Foundations and Applications of Variational and Perturbation Methods

351 Pages·2009·3.015 MB·English

by  S Raj Vatsya (ed)

F A
OUNDATIONS AND PPLICATIONS OF
V P
ARIATIONAL AND ERTURBATION
METHODS

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F A
OUNDATIONS AND PPLICATIONS OF
V P
ARIATIONAL AND ERTURBATION
METHODS

S. RAJ VATSYA

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New York
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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
Vatsya, S. Raj.
Foundations and applications of variational and perturbation methods / S. Raj Vatsya.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-60741-414-8 (eBook)
1. Perturbation (Mathematics) 2. Perturbation (Quantum dynamics) 3. Variational principles. I.
Title.
QA871.V285 2009
515'.392--dc22
2008050554

Published by Nova Science Publishers, Inc. (cid:31)(cid:31)(cid:31)(cid:31) New York
(cid:0)
CONTENTS

Preface vii
Acknowledgements ix

I. Foundations 1
Chapter 1 Integration and Vector Spaces 3
1. I. Preliminaries 3
1. II. Integration 6
1. III. Vector Spaces 15
Chapter 2 Operators in Vector Spaces 23
2.I. Operators in Banach Spaces 23
Chapter 3 Variational Methods 63
3.I. Formulation 63
3.II. Convergence 74
3.III. PadÉ Approximants 99
3.IV. Monotonic Convergence 118
Chapter 4 Perturbation Methods 127
4.I. Perturbed Operator 127
4.II. Spectral Perturbation 136
4.III. Spectral Differentiation 165
4.IV. Iteration 173

II. Applications 181
Chapter 5 Matrices 183
5.I. Tridiagonal Matrices 183
5.II. Structured Matrices 193
5.III. Conjugate Residual-Like Methods 202
Chapter 6 Atomic Systems 215
6.I. Preliminaries 215
6.II. Eigenvalues and Critical Points 216
vi Contents
6.III. Scattering 223
Chapter 7 Supplementary Examples 251
7.I. Ray Tomography 251
7.II. Maxwell’s Equations 254
7.III. Positivity Lemma for the Elliptic Operators 264
7.IV. Transport and Propagation 289
7.V. Quantum Theory 310
References 327
Index 331
PREFACE

Variational and perturbation methods constitute the basis of a variety of numerical
techniques to solve a wide range of problems in the physical sciences and applied
mathematics. Many of the practicing researchers have limited familiarity with the
mathematical foundations of these methods. This literature is scattered and often concentrates
on mathematical subtleties without associating them with the physical phenomenon, limiting
its accessibility. Another impediment to the dissemination of the rigorous results is a lack of
examples illustrating them. Computationally oriented texts present them as ad hoc, problem
specific procedures. Present text is aimed at presenting these and other related methods in a
unified, coherent framework. Pertaining results, e.g., the convergence and bound properties,
are obtained rigorously and illustrated by copious use of examples drawn from various areas
of physics, chemistry and engineering disciplines. The material provides sufficient
information to researchers in scientific disciplines to apply the mathematical results properly
and to mathematicians who may wish to use such techniques to develop solution schemes for
scientific problems or analyze them for their properties.
Out of a number of mathematical subtleties addressed in the material covered, one
deserves mention. Processes in the interior of a physical system respond in a fundamental
way to the conditions imposed at its boundary, which are often experimentally controllable or
determinable. In the models representing such phenomena, the boundary conditions determine
the mathematical character of equations in an equally fundamental way. For the methods to
solve such equations to be well founded and reliable, it is essential that these mathematical
properties be adequately taken into account. Effort is made to explain the impact of this and
other similar conditions on the mathematical properties, methods and physical phenomena.
While the mathematical rigor is not compromised, the material is kept focused on solving
physical problems of interest. To this end, only the essential areas of abstract mathematics are
covered and excessive abstraction is avoided. Mathematical concepts are developed assuming
the background of the reader expected of an advanced undergraduate student in science and
engineering programs. The concepts and their contents are clarified with examples and
comments to facilitate the understanding.
This book developed from the notes for an interdisciplinary course, “Applied Analysis,”
which the author developed and taught at York University. The course attracted advanced
undergraduate and graduate students from the physical sciences and mathematics
departments. The material was continuously updated in view of the new developments. Some
non-standard applications of the variational and perturbation methods that have not yet been
viii S. Raj Vatsya
included in the instructional texts and courses have also been included. Effort is made to
render this a suitable textbook for a course for the students with background in scientific and
mathematical disciplines, as well as a suitable reference item for researchers in a broad range
of areas. The material can also supplement other courses in approximation theory, numerical
analysis theory and functional analysis.
First two chapters concentrate on the mathematical topics needed for later developments.
This material is introduced and developed from commonly familiar grounds for accessibility,
maintaining the essentials of the concepts. These topics are covered in standard texts.
Therefore the details and the proofs are provided only when considered instructive. A reader
familiar with them may skip to the later chapters and refer back to these as needed. A reader
interested in exploring these topics in more detail may consult standard reference material.
While a vast amount of suitable literature is available, the following classic texts are still quite
prolific sources of information: M. H. Stone, Linear transformations in Hilbert space and
their applications to analysis, Am. Math. Soc. New York (1932); F. Riesz and B. Sz.-Nagy,
Functional analysis, Translated by L. F. Boron, Ungar, New York (1955); and T. Kato,
Perturbation theory for linear operators, Springer-Verlag, New York (1980); in modern
analysis, and R. Courant and D. Hilbert, Methods of mathematical physics, Interscience, New
York (1953); in classical analysis. Chapters 3 and 4 cover detailed mathematical foundations
of the variational and perturbation methods.
Remainder of the text is devoted to worked out examples from various areas of physics,
chemistry and engineering disciplines. The examples illustrate the techniques that can be used
to select a suitable method and verify the conditions for its validity and thus, establish its
reliability. Algebraic details and numerical applications to specific problems are covered
extensively in the existing literature. Applications part in the present text covers the ground
between the computational aspects of the methods and the results developed in foundations
part, where there is a dearth, partly due to a lack of the organized foundational literature.
The material is organized with mathematical classification, each class applicable to a
variety of problems. Applications part concentrates on concrete problems and reduces each to
one of the topics studied in the foundations part, thereby developing a solution scheme and
illustrating the applications of the results. The material is presented in precise terms unless
descriptive text is deemed necessary for explanations. Supplementary remarks are included
for further clarifications and to isolate parts of the text for later reference. An argument is
described in detail the first time it is needed. If a similar argument is used for other proofs
later, it is outlined relatively briefly to avoid excessive duplication. A term defined for the
first time is bolded. Proofs and definitive statements such as the remarks and examples, in
danger of confusion with other text, are concluded with a thick period•

S. R. Vatsya
Formerly: S. R. Singh
August 2008
ACKNOWLEDGEMENTS

The author is grateful to several researchers, particularly Professors Huw Pritchard and
John Nuttall, for bringing numerous problems to his attention. Thanks are due to them and his
students for persistently encouraging him to write a text covering this material. Also,
constructive advice of Dr. Mile Ostojic and Shafee Ahamed was greatly appreciated.