Table Of ContentTitelei_Masujima 23.12.2004 9:18 Uhr Seite 3
Michio Masujima
Applied Mathematical Methods
in Theoretical Physics
WILEY-VCH Verlag GmbH & Co. KGaA
Titelei_Masujima 23.12.2004 9:18 Uhr Seite 4
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Contents
Preface IX
Introduction 1
1 FunctionSpaces,LinearOperatorsandGreen’sFunctions 5
1.1 FunctionSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 OrthonormalSystemofFunctions . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 LinearOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 EigenvaluesandEigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 TheFredholmAlternative . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Self-adjointOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Green’sFunctionsforDifferentialEquations . . . . . . . . . . . . . . . . . 16
1.8 ReviewofComplexAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9 ReviewofFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 IntegralEquationsandGreen’sFunctions 33
2.1 IntroductiontoIntegralEquations . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Relationship of Integral Equations with Differential Equations and Green’s
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Sturm–LiouvilleSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Green’sFunctionforTime-DependentScatteringProblem . . . . . . . . . . 48
2.5 Lippmann–SchwingerEquation . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6 ProblemsforChapter2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 IntegralEquationsofVolterraType 63
3.1 IterativeSolutiontoVolterraIntegralEquationoftheSecondKind . . . . . 63
3.2 SolvablecasesofVolterraIntegralEquation . . . . . . . . . . . . . . . . . 66
3.3 ProblemsforChapter3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 IntegralEquationsoftheFredholmType 75
4.1 IterativeSolutiontotheFredholmIntegralEquationoftheSecondKind . . 75
4.2 ResolventKernel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Pincherle–GoursatKernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 FredholmTheoryforaBoundedKernel. . . . . . . . . . . . . . . . . . . . 86
4.5 SolvableExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
VI Contents
4.6 FredholmIntegralEquationwithaTranslationKernel . . . . . . . . . . . . 95
4.7 SystemofFredholmIntegralEquationsoftheSecondKind . . . . . . . . . 100
4.8 ProblemsforChapter4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 Hilbert–SchmidtTheoryofSymmetricKernel 109
5.1 RealandSymmetricMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 RealandSymmetricKernel . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 BoundsontheEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 RayleighQuotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5 CompletenessofSturm–LiouvilleEigenfunctions . . . . . . . . . . . . . . 129
5.6 GeneralizationofHilbert–SchmidtTheory . . . . . . . . . . . . . . . . . . 131
5.7 GeneralizationofSturm–LiouvilleSystem . . . . . . . . . . . . . . . . . . 138
5.8 ProblemsforChapter5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6 SingularIntegralEquationsofCauchyType 149
6.1 HilbertProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 CauchyIntegralEquationoftheFirstKind . . . . . . . . . . . . . . . . . . 153
6.3 CauchyIntegralEquationoftheSecondKind . . . . . . . . . . . . . . . . 157
6.4 CarlemanIntegralEquation . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5 DispersionRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.6 ProblemsforChapter6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7 Wiener–HopfMethodandWiener–HopfIntegralEquation 177
7.1 TheWiener–HopfMethodforPartialDifferentialEquations . . . . . . . . . 177
7.2 HomogeneousWiener–HopfIntegralEquationoftheSecondKind . . . . . 191
7.3 GeneralDecompositionProblem . . . . . . . . . . . . . . . . . . . . . . . 207
7.4 InhomogeneousWiener–HopfIntegralEquationoftheSecondKind . . . . 216
7.5 ToeplitzMatrixandWiener–HopfSumEquation . . . . . . . . . . . . . . . 227
7.6 Wiener–HopfIntegralEquationoftheFirstKindandDualIntegralEquations 235
7.7 ProblemsforChapter7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8 NonlinearIntegralEquations 249
8.1 NonlinearIntegralEquationofVolterratype . . . . . . . . . . . . . . . . . 249
