\,du$$ we deduce $$\eqalign{ \int_{-\infty}^\infty e^{itx^2}f(x)\,dx&= e^{\ffra \pi i,4}\sqrt{{\pi\over t}}\bra{f(0)+ \int_{-\infty}^\infty \hat f(\xi) \int_0^1 \bra{{-\pi^2 i\xi^2\over t}} e^{-\pi^2 iu\xi^2/t}\,du \,d\xi}\cr &= e^{\ffra \pi i,4}\sqrt{{\pi\over t}}\bra{f(0)+ {i\over 4t} \int_{-\infty}^\infty \widehat {f''}(\xi) \int_0^1 e^{-\pi^2 iu\xi^2/t}\,du \,d\xi}\cr &= e^{\ffra \pi i,4}\sqrt{{\pi\over t}}f(0)+ {i\over 4t} \int_{-\infty}^\infty {f''}(x) \int_0^1 e^{itx^2/u}\,{du\over u^{\frac 1,2}} \,dx\cr &= e^{\ffra \pi i,4}\sqrt{{\pi\over t}}f(0)+ {i\over 4t^{\frac 3,2}} \int_{-\infty}^\infty {f''}(x) g_t(x) \,dx \cr}$$ for $$ g_t(x)= \int_0^1 e^{itx^2/u}\bra{t\over u}^{\frac 1,2} \,du$$ Note that $g_t(x)=t^{\frac 1,2}g_1\bra{t^{\frac 1,2}x}$, and $$\eqalign{ g_1(x)&=\left.i\,e^{ix^2/u}{u^{\frac 3,2}\over x^2}\right|_0^1 -\Fra 3i,2\int_0^1e^{ix^2/u}{u^{\frac 1,2}\over x^2}\,du\cr }$$ hence $|g_1(x)|\le \fra 2,{|x|^2}$ and thus, $g_1$ is integrable. Furthermore $$\lnorm{g_t}_1=\lnorm{g_1}_1=\bigo{1}$$ and $|g_t(x)|\le 2 |x|^{-2}t^{-\frac 1,2}$. \vskip1em We also consider one--sided integrals of the form $$ \int_{0}^\infty e^{itx^2}f(x)\,dx $$ Define $$f^+(x)=\cases{f(x)&if $x\ge 0$\cr f(-x)&if $x\le 0$\cr}$$ and consider $f_\epsilon=f^+\ast\varphi_\epsilon$, for a suitable approximation to the identity $\varphi_\epsilon$. Using our previous identity, we obtain $$\eqalignno{ \int_0^\infty e^{itx^2}f(x)\,dx&= \oh \lim_{\epsilon\to 0}\int_{-\infty}^\infty e^{itx^2}f_\epsilon(x)\,dx\cr &= \oh e^{\ffra \pi i,4}\sqrt{{\pi\over t}}f(0)+ {i\over 8t^{\frac 3,2}} \lim_{\epsilon\to 0} \int_{-\infty}^\infty {f_\epsilon''}(x) g_t(x) \,dx\cr &= \oh e^{\ffra \pi i,4}\sqrt{{\pi\over t}}f(0)+ {i\over 4t^{\frac 3,2}} \int_{0}^\infty {f''}(x) g_t(x) \,dx \cr}$$ The last step follows since $g_t$ is integrable and both functions $g_t$ and $f^+$ are even. \title{Definition:} Let $\phi$ such that $$|\phi^{(n)}(x)|\le C_n,\quad (0\le n\le 5),\qquad \qquad c_0=\inf|\phi''(x)|>0$$ for $x$ in a certain interval which will be clear in our applications. We denote by $$c_0^+=\min(1,c_0), \qquad B(\phi)=\bra{{1+\norm\phi_{C^5}\over c_0^+}}^{54}.$$ \lem{\lema--(Stationary Phase Lemma)} Let $f(x)\in C^2_0(\reals{})$, such that $$ |f'(x)|\le\cases{ N_1&if $|x|\le L$\cr N_2&if $|x|> L$\cr}\qquad (N_1\le N_2)$$ and $$ |f''(x)|\le\cases{ M_1&if $|x|\le L$\cr M_2&if $|x|> L$\cr}\qquad (M_1\le M_2)$$ and let $\phi(x)$ such that $\phi(0) =\phi'(0)=0$, $\abs{\phi^{(n)}(x)}\le C_n$ for $0\le n\le 5$, and $\phi''(x)\ge c_0>0$ for all $x$ in the support of $f$. Then $$\displaylines{ \quad\abs{ \int_{-\infty}^\infty e^{it\phi(x)}f(x)\,dx- \bra{{2\pi\over |t|\phi''(0)}}^{\frac 1,2}e^{\sign(t)\,\fra \pi i,4}f(0)} \hfill\cr \hfill \le A\,B(\phi)\,t^{-\frac 3,2}\bra{\norm f_\infty+ N_1+ {N_2\over t^{\frac 1,2}L}+ M_1+ {M_2\over t^{\frac 1,2}L}} \qquad\qquad\llap\eqaa\cr}$$ Similarly, $$\displaylines{ \quad\abs{ \int_{0}^\infty e^{it\phi(x)}f(x)\,dx- \bra{{\pi\over 2|t|\phi''(0)}}^{\frac 1,2}e^{\sign(t)\,\fra \pi i,4}f(0)} \hfill\cr \hfill \le A\,B(\phi)\,t^{-\frac 3,2}\bra{\norm f_\infty+ \,N_1+ \,{N_2\over t^{\frac 1,2}L}+ M_1+ {M_2\over t^{\frac 1,2}L}} \qquad\qquad\llap\eqab\cr}$$ We also have the usual $L$--independent estimates $$\displaylines{ \quad\abs{ \int_{-\infty}^\infty e^{it\phi(x)}f(x)\,dx- \bra{{2\pi\over |t|\phi''(0)}}^{\frac 1,2}e^{\sign(t)\,\fra \pi i,4}f(0)} \hfill\cr \hfill \le A\,B(\phi)\,t^{-\frac 3,2}\bra{\norm f_\infty+ \lnorm{f'}_1+\lnorm{ f''}_1} \qquad\qquad \llap\eqac\cr} $$ and $$\displaylines{ \quad\abs{ \int_{0}^\infty e^{it\phi(x)}f(x)\,dx- \bra{{\pi\over 2|t|\phi''(0)}}^{\frac 1,2}e^{\sign(t)\,\fra \pi i,4}f(0)} \hfill\cr \hfill \le A\,B(\phi)\,t^{-\frac 3,2}\bra{\norm f_\infty+ \lnorm{f'}_1+\lnorm{ f''}_1}\qquad\qquad \llap\eqad\cr} $$ where $A$ is a universal constant, and $B(\phi) $ is as defined above for $x$ in the support of $f$. Here, $\sign(t)$ stands for the function which equals $1$ if $t>0$ and $-1$ if $t<0$. \proof It will obviously be enough to consider the case $t>0$. Consider the change of variables given by $$u(x)=x\,\sqrt{{\phi(x)\over x^2}}$$ and its inverse $z(u)$. We begin by obtaining regularity properties of $u$ and $z$. \vskip1em Let $k\ge 1$. In what follows, $A_k$ will denote a collection of universal constants depending only on $k$. First, we consider $|x|\le 1$ and define $$\phi_1(x)=x^{-2}\,\phi(x).$$ Since $$\phi_1(x)=\int_0^1\int_0^1\phi''(s\,t\,x)\,s\,dt\,ds$$ we have $$\norm{\phi_1}_{C^k}\le A_k\,\norm{\phi}_{C^{k+2}}.$$ Next, define $$\phi_2(x)=\sqrt{\phi_1(x)}, \qquad |x|\le 1$$ and note that $${d^k\phi_2(x)\over dx^k}={\displaystyle \sum_{\ttower{1\cdot i_1+\cdots+p\cdot i_p=k} {i_j\ge 0}} c_{i_1,\ldots,i_p}^{(k)} \>\bra{\phi_1'(x)}^{i_1}\cdot \cdots \cdot\bra{\phi_1^{(p)}(x)}^{i_p} \over \phi_1^{k-\ffra 1,2}(x)}$$ which can easily be checked by induction. As a result, we have $$\norm{\phi_2}_{C^k} \le A_k\,{\bra{1+\norm{\phi_1}_{C^k}}^k \over c_0^{k-\ffra 1,2}} \le A_k\,{\bra{1+\norm{\phi}_{C^{k+2}}}^k \over c_0^{k-\ffra 1,2}} $$ where we have used the fact that $\phi_1(x)\ge \oh\,c_0$. Therefore, since $u(x)=x\cdot\phi_2(x)$, we conclude that $$\abs{{d^ku(x)\over dx^k}} \le A_k\,{\bra{1+\norm{\phi}_{C^{k+2}}}^k \over {c^+_0}^{k-\ffra 1,2}}, \qquad {\rm when\ }|x|\le 1.$$ When $|x|>1$ we obviously have that $${d^ku(x)\over dx^k}=\sum_{i=1}^k\phi^{\ffra 1,2-i}(x) \sum_{\ttower{1\cdot i_1+\cdots+p\cdot i_p=k} {i_j\ge 0}} c^{i,k}_{i_1,\ldots,i_p}\, \Bracket{\phi'(x)}^{i_1}\cdot \cdots\cdot \Bracket{\phi^{(p)}(x)}^{i_1}, \qquad |x|>1$$ hence $$|u^{(k)}(x)| \le A_k\, {\bra{1+\norm{\phi}_{C^{k}}}^k \over {c^+_0}^{k-\ffra 1,2}}, \qquad \quad |x|>1$$ so altogether we obtain $$\norm{u}_{C^k} \le A_k\,{\bra{1+\norm{\phi}_{C^{k+2}}}^k \over {c^+_0}^{k-\ffra 1,2}}.$$ Finally, since $$\displaylines{ \quad z'\bracket{u(x)}\cdot u'(x)=1\hfill\cr \quad {d^kz\over du^k}\bracket{u(x)}\cdot\bra{u'(x)}^k =\sum_{p=1}^{k-1} \,K_{k,p}\,{d^pz\over du^p}\bracket{u(x)}\hfill\cr \hfill\cdot \sum_{\ttower{1\cdot i_1+\cdots+q\cdot i_q=k+1-p}{i_j\ge 0}} c^{k,p}_{i_2,\ldots,i_q} \>\Bracket{u'(x)}^{i_1}\cdot\cdots\cdot \Bracket{u^{(q)}(x)}^{i_q}, \qquad k\ge 2\cr }$$ and $$|u'(x)|=\Fra 1,2\abs{\phi'(x)\over\sqrt{\phi(x)}} \ge c,\qquad c \eqbydef{c_0\over \sqrt{C_2/2}}$$ we obtain by induction that $$\abs{d^kz(u)\over du^k}\le A_k\,\bra{1+C_2\over c^+_0}^{k^2}\,\bra{1+\norm u_{C^k}}^{k^2}.$$ With our previous estimate for $\norm u_{C^k}$ we then conclude that $$\abs{d^kz(u)\over du^k}\le A_k\,\bra{{1+\norm \phi_{C^{k+2}}\over c^+_0}}^{2k^3}.$$ Then $$\int_{-\infty}^\infty e^{it\phi(x)}f(x)\,dx= \int_{-\infty}^\infty e^{itu^2}\tilde f(u)\,du$$ for $$\tilde f(u)=f(z(u))\cdot z'(u)$$ Note that $\tilde f(0)=f(0)\cdot \sqrt{{2\over \phi''(0)}}$. Since $z'(u)\le c^{-1}$, $$\abs{f'(z(u))}\le\cases{ N_1&if $|u|\le c L$\cr N_2&otherwise\cr} , \qquad \abs{f''(z(u))}\le\cases{ M_1&if $|u|\le c L$\cr M_2&otherwise\cr} . $$ As a result, using stationary phase, we arrive at $$\abs{ \int_{-\infty}^\infty e^{it\phi(x)}\,f(x)\,dx- \bra{{2\pi\over t\phi''(0)}}^{\frac 1,2}\,e^{\ffra \pi i,4}\,f(0)} \le t^{-\frac 3,2}\bra{\abs{I_1}+\abs{I_2}+\abs{I_3}} $$ for $$\eqalignno{ I_1&=\int g_t(u)\,f(z(u))\,z'''(u)\,du\cr I_2&=\int g_t(u)\,f'(z(u))\,z'(u)\,z''(u)\,du\cr I_3&=\int g_t(u)\,f''(z(u))\,\bra{z'(u)}^3\,du.\cr }$$ Now, $$\eqalignno{ \abs{I_1}&\le \norm{z}_{C^3}\,\lnorm f_\infty\int\abs{g_t(u)}\,du \le A\cdot\bra{{1+\norm\phi_{C^5}\over c^+_0}}^{54}\cdot \lnorm f_\infty\cr}$$ Next, $$\eqalignno{ \abs{I_2}&\le A\,\bra{{C_0\over \sqrt{c_0}}}\cdot\bra{1+\norm\phi_{C^4}\over c^+_0}^{16} \cdot\bra{ N_1\int_{|u|\le cL}\abs{ g_t(u)}\,du +2\,N_2\int_{|u|\ge cL} t^{-\frac 1,2}u^{-2}\,du}\cr &\le A\, \bra{1+\norm\phi_{C^5}\over c^+_0}^{54} \cdot\bra{ N_1\lnorm{g_1}_1 +{2\,N_2\over t^{\frac 1,2} L}}\cr}$$ Finally, $$\eqalignno{ \abs{I_3}&\le M_1c^{-3}\int_{|u|\le cL} \abs{g_t(x)}\,dx +2M_2c^{-3}\int_{|u|\ge cL} t^{-\frac 1,2}u^{-2}\,du\cr &\le M_1c^{-3}\lnorm{g_1}_1+{2M_2\over c^3 t^{\frac 1,2} L}\cr}$$ which proves the first claim in our lemma. The one--sided integral is estimated in the same way. The $L$--independent estimates are obtained in a similar manner, except that integrals $I_2$ and $I_3$ in this case are estimated directly by $$ \abs{I_2}\le \bra{{C_0\over \sqrt{c_0}}}\cdot\bra{1+\norm\phi_{C^4}\over c^+_0}^{16} \lnorm{f'}_\infty \lnorm{g_t}_1,\qquad \abs{I_3}\le c^{-3}\lnorm{f''}_\infty \lnorm{g_t}_1. $$ The one--sided estimate in this case is also analogous. \endpf This lemma will be complemented with the following trivial results: \lem{\lemb} Let $f\in C^2_0(\reals{})$, and $\phi$ such that $|\phi'(x)|\ge d$ for all $x$ in the support of $f$. Then $$\eqalignno{ \abs{\int_{\reals{}}e^{it\phi(x)}\,f(x)\,dx}&\le t^{-1}\bra{{\lnorm {f'}_1\over d}+ {\lnorm {f\cdot \phi''}_1\over d^{2}} }&\eqha\cr \abs{\int_{\reals{}}e^{it\phi(x)}\,f(x)\,dx}&\le t^{-1}\bra{\lnorm {f\cdot\phi''}_1\over d^{2}}+ 4\,t^{-2}\bra{{\lnorm {f''}_1\over d^{2}} +{\lnorm{f' \cdot \phi''}_1\over d^3}}&\eqhb\cr \abs{\int_{\reals{}}e^{it\phi(x)}\,f(x)\,dx}&\le 10\,t^{-2}\biggl({\lnorm {f''}_1\over d^{2}} +{\lnorm{f'\cdot \phi''}_1\over d^3} +{\lnorm{f\cdot \phi'''}_1\over d^3} +{\lnorm{f'\cdot(\phi'')^2}_1\over d^4} \biggr)&\eqhc\cr }$$ \proof Integration by parts yields $$\eqalignno{ {\int_{\reals{}}e^{it\phi(x)}\,f(x)\,dx}& = -{1\over it}\int_{\reals{}}e^{it\phi(x)}{d\over dx}\bra{{f(x) \over \phi'(x)}}\,dx\cr &= {1\over it}\int_{\reals{}}e^{it\phi(x)}\bra{{f'(x) \over \phi'(x)}}\,dx -{1\over it}\int_{\reals{}}e^{it\phi(x)}\bra{{f(x)\,\phi''(x) \over {\phi'(x)}^2}}\,dx\qquad\qquad\hfill&\eqi\cr}$$ This yields {\eqha}. For {\eqhb} we perform another integration by parts to the first integral above, which equals $${1\over t^2}\int_{\reals{}}e^{it\phi(x)}\bra{{f''(x) \over \bra{\phi'(x)}^2}}\,dx +{2\over t^2}\int_{\reals{}}e^{it\phi(x)}\bra{{f'(x)\,\phi''(x) \over \bra{\phi'(x)}^3}}\,dx$$ which yields {\eqhb}. For {\eqhc}, we integrate by parts also the last integral in {\eqi}, which gives $$-t^{-2}\int\bra{{f'(x)\,\phi''(x)\over\phi'(x)^3} +{f(x)\,\phi'''(x)\over \phi'(x)^3} -3\,{f(x)\,{\phi''(x)}^2\over\phi'(x)^4} }\,e^{it\phi(x)}\,dx$$ as needed. \endpf \lem{\lemr} Let $f\in C_0^2(\ooint a,b)$ and $\phi$ such that $\phi''(x)\ge c_0>0$, and $\phi'(x)\ne 0$ for $x\in\ccint a,b$. Then, $$\abs{\int_a^be^{it\phi(x)}\,f(x)\,dx} \le t^{-1}\cdot|b-a|\cdot\bra{ {\norm{f''}_\infty\over c_0}+ {\norm{f}_\infty\cdot\norm{\phi''}_\infty\over c_0^2} }$$ \endt \title{Remark:} The point in this result is that the estimate is independent of $\inf|\phi'(x)|$. \proof $f$ vanishes at $a$ at order 2, which implies $$|f(x)|\le \norm {f''}_\infty\cdot|x-a|^2,\qquad |f'(x)|\le \norm {f''}_\infty\cdot|x-a|.