J-matrix method of scattering in one dimension: The nonrelativistic theory

TL;DR

This paper develops a nonrelativistic scattering theory in one dimension using the J-matrix method, revealing complex structures and demonstrating its potential as an alternative to traditional approaches.

Contribution

It introduces a novel 1D scattering formulation based on the J-matrix method with finite-range potentials, highlighting its structure and accuracy.

Findings

The 1D formulation shows a rich, non-trivial structure.

The method accurately models scattering with finite-range potentials.

Examples demonstrate the utility of the approach.

Abstract

We formulate a theory of nonrelativistic scattering in one dimension based on the J-matrix method. The scattering potential is assumed to have a finite range such that it is well represented by its matrix elements in a finite subset of a basis that supports a tridiagonal matrix representation for the reference wave operator. Contrary to our expectation, the 1D formulation reveals a rich and highly non-trivial structure compared to the 3D formulation. Examples are given to demonstrate the utility and accuracy of the method. It is hoped that this formulation constitutes a viable alternative to the classical treatment of 1D scattering problem and that it will help unveil new and interesting applications.