Integral representation of one dimensional three particle scattering for delta function interactions

TL;DR

This paper derives an integral representation for the three-particle scattering problem on a line with delta function interactions, providing an exact solution using hyperspherical coordinates and special functions.

Contribution

It introduces a novel integral representation and solution method for three-particle scattering with delta interactions, extending previous approaches and enabling potential generalizations.

Findings

Exact solution for three-particle scattering with delta interactions

Integral representation involving Bessel and pseudo-Sturmian functions

Derivation of the scattering matrix from the recurrence relation

Abstract

The Schr\"{o}dinger 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 Schr\"{o}dinger 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.