Applied Analysis by the Hilbert Space Method: An Introduction with Applications to the Wave, Heat, and Schrödinger Equations (Dover Books on Mathematics)

Description

Numerous worked examples and exercises highlight this unified treatment of the Hermitian operator theory in its Hilbert space setting. Its simple explanations of difficult subjects make it accessible to undergraduates as well as an ideal self-study guide. Featuring full discussions of first and second order linear differential equations, the text introduces the fundamentals of Hilbert space theory and Hermitian differential operators. It derives the eigenvalues and eigenfunctions of classical Hermitian differential operators, develops the general theory of orthogonal bases in Hilbert space, and offers a comprehensive account of Schrödinger's equations. In addition, it surveys the Fourier transform as a unitary operator and demonstrates the use of various differentiation and integration techniques. Samuel S. Holland, Jr. is a professor of mathematics at the University of Massachusetts, Amherst. He has kept this text accessible to undergraduates by omitting proofs of some theorems but maintaining the core ideas of crucially important results. Intuitively appealing to students in applied mathematics, physics, and engineering, this volume is also a fine reference for applied mathematicians, physicists, and theoretical engineers. Title Page Copyright Page Dedication Preface A Note on Method Table of Contents Chapter 1. First Order Linear Differential Equations 1.1. The Equation a(x)y′ + b(x)y = h(x) 1.2. First Order Linear Differential Expressions; the Kernel 1.3. Finding a Particular Solution by Variation of Parameters 1.4. Power Series Review 1.5. The Initial Value Problem for a First Order Linear Differential Equation Chapter 2. Second Order Linear Differential Equations 2.1. Basic Concepts of Linear Algebra for Function Spaces 2.2. The Initial Value Problem for a Second Order Linear Homogeneous Differential Equation 2.3. Dimension of the Kernel; General Solution; Abel’s Formula 2.4. Kernel of Constant-Coefficient Expressions 2.5. The Classical Linear Oscillator 2.6. Guessing a Particular Solution to a Constant-Coefficient Equation 2.7. Particular Solution by Variation of Parameters 2.8. The Kernel of Legendre’s Differential Expression 2.9. The Kernels of Other Classical Expressions 2.10. Dirac’s Delta Function and Green’s Functions Appendix 2.A. Second Order Linear Differential Equations in the Complex Domain Chapter 3. Hilbert Space 3.1. The Vibrating Wire 3.2. Fourier Series 3.3. Fourier Sine and Cosine Series 3.4. Fourier Series over Other Intervals 3.5. The Vibrating Wire, Revisited 3.6. The Inner Product 3.7. Schwarz’s Inequality 3.8. The Mean-Square Metric; Orthogonal Bases 3.9. L2 Spaces 3.10. Hilbert Space Chapter 4. Linear Second Order Differential Operators in L2 Spaces and Their Eigenvalues and Eigenfunctions 4.1. Compatibility 4.2. Eigenvalues and Eigenfunctions 4.3. Hermitian Operators 4.4. Some General Operator Theory 4.5. The One-Dimensional Laplacian 4.6. Legendre’s Operator and Its Eigenfunctions, the Legendre Polynomials 4.7. Solving Operator Equations with Legendre’s Operator 4.8. More on Legendre Polynomials: Rodrigues’ Formula, the Recursion Relation, and the Generating Function 4.9. Hermite’s Operator and Its Eigenfunctions, the Hermite Polynomials 4.10. Solving Operator Equations with Hermite’s Operator 4.11. More on Hermite Polynomials: Rodrigues’ Formula, the Recursion Relation, and the Generating Function Appendix 4.A. Mathematical Aspects of Differential Operators in L2 Spaces Chapter 5. Schrödinger’s Equations in One Dimension 5.1. The Wave Equation by the Hilbert Space Method 5.2. The Heat Equation by the Hilbert Space Method 5.3. Quanta as Eigenvalues—the Time-Independent Schrödinger Equation in One Dimension 5.4. Interpretation of the Ψ Function. The Time-Dependent Schrödinger Equation in One Dimension 5.5. The Quantum Linear Oscillator 5.6. Solution of the Time-Dependent Schrödinger Equation 5.7. A Brief History of Matrix Mechanics 5.8. A General Formulation of Quantum Mechanics: States 5.9. A General Formulation of Quantum Mechanics: Observables Chapter 6. Bessel’s Operator and Bessel Functions 6.1. The Wave Equation and Other Equations in Higher Dimensions; Polar Coordinates 6.2. Bessel’s Equation and Bessel’s Operator of Order Zero 6.3. J0(x): The Bessel Function of the First Kind of Order Zero 6.4. J0(x) Calculating Its Values and Finding Its Zeros 6.5. The Eigenvalues and Eigenfunctions of Bessel’s Operator of Order Zero 6.6. The Vibrating Drumhead, the Heated Disk, and the Quantum Particle Confined to a Circular Region (the θ Independent Case) 6.7. θ Dependence: Bessel’s Equation and Bessel’s Operator of Integral Order p 6.8. Jp(x): The Bessel Functions of the First Kind of Integral Order p 6.9. The Eigenvalues and Eigenfunctions of Bessel’s Operator of Integral Order p 6.10. Project on Bessel Functions of Nonintegral Order Appendix 6.A. Mathematical Theory of Bessel’s Operator Chapter 7. Eigenvalues of the Laplacian, with Applications 7.1. The Laplacian as a Hilbert Space Operator 7.2. Differential Forms, the Stokes Theorem, and Integration by Parts in Two Variables 7.3. The Laplacian Is Hermitian 7.4. General Facts About the Eigenvalues of the Laplacian 7.5. Eigenvalues of the Rectangle 7.6. Eigenvalues of the Disk 7.7. The Laplacian on the Sphere 7.8. Eigenvalues of the Sphere; Spherical Harmonics 7.9. The Hydrogen Atom 7.10. Project on Laguerre’s Operator and Laguerre Polynomials 7.11. Laplace’s Equation and Harmonic Polynomials Appendix 7.A. The Legendre, Laguerre, and Schrödinger Operators Chapter 8. The Fourier Transform 8.1. Complex Methods in Fourier Series; the Fourier Transform 8.2. Plancherel’s Theorem; Examples of Fourier Transforms 8.3. Fourier Sine and Cosine Transforms 8.4. The Fourier Transform Is a Unitary Operator on L2(— ∞, ∞). 8.5. The Fourier Transform Converts Differentiation into Multiplication by the Independent Variable 8.6. The Eigenvalues and Eigenfunctions of the Fourier Transform Appendix 8.A. ∫ ∞ –∞((sin x)/x)dx = π Index of Symbols Index