$$ So, $$\phi'(x)\ge c_0\cdot(x-a).$$ The lemma follows trivially by integration by parts, since $${\int_a^be^{it\phi(x)}\,f(x)\,dx} ={i\over t}\int_a^b\bra{{f'(x)\over\phi'(x)} -{f(x)\cdot\phi''(x)\over {\phi'(x)}^2}} e^{it\phi(x)}\,dx.\eqno\mathendpf$$ The following is a trivial variant of the usual Van der Corput lemmas. \lem{\leme} Let $f$ be differentiable in $\ccint a,b$, and $\phi$ such that $\phi''(x)\ge c_0>0$ for $x\in\ccint a,b$ Then, $$ \abs{\int_a^b e^{it\phi(x)}\,f(x)\,dx} \le 8\, t^{-\frac 1,2} c_0^{-\frac 1,2}\bra{1+\norm f_\infty+ \lnorm{f'}_1} $$ \proof Let $R=t^{-\frac 1,2}c_0^{\frac 1,2}$, and consider $$E_1=\{x\>:\> |\phi'(x)|\ge R \},\qquad E_2=\{x\>:\> |\phi'(x)|0$: of course, the same argument will work if we assumed $\phi''(x)>c_0$, with only a few signs being flipped, but we choose this sign in our non--degeneracy condition because it is exactly the one satisfied by the function $\phi$ in our application to the sum $\Psi_Q(Z)$. We also assume the bounds $$\abs{\phi^{(n)}(x)}\le C_n,\qquad 0\le n\le 5, \qquad{\rm for\ }x\in\ccint a,b$$ where $|b-a|$ is bounded by a universal constant, and define $$S(\lambda)=\sum_{l\in({\bf Z}+\gamma)\cap\ccint a\cdot\lambda,{b\cdot\lambda}} f\bra{l\over\lambda}\mu\bra{\lambda\phi({l\over \lambda})}$$ where $\gamma$ is a real number. \vskip1em In our applications, we will be concerned with the following two situations: on the one hand, we will have functions $f$ and $\phi$ independent of $\lambda$; this simplifies some estimates, but the amplitude function $f$ does not vanish at the endpoint $b$, which gives origin to a certain diophantine analysis of the phase $\phi$. On the other hand, we will have to deal with functions $f$ and $\phi$ which depend on $\lambda$, which will force us to keep track of error terms in a careful way: furthermore, there is no obvious multiscale analysis in the problem and we thus have to analyze blow up manually. However, in this case the amplitude function is supported inside $\ccint a,b$ which avoids diophantine discussions. \vskip1em We summarize both cases as follows. \title{Case I:} $f\in C^\infty_0(\ocint a,b)$. In this case, we shall impose that the bounds satisfied by $\phi$ and $f$ are universal, i.e, independent of $\lambda$. The obvious singularity in the sum appearing around $l=b\lambda$ will give rise to a purely arithmetic behavior of the sum. \vskip1em \title{Case II:} $f\in C^\infty_0(\ooint a,b)$. In this case, the functions $\phi$ and $f$ will depend on $\lambda$ in the sense that the bounds satisfied by $\phi$ will grow (slowly) as a function of $\lambda$. We will thus keep track carefully of the dependence of our error bounds in terms of the regularity assumptions of $f$ and $\phi$. The absence of singularities in this case will make the study of the sums purely analytical. \vskip1em We wish to understand the behavior of $S(\lambda)$ for large $\lambda$ in both cases. \Title{Case I} As mentioned above, $f$ and $\phi$ will satisfy universal bounds for its derivatives of the type $$\lnorm{\phi'(x)}_{C^5}\le C,\qquad -\phi''(x)\ge c_0, \qquad \lnorm f_{C^2}\le C$$ for constants $C$ and $c_0$ independent of $\lambda$. As a consequence, we will not keep track of the dependence of constants on the regularity properties of either $f$ or $\phi$, and the constant $C$ will be ubiquitously used to denote a universal constant depending on the regularity properties of $f$ and $\phi$ as stated above. Another constant will play a role, though, which is $\phi'(b)$ in the case that it is a rational number $\fra p,q$: in this case, some constants will depend on $q$, and this dependence will be made explicit. \vskip1em Let $\varphi(x)$ supported on $\ooint -\infty,b$, identically equal to 1 on $\ccint a,{b- \lambda^{-\ffra 1,2-\epsilon}}$, for $$\epsilon=\fra 1,{20}$$ $\varphi$ as smooth as possible. We denote by $$I_\lambda=\ccint{b-\lambda^{-\ffra 1,2-\epsilon}},b$$ the set where $\varphi'$ is supported. It is clear that $$S(\lambda)=\sum_{l\in{\bf Z}+\gamma}f\bra{{l\over \lambda}}\mu\bra{\lambda \phi({l\over \lambda})}\varphi(\lambda^{-1} l)+\bigo{ \lnorm f_\infty\lambda^{\ffra 1,2-\epsilon}}$$ and that in the new sum above, only finitely many terms are non--zero. Moreover, $\mu\bra{\lambda\phi({x/ \lambda})}(f\cdot\varphi)(\lambda^{-1} x)$ is a piecewise smooth function of compact support. We set $$\varphi_f(x)=(f\cdot\varphi)(x)$$ which satisfies $\norm{\varphi_f}_\infty\le\norm f_\infty$, and $$\abs{\varphi_f'(x)}\le \cases{ C &if $x\notin I_\lambda$\cr \lambda^{\ffra 1,2+\epsilon} &if $x\in I_\lambda$\cr} \qquad \abs{\varphi_f''(x)}\le \cases{ C &if $x\notin I_\lambda$\cr \lambda^{1+2\epsilon} &if $x\in I_\lambda$\cr} \eqno\eqf $$ The Poisson summation formula yields $$\eqalignno{ \sum_{l\in{\bf Z+\gamma}}\mu\bra{\lambda \phi({l\over \lambda})}\varphi_f(\lambda^{-1} l) &=\sum_le^{2\pi il\gamma}\int_{-\infty}^\infty \mu\bra{\lambda \phi({x\over\lambda})}\varphi_f(\lambda^{-1} x)e^{-2\pi ixl}\,dx\cr &=\sum_le^{2\pi il\gamma}\int_{\lambda a}^{\lambda b}\mu\bra{\lambda \phi({x\over\lambda})}\varphi_f(\lambda^{-1} x)e^{-2\pi ixl}\,dx\cr &=\sum_{{l\in{\bf Z}}\atop{n\ne 0}}\hat\mu(n) e^{2\pi il\gamma}\int_{\lambda a}^{\lambda b} e^{2\pi i\bra{\lambda n\phi(x/\lambda)-xl}}\varphi_f(\lambda^{-1} x)\,dx\cr &=\lambda\sum_{{l\in{\bf Z}}\atop{n\ne 0}}\hat\mu(n) e^{2\pi il\gamma}\int_{a}^b e^{2\pi i\lambda\bracket{n\phi(x)-xl}}\varphi_f(x)\,dx . \cr }$$ We will show below that the sum is absolutely convergent, due to the fast decrease of $\hat \mu$ assumed in \eqx, and the fast decrease of the integrals; therefore, the infinite sum can be taken in any order we like. \vskip2em Define $$I(n,l)=e^{2\pi il\gamma}\int_{a}^b e^{2\pi i\lambda\bracket{n\phi(x)-xl}}\varphi_f(x)\,dx$$ For integers $n$ and $l$, define $x_{n,l}$ as the unique point (when it exists) satisfying $\phi'(x_{n,l})=\fra l,n$. Note that $$c_0^{-1}\abs{\fra l,n-\fra l',{n'}} \ge\abs{x_{n,l}-x_{n',l'}}\ge \lnorm{\phi''}_\infty^{-1} \abs{\fra l,n-\fra l',{n'}} $$ Define also $$\theta(n,l)=n\cdot \phi(x_{n,l})-l\cdot x_{n,l}$$ and $$\sigma_\phi(n,l)=\lambda^{\frac 1,2}{ 1\over \abs{n\cdot\phi''(x_{n,l})}^{\frac 1,2}}\> e^{-\sign(n)\,\fra\pi i,4+ 2\pi i\Bracket{\lambda\theta(n,l)+\gamma\cdot l}}$$ We write $\sigma_\phi$ to point out that $\sigma $ depends only on $\phi$: the amplitude $f$ does not appear. We begin with the following crude estimate, which is a trivial consequence of Lemma~{\leme}. \lem{\lemd} $$\abs{I(n,l)}\le C\, \lambda^{-\frac 1,2}\cdot |n|^{-\frac 1,2}.$$ \endt This already implies that only the terms appearing for small $n$ play a role in our sum. \theorem{\thmb} With the previous notation, we have $$S(\lambda)= \bra{\sum_{{\scriptstyle n\ne 0} \atop{{{\scriptstyle l\in{\bf Z}}}\atop {\scriptstyle x_{n,l}\in\ocint a,b}}} \hat\mu(n)\cdot \ep_{n,l}\cdot f(x_{n,l})\cdot\sigma(n,l)}+A(\lambda)$$ where $\ep_{n,l}= 1$, unless $\phi'(b)=\fra l,n$ when $\ep_{n,l}= \oh$, and $$A(\lambda)=\littleo{\lambda^{\frac 1,2}}$$ If $\phi'(b)=\fra p,q$, then we have $$A(\lambda)=\bigo{C_q\lambda^{\ffra 1,2-\gamma}},\qquad \gamma>0.