Allthere is to know about functional analysis, integral equations and calculus of variations in a single volume.
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.
Preface.Introduction.
1 Function Spaces, Linear Operators and Greens 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 Greens Functions for Differential Equations.
1.8 Review of Complex Analysis.
1.9 Review of Fourier Transform.
2 Integral Equations and Greens Functions.
2.1 Introduction to Integral Equations.
2.2 Relationship of Integral Equations with Differential Equations and Greens Functions.
2.3 SturmLiouville System.
2.4 Greens Function for Time-Dependent Scattering Problem.
2.5 LippmannSchwinger 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 PincherleGoursat 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 HilbertSchmidt 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 SturmLiouville Eigen functions.
5.6 Generalization of HilbertSchmidt Theory.
5.7 Generalization of SturmLiouville 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 WienerHopf Method and WienerHopf Integral Equation.
7.1 The WienerHopf Method for Partial Differential Equations.
7.2 Homogeneous WienerHopf Integral Equation of the Second Kind.
7.3 General Decomposition Problem.
7.4 Inhomogeneous WienerHopf Integral Equation of the Second Kind.
7.5 Toeplitz Matrix and WienerHopf Sum Equation.
7.6 WienerHopf 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.
9.2 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 WeierstrassErdmann Corner Relation.
9.9 Problems for Chapter 9.
10 Calculus of Variations: Applications.
10.1 Feynmans Action Principle in Quantum Mechanics.
10.2 Feynmans Variational Principle in Quantum Statistical Mechanics.
10.3 SchwingerDyson Equation in Quantum Field Theory.
10.4 SchwingerDyson Equation in Quantum Statistical Mechanics.
10.5 Weyls Gauge Principle.
10.6 Problems for Chapter 10.
Bibliography.
Index.