8.2 NonlinearIntegralEquationofFredholmType . . . . . . . . . . . . . . . . 253
8.3 NonlinearIntegralEquationofHammersteintype . . . . . . . . . . . . . . 257
8.4 ProblemsforChapter8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9 CalculusofVariations: Fundamentals 263
9.1 HistoricalBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.3 EulerEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.4 GeneralizationoftheBasicProblems . . . . . . . . . . . . . . . . . . . . . 272
9.5 MoreExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
9.6 DifferentialEquations,IntegralEquations,andExtremizationofIntegrals. . 278
9.7 TheSecondVariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Contents VII
9.8 Weierstrass–ErdmannCornerRelation . . . . . . . . . . . . . . . . . . . . 297
9.9 ProblemsforChapter9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10 CalculusofVariations: Applications 303
10.1 Feynman’sActionPrincipleinQuantumMechanics . . . . . . . . . . . . . 303
10.2 Feynman’sVariationalPrincipleinQuantumStatisticalMechanics . . . . . 308
10.3 Schwinger–DysonEquationinQuantumFieldTheory . . . . . . . . . . . . 312
10.4 Schwinger–DysonEquationinQuantumStatisticalMechanics . . . . . . . 329
10.5 Weyl’sGaugePrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
10.6 ProblemsforChapter10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Bibliography 365
Index 373
Preface
This book on integral equations and the calculus of variations is intended for use by senior
undergraduatestudentsandfirst-yeargraduatestudentsinscienceandengineering. Basicfa-
miliaritywiththeoriesoflinearalgebra,calculus,differentialequations,andcomplexanalysis
onthemathematicsside,andclassicalmechanics,classicalelectrodynamics,quantummecha-
nicsincludingthesecondquantization,andquantumstatisticalmechanicsonthephysicsside,
isassumed.Anotherprerequisiteforthisbookonthemathematicssideisasoundunderstand-
ingoflocalandglobalanalysis.
This book grew out of the course notes for the last of the three-semester sequence of
Methods of Applied Mathematics I (Local Analysis), II (Global Analysis) and III (Integral
EquationsandCalculusofVariations)taughtintheDepartmentofMathematicsatMIT.About
two-thirds of the course is devoted to integral equations and the remaining one-third to the
calculusofvariations. ProfessorHungChengtaughtthecourseonintegralequationsandthe
calculus of variations every other year from the mid 1960s through the mid 1980s at MIT.
Sincethen,youngerfacultyhavebeenteachingthecourseinturn. Thecoursenotesevolved
intheinterveningyears. Thisbookistheculminationofthesejointefforts.
Therewillbetheobviousquestion: Whyyetanotherbookonintegralequationsandthe
calculus of variations? There are already many excellent books on the theory of integral
equations. No existingbook, however, discussesthe singular integralequations in detail; in
particular, Wiener–Hopfintegralequations andWiener–Hopfsumequations withthenotion
oftheWiener–Hopfindex. Inthisbook,thenotionoftheWiener–Hopfindexisdiscussedin
detail.
Thisbookisorganizedasfollows. InChapter1wediscussthenotionoffunctionspace,
thelinearoperator, theFredholmalternativeandGreen’sfunctions,topreparethereaderfor
thefurtherdevelopmentofthematerial. InChapter2wediscussafewexamplesofintegral
equations and Green’s functions. In Chapter 3 we discuss integral equations of the Volterra
type.InChapter4wediscussintegralequationsoftheFredholmtype.InChapter5wediscuss
theHilbert–Schmidttheoriesofthesymmetrickernel. InChapter6wediscusssingularinte-
gralequationsoftheCauchytype. InChapter7,wediscusstheWiener–Hopfmethodforthe
mixedboundary-valueprobleminclassicalelectrodynamics,Wiener–Hopfintegralequations,
andWiener–Hopfsumequations;thelattertwotopicsbeingdiscussedintermsofthenotion
oftheindex. InChapter8wediscussnonlinearintegralequationsoftheVolterra,Fredholm
and Hammerstein type. In Chapter9 wediscussthe calculusofvariations, in particular, the
secondvariations,theLegendretestandtheJacobitest,andtherelationshipbetweenintegral
equations and applications of the calculus of variations. In Chapter 10 we discuss Feyn-
man’s actionprinciple inquantummechanicsandFeynman’svariationalprinciple, asystem
X Preface
oftheSchwinger–Dysonequationsinquantumfieldtheoryandquantumstatisticalmechanics,
Weyl’sgaugeprincipleandKibble’sgaugeprinciple.
