The Padé Approximant in Theoretical Physics

383 Pages·1970·4.25 MB·English

by  GEORGE A. BAKER and JOHN L. GAMMEL (Eds.)

The Pad; Approximant in
Theoretical Physics
Edited by
GEORGE A. BAKER, JR. Brookhaven National Laboratory
Upton, Long Island, New York
and
JOHN L. GAMMEL LOS Alamos Scientific Laboratory
University of California
Los Alamos, New Mexico
@
Academic Press 1970 New York and London
COPYRIGHT 8 1970, BY ACADEMIPCR ESS, INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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United Kingdom Edition published by
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AMS 1970 Subject Classification 41A20
PRINTED IN THE UNITED STATES OF AMERICA
Contents
List of Contributors ix
Preface xi
1. The Pad6 Approximant Method and
Some Related Generalizations
George A. Baker, Jr.
I. Introduction 1
11. The Theory of the Pad6 Approximant Method 2
111. Generalized Approximation Procedures 20
IV. Applications 29
References 38
2. Application of Pad6 Approximants to Dispersion
Force and Optical Polarizability Computations
P. W.L anghoff and M. Karplus
I. Introduction 41
11. Formalism 43
111. Computation Procedures 53
IV. Applications 61
V. Comparisions with Related Methods 86
VI. Concluding Remarks 94
References 95
3. Bounds for Averages Using Moment Constraints
J. C. Wheeler and R. G.G ordon
I. Introduction 99
11. The Moments Problem 102
111. Applications 113
IV. Conclusion 123
Appendix A 124
Appendix B 126
References 128
vi Contents
4. Turbulent Diffusion: Evaluation of Primitive
and Renormalized Perturbation Series by Pad6
Approximants and by Expansion of Stieltjes
Transforms into Contributions from
Continuous Orthogonal Functions
Robert H. Kraichnan
I. Introduction 129
11. Continuous Orthogonal Expansion of Stieltjes
Transforms 131
111. Examples Comparing Continuous Orthogonal
Expansion and Pad6 Approximants 134
IV. Diffusion by a Random Velocity Field 139
V. Perturbation Expansions for the Diffusion
Problem 144
VI. Approximants to the Primitive Perturbation
Expansions 150
VII. Approximants to the Irreducible Expansion 156
References 169
5. Pad6 Approximants and Linear Integral
Equations
J. S. R. Chisholm
I. Introduction 171
11. Properties of Linear Integral Equations 172
111. Integral Equations and Pad6 Approximants 176
IV. The Integral Equation for Potential Scattering 178
References 182
6. Series of Derivatives of &Functions
J. S. R. Chisholm and A. K. Common
I. Introduction 183
11. An Example 185
111. Expressions for g(k) 186
IV. Approximants to g(k) 189
V. Conclusions 194
References 195
7. Hilbert Space and the Pad6 Approximant
D. Masson
I. Introduction 197
11. Extended Series of Stieltjes 198
111. Symmetric Operators and the Pad6 Approximant 202
IV. Method of Moments 206
V. Discussion 216
References 217
Contents vii
8. The Connection of Pad6 Approximants with
Stationary Variational Principles and the
Convergence of Certain Pad6 Approximants
J. Nuttall
I. Introduction 219
11. Derivation of Pad6 Approximants from
Stationary Variational Principles 220
111. Convergence of Pad6 Approximants 225
IV. The Convergence of Pad6 Approximants to
Convergent Power Series 221
References 229
9. Approximate N/D Solutions Using Pad6
Approximants
D. Masson
I. Introduction 23 1
11. Method 23 1
111. Proof of Unitarity and Convergence 233
IV. Discussion 239
References 239
10. The Solution of the N/D Equations Using the
Pad6 Approximant Method
A. K. Common
I. Introduction 24 1
11. The Integral Equations for Nand D and
