A regularized Lagrange-mesh method based on an orthonormal Lagrange-Laguerre basis

TL;DR

This paper introduces a regularized Lagrange-mesh method using orthonormal Lagrange-Laguerre basis functions, improving accuracy in solving radial quantum potentials with singularities, and demonstrates its effectiveness in calculating bound states and phase shifts.

Contribution

A new regularized Lagrange-mesh method based on orthonormal functions that enhances accuracy for singular potentials in quantum problems.

Findings

Accurate bound-state energies for Coulomb and harmonic oscillator potentials.

Simple rule to predict when singularities affect accuracy.

Efficient phase shift calculations with few mesh points.

Abstract

The Lagrange-mesh method is an approximate variational approach having the form of a mesh calculation because of the use of a Gauss quadrature. Although this method provides accurate results in many problems with small number of mesh points, its accuracy can be strongly reduced by the presence of singularities in the potential term. In this paper, a new regularized Lagrange-Laguerre mesh, based on \textit{exactly} orthonormal Lagrange functions, is devised. It is applied to two solvable radial potentials: the harmonic-oscillator and Coulomb potentials. In spite of the singularities of the Coulomb and centrifugal potentials, accurate bound-state energies are obtained for all partial waves. The analysis of these results and a comparison with other Lagrange-mesh calculations lead to a simple rule to predict in which cases a singularity does induce or not a significant loss of accuracy in Lagrange-mesh calculations. In addition, the Lagrange-Laguerre-mesh approach is applied to the evaluation of phase shifts via integral relations. Small numbers of mesh points suffice to provide very accurate results.