J-matrix method for calculations of three-body Coulomb wave functions and cross sections of physical processes
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dicated to the memory of Victor Andreevich Knyr
Jmatrix Method for Calculations of ThreeBody Coulomb Wave Functions and Cross Sections of Physical Processes Yu. V. Popova, S. A. Zaytsevb, and S. I. Vinitskyc a
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991 Russia email: [email protected] b Pacific National University, Khabarovsk, 680035 Russia cJoint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia email: [email protected]
Abstract—The review is devoted to a widely known method of numerical solution to the threebody Cou lomb problem, namely, the Jmatrix method. Special attention is paid to ways of solving the Lippmann– Schwinger integral equation without attraction of pseudostates. Difficulties related to the formulation of the integral equation in spherical coordinates, leading to the divergence of its integral part if the wave function is calculated with two asymptotically free electrons, are demonstrated. In addition, the relation between exact and approximate solutions turns out to be unclear if the matrix of a residual potential is restricted to a finite number of basis functions, with the latter being increased. It is shown that, in principle, these problems can be avoided by reformulating a problem in parabolic coordinates. DOI: 10.1134/S1063779611050042
CONTENTS INTRODUCTION 1. MATRIX DIFFERENTIAL EQUATIONS: PSEUDOSTATES 1.1. Mathematical Formalism for Calculating a Spectral Function with a Single Asymptotically Free Electron in the Field of an Ion 1.2. Two Asymptotically Free Electrons 2. FADDEEV–MERKURIEV MATRIX INTEGRAL EQUATIONS 2.1. Integral Equation for the Wave Function of Two Asymptotically Free Electrons in the Field of a Nucleus 2.2. Matrix Integral Equations and Pseudostates 2.3. Single Ionization of a Helium Atom 2.4. Double Ionization of a Helium Atom 2.5. Problems and Their Discussion 3. PARABOLIC COORDINATES 3.1. Formulation of the Problem 3.2. Discrete Analogue of the Lippmann–Schwinger Equation 3.3. Matrix of One and TwoDimensional Coulomb Green’s Functions 3.4. Matrix of a SixDimensional Coulomb Green’s Function
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3.5. Discussion of Results of Section 3 CONCLUSIONS APPENDIX REFERENCES
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INTRODUCTION The remarkable book by V.P. Zhigunov and B.N. Zakhariev Methods of Strongly Coupled Channels in Quantum Scattering Theory [1] appeared slightly less than 40 years ago. In this book, in addition to an intro duction to the methods of nonrelativistic quantum scattering theory that pave the ways for numerically solving a multiparticle Schrödinger equation, a review of the stateoftheart in this field of scientific research at that time was presented. In a certain sense, this review reflects on the development of those methods and approaches that took place since then in the field of atomic physics wherein Coulomb potentials domi nate, with a specific impact on the scattering theory of several quantum particles. Approaches used lately worldwide and th
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