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Physical Review A

Nonuniqueness of solutions to the Lippmann-Schwinger equation in a soluble three-body model

E. Gerjuoy, Sadhan K. Adhikari

Published Apr 1, 1985• 13 citations
Publisher

Abstract

It is generally accepted that with suitable boundary conditions the Lippmann-Schwinger (LS) integral equation is equivalent to the Schr\"odinger equation, even in systems of more than two particles. It also is generally accepted that although in two-particle systems the scattering solutions to the LS equation are uniquely defined by that equation without need for boundary conditions, the same cannot be said of systems containing more than two particles; in such many-particle systems, it is generally agreed, unique scattering solutions to the LS equation are not obtained unless boundary conditions are imposed. However, these assertions have not been verified heretofore in any system of three or more particles, because exact closed-form solutions of the Schr\"odinger or LS equation for such systems are so difficult to achieve. We have examined these questions in a one-dimensional three-body model first discussed by McGuire, wherein three equal-mass particles interact via equal and finite strength attractive \ensuremath{\delta}-function potentials. With this model, the scattering solutions to the Schr\"odinger equation can be written exactly, in closed form. Thereby we are able to demonstrate explicitly that in this model the scattering solutions to the Schr\"odinger equation do satisfy the LS equation; we also demonstrate explicitly that the LS equation's scattering solutions really are nonunique unless boundary conditions are imposed. This latter result strongly suggests that, in three-particle systems at any rate, recent criticisms of the aforementioned generally accepted nonuniqueness thesis are not well taken.

DOI
10.1103/physreva.31.2005

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It is generally accepted that with suitable boundary conditions the Lippmann-Schwinger (LS) integral equation is equivalent to the Schr"odinger equation, even in systems of more than two particles. It also is generally accepted that although in two-particle systems the scattering solutions to the LS equation are uniquely defined by that equation without need for boundary conditions, the same cannot be said of systems containing more than two particles; in such many-particle systems, it is generally agreed, unique scattering solutions to the LS equation are not obtained unless boundary conditions are imposed. However, these assertions have not been verified heretofore in any system of three or more particles, because exact closed-form solutions of the Schr"odinger or LS equation for such systems are so difficult to achieve. We have examined these questions in a one-dimensional three-body model first discussed by McGuire, wherein three equal-mass particles interact via equal and finite strength attractive \ensuremath{\delta}-function potentials. With this model, the scattering solutions to the Schr"odinger equation can be written exactly, in closed form. Thereby we are able to demonstrate explicitly that in this model the scattering solutions to the Schr"odinger equation do satisfy the LS equation; we also demonstrate explicitly that the LS equation's scattering solutions really are nonunique unless boundary conditions are imposed. This latter result strongly suggests that, in three-particle systems at any rate, recent criticisms of the aforementioned generally accepted nonuniqueness thesis are not well taken.
The boundary condition on the solution to the nonrelativistic time-independent Schr\"odinger equation for arbitrarily complicated rearrangements of spinless particles is carefully examined. For real energies EEE it is shown that the outgoing boundary condition on the scattered wave $\ensuremath{\phi}$ need not imply that $\ensuremath{\phi}$ is "every-where outgoing." This and similar considerations make apparent the fact remarked by Foldy and Tobocman, namely that the Lippmann-Schwinger integral equation need not have a unique solution for real energies. The relationship of this result to the added fact that solutions to the Lippmann-Schwinger integral equation are unique for complex energies $E+i\ensuremath{\epsilon}$, $\ensuremath{\epsilon}>0$ is discussed, as is also the relationship of the usual operator manipulations to the outgoing boundary condition.
The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.
In this paper it is shown that several cases of one-dimensional N-body problems are exactly soluble. The first case describes the motion of three one-dimensional particles of arbitrary mass which interact with one another via infinite-strength, repulsive delta-function potentials. It is found in this case that the stationary-state solution of the scattering of the three particles is analogous to an electro-magnetic diffraction problem which has already been solved. The solution to this analogous electro-magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite- but equal-strength delta-function potentials is exactly soluble. This example exhibits rearrangement and bound-state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite- but equal-strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N-particle bound state do occur. These rearrangements and the N-particle bound state are illustrated by means of a series of sample calculations.
Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.
We extend a method previously presented for the three-body problem with attractive interparticle potentials to the case of repulsive interparticle potentials, with periodic boundary conditions on each particle. As before, we decompose the wave function into three parts, and from the Schrödinger equation write equations for these parts. We observe that, in one-dimension and for δ-function potentials,these equations can be easily solved numerically, and we present these solutions for potential“strengths” ranging from zero to infinity. We then discuss a more general method for solving these equations, which method involves expansions in a certain set of two-body functions. This general method is not necessarily limited to either one dimension or δ-function potentials, but, as a check on it, we do apply it to that case and get good agreement with the previous numerical results. As a further exploration of the method, we apply it to square well potentials in one dimension. In an Appendix we discuss the set of two-body functions that we use.
A self-contained discussion of integral equations of scattering is presented in the case of centrally symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green’s function, Lippmann–Schwinger integral equations of scattering for wave function and transition operator, optical theorem, and unitarity relation. We illustrate the present approach with a Dirac delta potential.