Applied Mathematical Methods In Theoretical Physics

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.

Michio Masujima, born in 1947, studied physics and mathematics at the Massachusetts Institute of Technology and Stanford University. He received his PhD in mathematics from the MIT in 1983. Dr. Masujima worked for many years at the NEC Fundamental Research Laboratory in Japan, where he was in charge of computational physics, and later as a lecturer at the NEC Junior Technical College, where he was responsible for the subjects mathematics and physics. Dr. Masujima works currently in private enterprise.

Preface.

Introduction.

1 Function Spaces, Linear Operators, and Green's Functions.

1.1 Function Spaces.

1.2 Orthonormal System of Functions.

1.3 Linear Operators.

1.4 Eigenvalues and Eigenfunctions.

1.5 The Fredholm Alternative.

1.6 Self-Adjoint Operators.

1.7 Green's Functions for Differential Equations.

1.8 Review of Complex Analysis.

1.9 Review of Fourier Transform.

2 Integral Equations and Green's Functions.

2.1 Introduction to Integral Equations.

2.2 Relationship of Integral Equations with Differential Equations and Green's Functions.

2.3 Sturm–Liouville System.

2.4 Green's Function for Time-Dependent Scattering Problem.

2.5 Lippmann–Schwinger Equation.

2.6 Scalar Field Interacting with Static Source.

2.7 Problems for Chapter 2.

3 Integral Equations of the Volterra Type.

3.1 Iterative Solution to Volterra Integral Equation of the Second Kind.

3.2 Solvable Cases of the Volterra Integral Equation.

3.3 Problems for Chapter 3.

4 Integral Equations of the Fredholm Type.

4.1 Iterative Solution to the Fredholm Integral Equation of the Second Kind.

4.2 Resolvent Kernel.

4.3 Pincherle–Goursat Kernel.

4.4 Fredholm Theory for a Bounded Kernel.

4.5 Solvable Example.

4.6 Fredholm Integral Equation with a Translation Kernel.

4.7 System of Fredholm Integral Equations of the Second Kind.

4.8 Problems for Chapter 4.

5 Hilbert–Schmidt Theory of Symmetric Kernel.

5.1 Real and Symmetric Matrix.

5.2 Real and Symmetric Kernel.

5.3 Bounds on the Eigenvalues.

5.4 Rayleigh Quotient.

5.5 Completeness of Sturm–Liouville Eigenfunctions.

5.6 Generalization of Hilbert–Schmidt Theory.

5.7 Generalization of the Sturm–Liouville System.

5.8 Problems for Chapter 5.

6 Singular Integral Equations of the Cauchy Type

6.1 Hilbert Problem.

6.2 Cauchy Integral Equation of the First Kind.

6.3 Cauchy Integral Equation of the Second Kind.

6.4 Carleman Integral Equation.

6.5 Dispersion Relations.

6.6 Problems for Chapter 6.

7 Wiener–Hopf Method and Wiener–Hopf Integral Equation.

7.1 The Wiener–Hopf Method for Partial Differential Equations.

7.2 Homogeneous Wiener–Hopf Integral Equation of the Second Kind.

7.3 General Decomposition Problem.

7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind.

7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation.

7.6 Wiener–Hopf Integral Equation of the First Kind and Dual Integral Equations.

7.7 Problems for Chapter 7.

8 Nonlinear Integral Equations.

8.1 Nonlinear Integral Equation of the Volterra Type.

8.2 Nonlinear Integral Equation of the Fredholm Type.

8.3 Nonlinear Integral Equation of the Hammerstein Type.

8.4 Problems for Chapter 8.

9 Calculus of Variations: Fundamentals.

9.1 Historical Background.

9.2 Examples.

9.3 Euler Equation.

9.4 Generalization of the Basic Problems.

9.5 More Examples.

9.6 Differential Equations, Integral Equations, and Extremization of Integrals.

9.7 The Second Variation.

9.8 Weierstrass–Erdmann Corner Relation.

9.9 Problems for Chapter 9.

10 Calculus of Variations: Applications.

10.1 Hamilton–Jacobi Equation and Quantum Mechanics.

10.2 Feynman's Action Principle in Quantum Theory.

10.3 Schwinger's Action Principle in Quantum Theory.

10.4 Schwinger–Dyson Equation in Quantum Field Theory.

10.5 Schwinger–Dyson Equation in Quantum Statistical Mechanics.

10.6 Feynman's Variational Principle.

10.7 Poincare Transformation and Spin.

10.8 Conservation Laws and Noether's Theorem.

10.9 Weyl's Gauge Principle.

10.10 Path Integral Quantization of Gauge Field I.

10.11 Path Integral Quantization of Gauge Field II.

10.12 BRST Invariance and Renormalization.

10.13 Asymptotic Disaster in QED.

10.14 Asymptotic Freedom in QCD.

10.15 Renormalization Group Equations.

10.16 Standard Model.

10.17 Lattice Gauge Field Theory and Quark Confinement.

10.18 WKB Approximation in Path Integral Formalism.

10.19 Hartree–Fock Equation.

10.20 Problems for Chapter 10.

References.

Index.