Applied Mathematical Methods in Theoretical Physics
This advanced textbook is divided into two parts: The first on integral equations and the second on the calculus of variations. It begins with a short introduction to functional analysis, including a short review of complex analysis, before continuing a systematic discussion of different types of equations, such as Volterra integral equations, singular integral equations of Cauchy type, integral equations of the Fredholm type, with a special emphasis on Wiener-Hopf integral equations and Wiener-Hopf sum equations.
After a few remarks on the historical development, the second part starts with 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 over 150 problems and exercises - many from such branches of physics as quantum mechanics, quantum statistical mechanics, and quantum field theory - together with outlines of the solutions in each case. Detailed solutions are given, supplementing the materials discussed in the main text, allowing problems to be solved making direct use of the method illustrated. The original references are given for difficult problems. The result is complete coverage of the mathematical tools and techniques used by physicists and applied mathematicians
Intended for senior undergraduates and first-year graduates in science and engineering, this is equally useful as a reference and self-study guide.
1 Function Spaces, Linear Operators and Green’s Functions.
1.1 Function Spaces.
1.2 Ortho normal System of Functions.
1.3 Linear Operators.
1.4 Eigen values and Eigen functions.
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 Problems for Chapter 2.
3 Integral Equations of Volterra Type.
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind.
3.2 Solvable cases of 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 Eigen values.
5.4 Rayleigh Quotient.
5.5 Completeness of Sturm–Liouville Eigen functions.
5.6 Generalization of Hilbert–Schmidt Theory.
5.7 Generalization of Sturm–Liouville System.
5.8 Problems for Chapter 5.
6 Singular Integral Equations of 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 Volterra type.
8.2 Nonlinear Integral Equation of Fredholm Type.
8.3 Nonlinear Integral Equation of Hammerstein type.
8.4 Problems for Chapter 8.
9 Calculus of Variations: Fundamentals.
9.1 Historical Background.
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 Feynman’s Action Principle in Quantum Mechanics.
10.2 Feynman’s Variational Principle in Quantum Statistical Mechanics.
10.3 Schwinger–Dyson Equation in Quantum Field Theory.
10.4 Schwinger–Dyson Equation in Quantum Statistical Mechanics.
10.5 Weyl’s Gauge Principle.
10.6 Problems for Chapter 10.