.: ' :,. QUANTUM SCATTERING THEORY FOR SEVERAL PARTICLE SYSTEMS
Mathematical Physics and Applied Mathematics Editors: M. Plato, Universite de Bourgogne, Dijon, France The titles published in this series are listed at the end of this volume. Volume 11
Quantum Scattering Theory for Several Particle Systems by L.D. FADDEEV and S.P. MERKURIEV t Institute o/theoretical Physics, University 0/ St. Petersburg, St. Petersburg, Russia t Professor S.P. Merkuriev died on May 18, 1993 SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data Merkur'ev, S. P. (Stanislav Petrovich) [Kvantovafa teoriia rasseianlla dlia sistem neskol'kikh chastlts. English] Quantum scattering theory for several particle systems I by L.D. Faddeev and S.P. Merkuriev. p. cm. -- (Mathematical physics and applied mathematlcs ; v. 11 ) Translation of: Kvantovafa teori fa rassefani fa dl fa sistem neskol'kikh chastits. Author's names in reverse order in original Russian ed. Includes index. ISBN 978-90-481-4305-4 ISBN 978-94-017-2832-4 (ebook) DOI DOl 10.1007/978-94-017-2832-4 1. Scattering (Physics) 2. Quantum theory. 3. Few-body problem. I. Faddeev, L. D, II. Title. III. Series. QC794.6.S3M4713 1993 539.7'58--dc20 93-11377 ISBN 978-90-481-4305-4 This is the translation of the original Russian work, Kvantovaja teoria rasseivania dlja sistem neskolkih pastid, Published by Nauka Publishers, Moscow, 1985. Printed on acid-free paper All Rights Reserved 1993 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents Introduction xi 1 General Aspects of the Scattering Problem 1 1.1 Formulation of the Problem.... 1 1.2 Kinematics............. 7 1.2.1 Subsystems and partitions. 7 1.2.2 Reduced coordinates.... 10 1.2.3 The momentum space... 15 1.3 Fundamental Concepts of Dynamics 17 1.3.1 The energy operators 17 1.3.2 Clusters... 20 1.3.3 Channels of reactions 21 1.4 Wave Operators........ 23 1.4.1 Definition of wave operators. 23 1.4.2 Existence of wave operators 25 1.5 Properties of the Wave Operators. 30 1.6 The Scattering Operator...... 35 2 Stationary Approach to Scattering Theory 39 2.1 Resolvent and Wave Operators....... 39 ~.2 Singularities of the Resolvent. Neutral Particles. 43 2.2.1 Equations of perturbation theory. 43 2.2.2 Operators Ra/o (z).. 44 2.2.3 Non-connected parts 46 2.2.4 Pole singularities.. 48 2.3 Poles of Resolvent and Waves Operators 52 v
VI CONTENTS 2.3.1 Kernels of wave operators 2.3.2 The scattering operator. 2.3.3 Integral representations. 2.4 Singularities of Resolvent for Charged Particles 2.4.1 Resolvent and wave operators. 2.4.2 The scattering operator. 52 54 56 57 57 59 3 The Method of Integral Equation 61 3.1 Integral Equations for the Two-Body T-matrix..... 62 3.2 Compact Integral Equations for Three-Particle Systems 71 3.2.1 Derivation of compact equations........ 71 3.2.2 Properties of solutions of compact equations.. 76 3.3 Integral Equations for Resolvent and Wave Operators 3.3.1 Components of the resolvent... 3.3.2 Components of wave operators.. 3.3.3 Integral equations for components 3.3.4 Kernels of the scattering operator 3.4 Examples.................. 3.4.1 Scattering on rigid centre... 3.4.2 Discrete spectrum in neighbourhood of zero. 3.5 Compact Integral Equations for N-particle Systems. 100 3.5.1 Difficulty of the problem.... 100 3.5.2 Compact equations for four-particle systems. 104 3.5.3 N-particle problem.... 107 3.5.4 Singularities of the kernels MA 2 B 2 3.6 Charged Particles.... 3.6.1 Two-particle system.. 3.6.2 Three charged particles 4 Configuration Space. Neutral Particles 4.1 Two-particle System. 4.1.1 Wave functions 4.1.2 Green function 4.2 Coordinate Asymptotics of Wave Functions of Three-body 87 87 88 90 92 94 94 97 111 114 114 116 123 123 123 128 System............................... 132
CONTENTS vii 4.2.1 Incident and scattered waves 132 4.2.2 The wave functions 'Yo( X, P) 136 4.2.3 Boundary values problems.. 139 4.3 Contribution of Elementary Two-particle Collisions. 144 4.3.1 Single and double collisions of classical particles 144 4.3.2 Asymptotics of the functions P a and Pa,t3 146 4.3.3 Exponential decreasing of eigenfunctions. 154 4.4 The Green Function......... 157 4.4.1 Singularities and asymptotics 157 4.4.2 Iterations of R~1....... 159 4.4.3 Compact equations...... 165 4.5 Differential Equations for Components of N -body Wave Functions 167 4.5.1 Formal derivations of differential equations for threeand four-particle systems... 167 4.5.2 Differential equations for components of the resolvent for the four-body system................. 170 4.5.3 Differential equations for components for the N-body system............... 173 4.5.4 Asymptotic boundary conditions 177 4.6 Rapidly Oscillating Integrals.... 182 5 Charged Particles in Configuration Space 189 5.1 Two Charged Particles. 189 5.1.1 Wave functions. 190 5.1.2 Green function. 194 5.1.3 Superposition of Coulomb and short-range potentials. 196 5.1.4 Angular singularities of the scattering amplitude... 198 5.1.5 The Coulomb potential in R3N-3 202 5.2 Coordinate Asymptotics of Wave Functions for a System of Three Charged Particles....... 207 5.2.1 The eikonal approximation. 207 5.2.2 Plane eikonal Z, Z = (1), X) 211 5.2.3 The spherical eikonallxi 211 5.2.4 The single eikonal Zoe 212