$$ If, however, $\phi'(b)$ is irrational, the {\it o}--term depends on the diophantine properties of $\phi'(b)$. In any case, $|A(\lambda)|\le C\,\sqrt\lambda$ for $C$ which only depends on $\norm f_{C^2}$ and $B(\phi)$. \proof Note that there are three types of pairs $(n,l)$: those for which $\phi'(x)=\fra l,n$ for some $x=x_{n,l}\in\coint a,b$, those such that $\phi'(x)$ never equals $\fra l,n$ for any $x\in\coint a,b$, and those (if any) for which $\fra l,n$ equals $\phi'(b)$ ($\phi'(a)$ will play no role here since $f$ vanishes to infinite order at $a$). We need to deal with these cases separately, and we thus write $$S(\lambda)= S_1(\lambda)+ S_2(\lambda)+ S_3(\lambda)$$ where $$\eqalignno{ S_1\bra{\lambda}&=\lambda\sum_{\phi'(a)\le\fra l,n< {\phi'(b)}} \hat\mu(n)\cdot I(n,l) \cr S_2\bra{\lambda}&=\lambda\sum_{\ffra l,n={\phi'(b)}} \hat\mu(n)\cdot I(n,l) \cr S_3\bra{\lambda}&=\lambda\sum_{\ffra l,n\notin\ccint \phi'(a),{\phi'(b)}} \hat\mu(n)\cdot I(n,l) . \cr} $$ \title{Sum $S_1$:} For every term in this sum, the integrand in $I(n,l)$ has a stationary point $x_{n,l}$. Our stationary phase analysis then shows, using {\eqf}, that $$ \abs{\lambda\cdot I(n,l)-f(x_{n,l})\cdot\sigma_{n,l}}\le \lambda\cdot E_{n,l},$$ $$E_{n,l}= C\cdot (\lambda |n|)^{-\frac 3,2}\bra{1+\min\bra{ \lambda^{1+2\epsilon}\>, \> {\lambda^{\ffra 1,2+2\epsilon} \over |n|^{\frac 1,2}\cdot \dist\bra{x_{n,l}\,,\,I_\lambda}}}} \eqno\eqy $$ where the $\min$ appears as the best of estimates {\eqaa} and {\eqac} above. \vskip1em The terms in $S_1$ will be grouped into three categories. First, those for which $x_{n,l}$ falls far from $b$, and second, those for which $x_{n,l}$ falls near b. Within the second class, we will have to consider separately those that appear only when $n$ is large, and those with $n$ small. \vskip1em Fix $n$ in the sum above. For each $n$, the number of terms in the sum in $l$ is at most $C|n|$. And of those terms, the number of $l$ for which $x_{n,l}$ falls within $d$ of $I_{\lambda}$ is bounded by at most $1+C|n|(d+\lambda^{-\ffra 1,2- \epsilon})$. If, say, $\phi'(b)=\fra p,q$ is rational, and we take $d>\lambda^{-\frac 1,2}$, we have $$d\ge\abs{b-x_{n,l}} \ge C_2^{-1}\,\abs{\fra l,n-\fra p,q} \ge {C_2^{-1}\over |n\,q|}$$ which means that, if $$|n|<{\norm{\phi}_{C^2}^{-1}} q^{-1}d^{-1}$$ then there are no $l$ such that $x_{n,l}$ falls within $d$ if $I_\lambda$. Similarly, if $\phi'(b)$ is irrational, the number of such $l$ is at most 1 for $$|n|<{\lnorm{\phi}_{C^2}}^{-1} d^{-1}$$ if $d>\lambda^{-\ffra 1,2-\epsilon}$. We denote this unique $l$ (when it exists) by $l_0(n,d,\lambda)$, and we denote by $n_0(d,\lambda)$ the smallest $|n|$ for which $n$ has such an $l_0$. \vskip1em After all this, we choose $$d=\lambda^{-7\epsilon},\qquad c^\sharp=\cases{ \vpush{\lnorm{\phi}_{C^2}^{-1}} & if $\phi'(b)$ is irrational\cr \vpush{\lnorm{\phi}_{C^2}^{-1}\, q^{-1}} & if $\phi'(b)=\fra p,q$\cr}$$ and break up $$\displaylines{ \quad\lambda^{-1}S_1(\lambda)= \sum_{{\scriptstyle |n|\le c^\sharp d^{-1}}\atop{\scriptstyle d(x_{n,l},I_\lambda)\ge d}} \hat\mu(n)I(n,l) + \sum_{\ttower {|n|> c^{\sharp}d^{-1}} {{\rm all\ } l}} \hat\mu(n)I(n,l)\hfill\llap{\eqg}\cr \hfill + \sum_{c^\sharp d^{-1}> |n|\ge n_0(d,\lambda)} \hat\mu(n)I\bra{n,l_0(n,d,\lambda)}\quad\cr} $$ where it is understood that if $\phi'(b)$ is rational, the sum in the third term above is null, and the sum in $n$ also in the third term is extended only to those $n$ with a corresponding $l_0(n,d,\lambda)$. For the first term we use \eqaa\ to obtain $$E_{n,l}\le C\cdot(\lambda|n|)^{-\frac 3,2} \bra{1+|n|^{-\frac 1,2}\lambda^{\ffra 1,2+9\epsilon}}$$ Since $|n|\le c^\sharp\,\lambda^{7\epsilon}$, we obtain (recall $\ep=\fra 1,{20}$), $$E_{n,l}\le C|n|^{-2}\lambda^{-1+9\epsilon}$$ and, using again $\epsilon=\fra 1,{20}$, we obtain $$\sum_{{\scriptstyle |n|\le c^{\sharp}d^{-1}}\atop{\scriptstyle d(x_{n,l},I_\lambda)\ge d}} \lambda\cdot \hat\mu(n)\cdot I(n,l)= \sum_{{\scriptstyle |n|\le c^{\sharp}d^{-1}}\atop{\scriptstyle d(x_{n,l},I_\lambda)\ge d}} \hat\mu(n)\cdot f(x_{n,l})\cdot\sigma_{n,l}+ \bigo{\lambda^{{\ffra 1,2}-\epsilon}\sum_{n\ne 0}\abs{{\hat \mu(n)\over n}}}$$ For the second term in {\eqg}, we use the trivial estimate \eqab\ in {\eqy} to conclude that $$E_{n,l}={\cal O}\bra{\lambda^{-\ffra 1,2+2\epsilon}|n|^{-\frac 3,2}}$$ and since $|n|\ge c^\sharp \lambda^{7\epsilon}$, $$E_{n,l}={\cal O}_q \bra{\lambda^{-\ffra 1,2-\epsilon}|n|^{-\frac 15,{14}}}$$ and we obtain $$\sum_{|n|> c^\sharp d^{-1}} \lambda\cdot \hat\mu(n)I(n,l)= \sum_{|n|> c^{\sharp}d^{-1}} \hat\mu(n)\cdot f(x_{n,l})\cdot\sigma_{n,l}+ {\cal O}_q\bra{\lambda^{{\ffra 1,2}-\epsilon}\sum_{n\ne 0}{\abs{\hat \mu(n)}\over |n|^{\frac 15,{14}}}}.$$ Note that, here, we could simply have used Lemma~{\leme} to conclude that both $I(n,l)$ and $\sigma_{n,l}$ give a negligible contribution, but this would have required, either, to use the stronger assumption that $\sigma>\fra 3,2$, or to obtain an error estimate which depends on the value of the sum {\eqx}. Finally, for the third term, it is clear that $$\lim_{\lambda\to\infty} n_0\bra{\lambda^{-7\epsilon},\lambda}=\infty$$ which, using Lemma~{\lemd}, implies that $$\eqalignno{ \abs{\sum_{|n|\ge n_0(d,\lambda)} \hat\mu(n)I\bra{n,l_0(n,d,\lambda)}} &\le C \sum_{|n|\ge n_0(d,\lambda)} \abs{\hat\mu(n)}(\lambda|n|)^{-\frac 1,2}\cr &\le C\lambda^{-\frac 1,2}\abs{n_0(d,\lambda)}^{-\sigma+\ffra 1,2}\cr &=\littleo{\lambda^{-\frac 1,2}}\cr}$$ Similarly, observe that $$\eqalignno{ \abs{\sum_{|n|\ge n_0(d,\lambda)} \hat\mu(n)\cdot f(x_{n,l})\cdot\sigma_{n,l_0(n,d,\lambda)}} &\le C\lambda^{\frac 1,2}\abs{n_0(d,\lambda)}^{-\sigma+\ffra 1,2}\cr &=\littleo{\lambda^{\frac 1,2}}\cr}$$ Therefore, we can conclude that $$S_1(\lambda)=\sum_{\ffra l,n\in\coint\phi'(a),{\phi'(b)}} \hat\mu(n)\cdot f(x_{n,l})\cdot\sigma_{n,l}+\littleo{\lambda^{\frac 1,2}}$$ \vskip1em \title{Sum $S_2$:} If $\phi'(b)$ is irrational, this sum is empty. We thus assume that $\phi'(b)$ is rational. If we tried to proceed as we did for $S_1$, we find that $E_{n,l}$ is too big, and this has no remedy. This is so because we would be comparing $I(n,l)$ with the wrong thing: it is not $f(x_{n,l})\cdot\sigma_{n,l}$ what we should look at, but $\oh f(x_{n,l})\sigma_{n,l}$ instead. We proceed as follows: Say $\phi'(b)=\fra l,n$. We have, by {\eqad}, that $$\eqalignno{ e^{2\pi i l\gamma} \int_0^1e^{2\pi i \lambda(n\phi(x)-lx)}\varphi_f(x)\,dx &= e^{2\pi i l\gamma} \int_0^1e^{2\pi i \lambda(n\phi(x)-lx)}f(x)\,dx+ \bigo{\lambda^{-\ffra 1,2-\epsilon}}\cr &={1\over 2\lambda} f(x_{n,l})\cdot \sigma_{n,l}+ \bigo{(|n|\lambda)^{-\frac 3,2}}+ \bigo{\lambda^{-\ffra 1,2-\epsilon}}\cr}$$ which yields $$S_2(\lambda)={\oh}\sum_{\ffra l,n=\phi'(b)} \hat\mu(n)\cdot f(x_{n,l})\cdot \sigma_{n,l}+\bigo{\lambda^{\ffra 1,2-\epsilon}}.$$ \title{Sum $S_3$:} As for $S_1$, we deal separately with those $l$ and $n$ for which $\phi'(x)-\fra l,n$ is small or large, and for those for which it is small, we distinguish between small and large $n$. \vskip1em When $|\phi'(x)-\fra l,n|>d$ for all $x\in\ccint a,b$, we use {\eqhc} to obtain $$I(n,l)=\bigo{{\lambda^{-\ffra 3,2+\epsilon}\over |n|^2d^2}+{\lambda^{-2}\over |n|^2d^3}+{\lambda^{-2}\over |n|^2d^4}}$$ which implies $$\abs{ \sum_{ (n,l):|\phi'(x)-\ffra l,n|>d} \hat\mu(n)\cdot I(n,l)} \le C\lambda^{-\frac 3,2} \sum_{ {\rm all\ }n\ne 0} \abs{{\hat\mu(n)\over n}}\bra{ {\lambda^\epsilon\over d}+ {1\over\lambda^{\frac 1,2}d^2}+ {1\over\lambda^{\frac 1,2}d^3}}$$ If we now set $d=\lambda^{-7\epsilon}$ we obtain $$\sum_{ (n,l):|\phi'(x)-\ffra l,n|>d} \hat\mu(n)\cdot I(n,l)=\bigo{\lambda^{-\ffra 1,2-\epsilon}}$$ When $0<|\phi'(b)-\fra l,n|\le d$, for a fixed $n$ there are at most $1+\abs n\,d$ terms in the sum. And, as before, if $\phi'(b)$ is rational and $|n|d^{-1} }} \hat\mu(n)\cdot I(n,l) \qquad {\rm and} \qquad \sum_{|n|\ge n_0(d,\lambda)} \hat\mu(n)\cdot I\bra{n,l_0(n,d,\lambda)} . $$ The first sum above is trivially controlled by \eqhc, which implies $$I(n,l)=\bigo{\lambda^{-2+\ffra 7,5}|n|^{-2}}$$ hence $$ \sum_{{\scriptstyle (n,l):|\phi'(x)-\ffra l,n|>d} \atop {\scriptstyle |n|>d^{-1} }} \hat\mu(n)I(n,l)= \bigo{\lambda^{-\ffra 1,2-\epsilon}}$$ For the second term, note as before that $\lim_{\lambda\to\infty} n_0(d,\lambda)=\infty$, and therefore, using Lemma~{\leme}, we get $$\eqalignno{ \abs{\sum_{|n|\ge n_0(d,\lambda)} \hat\mu(n)\cdot I\bra{n,l_0(n,d,\lambda)}} &\le C \sum_{|n|\ge n_0(d,\lambda)} \abs{\hat\mu(n)}\cdot (\lambda|n|)^{-\frac 1,2}\cr &\le C\lambda^{-\frac 1,2}n_0^{-\sigma+\ffra 1,2}\cr &=\littleo{\lambda^{-\frac 1,2}}\cr}$$ All this implies that $$S_3(\lambda)=\littleo{\lambda^{\frac 1,2}}$$ and the theorem follows. \endpf \Title{Case II} In this case we will not need $\varphi(x)$ since $f$ is compactly supported and smooth. In fact, $\mu\bra{\lambda\phi({x/ \lambda})}f(\lambda^{-1} x)$ is a piecewise smooth function of compact support, and by the Poisson summation formula, as before, $$\eqalignno{ \sum_{l\in{\bf Z}+\gamma}\mu\bra{\lambda \phi({l\over \lambda})}f(\lambda^{-1} l) &=\lambda\sum_{{l\in{\bf Z}}\atop{n\ne 0}}\hat\mu(n) \cdot e^{2\pi il\gamma}\int_{a}^b e^{2\pi i\lambda\bracket{n\phi(x)-xl}}f(x)\,dx\cr }$$ \vskip2em Define, as before, $$I(n,l)=e^{2\pi il\gamma}\int_{a}^b e^{2\pi i\lambda\bracket{n\phi(x)-xl}}f(x)\,dx$$ and $x_{n,l}$ as the unique point (if it did exit) satisfying $\phi'(x_{n,l})=\fra l,n$. Also as before, we have $$c_0^{-1}\abs{\fra l,n-\fra l',{n'}} \ge\abs{x_{n,l}-x_{n',l'}}\ge \lnorm{\phi''}_\infty^{-1} \abs{\fra l,n-\fra l',{n'}} $$ Define also $\theta(n,l)$ and $\sigma(n,l)$ exactly as in Case~I. \theorem{\thmc} With the previous notation, we have $$S(\lambda)= \sum_{{\scriptstyle{n\ne 0}\atop{\scriptstyle l\in {\bf Z}}}} \hat\mu(n) \cdot \sigma(n,l)+\bigo{B(\phi)\cdot \norm f_{C^2}\cdot\bra{1+\norm {f'}_{\infty}} }$$ \proof In this case now there are only two types of pairs $(n,l)$: those for which $\phi'(x)=\fra l,n$ for some $x=x_{n,l}\in\ccint a,b$, and those such that $\phi'(x)$ never equals $\fra l,n$, and thus we write $$S(\lambda)= S_1(\lambda)+ S_2(\lambda)$$ where $$\eqalignno{ S_1\bra{\lambda}&=\lambda\sum_{\ffra l,n\in\ccint \phi'(a),{\phi'(b)}} \hat\mu(n)\cdot I(n,l) \cr S_2\bra{\lambda}&=\lambda\sum_{\ffra l,n\notin\ccint \phi'(a),{\phi'(b)}} \hat\mu(n)\cdot I(n,l) . \cr} $$ \title{Sum $S_1$:} We proceed as in the previous section, $$ \abs{\lambda\cdot I(n,l)-f(x_{n,l})\cdot \sigma_{n,l}}\le \lambda\cdot E_{n,l}$$ where, by {\eqac}, $E_{n,l}$ is given now by $$E_{n,l}= C\cdot B(\phi)\cdot\lnorm{f}_{C^2}\cdot (\lambda |n|)^{-\frac 3,2} $$ For each $n$, the number of terms in the sum is at most $(3+C_1)|n|$. Therefore, we can conclude that $$S_1(\lambda)=\lambda\sum_{\ffra l,n\in\ccint\phi'(a),{\phi'(b)}} \hat\mu(n)\cdot f(x_{n,l})\cdot \sigma_{n,l}+\bigo{C^+_1\cdot B(\phi)\cdot \norm f_{C^2} \cdot\lambda^{-\frac 1,2}}$$ for $$C_1^+=\max(1,C_1)$$ \vskip1em \title{Sum $S_2$:} By Lemma~{\lemr}, we have $$\abs{I(n,l)}\le C\,\norm f_{C^2}\,B(\phi)\,\bracket{\lambda |n|}^{-1}$$ which we use when $|l|\le 2 C_1^+|n|^{1-\delta}$, for $\delta>0$, to obtain $$\sum_{\{(n,l):\,|l|\le 2 C_1^+|n|^{1-\delta}\}} \abs{\hat\mu(n)\cdot I(n,l)} \le C \lambda^{-1}\,\norm f_{C^2}\,B(\phi)\cdot C_1^+ \,\sum_{n\ne 0} { \abs{\hat \mu(n)} \over |n|^{\delta} }$$ \vskip1em Outside of this range, we have $$\abs{\phi'(x)-\fra l,n}\ge{\fra 1,2}\abs{l\over n} \qquad{\rm for\ all\ }x\in\ccint a,b.