基本説明
Complete coverage of mathematical tools and techniques used by physicists and applied mathematicians. Features systematic discussion of the singular Cauchy type and Wiener-Hopf type integral equations.
Full Description
All there is to know about functional analysis, integral equations and calculus of variations in one handy volume, written for the specific needs of physicists and applied mathematicians.
The new edition of this handbook starts with a short introduction to functional analysis, including a review of complex analysis, before continuing a systematic discussion of different types of integral equations. After a few remarks on the historical development, the second part provides an introduction to the calculus of variations and the relationship between integral equations and applications of the calculus of variations. It further covers applications of the calculus of variations developed in the second half of the 20th century in the fields of quantum mechanics, quantum statistical mechanics and quantum field theory.
Throughout the book, the author presents a wealth of problems and examples often with a physical background. He provides outlines of the solutions for each problem, while detailed solutions are also given, supplementing the materials discussed in the main text. The problems can be solved by directly applying the method illustrated in the main text, and difficult problems are accompanied by a citation of the original references.
Highly recommended as a textbook for senior undergraduates and first-year graduates in science and engineering, this is equally useful as a reference or self-study guide.
Contents
Preface xi
Introduction xv
1 Function Spaces, Linear Operators, and Green's Functions 1
1.1 Function Spaces 1
1.2 Orthonormal System of Functions 3
1.3 Linear Operators 5
1.4 Eigenvalues and Eigenfunctions 7
1.5 The Fredholm Alternative 9
1.6 Self-Adjoint Operators 12
1.7 Green's Functions for Differential Equations 14
1.8 Review of Complex Analysis 18
1.9 Review of Fourier Transform 25
2 Integral Equations and Green's Functions 31
2.1 Introduction to Integral Equations 31
2.2 Relationship of Integral Equations with Differential Equations and Green's Functions 37
2.3 Sturm-Liouville System 43
2.4 Green's Function for Time-Dependent Scattering Problem 47
2.5 Lippmann-Schwinger Equation 51
2.6 Scalar Field Interacting with Static Source 62
2.7 Problems for Chapter 2 67
3 Integral Equations of the Volterra Type 105
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind 105
3.2 Solvable Cases of the Volterra Integral Equation 108
3.3 Problems for Chapter 3 112
4 Integral Equations of the Fredholm Type 117
4.1 Iterative Solution to the Fredholm Integral Equation of the Second Kind 117
4.2 Resolvent Kernel 120
4.3 Pincherle-Goursat Kernel 123
4.4 Fredholm Theory for a Bounded Kernel 127
4.5 Solvable Example 134
4.6 Fredholm Integral Equation with a Translation Kernel 136
4.7 System of Fredholm Integral Equations of the Second Kind 143
4.8 Problems for Chapter 4 143
5 Hilbert-Schmidt Theory of Symmetric Kernel 153
5.1 Real and Symmetric Matrix 153
5.2 Real and Symmetric Kernel 155
5.3 Bounds on the Eigenvalues 166
5.4 Rayleigh Quotient 169
5.5 Completeness of Sturm-Liouville Eigenfunctions 172
5.6 Generalization of Hilbert-Schmidt Theory 174
5.7 Generalization of the Sturm-Liouville System 181
5.8 Problems for Chapter 5 187
6 Singular Integral Equations of the Cauchy Type 193
6.1 Hilbert Problem 193
6.2 Cauchy Integral Equation of the First Kind 197
6.3 Cauchy Integral Equation of the Second Kind 201
6.4 Carleman Integral Equation 205
6.5 Dispersion Relations 211
6.6 Problems for Chapter 6 218
7 Wiener-Hopf Method and Wiener-Hopf Integral Equation 223
7.1 The Wiener-Hopf Method for Partial Differential Equations 223
7.2 Homogeneous Wiener-Hopf Integral Equation of the Second Kind 237
7.3 General Decomposition Problem 252
7.4 Inhomogeneous Wiener-Hopf Integral Equation of the Second Kind 261
7.5 Toeplitz Matrix and Wiener-Hopf Sum Equation 272
7.6 Wiener-Hopf Integral Equation of the First Kind and Dual Integral Equations 281
7.7 Problems for Chapter 7 285
8 Nonlinear Integral Equations 295
8.1 Nonlinear Integral Equation of the Volterra Type 295
8.2 Nonlinear Integral Equation of the Fredholm Type 299
8.3 Nonlinear Integral Equation of the Hammerstein Type 303
8.4 Problems for Chapter 8 305
9 Calculus of Variations: Fundamentals 309
9.1 Historical Background 309
9.2 Examples 313
9.3 Euler Equation 314
9.4 Generalization of the Basic Problems 319
9.5 More Examples 323
9.6 Differential Equations, Integral Equations, and Extremization of Integrals 326
9.7 The Second Variation 330
9.8 Weierstrass-Erdmann Corner Relation 345
9.9 Problems for Chapter 9 349
10 Calculus of Variations: Applications 353
10.1 Hamilton-Jacobi Equation and Quantum Mechanics 353
10.2 Feynman's Action Principle in Quantum Theory 361
10.3 Schwinger's Action Principle in Quantum Theory 368
10.4 Schwinger-Dyson Equation in Quantum Field Theory 371
10.5 Schwinger-Dyson Equation in Quantum Statistical Mechanics 385
10.6 Feynman's Variational Principle 395
10.7 Poincare Transformation and Spin 407
10.8 Conservation Laws and Noether's Theorem 411
10.9 Weyl's Gauge Principle 418
10.10 Path Integral Quantization of Gauge Field I 437
10.11 Path Integral Quantization of Gauge Field II 454
10.12 BRST Invariance and Renormalization 468
10.13 Asymptotic Disaster in QED 475
10.14 Asymptotic Freedom in QCD 479
10.15 Renormalization Group Equations 487
10.16 Standard Model 499
10.17 Lattice Gauge Field Theory and Quark Confinement 518
10.18 WKB Approximation in Path Integral Formalism 523
10.19 Hartree-Fock Equation 526
10.20 Problems for Chapter 10 529
References 567
Index 573