viii CONTENTS 5.2.5 The double eikonal ZOt{3 214 5.2.6 Asymptotics of the function 'Po 215 5.2.7 The wave functions 'PA(X,PA) 230 5.3 The Asymptotics of 'Po in Forward Direction 236 5.3.1 Distorted plane waves........ 236 5.3.2 The forward scattering........ 240 5.3.3 The asymptotics of the function I]i F 243 5.3.4 Plane waves in not.......... 246 5.4 Asymptotics of the Function 'Po in Singular Directions n~o) and n~oj.......... 251 5.4.1 The direction n~o) 251 5.4.2 The direction n~oj... 254 5.4.3 The parabolic equation in forward scattering region 257 5.4.4 The eigenfunctions.................. 260 5.5 Compact Equations in Configuration Space........ 265 5.5.1 5.5.2 5.5.3 5.5.4 Integral equations for components of the resolvent The operator Ra(z).... The Green Function of the Operator HOt The Green function R(X, X', z).. 265 266 272 278 5.6 Boundary Conditions for Wave Functions 282 5.6.1 Definition of the wave functions. 282 5.6.2 N charged particles........ 286 6 Mathematical Foundation of the Scattering Problem 289 6.1 The System of Two Particles 289 6.1.1 Neutral particles... 290 6.1.2 Charged particles... 293 6.1.3 The scattering operator 295 6.2 Continuous Spectrum ofthe Hamiltonian of the Three-Particle System................... 298 6.2.1 System of three neutral particles 299 6.2.2 Charged particles......... 304 6.2.3 The scattering operator..... 306 6.2.4 Coulomb rapidly oscillating integrals. 311
CONTENTS ix 6.3 Justification of the Non-Stationary Formulation of the Scattering Problem....... 316 6.3.1 The two-particle system. 316 6.3.2 The three-particle system 319 6.3.3 Charged particles. 321 7 Some Applications 323 7.1 Partial Waves in Two-Body Systems 323 7.1.1 Schrodinger equation..... 324 7.1.2 Charged particles....... 325 7.1.3 Low energy behaviour of Coulomb amplitudes. 327 7.1.4 Partial T-matrix...... 330 7.2 Partial Equations for Components.... 334 7.2.1 Bispherical basis.... 334 7.2.2 7.2.3 Numerical solution of the scattering problem Charged particles............. 341 347 7.3 Integral Equations for Separable Potentials.. 352 7.3.1 Compact equations in bispherical basis. 352 7.3.2 Separable potentials........... 353 7.3.3 Superposition of Coulomb and a separable potentials. 356 7.4 Cluster Integrals.......... 358 7.4.1 Formulation of the problem............. 358 7.4.2 The second cluster integral............. 360 7.4.3 Preparatory formulae for the N-th cluster integral 364 7.4.4 Trace formula for three-particle systems 367 7.4.5 Calculation of Ll(l)(E) 373 7.4.6 A simple model. 383 8 Comments on Literature 389 Bibliography 395 Index 401
Introduction The last decade witnessed an increasing interest of mathematicians in problems originated in mathematical physics. As a result of this effort, the scope of traditional mathematical physics changed considerably. New problems especially those connected with quantum physics make use of new ideas and methods. Together with classical and functional analysis, methods from differential geometry and Lie algebras, the theory of group representation, and even topology and algebraic geometry became efficient tools of mathematical physics. On the other hand, the problems tackled in mathematical physics helped to formulate new, purely mathematical, theorems. This important development must obviously influence the contemporary mathematical literature, especially the review articles and monographs. A considerable number of books and articles appeared, reflecting to some extend this trend. In our view, however, an adequate language and appropriate methodology has not been developed yet. Nowadays, the current literature includes either mathematical monographs occasionally using physical terms, or books on theoretical physics focused on the mathematical apparatus. We hold the opinion that the traditional mathematical language of lemmas and theorems is not appropriate for the contemporary writing on mathematical physics. In such literature, in contrast to the standard approaches of theoretical physics, the mathematical ideology must be utmost emphasized and the reference to physical ideas must be supported by appropriate mathematical statements. Of special importance are the results and methods that have been developed in this way for the first time. This monograph is intended to present an example of an up-to-date and difficult problem of theoretical physics, namely the quantum mechanical problem of scattering of N particles. We have made this choice for sevxi