$$ Therefore, {\eqhb} implies $$\eqalignno{ \abs{I(n,l)}&\le {C \,\norm f_{C^2}\,B(\phi) \over \lambda^2\,|n|^2} \,\bra{ {n^2\over l^2}+ +{|n|^3\over |l|^3} +{n^4\over l^4}} \cr }$$ Therefore, $$\sum_{\{(n,l):\,|l|>2 C_1^+|n|^{1-\delta}\}} \abs{\hat\mu(n)\cdot I(n,l)} \le C \lambda^{-2}\,\norm f_{C^2}\,B(\phi)\,\sum_{n\ne 0} { \abs{\hat \mu(n)} \over |n|^{1-4\delta} }\eqno\mathendpf$$ \Title{3. Energy Asymptotics} We plan to apply our previous estimates to the function $$\Psi_C(Z)=2\pi\cdot Z^{\frac 4,3}\cdot \sum_{l\in({\bf Z}+\ffra 1,2)\cap \ccint 1,{a^{-\frac 1,2}\cdot Z^{\frac 1,3}\cdot\Omega_c}} \eta\bra{l\cdot Z^{-\frac 1,3}}\>\cdot\> \mu{\bra{Z^{\frac 1,3}\phi\bracket{l\cdot Z^{-\frac 1,3}}}}$$ where $$\eqalignno{ \mu(x)&={\rm dist}(x,{\bf Z})^2-\fra 1,{12}\cr \phi(\Omega) &= {1\over\pi}\int\bra{{V^1_{\rm TF}(r)-{\Omega^2\over r^2}}}_+^{\frac 1,2} \,dr = {a^{-\frac 1,2} \over\pi}\int\bra{{y(r)\over r}-{a\cdot\Omega^2\over r^2}}_+^{\frac 1,2} \,dr\cr \eta(\Omega)&={\Omega\over P(\Omega)}\cr P(\Omega)&= \int \bra{V_{\rm TF}^1(r)-{\Omega^2\over r^2}}_+^{-\frac 1,2}\,dr= a^{-\frac 3,2} \int_{r_1(a^{\frac 1,2}\,\Omega)}^{r_2(a^{\frac 1,2}\,\Omega)} \bra{{y(r)\over r}-{a\cdot \Omega^2\over r^2}}^{-\ffra 1,2} \,dr\cr}$$ Here, $r_i(\Omega)$ are the two points where $y(r)/r$ equals $\Omega^2/r^2$ (see below) and $ \Omega_c $ is the supremum of the $\Omega$ for which $${y(r)\over r}-{\Omega^2\over r^2}$$ is positive somewhere. The crucial result we need is the non--vanishing of the second derivative of $\phi$. This was proved in {\fs7}. Because of its vital importance, we display it explicitly: \theorem{\thmfs} There exists a number $c_0$ such that $$-\phi''(\Omega)\ge c_0>0\qquad{\rm for\ all\ } \Omega\in\ooint 0,{\Omega_c}$$ \endt We will first recall some known results (which appear, for example, in {\fs7}, {\Hille} and {\Hughes}) which we will need here. After that, we will complement them with further properties of $\phi$ and $P$, some of which are taking from similar estimates appearing in {[FS2---8]}. \title{Review of earlier results.} If we set $u(r)=r\cdot y(r)$, then $u$ has a unique maximum at $r=r_c$, where $r_c\sim 2.1$. We set $\Omega_c^2=u(r_c)$. Then, $u$ is increasing on $\ccint 0,{r_c}$ and decreasing on $\ooint r_c,\infty$. This is a crucial fact whose proof goes back to Sommerfeld, and can be found in {\Hughes}. Around 0, $u$ satisfies the expansions $$u(x)=\sum_{n=2}^\infty u_n \, x^{\frac n,2},\qquad u_2=1,\quad u_3=0,\quad u_4,\sim -1.588.$$ Rigorous numerical bounds for $u_4$ can be found in {\fs7}. However, it is easy to see analytically that $u_4<0$. We also have $$\sum_{n=2}^\infty |u_n|\cdot \rho_0^n<\infty, \qquad \rho_0>0,$$ therefore, the function $$f(z)=\sum_{n=2}^\infty u_nz^n$$ is analytic in a small neighborhood around 0. Around $\infty$, we have the expansion $$u(x)={144\over x^2}\sum_{n=0}^\infty b_n\, x^{-\fra n\alpha,2} , \qquad\quad b_0=1, \quad b_1\sim -13, \qquad \alpha=\Fra\sqrt{73}-7,2\sim 0.772.$$ Again, rigorous numerical bounds for $b_1$ are found in {\fs7}, and it can be seen analytically that $b_1<0$. We also have that $$\sum_{n=0}^\infty |b_n|\,\rho_1^n<\infty, \qquad \rho_1>0,$$ and, as a result, we have $$u(x)={144\over x^2}\>f\bra{x^{-\frac \alpha,2}}\eqno\eqz$$ for a function $f$ analytic in a neighborhood of 0. Given any $\Omega\in\ooint 0,{\Omega_c}$, there exist two numbers, $r_1(\Omega)\le r_2(\Omega)$ where $u$ equals $\Omega^2$. We then have \lem{\lemg} The following formulas hold: $$\phi(\Omega)={a^{-\frac 1,2}\over\pi}\cdot F\bra{a^{\frac 1,2}\Omega}$$ where $$\eqalignno{ F(\Omega)&=\int\bra{u(x)-\Omega^2}_+^{\frac 1,2}\,{dx\over x}\cr F'(\Omega)&=-\Omega\int\bra{u(x)-\Omega^2}_+^{-\frac 1,2}\,{dx\over x}\cr F''(\Omega)&= -\lim_{\delta\to 0} \bra{\int_ {r_1(\Omega)+\delta} ^{r_2(\Omega)-\delta} \bra{u(x)-\Omega^2}^{-\frac 3,2}y(x)\,dx +c(\Omega)\delta^{-\frac 1,2}}\cr}$$ where $c(\Omega)$ is uniquely specified by requiring the finiteness of the limit. Moreover, if $b$ is any number less than $r_2(\Omega)$, then $${d^2\over d\Omega^2} \int_{r_1(\Omega)}^b\bra{u(x)-\Omega^2}_+^{\frac 1,2}\,{dx\over x} $$ equals $$ -\lim_{\delta\to 0} \bra{\int_ {r_1(\Omega)+\delta} ^{b} \bra{u(x)-\Omega^2}^{-\frac 3,2}y(x)\,dx +c_1(\Omega)\delta^{-\frac 1,2}}$$ again, for a constant $c_1$ that makes the limit finite. The corresponding symmetric case also holds. Furthermore, $F$ can be extended as an analytic function to a complex neighborhood of $\ocint 0,{\Omega_c}$. However, 0 is an essential singularity of $\phi$ (or $F$), and, moreover, $$\lim_{\Omega\to 0}\phi''(\Omega)\cdot\Omega^{\gamma}=\kappa , \qquad\gamma={9-\sqrt{73}\over 2}>0,$$ where $\kappa$ is a strictly negative real number. \endt A consequence of this which is of importance to us is that although $\phi$ and $\phi'$ remain bounded as we approach 0, the second derivative blows up slowly, and third will blow up much faster. In other words, $\phi$ does not satisfy sensible non--degenerate multiscale analysis bounds. \title{Further background results.} Here we will obtain growth and regularity properties of the functions $\phi$ and $P$ above. We define $$g_\gamma(x)=\int_1^{x^{-2}}\bra{t-1}^{-\frac 1,2} \,t^\gamma\,dt,\qquad{\rm for\ }0-\oh$, then $${x^{-2\gamma-1}-1\over |\gamma+\oh|}\le {g_\gamma(x)}\le \bra{100+{100\over|\gamma+\oh|}}x^{-2\gamma-1}\eqno\eqeb$$ where $0\oh t$ for $t>2$ to write $$\int_1^{x^{-2}}t^{\gamma-\ffra 1,2}\,dt\le g_\gamma(x)\le 2^{|\gamma|+3}+ 2\int_2^{x^{-2}}t^{\gamma-\ffra 1,2}\,dt$$ which implies {\eqeb} after using the fact that $2^{|\gamma|+\ffra 1,2} \le 4\,x^{-2\gamma-1}$. \endpf \lem{\lemh}Define $$f(\Omega)= \bra{\Omega_\ep^2-\Omega^2}^{-\frac 1,2}$$ Then, $|f^{(k)}(\Omega)|\le C_k\,\Omega_\ep^{-1-k}$ for $\Omega\le \oh\Omega_\ep$ and $k\ge 0$. \proof $$f(\Omega)=\Omega_\ep^{-1} g\bra{\Omega\over\Omega_\ep}, \qquad {\rm for\ } g(x)=\bra{1-x^2}^{-\frac 1,2}.\eqno\eqd$$ Done. \endpf \lem{\lemi} Given $\beta>0$, $\delta>0$, $\Omega_\ep>0$, $\tau$, $d$ and $w_n$ for $n=0,1,2,\ldots$, let $$f(\Omega)= \sum_{n=0}^\infty w_n\,m_n(\Omega),\qquad m_n(\Omega)=\Omega^{2\gamma_n+d}g_{\gamma_n}\bra{ {\Omega\over\Omega_\ep}} $$ where $\gamma_n=\tau+\beta\cdot n$ and $|\gamma_n +\oh|>\delta$ for all $n\ge 0$. Assume that $\sum w_nz^n$ has a radius of convergence $\rho$ and $\Omega_\ep^{2\beta}\le\oh\rho$. Then, $$\abs{{d^k f\over d\Omega^k}(\Omega)}\le \cases{ \vphantom{\biggl(} C\, \Omega^{2\tau+d-k} &if $\tau<-\oh$\cr \vphantom{\biggl(} C\, \Omega^{d-1-k} &if $\tau>-\oh$\cr} \qquad\qquad{\rm when\ }\Omega\le\oh\Omega_\ep$$ for a certain constant $C$ which depends on everything except $\Omega$. \proof Let's consider first those $n$ such that $\gamma_n>-\oh$. In this case, using Lemma~{\lemc} we obtain $$\eqalignno{ \abs{{d^km_n\over d\Omega^k}(\Omega)}&\le \sum_{l=0}^kC(k;\delta)\cdot(n+1)^{k-l} \cdot \Omega^{2\gamma_n+d-(k-l)}\cdot \Omega_\ep^{-l}\cdot \abs{{d^lg_{\gamma_n}(\Omega)\over d\Omega^l} \bra{\Omega\over\Omega_\ep}}\cr &\le C(k;\delta)\cdot (n+1)^k\cdot \Omega_\ep^{2\gamma_n+1}\cdot \Omega^{d-1-k}\cr}$$ If, on the other hand, $\gamma_n<-\oh$, the $l=0$ term above has to be estimated by $$C(k;\delta)\cdot (n+1)^k\cdot\Omega^{2\gamma_n+d-k}$$ and we obtain $$\eqalignno{ \abs{{d^km_n\over d\Omega^k}(\Omega)}&\le C(k;\delta)\cdot (n+1)^k\cdot \bra{ \Omega_\ep^{2\gamma_n+1}\cdot \Omega^{d-1-k}+ \Omega^{2\gamma_n+d-k}}\cr &\le 2\,C(k;\delta)\cdot (n+1)^k\cdot \Omega^{2\gamma_n+d-k}\cr}$$ because $\Omega\le\Omega_\ep$. Since we can only have $\gamma_n<-\oh$ for finitely many $n$, we conclude that $$\sum_{n=0}^\infty w_n \, m_n^{(k)}(\Omega)={d^k f\over d\Omega^k}(\Omega)$$ where the sum converges absolutely, and we obtain the required estimate. \endpf This ends our presentation of calculus results. In what follows we will develop the regularity bounds for $\phi$ first and then $P$. \lem{\lemf} For constants $C_n$ and $c>0$ we have $$|\phi(t)|\le C_0,\qquad |\phi'(t)|\le C_1,\qquad \abs{{d^n\phi\over d\Omega^n}}\le C_n\,\Omega^{-n+1+\alpha}\quad (n\ge 2)$$ and $$-\phi''(\Omega)\ge c\cdot\Omega^{-1+\alpha}.