Applied Mathematical Methods in Theoretical Physics - Masujima M..pdf

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267176668 UNPDF
Michio Masujima
Applied Mathematical Methods
in Theoretical Physics
WILEY-VCH Verlag GmbH & Co. KGaA
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K. Schmidt
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ISBN-13: 978- 3-527-40534-3
ISBN-10: 3-527-40534-8
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Contents
Preface
IX
Introduction
1
1 Function Spaces, Linear Operators and Green’s Functions 5
1.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Orthonormal System of Functions . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 LinearOperators................................ 8
1.4 EigenvaluesandEigenfunctions........................ 11
1.5 The Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Self-adjointOperators............................. 15
1.7 Green’s Functions for Differential Equations . . . . . . . . . . . . . . . . . 16
1.8 ReviewofComplexAnalysis......................... 21
1.9 ReviewofFourierTransform......................... 28
2 Integral Equations and Green’s Functions 33
2.1 Introduction to Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Relationship of Integral Equations with Differential Equations and Green’s
Functions.................................... 39
2.3 Sturm–Liouville System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Green’s Function for Time-Dependent Scattering Problem . . . . . . . . . . 48
2.5 Lippmann–Schwinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6 ProblemsforChapter2 ............................ 57
3 Integral Equations of Volterra Type 63
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind . . . . . 63
3.2 SolvablecasesofVolterraIntegralEquation ................. 66
3.3 ProblemsforChapter3 ............................ 71
4 Integral Equations of the Fredholm Type 75
4.1 Iterative Solution to the Fredholm Integral Equation of the Second Kind . . 75
4.2 ResolventKernel................................ 78
4.3 Pincherle–GoursatKernel........................... 81
4.4 Fredholm Theory for a Bounded Kernel . . . . . . . . . . . . . . . . . . . . 86
4.5 SolvableExample............................... 93
VI
Contents
4.6 Fredholm Integral Equation with a Translation Kernel . . . . . . . . . . . . 95
4.7 System of Fredholm Integral Equations of the Second Kind . . . . . . . . . 100
4.8 ProblemsforChapter4 ............................ 101
5 Hilbert–Schmidt Theory of Symmetric Kernel 109
5.1 RealandSymmetricMatrix.......................... 109
5.2 RealandSymmetricKernel.......................... 111
5.3 Bounds on the Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 RayleighQuotient............................... 126
5.5 Completeness of Sturm–Liouville Eigenfunctions . . . . . . . . . . . . . . 129
5.6 GeneralizationofHilbert–SchmidtTheory.................. 131
5.7 Generalization of Sturm–Liouville System . . . . . . . . . . . . . . . . . . 138
5.8 ProblemsforChapter5 ............................ 144
6 Singular Integral Equations of Cauchy Type 149
6.1 HilbertProblem................................ 149
6.2 CauchyIntegralEquationoftheFirstKind.................. 153
6.3 Cauchy Integral Equation of the Second Kind . . . . . . . . . . . . . . . . 157
6.4 CarlemanIntegralEquation.......................... 161
6.5 DispersionRelations.............................. 166
6.6 ProblemsforChapter6 ............................ 173
7 Wiener–Hopf Method and Wiener–Hopf Integral Equation 177
7.1 The Wiener–Hopf Method for Partial Differential Equations . . . . . . . . . 177
7.2 Homogeneous Wiener–Hopf Integral Equation of the Second Kind . . . . . 191
7.3 General Decomposition Problem . . . . . . . . . . . . . . . . . . . . . . . 207
7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind . . . . 216
7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation . . . . . . . . . . . . . . . 227
7.6 Wiener–Hopf Integral Equation of the First Kind and Dual Integral Equations 235
7.7 ProblemsforChapter7 ............................ 239
8 Nonlinear Integral Equations 249
8.1 Nonlinear Integral Equation of Volterra type . . . . . . . . . . . . . . . . . 249
8.2 Nonlinear Integral Equation of Fredholm Type . . . . . . . . . . . . . . . . 253
8.3 Nonlinear Integral Equation of Hammerstein type . . . . . . . . . . . . . . 257
8.4 ProblemsforChapter8 ............................ 259
9 Calculus of Variations: Fundamentals 263
9.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.2 Examples.................................... 267
9.3 EulerEquation................................. 267
9.4 GeneralizationoftheBasicProblems..................... 272
9.5 MoreExamples ................................ 276
9.6 Differential Equations, Integral Equations, and Extremization of Integrals . . 278
9.7 TheSecondVariation ............................. 283
Contents
VII
9.8 Weierstrass–ErdmannCornerRelation .................... 297
9.9 ProblemsforChapter9 ............................ 300
10 Calculus of Variations: Applications 303
10.1 Feynman’s Action Principle in Quantum Mechanics . . . . . . . . . . . . . 303
10.2 Feynman’s Variational Principle in Quantum Statistical Mechanics . . . . . 308
10.3 Schwinger–Dyson Equation in Quantum Field Theory . . . . . . . . . . . . 312
10.4 Schwinger–Dyson Equation in Quantum Statistical Mechanics . . . . . . . 329
10.5 Weyl’sGaugePrinciple ............................ 339
10.6 ProblemsforChapter10............................ 356
Bibliography
365
Index
373

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