AsubstantialportionofChapter10istakenfrommymonograph,“PathIntegralQuanti-
zationandStochasticQuantization”,Vol. 165,SpringerTractsinModernPhysics,Springer,
Heidelberg,publishedintheyear2000.
AreasonableunderstandingofChapter10requiresthereadertohaveabasicunderstand-
ingofclassicalmechanics,classicalfieldtheory,classicalelectrodynamics,quantummecha-
nics including the second quantization, and quantum statistical mechanics. For this reason,
Chapter10canbereadasasidereferenceontheoreticalphysics,independentlyofChapters1
through9.
Theexamplesaremostlytakenfromclassicalmechanics, classicalfieldtheory, classical
electrodynamics, quantummechanics, quantumstatisticalmechanicsandquantumfield the-
ory. Mostofthemare worked out indetailtoillustrate the methods ofthe solutions. Those
exampleswhicharenotworkedoutindetailareeitherintendedtoillustratethegeneralmeth-
odsofthesolutionsoritislefttothereadertocompletethesolutions.
At the end of each chapter, with the exception of Chapter 1, problem sets are given for
soundunderstandingofthecontentofthemaintext. Thereaderisrecommendedtosolveall
the problems at the end of each chapter. Many of the problems were created by Professor
HungChengoverthepastthreedecades.Theproblemsduetohimaredesignatedbythenote
‘(Due to H. C.)’. Some of the problems are thoseencounteredby ProfessorHung Chengin
thecourseofhisownresearchactivities.
Mostoftheproblemscanbesolvedbythedirectapplicationofthemethodillustratedinthe
maintext. Difficultproblemsareaccompaniedbythecitationoftheoriginalreferences. The
problemsforChapter10aremostlytakenfromclassicalmechanics,classicalelectrodynamics,
quantummechanics,quantumstatisticalmechanicsandquantumfieldtheory.
Abibliographyisprovidedattheendofthebookforanin-depthstudyofthebackground
materialsinphysics,besidethestandardreferencesonthetheoryofintegralequationsandthe
calculusofvariations.
The instructor can cover Chapters 1 through 9 in one semester or two quarters with a
choiceofthetopicofhisorherowntastefromChapter10.
I would like to express many heart-felt thanks to Professor Hung Cheng at MIT, who
appointed me as his teaching assistant for the course when I was a graduate student in the
DepartmentofMathematicsatMIT,forhispermissiontopublishthisbookundermysingle
authorship and also for his criticism and constant encouragement without which this book
wouldnothavematerialized.
IwouldliketothankProfessorFrancisE.LowandProfessorKersonHuangatMIT,who
taught me many of the topics within theoretical physics. I would like to thank Professor
RobertoD.PecceiatStanfordUniversity,nowatUCLA,whotaughtmequantumfieldtheory
anddispersiontheory.
I would like to thankProfessor Richard M.Dudleyat MIT, who taught me real analysis
andtheoriesofprobabilityandstochasticprocesses. IwouldliketothankProfessorHerman
Chernoff,thenatMIT,nowatHarvardUniversity,whotaughtmemanytopicsinmathematical
statisticsstartingfrommultivariatenormalanalysis,forhissupervisionofmyPh.D.thesisat
MIT.
