# Properties of a potential energy matrix in oscillator basis

## TL;DR

This paper analyzes the properties of potential energy matrices in an oscillator basis, revealing their eigenvalues and eigenfunctions, and connecting them to coordinate space potential values and scattering methods.

## Contribution

It provides a detailed study of potential energy matrices in oscillator bases, including eigenvalue correspondence to coordinate space and links to scattering theory methods.

## Key findings

- Eigenvalues match potential energy at specific points in coordinate space.
- Eigenfunctions are expansion coefficients of spherical Bessel functions.
- Close relation established between separable approximation and J-matrix method.

## Abstract

Matrix elements of potential energy are examined in detail. We consider a model problem - a particle in a central potential. The most popular forms of central potential are taken up, namely, square-well potential, Gaussian, Yukawa and exponential potentials. We study eigenvalues and eigenfunctions of the potential energy matrix constructed with oscillator functions. It is demonstrated that eigenvalues coincide with the potential energy in coordinate space at some specific discrete points. We establish approximate values for these points. It is also shown that the eigenfunctions of the potential energy matrix are the expansion coefficients of the spherical Bessel functions in a harmonic oscillator basis. We also demonstrate a close relation between the separable approximation and $L^{2}$ basis (J-matrix) method for the quantum theory of scattering.