$$ \proof As in Lemma~{\lemg}, in order not to bother with the presence of the constant $a$, we will prove this result for the function $F$ instead. The inequalities for $\phi $ and $\phi'$ are obvious. For the higher derivatives, the bounds outside a neighborhood of 0 are a direct consequence of the analytic extension of $F$ to a complex neighborhood of $\ocint 0,{\Omega_c}$, which is Corollary~1.3 in {\fs7}. We are thus only left with proving the bounds in an arbitrarily small neighborhood to the right of 0, given by $\ooint 0,{\bar\Omega_\ep}$, for a small universal number $\bar\Omega_\ep$ to be picked up later in the proof. Arguing as in formula (4.1abc) {\fs7}, using Lemma~{\lemg}, we write $$-F''(t)=I_1+I_2+I_3$$ for $$\eqalignno{ I_1&=\int_a^b\bra{u(r)-\Omega^2}^{-\frac 3,2}y(r)\,dr\cr I_2&=\lim_{\delta\to 0}\bra{\int_{r_1(\Omega)+\delta}^a \bra{u(r)-\Omega^2}^{-\frac 3,2}y(r)\,dr-G_1(\Omega)\delta^{-\frac 1,2}} \cr I_3&=\lim_{\delta\to 0}\bra{\int^{r_2(\Omega)-\delta}_b \bra{u(r)-\Omega^2}^{-\frac 3,2}y(r)\,dr-G_2(\Omega)\delta^{-\frac 1,2}} \cr }$$ with $G_i$ such that the limit is finite, and $a$ and $b$ any numbers such that $r_1(\Omega)0$ (Equation (4.20) in {\fs7}). This allows us to put $$I_3(\Omega)=f_1(\Omega)+f_2(\Omega)$$ where $$\eqalignno{ f_1(\Omega)&=\sum_{n=1}^\infty 2\,h_n\,m_n(\Omega),\quad m_n(\Omega)= \Omega^{-1+n\alpha}\int_1^{\Omega^{-2}\Omega_\ep^2}(t-1)^{-\frac 1,2} t^{-1+\ffra n\alpha,2}\,dt&\eqk\cr f_2(\Omega)&= -2\bra{\Omega_\ep^2-\Omega^2}^{-\frac 1,2} w(\Omega_\ep^2)\cdot \Omega_\ep^2 . \cr}$$ If we make sure that $$\Omega_\ep^\alpha\le\oh\,\rho_h\eqno{\eqc}$$ we can invoke Lemma~{\lemi}, with $\tau=-1+\fra\alpha,2$, which is less than $-\oh$, $\beta=\fra \alpha,2$ and $d=1$ to obtain $$\abs{{d^kf_1\over d\Omega^k}(\Omega)}\le C(\Omega_\ep;k)\cdot \Omega^{-1+\alpha-k}.$$ \vskip1em This ends the proof of all upper bounds in the statement of the lemma. For the lower bound for $-\phi''$, we use the notation in {\eqk} to write $$ f_1(\Omega)=2h_1\cdot m_1(\Omega)+\tilde f_1(\Omega) , \qquad \tilde f_1(\Omega)=\sum_{n=2}^\infty 2\,h_n\,m_n(\Omega)$$ Applying Lemma~{\lemc} to the first term above with $\gamma=-1+\fra\alpha,2<-\oh$, and Lemma~{\lemi} applied to $\tilde f_1(\Omega)$ with $\tau=-1+\alpha>-\oh$, we obtain $$ f_1(\Omega)\ge c\,h_1\Omega^{-1+\alpha} , \qquad |\tilde f_1(\Omega)|\le C(\Omega_\ep) . $$ Since all other terms in the break--up of $-\phi''$ remain bounded as $\Omega\to 0$, we conclude that $$-\phi''(\Omega)\ge c\Omega^{-1+\alpha},\qquad\Omega\le \bar\Omega_\ep$$ for $\bar\Omega_\ep\muchsmaller\Omega_\ep$, as required. \endpf We now turn our attention to $P$. In this case, rather than introducing a new function that allows us to do without the bothersome constant $a$, we will simply proceed as if $a$ did not appear in the definition of $P$. This simplification clearly does not change the result, except of course, that the details of the proof will not contain the $a$ dependence. The following result is a trivial adaptation of Lemma~1.2 in {\fs7} for $P$ instead of $\phi$. \lem{\lemj} We have $P\in C^\infty{\ooint 0,{\Omega_c}}$. Furthermore, $P$ admits an analytic extension to a neighborhood of $\Omega_c$. \proof Let $$H(\delta,\Omega)=\Omega\int_{r_1(\Omega)+\delta} ^{r_2(\Omega)-\delta}\bra{u(r)-\Omega^2}^{-\frac 1,2}\>{r\,dr}$$ Consider the analytic change of variables given by $$t(r)=\cases{\bra{\Omega_c^2-u(r)}^{\frac 1,2}&if $r\ge r_c$\cr -\bra{\Omega_c^2-u(r)}^{\frac 1,2}&if $r\le r_c$\cr}\eqno\sdd$$ Note that $t$ is smooth and strictly increasing in the range $\ooint 0,\infty$. We can therefore consider its inverse, $r(t)$, and use it to rewrite $$H(\delta,\Omega)=\Omega\int_{t_1(\delta,\Omega)}^{t_2(\delta,\Omega)}\bra{D^2 -t^2}^{-\frac 1,2}w(t) \,dt$$ where $$t_1=t(r_1+\delta),\qquad t_2=t(r_2-\delta) , \qquad D^2=\Omega_c^2-\Omega^2 , \qquad w(t)={r'(t) \cdot r(t)} . $$ Note that $w$ is smooth on $\ooint -\Omega_c,{\Omega_c}$, and that $$t_1=-D\bra{1+\tau_1(\delta)} , \qquad t_2=D\bra{1+\tau_2\bra{\delta}} , \qquad c\delta\le|\tau_i|\le C\delta\quad{\rm\ for\ }i=1,2 \eqno\sdb$$ uniformly on compact subsets of $\ooint -\Omega_c,{\Omega_c}$, which implies that $$H(\delta,\Omega)=\Omega\int_{D^{-1}t_1}^{D^{-1}t_2}\bra{1-t^2}^{-\frac 1,2} w(tD) \,dt$$ converges as $\delta\to 0$ uniformly to the $C^1$ function $$H(0,\Omega)=\Omega\int_{-1}^{1}\bra{1-t^2}^{-\frac 1,2} w(tD) \,dt=P(\Omega).\eqno\sde$$ To show analyticity around $\Omega_c$, note that $w(t)$ is analytic around 0; thus, it admits a convergent power series expansion given by $$w(t)=\sum_{n=0}^\infty w_n t^n , \qquad |t|\le\rho$$ which implies $$P(\Omega)=\Omega \sum_{n=0}^\infty w_n \cdot D^n\int_{-1}^{1}\bra{1-t^2}^{-\frac 1,2}t^n\,dt . $$ The integral corresponding to the odd terms in the sum is 0, which implies that in fact $$P(\Omega)=\Omega \sum_{n=0}^\infty w_{2n} \cdot D^{2n}\int_{-1}^{1}\bra{1-t^2}^{-\frac 1,2}t^{2n}\,dt$$ which defines an analytic function of $\Omega$ around $\Omega_c$, since $D^2$ now is analytic in $\Omega$. \endpf \lem{\lemk} For constants $C_n$ and $c>0$ we have $$\abs{{d^kP\over d\Omega^k}(\Omega)}\le C_k\cdot\Omega^{-3-k} , \qquad |P(\Omega)|\ge c\cdot\Omega^{-3} . $$ \proof Lemma~{\lemj} establishes our inequalities outside an arbitrarily small neighborhood of 0. For a neighborhood to the right of 0 given by $\ooint 0,{\bar\Omega_\ep}$ we proceed as before, setting $$f(\Omega)=I_1+I_2+I_3$$ for $$\eqalignno{ I_1&=\int_a^b\bra{u(r)-\Omega^2}^{-\frac 1,2}r\,dr\cr I_2&=\int_{r_1(\Omega)}^a \bra{u(r)-\Omega^2}^{-\frac 1,2}r\,dr \cr I_3&=\int^{r_2(\Omega)}_b \bra{u(r)-\Omega^2}^{-\frac 1,2}r\,dr \cr }$$ with $a$ and $b$ any numbers such that $r_1(\Omega)\cdot\> \mu\bra{Z^{\frac 1,3}\cdot\phi\bra{ \sqrt{ \maca }}} . $$ Define $l_{\rm max}$ as the largest $l$ appearing in the sum defining $\Psi_C$. Note: {\itemize \item{1.} Each term appearing in the definition of either $\Psi_Q$ above, or in $\Psi_C$, is bounded by a constant independent of $Z$; \item{2.} The sum in $\Psi_Q$ is taken over integers, while the sum in $\Psi_Q$ is taken over half--integers. \item{3.} The number of terms in either sum ($l_{\rm TF}$ for $\Psi_Q$ and $l_{\rm max}-\oh$ for $\Psi_C$) may differ slightly because in general, $l_{\rm TF}+\oh\ne l_{\rm max}$. } \vskip1em We show now that the number of terms in both sums differ by at most 1. $l_{\rm max}$ is the greatest element in ${\bf Z}+\oh$ which is less than or equal to $a^{-\frac 1,2}Z^{\frac 1,3}\cdot\Omega_c$. Similarly, $l_{\rm TF}$ is the largest integer that satisfies $$ l_{\rm TF}\,\bra{l_{\rm TF}+1} = \bra{l_{\rm TF}+\oh}\cdot \sqrt{1 -{1\over (2\cdot l_{\rm TF} +1)^2}}\>\le a^{-\frac 1,2}Z^{\frac 1,3}\cdot\Omega_c . $$ An immediate consequence of this is that $$c\cdot Z^{\frac 1,3}\le l_{\rm TF}\>,\>l_{\rm max}\le C\cdot Z^{\frac 1,3}$$ Then, for $Z$ large enough, $l_{\rm TF}+\oh\le a^{-\frac 1,2}Z^{\frac 1,3}\Omega_c+1$ which implies $$l_{\rm TF}+\oh\le l_{\rm max}+1.\eqno{\equ}$$ On the other hand, $$\eqalignno{ {l_{\rm max}-l_{\rm TF}}&\le a^{-\frac 1,2}\cdot Z^{\frac 1,3}\cdot\Omega_c - l_{\rm TF} . \cr}$$ Since we must have $$l_{\rm TF}+\fra 3,2>{a^{-\frac 1,2}\cdot Z^{\frac 1,3}\cdot \Omega_c\over\sqrt{1-\bra{2l_{\rm TF}+3}^{-2}}} $$ which implies $$ \eqalignno{ a^{-\frac 1,2}\cdot Z^{\frac 1,3}\cdot\Omega_c - l_{\rm TF}-\oh &\le \fra 3,2 . \cr} $$ we conclude that $l_{\rm max}-\oh-l_{\rm TF}<1$, or $$l_{\rm max}-\oh\le l_{\rm TF}\eqno\eqxa$$ Thus, using {\equ}, $$\abs{l_{\rm max}-\oh-l_{\rm TF}}\le 1\eqno\eqv$$ for $Z$ large enough. As a consequence of this, if we rewrite $$\eqalignno{ \Psi_C(Z)&=2\pi\cdot Z^{\frac 4,3}\cdot\sum_{l=1}^{l_{\rm max}-\oh} \eta\bra{(l+\oh)\cdot Z^{-\frac 1,3}} \cdot\mu\bra{Z^{\frac 1,3}\cdot\phi\bra{ (l+\oh)\cdot Z^{-\frac 1,3}}} \cr \tilde\Psi_Q(Z) &=2\pi\cdot Z^{\frac 4,3}\cdot\sum_{l=1}^{l_{\rm max}-\oh} {\tilde l\over P\bra{ \sqrt{ \maca }}}\>\cdot\> \mu\bra{Z^{\frac 1,3}\cdot\phi\bra{ \sqrt{ \maca }}} \cr} $$ which makes sense by {\eqxa}, and by noting that $l_{\rm max}$ in fact refers to a half--integer, we see that $$\abs{\Psi_Q(Z)-\tilde\Psi_Q(Z)}\le C\cdot Z^{\frac 4,3}$$ since, after all, the difference is at most one term of size at most $Z^{\frac 4,3}$. Next, we compare $\Psi_C$ and $\tilde \Psi_Q$ term by term. To this end, we observe that, since $\tilde l\ge Z^{-\frac 1,3}$, we have $$\eqalignno{ \abs{{1\over P(\tilde l)}-{1\over P\bra{\sqrt{\maca}}}} &\le{\abs{P(\tilde l)-P\bra{\sqrt{\maca}}}\over P(\tilde l)\cdot P\bra{\sqrt{\maca}}}\cr &\le C\cdot {\tilde l}\ ^6 \cdot {\tilde x}^{-4}\cdot \tilde l\cdot \bra{ 1- \sqrt{1-{1\over 4{\tilde l}^2\lambda^2}}}\cr \noalign{\hbox{for $\tilde x\in \ccint {\sqrt{\maca}},{\tilde l}$}} & \le C\cdot \tilde l\cdot\lambda^{-2}\cr}$$ which implies $$ \eqalignno{ \abs{\eta\bra{{l+\oh\over\lambda}}-{\tilde l\over P\bra{ \sqrt{\maca}}}} &\le \tilde l \cdot \abs{{1\over P\bra{{\tilde l}}}-{1\over P\bra{ \maca }}} \cr &\le C\cdot\tilde l\ ^2\cdot\lambda^{-2}\cr }$$ Similarly, since $\phi'$ is bounded, $$\eqalignno{ \abs{\phi\bra{{\tilde l}}-\phi\bra{ \sqrt{\maca }}} &\le C\cdot{\tilde l}\cdot\bra{1-\sqrt{1-\Fra 1,{4{\tilde l}^2\lambda^{-2}}}} \cr &\le C\cdot{ \tilde l}^{-1}\cdot\lambda^{-2}\cr}$$ and since $\mu$ is Lipschitz, we conclude that $$\eqalignno{ \abs{\mu\bra{\lambda\cdot \phi\bra{{l+\oh\over\lambda}}}-\mu\bra{ \lambda\phi\bra{ \sqrt{\maca }}}} &\le C\cdot {\tilde l}^{-1}\cdot\lambda^{-1}\cr}$$ Therefore, the terms indexed by $l$ in $\tilde\Psi_Q$ and $\Psi_C$ differ by at most $$C\cdot\bra{\lambda^{-2}\cdot{\tilde l}^2+ \sup_{x\in\ccint {\sqrt{\maca}},{\tilde l}}|\eta(x)|\cdot \lambda^{-1}\cdot{\tilde l}^{-1}}$$ which implies $$\eqalignno{ \abs{\Psi_Q(Z)-\tilde \Psi_C(Z)}&\le C\cdot Z^{\frac 4,3}\>\sum_{l=1}^{l_{\rm TF}}\bra{ Z^{-\frac 4,3}\cdot l^2+Z^{-\frac 4,3}\cdot l^3} \cr & \le C\cdot Z^{\frac 4,3}\cr}$$ } as required. \endpf In what follows, we will denote either of the sums $\Psi_C$ or $\Psi_Q$ simply by $\Psi$, since we now know that both are the same modulo errors of order $Z^{\frac 4,3}$. \vskip1em Define $\Omega_{n,l}$ such that $\phi'(\Omega_{n,l})=\fra l,n$, and $$\theta(n,l)=n\cdot \phi(\Omega_{n,l})-l\cdot\Omega_{n,l}$$ \theorem{\thmd} $$\Psi(Z)=\Psi_0(Z)+\littleo{Z^{\ffra 3,2}}$$ where $$\Psi_0(Z)=2\pi \cdot Z^{\frac 3,2}\sum_{n,l} {\eta(\Omega_{n,l})\cdot\hat\mu(n)\over \abs{n\cdot\phi''(\Omega_{n,l})}^{\frac 1,2}} \cdot e^{2\pi i\cdot Z^{\frac 