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MichioMasujima AppliedMathematicalMethods inTheoreticalPhysics Related Titles Vaughn,M.T. IntroductiontoMathematicalPhysics 2007 ISBN:978-3-527-40627-2 Trigg,G.L.(ed.) MathematicalToolsforPhysicists 2005 ISBN:978-3-527-40548-0 Boas,M.L. MathematicalMethodsin thePhysical Sciences InternationalEdition 2005 ISBN:978-0-471-36580-8 Tinker,M.,Lambourne,R. FurtherMathematicsforthePhysical Sciences 2000 ISBN:978-0-471-86691-6 Michio Masujima Applied Mathematical Methods in Theoretical Physics Second, Enlarged and Improved Edition TheAuthor AllbookspublishedbyWiley-VCHare carefullyproduced.Nevertheless,authors, Dr.MichioMasujima editors,andpublisherdonotwarrantthe Tokyo,Japan informationcontainedinthesebooks, includingthisbook,tobefreeoferrors. Readersareadvisedtokeepinmindthat statements,data,illustrations,procedural detailsorotheritemsmayinadvertentlybe inaccurate. LibraryofCongressCardNo.:appliedfor BritishLibraryCataloguing-in-Publication Data Acataloguerecordforthisbookisavailable fromtheBritishLibrary. Bibliographicinformationpublishedby theDeutscheNationalbibliothek DieDeutscheNationalbibliothek liststhispublicationintheDeutsche Nationalbibliografie;detailedbibliographic dataareavailableontheInternetat http://dnb.d-nb.de. 2009WILEY-VCHVerlagGmbH&Co. KGaA,Weinheim Allrightsreserved(includingthoseof translationintootherlanguages).Nopart ofthisbookmaybereproducedinany form–byphotoprinting,microfilm,orany othermeans–nortransmittedortranslated intoamachinelanguagewithoutwritten permissionfromthepublishers.Registered names,trademarks,etc.usedinthisbook, evenwhennotspecificallymarkedassuch, arenottobeconsideredunprotectedbylaw. Composition LaserwordsPrivateLimited, Chennai,India Printing StraussGmbH,Mo¨rlenbach Bookbinding Litges&DopfGmbH, Heppenheim CoverDesign K.Schmidt PrintedintheFederalRepublicofGermany Printedonacid-freepaper ISBN:978-3-527-40936-5 Tomygranddaughter,Honoka VII Contents Preface XI Introduction XV 1 FunctionSpaces,LinearOperators,andGreen’sFunctions 1 1.1 FunctionSpaces 1 1.2 OrthonormalSystemofFunctions 3 1.3 LinearOperators 5 1.4 EigenvaluesandEigenfunctions 7 1.5 TheFredholmAlternative 9 1.6 Self-AdjointOperators 12 1.7 Green’sFunctionsforDifferentialEquations 14 1.8 ReviewofComplexAnalysis 18 1.9 ReviewofFourierTransform 25 2 IntegralEquationsandGreen’sFunctions 31 2.1 IntroductiontoIntegralEquations 31 2.2 RelationshipofIntegralEquationswithDifferentialEquationsand Green’sFunctions 37 2.3 Sturm–LiouvilleSystem 43 2.4 Green’sFunctionforTime-DependentScatteringProblem 47 2.5 Lippmann–SchwingerEquation 51 2.6 ScalarFieldInteractingwithStaticSource 62 2.7 ProblemsforChapter2 67 3 IntegralEquationsoftheVolterraType 105 3.1 IterativeSolutiontoVolterraIntegralEquationoftheSecondKind 105 3.2 SolvableCasesoftheVolterraIntegralEquation 108 3.3 ProblemsforChapter3 112 AppliedMathematicalMethodsinTheoreticalPhysics,SecondEdition.MichioMasujima Copyright2009WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-40936-5 VIII Contents 4 IntegralEquationsoftheFredholmType 117 4.1 IterativeSolutiontotheFredholmIntegralEquationoftheSecond Kind 117 4.2 ResolventKernel 120 4.3 Pincherle–GoursatKernel 123 4.4 FredholmTheoryforaBoundedKernel 127 4.5 SolvableExample 134 4.6 FredholmIntegralEquationwithaTranslationKernel 136 4.7 SystemofFredholmIntegralEquationsoftheSecondKind 143 4.8 ProblemsforChapter4 143 5 Hilbert–SchmidtTheoryofSymmetricKernel 153 5.1 RealandSymmetricMatrix 153 5.2 RealandSymmetricKernel 155 5.3 BoundsontheEigenvalues 166 5.4 RayleighQuotient 169 5.5 CompletenessofSturm–LiouvilleEigenfunctions 172 5.6 GeneralizationofHilbert–SchmidtTheory 174 5.7 GeneralizationoftheSturm–LiouvilleSystem 181 5.8 ProblemsforChapter5 187 6 SingularIntegralEquationsoftheCauchyType 193 6.1 HilbertProblem 193 6.2 CauchyIntegralEquationoftheFirstKind 197 6.3 CauchyIntegralEquationoftheSecondKind 201 6.4 CarlemanIntegralEquation 205 6.5 DispersionRelations 211 6.6 ProblemsforChapter6 218 7 Wiener–HopfMethodandWiener–HopfIntegralEquation 223 7.1 TheWiener–HopfMethodforPartialDifferentialEquations 223 7.2 HomogeneousWiener–HopfIntegralEquationoftheSecond Kind 237 7.3 GeneralDecompositionProblem 