Preface XI
I would like to thank Dr. Ali Nadim for supplying his version of the course notes and
Dr. Dionisios Margetis at MIT for supplying examples and problems of integral equations
fromhiscoursesatHarvardUniversityandMIT.Theproblemsduetohimaredesignatedby
thenote‘(DuetoD.M.)’. IwouldliketothankDr.GeorgeFikiorisattheNationalTechnical
UniversityofAthensforsupplyingthereferencesontheYagi–Udasemi-infinitearrays.
Iwouldliketothankmyparents,MikioandHanakoMasujima,whomademyundergrad-
uatestudyatMITpossiblebytheirfinancialsupport. Ialsoverymuchappreciatetheirmoral
supportduringmygraduatestudentdaysatMIT.Iwouldliketothankmywife,Mari,andmy
son,Masachika,fortheirstrongmoralsupport,patienceandencouragementduringtheperiod
ofthewritingofthisbook,whenthe‘goinggottough’.
Lastly,IwouldliketothankDr.AlexanderGrossmannandDr.RonSchulzofWiley-VCH
GmbH & Co. KGaA for their administrative and legalassistancein resolving the copyright
problemwithSpringer.
MichioMasujima
Tokyo,Japan,
June,2004
Introduction
Manyproblemswithintheoreticalphysicsarefrequentlyformulatedintermsofordinarydif-
ferential equations or partial differential equations. We can often convert them into integral
equations with boundary conditions or with initial conditions built in. We can formally de-
veloptheperturbationseriesbyiterations. AgoodexampleistheBornseriesforthepotential
scatteringprobleminquantummechanics.Insomecases,theresultingequationsarenonlinear
integro-differentialequations. AgoodexampleistheSchwinger–Dysonequationinquantum
fieldtheoryandquantumstatisticalmechanics.Itisthenonlinearintegro-differentialequation,
and is exact and closed. It provides the starting point of Feynman–Dyson type perturbation
theoryinconfigurationspaceandinmomentumspace. Insomesingularcases,theresulting
equationsareWiener–Hopfintegralequations. Theseoriginatefromresearchontheradiative
equilibrium onthe surfaceofastar. Inthetwo-dimensionalIsingmodelandthe analysisof
theYagi–Udasemi-infinitearraysofantennas,amongothers,wehavetheWiener–Hopfsum
equation.
Thetheoryofintegralequationsisbestillustratedbythenotionoffunctionalsdefinedon
some function space. If the functionals involved are quadratic in the function, the integral
equationsaresaidtobelinearintegralequations,andiftheyarehigherthanquadraticinthe
function,theintegralequationsaresaidtobenonlinearintegralequations. Dependingonthe
formofthefunctionals,theresultingintegralequationsaresaidtobeofthefirstkind,ofthe
secondkind,orofthethirdkind. Ifthekernelsoftheintegralequationsaresquare-integrable,
theintegralequationsaresaidtobenonsingular,andifthekernelsoftheintegralequationsare
notsquare-integrable,theintegralequationsarethensaidtobesingular.Furthermore,depend-
ingonwhetherornottheendpointsofthekernelarefixedconstants,theintegralequationsare
saidtobeoftheFredholmtype, Volterratype, Cauchytype, orWiener–Hopftypes, etc. By
the discussionofthe variational derivativeof the quadratic functional, we canalsoestablish
the relationshipbetweenthe theoryofintegralequationsandthe calculusofvariations. The
integro-differentialequationscanbestbeformulatedinthismanner. Analogiesofthetheory
ofintegralequationswiththesystemoflinearalgebraicequationsarealsouseful.