1,3}\cdot\theta(n,l)+\pi i\Bracket{l-\fra\sign(n) ,4}}$$ Also, $\Psi_0$ satisfies the bound $$\abs{\Psi_0(Z)}\le C\,Z^{\fra 3,2}$$ for a constant $C$. \proof Construct now a partition of unity given by $\{U_\nu,\theta_\nu\}$ for $\nu=0,1,\cdots$, such that $$\quad\eqalign{ &U_\nu=\ccint a_\nu,{b_\nu},\qquad a_\nu=2^{-\nu-2}\cdot a^{-\frac 1,2}\cdot\Omega_c,\quad b_\nu=2^{-\nu}\cdot a^{-\frac 1,2}\cdot\Omega_c,\cr &d_\nu\eqbydef b_\nu-a_\nu,\quad d_\nu\sim a_\nu\sim b_\nu,\cr &\sum_{\nu}\theta(x)=\cases{1 &if $0\nu_1$, we argue directly as we did before before Theorem~{\thmb} to obtain that $$S_\nu=\lambda \sum_{{\scriptstyle n\ne 0}\atop{\scriptstyle l\in {\bf Z}}} \hat\mu(n)\cdot I(n,l) , \qquad I(n,l)= \int \bra{\theta_\nu\cdot\eta}(x)\,e^{2\pi i\lambda\bracket{n\phi(x)-lx}}\,dx . $$ In order to analyze $I(n,l)$, we use lemma~{\leme} to obtain $$\abs{I(n,l)}\le C\, \bracket{\lambda|n|}^{-\frac 1,2} \,\bra{d_\nu^{-1+\alpha}}^{-\frac 1,2}\cdot d_\nu^{3+\alpha}$$ which we will use when $|l|\le 4\,\norm{\phi'}_\infty\,|n|$, to obtain $$\abs{\sum_{|l|\le 4\,\norm{\phi'}_\infty\,|n|}\hat\mu(n)\cdot I(n,l)} \le C\,\lambda^{-\frac 1,2}\cdot d_\nu^{4+\ffra\alpha,2} \eqno\eqma$$ When $|l|\ge 4\,\norm{\phi'}_{\infty}\,|n|$, we have $|\phi'(x)-\fra l,n|\ge \fra |l|,{4|n|}$; thus, we apply {\eqhb} directly to $I(n,l)$ to obtain $$\abs{I(n,l)}\le \bracket{\lambda |n|}^{-1}\>{d_\nu^{4+\alpha}\over \bra{|l|\over 4|n|}^2} + 4\,\bracket{\lambda |n|}^{-2} \> \bra{{d_\nu^{3}\over \bra{|l|\over 4|n|}^2}+ {d_\nu^{3+\alpha}\over \bra{|l|\over 4|n|}^3}}$$ which implies that, for $n$ fixed, $$\abs{\sum_{\left\{ l \,:\, |l|\ge 4 \,\norm{\phi'}_\infty\, |n|\right\} }I(n,l)} \le C\, \bra{\lambda^{-1}\cdot d_\nu^{4+\alpha} +{\lambda^{-2}\cdot d_\nu^3\over |n|} +{\lambda^{-2}\cdot d_\nu^{3+\alpha}\over |n|}} $$ which finally implies $$\abs{\sum_{ |l|>4 \,\norm{\phi'}_\infty\, |n|}\hat\mu(n)\cdot I(n,l)} \le \lambda^{-1}\cdot d_\nu^3 \eqno\eqmb $$ Putting {\eqma} and {\eqmb} together, we obtain $$\abs{S_\nu}\le\lambda^{\frac 1,2}\cdot d_\nu^{3}$$ which implies $$\abs{\sum_{\nu>\nu_1}S_\nu}\le \lambda^{\frac 1,2}\cdot 2^{-3\,\nu_1}= \lambda^{\ffra 1,2-3\,\ep_1} . \eqno{\eqn}$$ Also, $$ \eqalignno{ \abs{\sum_{\nu>\nu_1} \> \sum_{(n,l):\>\Omega_{n,l}\in U_\nu} {\hat\mu(n)\>(\eta\cdot\theta_\nu)(\Omega_{n,l})\over |n\cdot\phi''(\Omega_{n,l})|^{\frac 1,2}} \> e^{2\pi i\lambda\theta(n,l)+\pi i\bracket{l-\ffra\sign(n),4}}} &\le C\,\sup_{x\in\cup_{\nu=\nu_1+1}^\infty U_\nu}|\eta(x)|\cr &=\bigo{ d_{\nu_1}^4}= \bigo{\lambda^{-4\,\ep_1}}.\cr}$$ and, putting {\eqr}, {\eqq}, {\eqp} and {\eqn} together, we obtain $$\Psi(Z)=\Psi_0(Z)+ \littleo{Z^{\frac 3,2}}\eqno\mathendpf$$ \Title{4. Lower Bounds} Theorem~{\thmd} told us two things: that $\Psi$ has a leading expression as a trigonometric sum, and that the size of this trigonometric sum, and therefore of $\Psi$, is {\it at most} of order $Z^{\frac 3,2}$. The question remains whether this bound is sharp or not. In related problems, such as the lattice point problem, sharp upper and lower bounds {\it on average} have been known for over fifty years (see {\Bleherb} and {\HeathBrown} for recent developments). This, in our context, would translate into the statement that indeed $Z^{\frac 3,2}$ is best possible. \vskip1em The aim of this section is to derive such estimates {\it on average} for the function $\Psi_0$. Classical ideas will work effortlessly after we show that a certain number is not 0. This number can be viewed as a certain (analytic, not necessarily arithmetic) $L$--function evaluated at the point $s=2$. Thus, it is not surprising that such a condition appears if one thinks, for example, about the lattice point problem for parabola. \vskip1em First, we will consider real values for $Z$, and then use this to study the case of interest, integer $Z$, as it relates to our original goal to understand the ground state energy of an atom of nuclear charge $Z$. \vskip1em We begin by defining $$ a_{n,l}={\hat\mu(n)\cdot \eta(\Omega_{n,l}) \over|\phi''(x_{n,l})\cdot n|^{\frac 1,2}} e^{\pi i(l-\ffra \sign n,4)} . $$ Let $\{\theta_\nu\}$ the set of all possible values of $\theta(n,l)$, selected such that $\theta_\nu\ne\theta_{\nu'}$ for $\nu\ne\nu'$. Define $$\eqalignno{ L\eqbydef L_{\eta,\phi}&=\sum_{\nu}\abs{\sum_{\theta(n,l)=\theta_\nu} a_{n,l}}^2 \cr}$$ which Lemma~{\lemm} will prove it to be non--zero; if, however, $L$ were equal to 0, then it is easy to see that in fact we would have that $$\Psi_0(Z)\equiv 0 \qquad{\rm all\ }Z . $$ In the meantime, we will simply assume $L\ne 0$. \vskip1em Define also $$\eqalignno{ C^\ast&=1+\bra{{48\cdot\norm\eta_\infty\cdot\norm{\phi'}_\infty \over c_0\cdot L}}^{\frac 2,3} \cr c^\sharp&=\inf\left\{\abs{\theta(n,l)-\theta(n',l')}\>:\> \theta(n,l)\ne\theta(n',l'){\rm\ and\ }|n|,|n'|\le C^\ast\right\} . \cr }$$ Therefore, setting as usual $\lambda=Z^{\frac 1,3}$, $$\eqalignno{ Z^{-3}\abs{\Psi_0(Z)}^2 &=\sum_{n,n',l,l'} a_{n,l}\cdot \overline{a_{n',l'}}\cdot e^{2\pi i\lambda\bra{\theta(n,l)-\theta(n',l')}} \cr &\ge \sum_{{\scriptstyle n,n',l,l'} \atop{\scriptstyle \theta(n,l)=\theta(n',l')}} a_{n,l}\cdot \overline{a_{n',l'}}\cr &\qquad +\sum_{\theta(n,l)\ne \theta(n',l')} a_{n,l}\cdot\overline{a_{n',l'}}\cdot e^{2\pi i\lambda\bra{\theta(n,l)-\theta(n',l')}} \cr \noalign{\vbox{\medskip The first term above is our $L$ defined above. In the second term, we separate the terms for small $|n|,|n'|$, which we keep untouched, and the ones for which {\it either} $n$ or $n'$ are large, which we estimate using the fact that $|l|\le \norm{\phi'}_\infty\cdot |n|$ and $|l'|\le \norm{\phi'}_\infty\cdot |n'|$, to obtain \medskip }} &\ge L -\sum_{{\scriptstyle \abs{\theta(n,l)-\theta(n',l')}>0}\atop {\scriptstyle |n|,|n'|\le C^\ast}} a_{n,l}\cdot\overline{a_{n',l'}}\cdot e^{2\pi i\lambda\bra{\theta(n,l)-\theta(n',l')}} \cr&\qquad -2c_0^{-1} \cdot\norm\eta_\infty^2\cdot\norm{\phi'}_\infty^2\cdot \sum_{{\scriptstyle |n|\ge C^\ast}\atop {\scriptstyle {\rm all\ } n'}} |n\cdot n'|^{-\frac 5,2} \cr &\ge L -\sum_{{\scriptstyle \abs{\theta(n,l)-\theta(n',l')}>0}\atop {\scriptstyle |n|,|n'|\le C^\ast}} a_{n,l}\cdot\overline{a_{n',l'}}\cdot e^{2\pi i\lambda\bra{\theta(n,l)-\theta(n',l')}} \cr&\qquad -24\cdot c_0^{-1} \cdot\norm\eta_\infty^2\cdot\norm{\phi'}_\infty^2\cdot \bra{C^\ast-1}^{-\frac 3,2} \cr }$$ which, by our choice of $C^\ast$, implies $$\eqalignno{ Z^{-3}\abs{\Psi_0(Z)}^2 &\ge\oh L -\sum_{{\scriptstyle \abs{\theta(n,l)-\theta(n',l')}>0}\atop {\scriptstyle |n|,|n'|\le C^\ast}} a_{n,l}\cdot\overline{a_{n',l'}}\cdot e^{2\pi i\lambda\bra{\theta(n,l)-\theta(n',l')}} . \cr }$$ Now we consider $Z_0\ge 1$ and $Z\le\oh Z_0$ and prepare to integrate both sides from $Z_0$ to $Z_0+Z$. For that, note that $dZ=3\lambda^2\,d\lambda$ and, if we set $\lambda_0=Z_0^{\frac 1,3}$, then $$\bra{Z_0+Z}^{\frac 1,3}=\lambda_0+\Lambda ,\qquad \Lambda\le {Z/Z_0^{\frac 2,3}} . $$ Also, $$\abs{\int_{a}^b\lambda^2e^{i\lambda\theta}\,d\lambda} \le {3b^2\over \theta} , \qquad a,b>0 . $$ Therefore, $$\displaylines{ \quad \int_{Z_0}^{Z_0+Z}\abs{\Psi_0(z)}^2\>{dz\over z^3} \hfill\cr \hfill \eqalign{ &\ge{Z\over 2}L - 36\,Z_0^{\frac 2,3} c_0^{-1} \cdot\norm\eta_\infty^2\cdot\norm{\phi'}_\infty^2\cdot \sum_{{\scriptstyle \abs{\theta(n,l)-\theta(n',l')}>0}\atop {\scriptstyle |n|,|n'|\le C^\ast}} {|n\cdot n'|^{-\frac 5,2} \over 2\pi \abs{\theta(n,l)-\theta(n',l')}} \cr &\ge{Z\over 2}L - 36\,Z_0^{\frac 2,3} { \norm\eta_\infty^2\cdot\norm{\phi'}_\infty^2 \over c_0\cdot c^\sharp} \sum_{|n|,|n'|\le C^\ast} |n\cdot n'|^{-\frac 5,2} \cr &\ge{Z\over 2}L\> -Z_0^{\frac 2,3} \, {18000\cdot \norm\eta_\infty^2\cdot\norm{\phi'}_\infty^2 \over c_0\cdot c^\sharp} . \cr } \quad\cr}$$ As a consequence, taking $Z/Z_0^{\frac 2,3}$ large depending on $L$, $c_0$, $c^\sharp$, $\norm\eta_\infty$ and $\norm{\phi'}_\infty$, but still $Z$ not larger than $\oh\,Z_0$, we have $$ \int_{Z_0}^{Z_0+Z}\abs{\Psi_0(z)}^2\>{dz\over z^3} \ge{Z\over 4}L . $$ Next, we turn to the non--vanishing of $L$. In preparation for the proof, set $$\alpha_1=\phi'(0) , \qquad \alpha_2=\phi' \bra{a^{-\frac 1,2}\cdot\Omega_c}$$ and note that $0>\alpha_1>\alpha_2$. Clearly, the $(n,l)$ that enter in the sum for $\Psi_0$ are determined by the lattice points in ${\bf Z}^2$ which fall in the double--cone $$\Gamma=\{(u,v):\quad \alpha_2\cdot \abs u\le -\abs v<\alpha_1\cdot \abs u, \qquad u\cdot v<0\}$$ Define $x_{u,v}$, for $(u,v)\in\Gamma$, as the unique point satisfying $$\phi'(x_{u,v})=\Fra v,n . $$ Note that $x_{u,v}$ is strictly positive. Then, we define $$\theta(u,v)=u\cdot \phi(x_{u,v})-v\cdot x_{u,v} , \qquad (u,v)\in\Gamma .