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

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 2 Copyright Page 3 Dedication 4 Preface 5 A Note on Method 7 Table of Contents 8 Chapter 1. First Order Linear Differential Equations 13 1.1. The Equation a(x)y′ + b(x)y = h(x) 13 1.2. First Order Linear Differential Expressions; the Kernel 17 1.3. Finding a Particular Solution by Variation of Parameters 22 1.4. Power Series Review 24 1.5. The Initial Value Problem for a First Order Linear Differential Equation 35 Chapter 2. Second Order Linear Differential Equations 40 2.1. Basic Concepts of Linear Algebra for Function Spaces 40 2.2. The Initial Value Problem for a Second Order Linear Homogeneous Differential Equation 47 2.3. Dimension of the Kernel; General Solution; Abel’s Formula 56 2.4. Kernel of Constant-Coefficient Expressions 62 2.5. The Classical Linear Oscillator 66 2.6. Guessing a Particular Solution to a Constant-Coefficient Equation 73 2.7. Particular Solution by Variation of Parameters 77 2.8. The Kernel of Legendre’s Differential Expression 80 2.9. The Kernels of Other Classical Expressions 84 2.10. Dirac’s Delta Function and Green’s Functions 87 Appendix 2.A. Second Order Linear Differential Equations in the Complex Domain 99 Chapter 3. Hilbert Space 110 3.1. The Vibrating Wire 110 3.2. Fourier Series 116 3.3. Fourier Sine and Cosine Series 128 3.4. Fourier Series over Other Intervals 135 3.5. The Vibrating Wire, Revisited 139 3.6. The Inner Product 147 3.7. Schwarz’s Inequality 160 3.8. The Mean-Square Metric; Orthogonal Bases 170 3.9. L2 Spaces 184 3.10. Hilbert Space 198 Chapter 4. Linear Second Order Differential Operators in L2 Spaces and Their Eigenvalues and Eigenfunctions 208 4.1. Compatibility 208 4.2. Eigenvalues and Eigenfunctions 222 4.3. Hermitian Operators 231 4.4. Some General Operator Theory 242 4.5. The One-Dimensional Laplacian 252 4.6. Legendre’s Operator and Its Eigenfunctions, the Legendre Polynomials 266 4.7. Solving Operator Equations with Legendre’s Operator 284 4.8. More on Legendre Polynomials: Rodrigues’ Formula, the Recursion Relation, and the Generating Function 294 4.9. Hermite’s Operator and Its Eigenfunctions, the Hermite Polynomials 306 4.10. Solving Operator Equations with Hermite’s Operator 320 4.11. More on Hermite Polynomials: Rodrigues’ Formula, the Recursion Relation, and the Generating Function 327 Appendix 4.A. Mathematical Aspects of Differential Operators in L2 Spaces 333 Chapter 5. Schrödinger’s Equations in One Dimension 352 5.1. The Wave Equation by the Hilbert Space Method 352 5.2. The Heat Equation by the Hilbert Space Method 359 5.3. Quanta as Eigenvalues—the Time-Independent Schrödinger Equation in One Dimension 369 5.4. Interpretation of the Ψ Function. The Time-Dependent Schrödinger Equation in One Dimension 378 5.5. The Quantum Linear Oscillator 388 5.6. Solution of the Time-Dependent Schrödinger Equation 400 5.7. A Brief History of Matrix Mechanics 406 5.8. A General Formulation of Quantum Mechanics: States 409 5.9. A General Formulation of Quantum Mechanics: Observables 421 Chapter 6. Bessel’s Operator and Bessel Functions 428 6.1. The Wave Equation and Other Equations in Higher Dimensions; Polar Coordinates 428 6.2. Bessel’s Equation and Bessel’s Operator of Order Zero 439 6.3. J0(x): The Bessel Function of the First Kind of Order Zero 446 6.4. J0(x) Calculating Its Values and Finding Its Zeros 453 6.5. The Eigenvalues and Eigenfunctions of Bessel’s Operator of Order Zero 465 6.6. The Vibrating Drumhead, the Heated Disk, and the Quantum Particle Confined to a Circular Region (the θ Independent Case) 475 6.7. θ Dependence: Bessel’s Equation and Bessel’s Operator of Integral Order p 485 6.8. Jp(x): The Bessel Functions of the First Kind of Integral Order p 491 6.9. The Eigenvalues and Eigenfunctions of Bessel’s Operator of Integral Order p 501 6.10. Project on Bessel Functions of Nonintegral Order 507 Appendix 6.A. Mathematical Theory of Bessel’s Operator 509 Chapter 7. Eigenvalues of the Laplacian, with Applications 520 7.1. The Laplacian as a Hilbert Space Operator 520 7.2. Differential Forms, the Stokes Theorem, and Integration by Parts in Two Variables 530 7.3. The Laplacian Is Hermitian 540 7.4. General Facts About the Eigenvalues of the Laplacian 549 7.5. Eigenvalues of the Rectangle 567 7.6. Eigenvalues of the Disk 579 7.7. The Laplacian on the Sphere 582 7.8. Eigenvalues of the Sphere; Spherical Harmonics 600 7.9. The Hydrogen Atom 621 7.10. Project on Laguerre’s Operator and Laguerre Polynomials 637 7.11. Laplace’s Equation and Harmonic Polynomials 647 Appendix 7.A. The Legendre, Laguerre, and Schrödinger Operators 655 Chapter 8. The Fourier Transform 664 8.1. Complex Methods in Fourier Series; the Fourier Transform 664 8.2. Plancherel’s Theorem; Examples of Fourier Transforms 670 8.3. Fourier Sine and Cosine Transforms 679 8.4. The Fourier Transform Is a Unitary Operator on L2(— ∞, ∞). 683 8.5. The Fourier Transform Converts Differentiation into Multiplication by the Independent Variable 692 8.6. The Eigenvalues and Eigenfunctions of the Fourier Transform 700 Appendix 8.A. ∫ ∞ –∞((sin x)/x)dx = π 710 Index of Symbols 712 Index 714

Numerous 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 intuitively appealing to students in applied mathematics, physics, and engineering. It is also a fine reference for professionals. 1990 edition.