252 7.4 InhomogeneousWiener–HopfIntegralEquationoftheSecond Kind 261 7.5 ToeplitzMatrixandWiener–HopfSumEquation 272 7.6 Wiener–HopfIntegralEquationoftheFirstKindandDualIntegral Equations 281 7.7 ProblemsforChapter7 285 8 NonlinearIntegralEquations 295 8.1 NonlinearIntegralEquationoftheVolterraType 295 8.2 NonlinearIntegralEquationoftheFredholmType 299 8.3 NonlinearIntegralEquationoftheHammersteinType 303 8.4 ProblemsforChapter8 305 Contents IX 9 CalculusofVariations:Fundamentals 309 9.1 HistoricalBackground 309 9.2 Examples 313 9.3 EulerEquation 314 9.4 GeneralizationoftheBasicProblems 319 9.5 MoreExamples 323 9.6 DifferentialEquations,IntegralEquations,andExtremizationof Integrals 326 9.7 TheSecondVariation 330 9.8 Weierstrass–ErdmannCornerRelation 345 9.9 ProblemsforChapter9 349 10 CalculusofVariations:Applications 353 10.1 Hamilton–JacobiEquationandQuantumMechanics 353 10.2 Feynman’sActionPrincipleinQuantumTheory 361 10.3 Schwinger’sActionPrincipleinQuantumTheory 368 10.4 Schwinger–DysonEquationinQuantumFieldTheory 371 10.5 Schwinger–DysonEquationinQuantumStatisticalMechanics 385 10.6 Feynman’sVariationalPrinciple 395 10.7 PoincareTransformationandSpin 407 10.8 ConservationLawsandNoether’sTheorem 411 10.9 Weyl’sGaugePrinciple 418 10.10 PathIntegralQuantizationofGaugeFieldI 437 10.11 PathIntegralQuantizationofGaugeFieldII 454 10.12 BRSTInvarianceandRenormalization 468 10.13 AsymptoticDisasterinQED 475 10.14 AsymptoticFreedominQCD 479 10.15 RenormalizationGroupEquations 487 10.16 StandardModel 499 10.17 LatticeGaugeFieldTheoryandQuarkConfinement 518 10.18 WKBApproximationinPathIntegralFormalism 523 10.19 Hartree–FockEquation 526 10.20 ProblemsforChapter10 529 References 567 Index 573 XI Preface Thisbookonintegralequationsandthecalculusofvariationsisintendedforuse byseniorundergraduatestudentsandfirst-yeargraduatestudentsinscienceand engineering.Basicfamiliaritywiththeoriesoflinearalgebra,calculus,differential equations,andcomplexanalysisonthemathematicsside,andclassicalmechanics, classicalelectrodynamics,quantummechanicsincludingthesecondquantization, and quantum statistical mechanics on the physics side is assumed. Another prerequisite on the mathematics side for this book is a sound understanding of localanalysisandglobalanalysis. Thisbookgrewoutofthecoursenotesforthelastofthethree-semestersequence of Methods of Applied Mathematics I (Local Analysis), II (Global Analysis) and III (Integral Equations and Calculus of Variations) taught in the Department of MathematicsatMIT.Abouttwothirdsofthecourseisdevotedtointegralequations andtheremainingonethirdtothecalculusofvariations.ProfessorHungCheng taughtthecourseonintegralequationsandthecalculusofvariationseveryother yearfromthemid-1960sthroughthemid-1980satMIT.Sincethen,youngerfaculty havebeenteachingthecourseinturn.Thecoursenotesevolvedintheintervening years.Thisbookistheculminationofthesejointefforts. Therewillbeanaturalquestion:Whynowanotherbookonintegralequationsand thecalculusofvariations?Thereexistmanyexcellentbooksonthetheoryofintegral equations. No existing book, however, discusses the singular integral equations in detail, in particular, Wiener–Hopf integral equations and Wiener–Hopf sum equationswiththenotionoftheWiener–Hopfindex.Inthisbook,thenotionof theWiener–Hopfindexisdiscussedindetail. This book is organized as follows. In Chapter 1, we discuss the notion of functionspace,thelinearoperator,theFredholmalternative,andGreen’sfunctions, preparingthereaderforthefurtherdevelopmentofthematerial.InChapter2,we discussafewexamplesofintegralequationsandGreen’sfunctions.InChapter3, wediscussintegralequationsoftheVolterratype.InChapter4,wediscussintegral equations of the Fredholm type. In Chapter 5, we discuss the Hilbert–Schmidt theoriesofsymmetrickernel.InChapter6,wediscusssingularintegralequations of the Cauchy type. In Chapter 7, we discuss the Wiener–Hopf method for the mixed boundary-value problem in classical electrodynamics, Wiener–Hopf integral equations,and Wiener–Hopfsumequations,the latter two topics being AppliedMathematicalMethodsinTheoreticalPhysics,SecondEdition.MichioMasujima Copyright2009WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-40936-5