Lecture Notes in Mathematics (cid:50)(cid:48)(cid:48)(cid:48) Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris Harry Yserentant Regularity and Approximability of Electronic Wave Functions ABC HarryYserentant TechnischeUniversitätBerlin InstitutfürMathematik Straßedes17.Juni136 10623Berlin Germany [email protected] ISBN:978-3-642-12247-7 e-ISBN:978-3-642-12248-4 DOI:10.1007/978-3-642-12248-4 SpringerHeidelbergDordrechtLondonNewYork LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2010927755 MathematicsSubjectClassification(2000):35J10,35B65,41A25,41A63,68Q17 (cid:176)c Springer-VerlagBerlinHeidelberg2010 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember(cid:57), (cid:49)(cid:57)(cid:54)(cid:53),initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:SPiPublisherServices Printedonacid-freepaper springer.com Preface The electronic Schro¨dinger equation describes the motion of N electrons under Coulombinteractionforcesinafieldofclampednuclei.Solutionsofthisequation depend on 3N variables, three spatial dimensions for each electron. Approximat- ingthesolutionsisthusinordinatelychallenging,anditisconventionallybelieved that a reduction to simplified models, such as those of the Hartree-Fock method ordensityfunctionaltheory,istheonlytenableapproach.Thisbookseekstocon- vincethereaderthatthisconventionalwisdomneednotbeironclad:theregularity ofthesolutions,whichincreaseswiththenumberofelectrons,thedecaybehavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximatedwith an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The present notes arose from lectures that I gave in Berlin during the academic year 2008/09 to introduce beginning graduate students of mathematics intothissubject.Theyarekeptonanintermediatelevelthatshouldbeaccessibleto anaudienceofthiskindaswellastophysicistsandtheoreticalchemistswithacor- respondingmathematicaltraining.Thetextrequiresa goodknowledgeofanalysis totheextenttaughtatGermanuniversitiesinthefirsttwoyearsofstudy,including LebesgueintegrationandsomebasicfactsonBanachandHilbertspaces(comple- tion,orthogonality,projectiontheorem,Lax-Milgramtheorem,weakconvergence), butnodeeperknowledgeofthetheoryofpartialdifferentialequations,offunctional analysis, or quantum theory. I thank everybody with whom I had the opportunity to discuss the topic during the past years, my coworkersboth from Tu¨bingenand Berlin, above all Jerry Gagelman, who read this text very carefully, found many inconsistencies, and to whom I owe many hints to improve my English, and par- ticularly my colleagues Hanns Ruder, who raised my awareness of the physical background,andReinholdSchneider,whogenerouslysharedallhisknowledgeand insightintoquantum-chemicalapproximationmethods.TheDeutscheForschungs- gemeinschaft supported my work through several projects, inside and outside the DFG-ResearchCenterMATHEON.IdedicatethisbooktomysonsKlausandMax. Berlin,September2009 HarryYserentant v Contents 1 IntroductionandOutline...................................................... 1 2 FourierAnalysis................................................................ 13 2.1 RapidlyDecreasingFunctions............................................ 13 2.2 IntegrableandSquareIntegrableFunctions ............................. 17 2.3 SpacesofWeaklyDifferentiableFunctions.............................. 20 2.4 FourierandLaplaceTransformation..................................... 25 3 TheBasicsofQuantumMechanics .......................................... 27 3.1 Waves,WavePackets,andWaveEquations ............................. 28 3.2 TheSchro¨dingerEquationforaFreeParticle ........................... 29 3.3 TheMathematicalFrameworkofQuantumMechanics................. 32 3.4 TheHarmonicOscillatorandItsEigenfunctions........................ 37 3.5 TheWeakFormoftheSchro¨dingerEquation ........................... 44 3.6 TheQuantumMechanicsofMulti-ParticleSystems.................... 46 4 TheElectronicSchro¨dingerEquation....................................... 51 4.1 TheHardyInequalityandtheInteractionEnergy....................... 52 4.2 SpinandthePauliPrinciple .............................................. 55 5 SpectrumandExponentialDecay............................................ 59 5.1 TheMinimumEnergyandtheIonizationThreshold.................... 60 5.2 DiscreteandEssentialSpectrum......................................... 61 5.3 TheRayleigh-RitzMethod ............................................... 71 5.4 TheLowerBoundoftheEssentialSpectrum............................ 78 5.5 TheExponentialDecayoftheEigenfunctions .......................... 