Applied Analysis by the Hilbert Space Method An Introduction with Applications to the Wave, Heat, and Schrödinger Equations

by Holland, Samuel S.

• ISBN13: 9780486458014
• ISBN10: 0486458016
• eBook ISBN(s): 9780486139296
• Format: Paperback
• Copyright: 2007-06-05
• Publisher: Dover Publications
• More Book Details

Summary

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. The author is a professor of mathematics at the University of Massachusetts, Amherst.

Author Biography

Samuel S. Holland, Jr. is a professor of mathematics at the University of Massachusetts, Amherst

Table of Contents

Preface	p. v
A Note on Method	p. vii
First Order Linear Differential Equations	p. 1
The Equation a(x)y[prime] + b(x)y = h(x)	p. 1
First Order Linear Differential Expressions; the Kernel	p. 4
Finding a Particular Solution by Variation of Parameters	p. 9
Power Series Review	p. 10
The Initial Value Problem for a First Order Linear Differential Equation	p. 18
Second Order Linear Differential Equations	p. 22
Basic Concepts of Linear Algebra for Function Spaces	p. 22
The Initial Value Problem for a Second Order Linear Homogeneous Differential Equation	p. 28
Dimension of the Kernel; General Solution; Abel's Formula	p. 35
Kernel of Constant-Coefficient Expressions	p. 39
The Classical Linear Oscillator	p. 43
Guessing a Particular Solution to a Constant-Coefficient Equation	p. 48
Particular Solution by Variation of Parameters	p. 51
The Kernel of Legendre's Differential Expression	p. 53
The Kernels of Other Classical Expressions	p. 57
Dirac's Delta Function and Green's Functions	p. 59
Second Order Linear Differential Equations in the Complex Domain	p. 69
Hilbert Space	p. 78
The Vibrating Wire	p. 78
Fourier Series	p. 83
Fourier Sine and Cosine Series	p. 94
Fourier Series over Other Intervals	p. 99
The Vibrating Wire, Revisited	p. 102
The Inner Product	p. 108
Schwarz's Inequality	p. 118
The Mean-Square Metric; Orthogonal Bases	p. 127
L[superscript 2] Spaces	p. 138
Hilbert Space	p. 149
Linear Second Order Differential Operators in L[superscript 2] Spaces and Their Eigenvalues and Eigenfunctions	p. 157
Compatibility	p. 157
Eigenvalues and Eigenfunctions	p. 168
Hermitian Operators	p. 175
Some General Operator Theory	p. 184
The One-Dimensional Laplacian	p. 191
Legendre's Operator and Its Eigenfunctions, the Legendre Polynomials	p. 202
Solving Operator Equations with Legendre's Operator	p. 217
More on Legendre Polynomials: Rodrigues' Formula, the Recursion Relation, and the Generating Function	p. 224
Hermite's Operator and Its Eigenfunctions, the Hermite Polynomials	p. 233
Solving Operator Equations with Hermite's Operator	p. 243
More on Hermite Polynomials: Rodrigues' Formula, the Recursion Relation, and the Generating Function	p. 248
Mathematical Aspects of Differential Operators in L[superscript 2] Spaces	p. 253
Schrodinger's Equations in One Dimension	p. 267
The Wave Equation by the Hilbert Space Method	p. 267
The Heat Equation by the Hilbert Space Method	p. 273
Quanta as Eigenvalues-the Time-Independent Schrodinger Equation in One Dimension	p. 282
Interpretation of the [Psi] Function. The Time-Dependent Schrodinger Equation in One Dimension	p. 289
The Quantum Linear Oscillator	p. 298
Solution of the Time-Dependent Schrodinger Equation	p. 308
A Brief History of Matrix Mechanics	p. 313
A General Formulation of Quantum Mechanics: States	p. 316
A General Formulation of Quantum Mechanics: Observables	p. 326
Bessel's Operator and Bessel Functions	p. 331
The Wave Equation and Other Equations in Higher Dimensions; Polar Coordinates	p. 331
Bessel's Equation and Bessel's Operator of Order Zero	p. 339
J[subscript 0](x): The Bessel Function of the First Kind of Order Zero	p. 345
J[subscript 0](x): Calculating Its Values and Finding Its Zeros	p. 350
The Eigenvalues and Eigenfunctions of Bessel's Operator of Order Zero	p. 359
The Vibrating Drumhead, the Heated Disk, and the Quantum Particle Confined to a Circular Region (the [theta]-Independent Case)	p. 367
[theta] Dependence: Bessel's Equation and Bessel's Operator of Integral Order p	p. 375
J[subscript p](x): The Bessel Functions of the First Kind of Integral Order p	p. 380
The Eigenvalues and Eigenfunctions of Bessel's Operator of Integral Order p	p. 387
Project on Bessel Functions of Nonintegral Order	p. 392
Mathematical Theory of Bessel's Operator	p. 393
Eigenvalues of the Laplacian, with Applications	p. 402
The Laplacian as a Hilbert Space Operator	p. 402
Differential Forms, the Stokes Theorem, and Integration by Parts in Two Variables	p. 410
The Laplacian Is Hermitian	p. 418
General Facts About the Eigenvalues of the Laplacian	p. 426
Eigenvalues of the Rectangle	p. 440
Eigenvalues of the Disk	p. 450
The Laplacian on the Sphere	p. 452
Eigenvalues of the Sphere; Spherical Harmonics	p. 466
The Hydrogen Atom	p. 481
Project on Laguerre's Operator and Laguerre Polynomials	p. 493
Laplace's Equation and Harmonic Polynomials	p. 500
The Legendre, Laguerre, and Schrodinger Operators	p. 507
The Fourier Transform	p. 514
Complex Methods in Fourier Series; the Fourier Transform	p. 514
Plancherel's Theorem; Examples of Fourier Transforms	p. 520
Fourier Sine and Cosine Transforms	p. 526
The Fourier Transform Is a Unitary Operator on L[superscript 2](- [infinity], [infinity])	p. 529
The Fourier Transform Converts Differentiation into Multiplication by the Independent Variable	p. 537
The Eigenvalues and Eigenfunctions of the Fourier Transform	p. 544
[Characters not reproducible]((sin x)/x)dx = [pi]	p. 551
Index of Symbols	p. 553
Index	p. 555
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