Table Of ContentApplied Analysis by the
Hilbert Space Method
An Introduction with Applications to the
Wave, Heat, and Schrödinger Equations
Samuel S. Holland, Jr.
University of Massachusetts
Amherst, Massachusetts
Dover Publications, Inc.
Mineola, New York
1
Copyright
Copyright © 1990 by Samuel S. Holland, Jr.
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2007, is a corrected republication of the work
originally published in the “Monographs and Textbooks in Pure and Applied Mathematics”
series by Marcel Dekker, Inc., New York, in 1990.
Library of Congress Cataloging-in-Publication Data
Holland, Samuel S., 1928–
Applied analysis by the hilbert space method : an introduction with applications to
the wave, heat, and Schrödinger equations / Samuel S. Holland, Jr.
p. cm.
Originally published: New York : M. Dekker, 1990, in series: Monographs and
textbooks in pure and applied mathematics.
Includes index.
eISBN 13: 978-0-486-13929-6
1. Differential equations. 2. Hilbert space. 3. Differential operators. I. Title.
QA372.H76 2007
515′.35—dc22
2006103551
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
2
To Chester Brown and John Byrne,
teachers of Physics and Algebra at Methuen High School,
Methuen, Massachusetts, 1945
3
Preface
This book, designed as a text for a one-year undergraduate applied
2
mathematics course, features Hermitian differential operators on L spaces
and the eigenvalues and eigenfunctions of these operators. The operator
theory, which is presented here in a format accessible to an undergraduate
student, allows for a conceptually unified modern exposition of most of the
commonly covered topics, such as: ordinary linear differential equations;
Fourier series; Legendre, Hermite, and Laguerre polynomials; Bessel
functions; spherical harmonics; the wave, heat, and Schrödinger equations;
2
and the Fourier transform. In addition, the L spaces and Hermitian
operators have independent value as important concepts of modern
mathematics, concepts of general application. No presently available
undergraduate textbook appears to offer a comparable treatment of these
important concepts.
The book can also serve as a reference for scientists who want an
elementary exposition of the basic theory of unbounded Hermitian
operators on Hilbert space.
The prerequisites for the use of this text are three semesters of calculus,
one semester of linear algebra, and basic physics. No prior training in
differential equations is required, as first and second order linear differential
equations are covered in the first two chapters. Because the book includes
the usual material on ordinary differential equations, students need not take
a separate course in that subject, so may take this course earlier in their
undergraduate careers.
A substantial number of the exercises involve numerical calculations.
All the computations required in these exercises are within range of a good
handheld programmable scientific electronic, calculator. In practice, many
students will know a scientific programming language, like BASIC, “C,”
FORTRAN, or PASCAL, and will have access to a computer. These
4
students will have little trouble writing the simple programs required for
these exercises. But the text does not require knowledge of a programming
language nor does it require access to a computer. All the numerical
exercises can be done on a good, programmable hand-held calculator. To
show the feasibility of this, I have included several sample programs for the
Hewlett–Packard HP-15C.
I should like to thank reviewers Peter J. Gingo of the University of
Akron, G. V. Ramanathan of the University of Illinois at Chicago, and
Eugene R. Speer of Rutgers for their insightful reviews and very helpful
comments.
Special thanks are due the many students who patiently suffered through
the various early experimental versions of this course. Whatever success
this book may achieve in making the hitherto recondite topic of unbounded
operators on Hilbert space accessible to today’s undergraduate may in large
measure be attributable to them. They were the arbiters who guided the
evolution of the presentation to its current form. There are far too many to
name individually, but I do want to single out Christopher A. Davis and Dae
Song Im, two students who made a special contribution in the Spring of
1984.
Also, my sincere thanks to Ms. Maria Allegra, Associate Acquisitions
Editor at Marcel Dekker, Inc.
This book has evolved from lecture notes prepared over the years for a
one-year course entitled “Applied Analysis,” given at the University of
Massachusetts. More often than not, the notes were produced lecture-by-
lecture and were passed out to the student hot off the presses. Working
under such frantic deadlines, Mrs. Marguerite Bombardier cheerfully,
swiftly, and accurately typed each section, most several times over, did the
figures and tables, and corrected my errors, all the while maintaining an
unfailing good humor. She also flawlessly typed the final manuscript.
Without Peg, there would not have been any book. To her, my deepest
gratitude.
Samuel S. Holland, Jr.
5
A Note on Method
One of the most troublesome features inherent in the theory of differential
2
operators on L spaces is the fact that these operators cannot be everywhere
defined. Providing an accessible explanation of how one defines the domain
of an unbounded operator is the principal challenge facing any expositor of
this theory. The generally used procedure is to first define the differential
operator on a convenient dense domain, verify its Hermitian character there,
then survey all possible selfadjoint extensions (if any). This complicated
process is impossible to explain to an undergraduate. Moreover, it also
appears as an almost insurmountable hurdle for a physicist or engineer who
wishes to get quickly to the applications. In this text I have used a more
direct method. For each of the classical differential expressions (Bessel,
Hermite, Laguerre, Laplace, Legendre), I begin with the maximal operator
associated with that expression. The concept of the maximal operator is
explained well in J. Weidmann, Linear Operators in Hilbert Spaces (Grad.
Texts in Math. 68, Springer-Verlag, New York, 1980, Section 8.4). When
this maximal operator is Hermitian, it is already selfadjoint (Weidmann’s
Theorem 8.22). Among the classical cases, this occurs only for Hermite’s
2
operator and for the Laplacian on L ( — ∞, ∞). (The proof for Hermite’s
operator is in the Appendix to my Chapter 4; that for the Laplacian in
Theorem 5.4 of Chapter 8.) For the other cases, when the maximal operator
is not Hermitian, I narrow the domain by the imposition of boundary
conditions to produce a Hermitian operator. This procedure seems to be
accessible and intuitively appealing to my intended undergraduate audience.
It also leads quickly to the applications. For the case of Legendre’s operator,
the connection of this approach and the classical process referred to above
has been worked out in detail by N. I. Akhiezer and I. M. Glazman in their
book Theory of Linear Operators in Hilbert Space (Frederick Ungar, New
York, 1963, vol II, Appendix II, §9).
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Contents
Preface
A Note on Method
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
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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
2
3.9. L Spaces
3.10. Hilbert Space
2
Chapter 4. Linear Second Order Differential Operators in L 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
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Appendix 4.A. Mathematical Aspects of Differential Operators
2
in L 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. J (x): The Bessel Function of the First Kind of Order Zero
0
6.4. J (x) Calculating Its Values and Finding Its Zeros
0
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. J (x): The Bessel Functions of the First Kind of Integral Order
p
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
9