Their Approximate Solution 242
111. Convergent Sequences of Approximate
Solutions to the NID Equations 245
IV. The p Bootstrap and Related Amplitudes 249
References 256
11. Pad6 Approximants as a Computational
Tool for Solving the Schriidinger and
Bethe-Salpeter Equations
Richard W.H aymaker and
Leonard Schlessinger
I. Introduction 251
11. Basic Philosophy and a Simple Example 258
111. Rational-Fraction Fitting 263
IV. Two-Channel Nonrelativistic Scattering 269
V. Bethe-Salpeter Equation 212
VI. Three-Body Scattering 280
References 281
viii Contents
12. Solution of the Bethe-Salpeter Equation
by Means of Pad6 Approximants
J. A. Tjon and H. M. Nieland
I. Introduction 289
11. The Integral Equations 290
111. The Numerical Method 295
IV. Results and Discussion 298
Appendix A 300
Appendix B 301
References 302
13. Application of the Pad6 Approximant to a
Coupled Channel Problem:
Meson-Baryon Scattering
J. L. Gamrnel, M. T. Menzel,
and J. J. Kubis
I. Introduction 303
11. The Pad6 Approximant to the Scattering
Amplitude in the Case of Coupled Channel
Potential Models 305
111. Formal Preliminaries to the Results 3 17
IV. Results for the (3, 3) State (x+--p and
K+--Z+ Elastic Scatterings and Reaction) 326
V. Study of a Potential Model 330
References 33 1
14. The Pad6 Approximant in the
Nucleon-Nucleon System
W.R . Wortman
I. Introduction 333
11. The Pad6 Approximant in Scattering Theory 334
111. The Nucleon-Nucleon System 339
IV. The Pad6 Approximant for N-N Scattering 349
V. Results 363
VI. Summary 365
References 367
Author Index 369
Subject Index 374
List of Contributors
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
GEORGE A. BAKER, JR. (l), Brookhaven National Laboratory, Upton, Long Island,
New York
J. S. R. CHISHOLM (171, 183), University of Kent at Canterbury, Canterbury, England
A. K. COMMON* (183, 241), University of Kent at Canterbury, Canterbury, England
3. L. GAMMEL (303), University of California, Los Alamos Scientific Laboratory, Los
Alamos, New Mexico
R. G. GORDON (99), Department of Chemistry, Harvard University, Cambridge, Mas-
sachusetts
RICHARD W. HAYMAKER7 (257), Department of Physics, University of California,
Santa Barbara, California
M. KARPLUS (41), Department of Chemistry, Harvard University, Cambridge, Mas-
sachusetts
ROBERT H. KRAICHNAN (129), Dublin, New Hampshire
J. J. KUBISX (303), Cambridge University, Cambridge, England
P. W. LANGHOFFI (41), Department of Chemistry, Harvard University, Cambridge,
Massachusetts
D. MASSON (197, 231). University of Toronto, Toronto, Canada
M. T. MENZEL (303), University of California, Los Alamos Scientific Laboratory, Los
Alarnos, New Mexico
H. M. NIELAND (289), University of Nijmegen, Nijmegen, The Netherlands
J. NUTTALL (219), Texas A & M University, College Station, Texas
LEONARD SCHLESSINGERI (257). Department of Physics, University of Illinois,
Urbana, Illinois
J. A. TJON (289), University of Utrecht, Utrecht, The Netherlands
J. C. WHEELER1 (99). Department of Chemistry, Harvard University, Cambridge,
Massachusetts
W. R. WORTMAN (333). University of California, Los Alamos Scientific Laboratory,
Los Alamos, New Mexico
*
Present address: CERN, Geneva, Switzerland.
t Present address: Laboratory of Nuclear Studies, Cornell University, Ithaca, New York.
X Present address: Department of Physics, Michigan State University, East Lansing,
Michigan.
1 Present address: Department of Chemistry, Indiana University, Bloomington, Indiana.
I Present address: Department of Physics and Astronomy, University of Iowa, Iowa
City, Iowa.