$$ The following is a trivial fact \lem{\lemn} On $\Gamma$ we have $$\nabla\theta(u,v)=\bra{\phi(x_{u,v}), -x_{u,v}}$$ \endt A consequence of this trivial fact is the result we mentioned above. \lem{\lemm} $L\ne 0$. \proof We will show that there is one $\theta_\nu$ which only has one $(n,l)\in\Gamma$ such that $\theta(n,l)=\theta_\nu$. This clearly shows that $L$ is not 0. \vskip1em Let $n_0$ be the smallest positive integer such that $(n,l)\in\Gamma$ for some $l$. In our case, this $l$ is negative. Choose the largest (negative) such $l$, which we denote by $l_0$. That is, $l_0$ is the largest negative integer that satisfies $l< \alpha_1\cdot n_0$. Then we claim that there is no other pair $(n',l')\in\Gamma$ such that $\theta(n_0,l_0)=\theta(n',l')$. Indeed, since $\theta(n,l)$ is strictly positive for $n>0$, and strictly negative for $n<0$, such $n'$ would also have to be positive. It cannot equal $n_0$ because, since we should have $l'\theta(n_0,l_0)$$ We must then have $n'>n_0$, which also implies $$l'<\alpha_1\cdot n'<\alpha_1\cdot n_0\quad (<0)$$ thus showing that $l'\le l$. But this is also impossible because, also by the previous lemma, there exists a pair $(\xi,\eta)$ on the segment joining $(n_0,l_0)$ to $(n',l')$, such that $$\theta(n',l')-\theta(n_0,l_0)=\phi(x_{\xi,\eta})\cdot (n'-n_0)-x_{\xi,\eta}\cdot(l'-l_0)$$ and this last expression is then strictly positive. \endpf We summarize all this in the following lemma \lem{\lems} There is a small constant $\kappa_0$ and a large constant $K$ such that $$\int_{Z_0}^{Z_0+Z}z^{-3}\abs{\Psi_0(z)}^2\,dz\ge \kappa_0 \cdot Z$$ whenever $Z\ge K\cdot Z_0^{\frac 2,3}$, and $Z_0\ge K$. \proof Our previous calculations show this result in the case that $Z\ge C\,Z_0^{\frac 2,3}$ but $Z\le \oh \, Z$, for a certain large constant $C$. For the general case, break up $$\int_{Z_0}^{Z_0+Z}z^{-3}\abs{\Psi_0(z)}^2\,dz =\sum_{n=0}^N \int_{Z_n}^{Z_{n+1}}z^{-3}\abs{\Psi_0(z)}^2\,dz$$ where $$Z_{n+1}=1.01\, Z_{n}\quad {\rm when\ }n=0,\dots,N-1; \qquad Z_{N+1}=Z_0+Z$$ and $N$ is chosen so that $1.01^{N+1}\,Z_0\le Z_0+Z<1.01^{N+2}\,Z_0$. Our previous calculations would apply to each of the integrals in the sum provided $$Z_n-Z_{n-1}\ge C\,Z_{n-1}^{\frac 2,3} ,\quad(n=1,\ldots,N); \qquad Z+Z_0\ge Z_{N}^{\frac 2,3}$$ which amounts to requiring $Z_0\ge 10^6$. \endpf This mean value information can be used to obtain information about the oscillating behavior of $\Psi_0$, as follows: Let $$I=\ccint Z_0,{Z_0+\hat CZ_0^{\frac 2,3}}=\bigcup I_j$$ for $$I_j=\ccint {Z_0+\hat c\cdot j\cdot Z_0^{\frac 2,3}} , {Z_0+\hat c\cdot (j+1)\cdot Z_0^{\frac 2,3}} $$ where $\hat C$ is large depending on $K$ and $\hat c$ is small depending on $\kappa_0$. \vskip1em Denote also, for any function $f$, $$m_j(f)=\inf_{x\in I_j}\abs{x^{-\frac 3,2}f(x)} . $$ \corol{\corolb} Given any $\ep>0$ small depending on $k_0$, any $\hat C$ and $\hat c$ as above, with the extra requirement that $\hat c$ is small depending on $\ep$, and $Z_0$ also large enough, there exists a constant $0<\alpha<1$ such that $m_j(\Psi_0)>\ep$ for at most $\alpha \hat C/\hat c$ of the $I_j$ . \proof Put $$Z^{-\frac 3,2}\Psi_0(Z)=F(Z)+E(Z)$$ such that $F(Z)$ contains only finitely many terms in the sum, and $|E(Z)|$ is always less than $\ep$. In particular, we have that in order that $m_j(\Psi_0)<\ep$ we must have $m_j(F)<2\ep$. It will therefore be enough to count how many of the $m_j(F)$ stay below $2\ep$ to obtain the conclusion of the lemma. \vskip1em Since both $\phi$ and $\phi'$ are bounded, we have the trivial bound $$\abs{F'(Z)}\le C_\ep\,Z^{-\frac 2,3}$$ for some constant $C_\ep$ which depends on $\ep$. Thus, if $m_j(\Psi_0)\le \ep$, we have $F(Z)\le 2\ep+\hat c\cdot C_\ep$ which implies $$\int_{I_j}|F(z)|^2\,dz\le 16\, |I_j|\cdot\bra{\ep^2+\hat c^2\cdot C_\ep^2} . $$ Therefore, if we denote by $$M={\rm number \ of\ }j{\rm\ such\ that \ }m_j(\Psi_0)<\ep$$ we use the trivial bound $\abs{F(z)}\le C$ for all $z$, for a universal constant $C$, to get $$\eqalignno{ \kappa_0\cdot\hat C\cdot Z_0^{\frac 2,3} &\le\int_{I}\abs{F(z)}^2\,dz\cr &\le \bra{{\hat C\over\hat c}-M}\cdot C\cdot \hat c\cdot Z_0^{\frac 2,3} + 16\,M\, \hat c\cdot Z_0^{\frac 2,3}\cdot\bra{\ep^2+\hat c^2\cdot C_\ep^2} \cr }$$ This implies that $$M\le\alpha{\hat C\over \hat c} \quad{\rm for\ } \alpha={C-\kappa_0\over C-16(\ep^2+\hat c^2\cdot C_\ep^2)} $$ By adjusting $\hat c$ depending on $\ep$ it is easy to make $\alpha<1$. \endpf A consequence of this corollary is another which shows that the size of $\Psi_0(Z)$ is at most $cZ^{\frac 3,2}$, for a small constant $c$, even when we restrict our attention to integer values of $Z$. \corol{\corolc} $$\liminf_{\ttower {Z\to\infty} {Z=1,2,3,\ldots}} \abs{Z^{-\frac 2,3}\cdot\Psi_0(Z)}\ne 0$$ \proof Apply the previous corollary to any $\ep>0$ small as required, and to any $\hat C$ large and $\hat c$ small also as needed, and then to infinitely $Z_0$ to conclude that there are infinitely many intervals of lengths going to infinity where $|x^{-\frac 3,2}\cdot\Psi_0(x)|$ is never smaller than $\ep$. \endpf \Title{5. The Classical Picture} \def\rmin{{r_{\rm min}}} \def\rmax{{r_{\rm max}}} \def\tmin{{t_{\rm min}}} \def\tmax{{t_{\rm max}}} In this Section we identify all quantities appearing in the expression for $\Psi_0$ in terms of data coming from the classical dynamics of a particle in the field created by the Thomas--Fermi potential. We begin with a brief review of elementary classical mechanics, which can be found, among many other places, in {\Arnold}. \vskip1em Consider a particle with mass $\oh$, in $\reals 3$, moving in a negative radial potential $-V(r)$, which for us, will equal $-V^1_{\rm TF}(r)$. The motion is planar, and can be described by the distance to the origin $r(t)$ and the angle $\varphi$, which satisfy the relations $$\dot \varphi={2M\over r^2}, \qquad \fra 1,4\bra{\dot r^2+r^2\dot\varphi^2}-V(r)=E$$ where $M$ is the angular momentum, and $E$ is the energy of the orbit. We begin assuming that the particle travels counter--clockwise in our frame of reference $(r,\varphi)$. The motion takes place between radii $r_{\rm min}$ and $r_{\rm max}$ given by the two solutions of the equation $$ -V(r)+{M^2\over r^2}=E.$$ This implies that all trajectories for a fixed energy and angular momentum can be obtained by rotation of a fixed one. \vskip1em At energy 0, we denote by $\tmin$ and $\tmax$ the times at which the particle passes through $\rmin$ and $\rmax$ respectively. The angle of motion $\varphi$ and the distance to the origin $r$ satisfy the equations $$ {dr\over dt}=2\sqrt{V(r)-{M^2\over r^2}},\qquad {d\varphi\over dr}={{M/r^2}\over\sqrt{ V(r)-{M^2\over r^2}}} . $$ As a consequence, the particle going from $r_{\rm min}$ to $r_{\rm max}$ sweeps out an angle given by $$M\int_{r_{\rm min}}^{r_{\rm max}}\bra{V(r)-{M^2\over r^2}} ^{-\frac 1,2}\,{dr\over r^2}= -\pi\phi'\bra{M} . $$ and the trajectory is clearly symmetric with respect to either $r_{\rm min}$ (or $r_{\rm max}$). Therefore, the angular momentum $M$ will give rise to a closed orbit if and only if $$-\phi'\bra{M}=\fra l,n \eqno{\eqw}$$ and in this case, $n$ represents the number of times the particle oscillates between successive $r_{\rm min}$ (or $\rmax$) before closing, and $l$ represents the winding number of the orbit around 0. Our initial assumption that the particle travels counter--clockwise means that $n,l\ge 0$. In our previous notation, we also have $$M=\Omega_{n,-l}$$ If $(l,n)=1$ (where $(,)$ denotes greatest common divisor), the orbit is usually called {\it primitive}. The period is given by $$\eqalignno{ T(M)&= 2n\cdot \int_\tmin^\tmax \,dt\cr &= n\int_{\rmin}^{\rmax} {dr\over\sqrt{V(r)-\fra M^2,{r^2}}}\cr &=n\cdot P\bra{M}. \cr}$$ In order to find the action $S$ along this closed trajectory, $$S=2n\cdot \int_{ \tmin}^\tmax \bra{{\rm Kinetic\ Energy\ }+V}\,dt , $$ we note that, since we are at energy 0, Kinetic Energy = $V(r)$, which implies $$ \eqalignno{ S&=4n\int_\rmin^\rmax {V(r)\over 2\sqrt{V(r)-\Fra {M^2},{r^2}}}\,dr\cr &=2n\bra{\int_\rmin^\rmax\sqrt{V-\Fra M^2,{r^2}}\,dr+ \int_\rmin^\rmax{M^2/r^2\over \sqrt{V-\Fra M^2,{r^2}}}\,dr }\cr &=2\pi\bra{n\phi\bra{M}+l\cdot M}\cr &=2\pi\bra{n\phi(\Omega_{n,-l})+l\cdot\Omega_{n,-l}} . \cr}$$ As a consequence, denoting by $S(M)$ as the action along a closed counter--clockwise trajectory at energy 0 with angular momentum $M$, we have $$2\pi \theta(n,-l)= S\bra{\Omega_{n,-l}}$$ When the particle travels clockwise, we agree that $S$, $T$, $n$ and $l$ change sign, but we keep $M\ge 0$. We have so far identified all terms in the definition of $\Psi_0$ except $n\cdot\phi''(\Omega_{n,l})$. For this one, consider a closed trajectory arising from angular momentum $M$, which gives rise to $n$ oscillations between successive $\rmax(M)$; for $\ep$ small, consider a trajectory with angular momentum $M+\ep$ which begins at $\rmax(M+\ep)$ and denote by $2\pi\alpha_M(\ep)$ the absolute value of the angle the particle forms between the initial position at $\rmax(M+\ep)$ and the position after $n$ oscillations also at $\rmax(M+\ep)$, where we take $\alpha$ between 0 and 1/2. Then $$D(M)\eqbydef\lim_{\ep\to 0}\ep^{-1}\alpha_M(\ep)=|n\cdot\phi''(M)|$$ It is clear now that the nonvanishing of the second derivative of $\phi$ translates into the fact that closed trajectories are isolated modulo the trivial symmetry given by the rotation group. \vskip1em The motion degenerates for the one circular orbit arising from $M_{\rm max}=a^{-\frac 1,2}\, \Omega_c$, the maximum angular momentum allowed in our system. In this case, we define the above classical variables simply in terms of $\phi'$ using the formulas we derived for the other trajectories. \theorem{\thme} $$\Psi_0(Z)=2\pi \cdot Z^{\frac 3,2} \cdot \sum_{\rm closed\ trajectories} \delta \,{n\, \hat\mu(n)\, M\over T} \cdot\abs{D(M)}^{-\frac 1,2} \cdot e^{i\Bracket{Z^{\frac 1,3} S-\pi\cdot (l+\fra{\sign n},4)}} $$ where circular trajectories appear in the sum only when they has associated a finite number of oscillations $n$. We have $\delta=1$ except for the circular trajectories, when $\delta=\oh$. The sum is absolutely convergent. \endt \title{Remark:} Note that the contribution of each trajectory depends on the particular frame of reference we take to compute $n$ and $l$, but the sum is independent of it. \title{Remark:} One might think of the different values for $\delta$ as follows: non--circular trajectories contribute fully to eigenvalues, while the circular ones, being right at the outskirts of the classically allowed region, contribute half as eigenvalues, half as resonances. \Title{6. Further Considerations} In this section we will compute the derivatives of $\phi$ at the ends of our interval of interest $\ccint 0,{a^{-\frac 1,2}\cdot\Omega_c}$. The derivative at $a^{-\frac 1,2}\cdot\Omega_c$ plays a role in the sense described on {\bf The Heart of the Matter}, since its rationality or irrationality translates in the appearance or absence of a certain contribution to $\Psi$ or size $Z^{\frac 3,2}$. The derivative at 0 does not play such a role since the amplitude vanishes there. \lem{\lemq} $$-{1\over\pi}F'\bra{\Omega_c}=\bra{1-\oh r_c^{\frac 3,2} \cdot y_c^{\frac 1,2}}^{-\frac 1,2} ={1\over\sqrt{1-\oh r_c\cdot \Omega_{c}}} \sim 1.9376783$$ \proof We use the change of variables $t(r)$ given by {\sdd}, and its inverse $r(t)$, to write $$-F'\bra{\Omega_c}=\Omega_c\cdot\int_{-1}^1 (1-t^2)^{-\frac 1,2}\cdot w(0)\,dt,$$ with $$w(0)={r'(0)\over r_c}={1\over r_c\cdot t'(r_c)}= {\sqrt 2\over r_c\cdot \abs{u''(r_c)}^{\frac 1,2}}.