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arXiv:math-ph/0406019 · Journal of Mathematical Physics

Integral representation of one-dimensional three particle scattering for δ function interactions

Alejandro Amaya-Tapia, G. Gasaneo, S. Yu. Ovchinnikov, J. H. Macek, S. Y. Larsen

Published Aug 6, 2004• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.1769613

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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.
The reformulation of the one-dimensional three-body problems with boundary condition and delta function interactions, based on the Sommerfeld–Maluzhinetz transformation, is presented. The argumentation is carried out as exemplified by two models–the exactly soluble model of two identical particles interacting through delta potential and each of which interacts with a third one through boundary condition interactions, and a model of two identical particles and a fixed wall, all interactions being of the delta function type. The problems are reduced to those of solving coupled systems of functional equations for the Sommerfeld transforms of the wavefunction. The functional properties of the transforms are then used to derive expressions relating them to the half off-shell extensions of the elastic and exchange probability amplitudes as defined in the Faddeev–Lovelace approach.
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.
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.
An algebraic formulation of the problem of three particles in one dimension is given, where the particles interact with delta function potentials of arbitrary strength and have almost arbitrary mass. An algebraic formulation is taken to mean that the steps implied from formulation to solution involve finite algebra. The canonical example is equal mass particles interacting with equal strength delta function potentials, where the Bethe ansatz holds and the solution involves only sums of products of matrices with elements that are rational functions of a complex variable. When the Bethe ansatz fails the Sommerfeld diffraction ansatz is satisfied if a condition of internal consistency is met. This condition of internal consistency requires the solution to a Riemann–Hilbert functional equation with an algebraic coefficient. The solution to this functional equation is an analytic, but not generally a meromorphic function. It is demonstrated that an asymptotic solution may be constructed within the domain of algebraic functions.
We derive some new equations for the wavefunction for the three-body scattering problem using the ``partial-wavefunction approach.'' All scattering processes are solutions of inhomogeneous coupled integral equations for the scattered parts of the total wavefunction. We show how the Pauli principle is incorporated when some or all of the particles are identical. This leads to a reduction in the number of independent scattered partial wavefunctions when some of the particles are identical. The kernels in the integral equations contain no δ functions but each scattered partial wavefunction does have a self-coupling term. We apply the method to the scattering of three identical bosons interacting through zero-range pair potentials. The method gives the same equation for this problem as obtained by the Faddeev T-matrix approach.