Applied Mathematical Methods In Theoretical Physics

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 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 The Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Green’s Functions for Differential Equations . . . . . . . . . . . . . . . . . 16
1.8 Review of Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9 Review of Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 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 Problems for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Integral Equations of Volterra Type 63
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind . . . . . 63
3.2 Solvable cases of Volterra Integral Equation . . . . . . . . . . . . . . . . . 66
3.3 Problems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Resolvent Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Pincherle–Goursat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Fredholm Theory for a Bounded Kernel . . . . . . . . . . . . . . . . . . . . 86
4.5 Solvable Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Fredholm Integral Equation with a Translation Kernel . . . . . . . . . . . . 95
4.7 System of Fredholm Integral Equations of the Second Kind . . . . . . . . . 100
4.8 Problems for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 Hilbert–Schmidt Theory of Symmetric Kernel 109
5.1 Real and Symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Real and Symmetric Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 Bounds on the Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5 Completeness of Sturm–Liouville Eigenfunctions . . . . . . . . . . . . . . 129
5.6 Generalization of Hilbert–Schmidt Theory . . . . . . . . . . . . . . . . . . 131
5.7 Generalization of Sturm–Liouville System . . . . . . . . . . . . . . . . . . 138
5.8 Problems for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6 Singular Integral Equations of Cauchy Type 149
6.1 Hilbert Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 Cauchy Integral Equation of the First Kind . . . . . . . . . . . . . . . . . . 153
6.3 Cauchy Integral Equation of the Second Kind . . . . . . . . . . . . . . . . 157
6.4 Carleman Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.6 Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9 Calculus of Variations: Fundamentals 263
9.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.3 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.4 Generalization of the Basic Problems . . . . . . . . . . . . . . . . . . . . . 272
9.5 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
9.6 Differential Equations, Integral Equations, and Extremization of Integrals . . 278
9.7 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
9.8 Weierstrass–Erdmann Corner Relation . . . . . . . . . . . . . . . . . . . . 297
9.9 Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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’s Gauge Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
10.6 Problems for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Bibliography 365
Index 373

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