I Present address: Department of Chemistry, University of California, San Diego; La
Jolla, California.
ix
Preface
An infinite series of some sort presents itself in many situations in
mathematical physics. Since nearly the beginning of mathematics, or more
properly, since the development of the Taylor series, the value of such
series has been subject to controversy. This book deals with the problem
of extracting maximum information from such series using existing mathe-
matical and physical information.
Too many physicists seem to hold the notion that a series must con-
verge to be useful. We are told’ that after a scientific meeting in which
Cauchy had presented his first research on series, Laplace hastened home
and remained there in seclusion until he had examined the series in
“Mtchanique Celeste.” Abel wrote: “On the whole, divergent series are
a deviltry, and it is a shame to base any demonstration upon them. By
using them one can produce any result he wishes, and they are the cause
of many calamities and paradoxes.” There is no doubt that the negative
point of view of Cauchy and Abel prevailed among mathematicians until
recent (18 90) times, and it still prevails among some physicists.
The mathematical researches of T. J. Stieltjes [Reserches sur les frac-
tions continues, Ann. Fac. Sci. Toulouse 8, J, 1-122 (1894); 9, A, 1-47
(1894)J established that at least for a class of divergent series (which are
most important in theoretical physics as several papers in this book show)
rigorous and useful sums can be obtained. The mathematical work with
which we are most concerned was done in about 1892 [H. Padt, Sur la
representation approchte d’une fonction par des fractiones rationelles,
Thesis, Ann. Ecole. Nor. 9, 1-93 (1892)l. A most useful book presenting
these works and recent developments is H. S. Wall’s “Continued Frac-
tions” (Van Nostrand, Princeton, New Jersey, 1948). [What seems an
almost separate development of the theory of divergent series is presented
by G. H. Hardy, “Divergent Series,” Oxford Univ. Press, London and New
York, 1949.1 One of the final sentences in Wall’s book reflects the present
situation: “It is difficult to appraise the significance of the Pad6 table in
the theory of power series. We feel that an appraisal must await further
and deeper investigations” (p. 410). Unfortunately, little attention is given
to this subject by mathematicians at the present time. It is ironic that
Borel, who started many of the researches in point set theory now so
popular, really had on the top of his mind the problem of divergent series
1 F. Cajori, “A History of Mathematics,” p. 337. MacMillan, New York, 1901.
xi
xii Preface
(A. Borel, “LeGons sur les Fonctions MonogBnes,” Gauthier-Villars, Paris,
1917).
Since about 1960, a small group of theoretical physicists has given
attention to the idea that the Pad6 approximant may be useful in summing
series which occur in theoretical physics. A review of work done up until
1964 is given by G. A. Baker, Jr. [Advan. Theoret. Phys. 1, 1 (1965)l. It is
shown that the class of series studied by Stieltjes occur frequently. The
theory has been most useful and most successful in the problem of critical
phenomena. In scattering theory and quantum field theory series of Stieltjes
also occur, but the papers on field theory in this volume leave much room
for future development.
The present volume is a collection of original papers by some of the
authors who have been particularly active in an attempt to sweep away
once and for all the notion that only a convergent series well inside its
radius of convergence or an asymptotic series of rapidly decreasing terms
is useful. There is other work2 which deserves attention and for which the
editors have been unable to obtain full length original articles for inclusion
in this volume. When it appears, the proceedings of the CargBse summer
school for 1970 should contain some of this work.
The editors and authors hope that this book will serve to bring the
Pad6 approximant into wider use and establish it as a tool to broaden the
powers of the mathematical physicist.
J. L. Basdevant, D. Bessis, and J. Zinn-Justin, Pad6 approximants in strong inter-
actions. Two body pion and kaon systems, Nuovo Cimento A 60, 185 (1969); D. Bessis
and M. Pusterla, Unitary Pad6 approximants in strong coupling field theory and appli-
cation to the calculation of the p- and fo-meson Regge trajectories, Nuovo Cimento A 54,
243 (1968).