82 6 ExistenceandDecayofMixedDerivatives.................................. 87 6.1 AModifiedEigenvalueProblem ......................................... 88 6.2 SpacesofFunctionswithHigh-OrderMixedDerivatives .............. 91 6.3 EstimatesfortheLow-OrderTerms,Part1.............................. 93 6.4 EstimatesfortheLow-OrderTerms,Part2.............................. 97 vii viii Contents 6.5 TheRegularityoftheWeightedEigenfunctions.........................103 6.6 AtomsasModelSystems.................................................111 6.7 TheExponentialDecayoftheMixedDerivatives.......................115 7 EigenfunctionExpansions ....................................................117 7.1 DiscreteRegularity........................................................118 7.2 Antisymmetry.............................................................123 7.3 HyperbolicCrossSpaces .................................................124 8 ConvergenceRatesandComplexityBounds................................127 8.1 TheGrowthoftheEigenvaluesinthe3d-Case..........................128 8.2 ADimensionEstimateforHyperbolicCrossSpaces ...................131 8.3 AnAsymptoticBound....................................................134 8.4 AProofoftheEstimateforthePartitionNumbers......................135 8.5 TheComplexityoftheQuantumN-BodyProblem .....................139 9 TheRadial-AngularDecomposition.........................................141 9.1 Three-DimensionalSphericalHarmonics................................142 9.2 TheDecompositionofN-ParticleWaveFunctions......................151 9.3 TheRadialSchro¨dingerEquation........................................154 9.4 AnExcursustotheCoulombProblem...................................159 9.5 TheHarmonicOscillator .................................................162 9.6 EigenfunctionExpansionsRevisited.....................................164 9.7 ApproximationbyGaussFunctions......................................165 9.8 TheEffectofScaling .....................................................167 Appendix:TheStandardBasisoftheSphericalHarmonics ..................170 References...........................................................................177 Index.................................................................................181 Chapter 1 Introduction and Outline Theapproximationofhigh-dimensionalfunctions,whethertheybegivenexplicitly orimplicitlyassolutionsofdifferentialequations,representsoneofthegrandchal- lengesofappliedmathematics.High-dimensionalproblemsariseinmanyfieldsof applicationsuch asdata analysisand statistics, butfirst ofallin the sciences. One of the most notorious and complicated problems of this type is the Schro¨dinger equation.The Schro¨dingerequationformsthe basis of quantummechanicsand is offundamentalimportanceforourunderstandingofatomsandmolecules.Itlinks chemistrytophysicsanddescribesasystemofelectronsandnucleithatinteractby Coulombattractionandrepulsionforces.AsproposedbyBornandOppenheimerin thenascencyofquantummechanics,theslowermotionofthenucleiismostlysepa- ratedfromthatoftheelectrons.ThisresultsintheelectronicSchro¨dingerequation, theproblemtofindtheeigenvaluesandeigenfunctionsoftheHamiltonoperator H = −1 ∑N Δ − ∑N ∑K Zν + 1 ∑N 1 (1.1) 2 i=1 i i=1ν=1|xi−aν| 2i,j=1|xi−xj| i(cid:2)=j writtendownhereindimensionlessformoratomicunits.Itactsonfunctionswith argumentsx ,...,x ∈R3,whichareassociatedwiththepositionsoftheconsidered 1 N electrons.TheaνarethefixedpositionsofthenucleiandthepositivevaluesZνthe chargesofthenucleiinmultiplesoftheabsoluteelectroncharge. The mathematical theory of the Schro¨dinger equation for a system of charged particles is today a central, highly developed part of mathematical physics. Start- ingpointwasKato’swork[48]inwhichheshowedthatHamiltonoperatorsofthe given form fit into the abstract framework that was laid by von Neumann [64] a short time after Schro¨dinger [73] set up his equation and Born and Oppenheimer [11]simplifiedit.AnimportantbreakthroughwastheHunziker-vanWinter-Zhislin theorem [46,90,98], which states that the spectrum of an atom or molecule con- sists ofisolatedeigenvaluesλ ≤λ<Σoffinitemultiplicitybetweena minimum 0 eigenvalueλ andaionizationboundΣandanessentialspectrumλ≥Σ.Themath- 0 ematicaltheoryoftheSchro¨dingerequationtraditionallycentersonspectraltheory. Of at least equal importance in the given context are the regularity properties of the eigenfunctions,whose study began with [49]. For newer developmentsin this H.Yserentant,RegularityandApproximabilityofElectronicWaveFunctions, 1 LectureNotesinMathematics2000,DOI10.1007/978-3-642-12248-4 1, (cid:2)c Springer-VerlagBerlinHeidelberg2010