The integral equation of Cauchy type has an interesting application to classical electro-
dynamics, namely, dispersion relations. Dispersion relations were derived by Kramers in
1927andKronigin1926,forX-raydispersionandopticaldispersion,respectively. Kramers-
Kronigdispersionrelationsareofverygeneralvaliditywhichonlydependsontheassumption
of the causality. The requirement of the causalityalone determines the region of analyticity
of dielectric constants. In the mid 1950s, these dispersion relations were also derived from
quantumfieldtheoryandappliedtostronginteractionphysics. Theapplicationofthecovari-
antperturbationtheorytostronginteractionphysicswasimpossibleduetothelargecoupling
AppliedMathematicsinTheoreticalPhysics.MichioMasujima
Copyright©2005Wiley-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40534-8
2 Introduction
constant. From the mid 1950s to the 1960s, the dispersion-theoretic approach to strong in-
teraction physics was the only realistic approach that provided many sum rules. To cite a
few, we have the Goldberger–Treiman relation, the Goldberger–Miyazawa–Oehme formula
and the Adler–Weisberger sum rule. In the dispersion-theoretic approach to strong interac-
tion physics, experimentally observed data were directly used in the sum rules. The situa-
tionchangeddramaticallyintheearly1970swhenquantumchromodynamics,therelativistic
quantumfieldtheoryofstronginteractionphysics,wasinventedbytheuseofasymptotically-
freenon-Abeliangaugefieldtheory.
Theregionofanalyticityofthescatteringamplitudeintheupper-halfk-planeinquantum
field theory, when expressedin terms of the Fourier transform, is immediate since quantum
fieldtheoryhasmicroscopiccausality. But, theregionofanalyticityofthescatteringampli-
tudeintheupper-halfk-planeinquantummechanics,whenexpressedintermsoftheFourier
transform, is not immediate since quantum mechanics does not have microscopic causality.
Weshallinvokethegeneralizedtriangularinequalitytoderivetheregionofanalyticityofthe
scatteringamplitudeintheupper-halfk-planeinquantummechanics.Thisregionofanalytic-
ityofthescatteringamplitudesintheupper-halfk-planeinquantummechanicsandquantum
fieldtheorystronglydependsonthefactthatthescatteringamplitudesareexpressedinterms
oftheFouriertransform. Whenanotherexpansionbasisischosen,suchastheFourier–Bessel
series,theregionofanalyticitydrasticallychangesitsdomain.
Inthestandardapplicationofthecalculusofvariationstothevarietyofproblemsintheo-
reticalphysics,wesimplywritetheEulerequationandarerarelyconcernedwiththesecond
variations;theLegendretestandtheJacobitest. Examinationofthesecondvariationsandthe
application of the Legendre testand the Jacobitest becomes necessaryin some cases of the
application of the calculus of variations theoretical physics problems. In order to bring the
developmentoftheoreticalphysicsandthecalculusofvariationsmuchcloser,somehistorical
commentsareinorderhere.
Euler formulated Newtonian mechanics by the variational principle; the Euler equation.
Lagrange began the whole field of the calculus of variations. He also introduced the notion
of generalized coordinates into classical mechanics and completely reduced the problem to
that of differential equations, which are presently known as Lagrange equations of motion,
with the Lagrangian appropriately written in terms of kinetic energy and potential energy.
Hesuccessfullyconvertedclassicalmechanicsintoanalyticalmechanicsusingthevariational
principle. Legendre constructed the transformation methods for thermodynamics which are
presently known as the Legendre transformations. Hamilton succeeded in transforming the
Lagrangeequationsofmotion,whichareofthesecondorder,intoasetoffirst-orderdifferen-
tialequationswithtwiceasmanyvariables.Hedidthisbyintroducingthecanonicalmomenta
which are conjugate to the generalized coordinates. His equations are knownas Hamilton’s
canonical equations of motion. He successfully formulated classical mechanics in terms of
the principle of least action. The variational principles formulated by Euler and Lagrange
applyonlytotheconservativesystem. Hamiltonrecognizedthattheprincipleofleastaction
inclassicalmechanicsandFermat’sprincipleofshortesttimeingeometricalopticsarestrik-
inglyanalogous,permittingtheinterpretationofopticalphenomenainmechanicaltermsand
vice versa. Jacobi quickly realized the importance of the work of Hamilton. He noted that
Hamiltonwasusingjustoneparticularsetofthevariablestodescribethemechanicalsystem
and formulated the canonical transformation theory using the Legendre transformation. He