xii INTRODUCTION eral reasons: First, the N-body problem is a traditional and difficult problem of mathematical physics. Its quantum analog has a number of interesting applications in atomic and nuclear physics. Second, the problem is very interesting from the mathematical point of view. The mathematical treatment of this problem has led to considerable enrichment of methods of classical and functional analysis. Finally, for many years both authors worked and continue to work in it, so that this book can reflect their views, approaches, and results. As a mathematical task, the problem considered is too cumbersome and, moreover, has not been completely solved yet. A strict presentation according to the standards of theoretical mathematics requires a large number of technical details which may often be very tedious. hence, in accordance with the aforementioned methodology, we will give a basic explanation of the formal level of the theory, without giving the full proofs. However, in contrast to theoretical physics papers, we admit the necessity of proofs and will explicitly show that the results can be justified. In this way the more mathematically oriented readers will be freed from tedious estimates and physicists will get ideas on the mathematical methods in situations they are familiar with. Therefore we hope that this book will be useful both for mathematicians and physicists. This book is mainly focused on the new and rapidly growing group of specialists working in contemporary mathematical physics. The book can be divided into three parts: the general formulation of the problem of N particle scattering, its justification on the basis of compact integral equations and description of basic objects of scattering theory: wave functions, their asymptotics and scattering amplitudes. Chapters 1 and 2 are devoted to the first part, the second part is contained mainly in Chapters 3 and 6, and partially in Chapter 4 and 5. Chapter 7 and the main part of Chapter 4 and 5 are devoted to the last circle of problems. In Chapter 1, the basic dynamical concepts are introduced, the wave and scattering operators are defined and their general properties are described. In Chapter 2, we turn to the stationary formalism of the scattering theory. here we describe general properties of the resolvent of the energy operator and we obtain expressions for the kernels of wave operators and scaterring operators in terms of singularities of the resolvent kernel. Chapter 3 is devoted to the method of
INTRODUCTION xiii integral equations. The Fredholm-type equations are obtained for few-body systems and properties of their resolvent kernels are investigated also in the momentum representation. The next two chapters are devoted to the study of wave functions in the configuration space: Chapter 4 treats neutral particles and Chapter 5 treats systems of charged particles. The Fredholm-type integral equations for systems of charged particles are obtained in Chapter 5 too. Some problems of the mathematical foundation of scattering theory are considered in Chapter 6. Here, the general features of the wave operators discussed in Chapter 1 are proved. A number of applications of the stationary scattering theory is discussed in Chapter 7. The aim of each chapter is more precisely described in short introduction at the beginning. Similar introductory comments are presented at the beginning of each section. We do not refer to any specific bibliographic sources in the main text. References of special importance are given at the and of the book. Finally, we introduce some basic notation which will be used throughout the book. The term variable and the letters x, y, X, k, p, P with or without indices denote vectors in N-dimensional space. The symbol( k, p) denotes the scalar product of two vectors k and p, P = (k, k), Ikl = (P)1/2, k is the unit vector in the direction of k: k = I~I' and dk and dk are the volume and surface elements on the unit sphere. The symbol J with no indication of the integration range means the integral over the whole range of integration variables. The letters a!, y, X denote vectors in the configuration space, k, p, P, q are vectors in the momentum space. Transformation from the coordinate representation to the momentum representation is carried out by Fourier transformation, to wit j(k) = (27l"tft/2j exp{-i(k,x)}f(x)dx. As it is common in the physical literature, we will often use the same symbols for functions and their Fourier transforms f( x) +-t f( k) with the argument indicating the actual case.