$$ Thus, $$-{1\over\pi}F'(\Omega_c) ={\Omega_c\over r_c\cdot \abs{\oh u''(r_c)}^{\frac 1,2}} =\abs{2y(r_c)\over r_c\cdot u''(r_c)}^{\frac 1,2} $$ since $${1\over\pi}\int_{-1}^1(1-t^2)^{-\frac 1,2}\,dt=1.$$ Manipulations using the identities $$u(x)=xy(x),\quad u'(x)=xy'(x)+y(x), \quad u''(x)=xy''(x)+2y'(x)=x^{\frac 1,2}y^{\frac 3,2}(x)+2y'(x)$$ and $$r_c\, y'(r_c)=-y(r_c)\qquad $$ yield our result. \endpf \lem{\lemp} $$-\lim_{\Omega\to 0}F'(\Omega)= \fra 3,2 \pi$$ \proof Let $r_0(\Omega)$ and $r_1(\Omega)$ be the two solutions of $u(r)=\Omega^2$. We study first the asymptotics of $r_0$ and $r_1$. For $r_0$, put $z=r_0^{\frac 1,2}$; then, for $$f(z)=u(z^2)=z^2-wz^4+\bigo{z^5}$$ we have that $f(z)=\Omega^2$. This implies that $z=\Omega+\bigo{\Omega^2}$ and thus $$r_0(\Omega)=\Omega^2+\bigo{\Omega^3}.$$ For $r_1$, since $u(r)$ decreases monotonically to $0$, $r_1(\Omega)\to\infty$. Since we have that $u(r)=144r^{-2}+\bigo{r^{-2-\alpha}}$, for $\alpha>0$, we get that $$r_1(\Omega)={12\over \Omega}+\littleo{\Omega^{-1}}.$$ \bigskip In order now to analyze $F'(\Omega)$, take $\epsilon$ be a small enough constant to be picked up later and rewrite $$\eqalignno{ g(\Omega)&=\int_{r_0(\Omega)}^\epsilon \bra{u(r)-\Omega^2}^{-\frac 1,2}\,{dr\over r}+ \int^{r_1(\Omega)}_\epsilon \bra{u(r)-\Omega^2}^{-\frac 1,2}\,{dr\over r}&\one\cr &=\int_{r_0(\Omega)}^\epsilon \bra{u'(r_0)(r-r_0)}^{-\frac 1,2}\,{dr\over r}+ \int^{r_1(\Omega)}_\epsilon \bra{u(r)-\Omega^2}^{-\frac 1,2}\,{dr\over r}\cr &\qquad+R_0(\Omega)\cr}$$ for $$\eqalignno{ R_0&= \int_{r_0}^\epsilon\bra{\bra{u(r)-\Omega^2}^{-\frac 1,2} -\bra{u'(r_0)(r-r_0)}^{-\frac 1,2}}\,{dr\over r}\cr }$$ We show first that $R_0=\bigo{1}$. Fix $\Omega$: $$\displaylines{ \quad\bra{u(r)-\Omega^2}^{-\frac 1,2} -\bra{u'(r_0)(r-r_0)}^{-\frac 1,2}=\hfill\cr \hfill=\sum_{n=1}^\infty c_n\bra{u'(r_0)(r-r_0)}^{-n-\frac 1,2}\cdot \bbracket{\bra{u(r)-u(r_0)}-u'(r_0)\bra{r-r_0}}^n \quad\cr}$$ Throughout this analysis, $c_n$ will denote a generic sequence of bounded constants. Note that $$\eqalignno{ \abs{u(r)-u(r_0)-u'(r_0)(r-r_0)}&\le \oh\sup_{0\le r\le \epsilon} \abs{u''(r)}\cdot\bra{r-r_0}^2\cr &\le \bra{y'(0)+\oh\epsilon^{\frac 1,2}} \cdot\bra{r-r_0}^2\cr &\le C_0\bra{r-r_0}^2\cr}$$ since $u''(r)=2y'(r)+ry''(r)$, $\abs{y'(r)}\le \abs{y'(0)}$ and $y''(r)\le r^{-\frac 1,2}$. Therefore, the sum converges uniformly for $|r-r_0|<\oh C_0$ and integrating with respect to $dr/r$ on $(r_0,\epsilon)$, for $\epsilon<\oh C_0$, we obtain $$\eqalignno{R_0&\le\sum_{n=1}^\infty |c_n|\cdot|u'(r_0)|^{-n-\frac 1,2} \int_{r_0}^\epsilon C_0^n\abs{r-r_0}^{n-\frac 1,2}\,{dr\over r}\cr &\le\sum_{n=1}^\infty |c_n|\cdot|u'(r_0)|^{-n-\frac 1,2} C_0^nr_0^{n-\frac 1,2}\int_{1}^{\frac\epsilon,{r_0}}(y-1)^{n-\frac 1,2} \,{dy\over y}\cr &\le\sum_{n=1}^\infty |c_n|\cdot|u'(r_0)|^{-n-\frac 1,2} C_0^nr_0^{n-\frac 1,2}\cdot\left. (y-1)^{n-\frac 1,2}\right|^{r_0^{-1}\epsilon}_1 \cr &\le\sum_{n=1}^\infty |c_n|\cdot|u'(r_0)|^{-n-\frac 1,2} C_0^n\epsilon^{n-\frac 1,2} \cr }$$ For $\Omega$ small enough, we can make $|u'(r_0)|<2$, and this will make the previous sum converge to $\bigo{1}$ for $\epsilon$ small, thus proving that $R_0$ is bounded. Recall now that $\Omega r_0(\Omega)^{-\frac 1,2}\to 1$, what implies $$\eqalignno{\Omega\cdot \int_{r_0}^\epsilon\bra{r-r_0}_+^{-\frac 1,2}\,{dr\over r} &= \Omega\cdot r_0^{-\frac 1,2}\int_1^{r_0^{-1}\epsilon} y^{-1}(y-1)^{-\frac 1,2}\,dy\cr &\to \int_1^\infty y^{-1}(y-1)^{-\frac 1,2}\,dy\cr &=\pi\cr}$$ which, with the fact that $u'\bra{r_0(\Omega)}\to 1$, implies that $$\eqalignno{ \lim_{\Omega\to 1}\Omega\int^\epsilon_{r_0} \bra{u(r)-\Omega^2}^{-\frac 1,2}\,{dr\over r} &= \lim_{\Omega\to 1}\Omega u'(r_0)^{-\frac 1,2}\int_{r_0(\Omega)}^\epsilon \bra{r-r_0(\Omega)}^{-\frac 1,2}\,{dr\over r} \cr&=\pi\cr}$$ which completes the analysis of the first integral in \one. \bigskip As for the other integral, we break it up into 5 pieces as follows: $$\eqalignno{ \int_\epsilon^{r_1}\bra{u(r)-\Omega}^{-\frac 1,2}\,{dr\over r} &=\int_\epsilon^M+\int_M^{r_1^{\ffra 99,{100}}}+ \int_{r_1^{\ffra 99,{100}}}^{r_1/2}+\int_{r_1/2}^{r_1-r_1^{\ffra 2,{3}}}+ \int_{r_1-r_1^{\ffra 2,{3}}}^{r_1}\cr &= I_1+I_2+I_3+I_4+I_5\cr}$$ It is clear that $I_1=\bigo{1}$, so we don't worry about it. For $I_2$, note that, if $M$ is large enough so that $u$ is decreasing from $M$ on, we have on its domain that $$\eqalignno{\abs{u(r)-\Omega^2}&\ge\abs{u\bra{r_1^{\ffra 99,{100}}} -u(r_1)}\cr &\ge C\bra{r_1^{\ffra 99,{100}}}^{-2}-cr_1^{-2} \cr &\ge cr_1^{-\ffra 99,{50}}\cr}$$ and thus $$\eqalignno{|I_2|&\le C\int_M^{r_1^{\ffra 99,{100}}} \abs{r_1^{\ffra 99,{50}}}^{\frac 1,2}\,{dr\over r}\cr &\le Cr_1^{\ffra 99,{100}}\log r_1\cr}$$ which does not contribute to the result. Similarly, for $I_5$, note that $$\eqalignno{\abs{u(r)-\Omega^2}& \ge \bra{\inf_{|r-r_1|\le r_1^{\frac 2,{3}}}|u'(r)|}\cdot |r-r_1|\cr &\ge Cr_1^{-3}|r-r_1|\cr}$$ thus $$\eqalignno{|I_5|&\le C\int_{r_1-r_1^{\frac 2,{3}}}^{r_1}r_1^{\frac 3,2} (r_1-r)^{-\frac 1,2}\,{dr\over r}\cr &\le Cr_1^{\frac 1,2}\int_{r_1-r_1^{\frac 2,{3}}}^{r_1} (r_1-r)^{-\frac 1,2}\,dr\cr &\le C r_1^{\frac 1,2}\cdot r_1^{\frac 1,{3}}\cr &\le C r_1^{\frac 5,{6}}\cr}$$ and again, it does not contribute to the total outcome. For $I_4$, note that $$\eqalignno{|u(r)-\Omega^2|&\ge \bra{\inf_{r\sim r_1} |u'(r)|}\cdot r_1^{\frac 2,{3}}\cr &\ge Cr_1^{-3}r_1^{\frac 2,{3}}\cr }$$ This implies two things: First, since $|u(r)-144r^{-2}|\le C r^{-2-\alpha}\le Cr_1^{-2-\alpha}$, for $\alpha = \oh(\sqrt{73}-7)>1-{2\over 3}$, we have that $144r^{-2}-\Omega^2>0$ on this range, for $\Omega$ small enough. And second, $$\eqalignno{\abs{\bra{u(r)-\Omega^2}^{-\frac 1,2}-\bra{ {144\over r^2}-\Omega^2}^{-\frac 1,2}}&\le\sum_{n=1}^\infty c_n|u(r)-\Omega^2|^{-n-\ffra 1,2}\bra{Cr^{-2-\alpha}}^n\cr &\le \sum_{n=1}^\infty c_n\bra{Cr_1^{3-\ffra 2,{3}}}^{n+\ffra 1,2} r^{-n(2+\alpha)}\cr}$$ and thus $$\eqalignno{ \int_{r_1/2}^{r_1-r_1^{\ffra 19,{30}}} \left|\bra{u(r)-\Omega^2}^{-\frac 1,2}-\bra{ {144\over r^2}-\Omega^2}^{-\frac 1,2}\right|\, {dr\over r} &\le\sum_{n=1}^\infty c_n\bra{Cr_1^{3-\ffra 2,{3}}}^{n+\ffra 1,2} r_1^{-n(2+\alpha)}\cr &\le Cr_1^{\ffra 7,{3}\cdot \ffra 3,2-(2+\alpha)}\cr &= \littleo{r_1(\Omega)}\cr}$$ Finally, for $I_3$, $$\displaylines{\quad\int_{r_1^{\ffra 99,{100}}}^{r_1/2} \left|\bra{u(r)-\Omega^2}^{-\frac 1,2}-\bra{ {144\over r^2}-\Omega^2}^{-\frac 1,2}\right|\,{dr\over r} \le\hfill\cr \hfill\eqalign{&\le\int_{r_1^{\ffra 99,{100}}}^{r_1/2} \sum_{n=1}^\infty c_n\bra{u(\fra r_1,2)-\Omega^2}^{-n-\ffra 1,2} (cr)^{-n(2+\alpha)}\,{dr\over r}\cr &\le \int_{r_1^{\ffra 99,{100}}}^{r_1/2} \sum_{n=1}^\infty c_n\abs{Cr_1^{-2}}^{-n-\ffra 1,2}\cdot r^{-n(2+\alpha)} \,{dr\over r}\cr &\le \sum_{n=1}^\infty c_n\abs{Cr_1^{2}}^{n+\ffra 1,2}\cdot r_1^{-\ffra 99,{100} n(2+\alpha)}\cr &\le Cr_1^{3}\cdot r_1^{-\ffra 99,{100} (2+\alpha)}\cr &= \littleo{r_1(\Omega)}\cr}\quad\cr}$$ Therefore, the second integral in {\one} agrees modulo $\littleo{ \Omega^{-1}}$ with $$\eqalignno{\int_{r_1^{\ffra 99,{100}}}^{r_1-r_1^{\frac 2,{3}}} \bra{{144\over r^2}-\Omega^2}^{-\frac 1,2}\,{dr\over r}&= \int_{r_1^{\ffra 99,{100}}}^{r_1-r_1^{\frac 2,{3}}} \Bracket{{144\over r^2}}^{-\frac 1,2} \bra{1-{\Omega^2r^2\over 144}}^{-\frac 1,2}\,{dr\over r}\cr &= \Omega^{-1}\int_{{\Omega\over 12} r_1^{\ffra 99,{100}}} ^{{\Omega\over 12}(r_1-r_1^{\ffra 19,{30}})} (1-y^2)^{-\frac 1,2} \,dy\cr}$$ and therefore $$\eqalignno{ \lim_{\Omega\to 0}\Omega\int_\epsilon^{r_1(\Omega)} \bra{u(r)-\Omega^2}^{-\frac 1,2}&= \int_0^1 (1-y^2)^{-\frac 1,2} \,dy\cr &={\pi\over 2}\cr}$$ which proves the lemma. \endpf \title{Acknowledgments.} This research is partially supported by a N.A.T.O research grant no. {CRG921184}. A. C\'ordoba is partially supported by a CICYT grant. C. Fefferman is partially supported by an NSF grant. L. Seco is partially supported by NSERC grants no. OGP0121848 and {EQPEQ336}, by a CICYT grant and by a Connaught Fellowship. \bibliography \BBArnoldBB \BBBleheraBB \BBBleherbBB \BBDiracBB \BBEnglertBB \BBEnglertSchwingeraBB \BBEnglertSchwingerbBB \BBFeffntBB \BBFeffSecaBB \BBFeffSecbBB \BBFeffSeccBB \BBFeffSecdBB \BBFeffSeceBB \BBFeffSecfBB \BBFeffSecgBB \BBfs7BB \BBFermiBB \BBGrahamKolesnikBB \BBGutzwillerBB \BBHeathBrownBB \BBHilleBB \BBHughesBB \BBHuxleyBB \BBLiebtfBB \BBLiebSimonBB \BBIvriiSigalBB \BBIwaniecMozzochiBB \BBMarchPlaskettBB \BBScottBB \BBSchwingerBB \BBSiedentopWeikardBB \BBSiedentopWeikardLBBB \BBSiedentopWeikardLBIIBB \BBSimonBB \BBThomasBB \end