Lagrange-mesh calculations in momentum space

TL;DR

This paper adapts the Lagrange-mesh method, originally for configuration space, to momentum space eigenequations, enabling accurate and easy computation of eigenfunctions and observables with minimal potential evaluations.

Contribution

It introduces a novel adaptation of the Lagrange-mesh method for momentum space, maintaining its simplicity and accuracy for solving eigenequations.

Findings

The method accurately computes eigenfunctions in momentum space.

Eigenvalues converge with small mesh sizes for Gaussian potentials.

Eigenvalues require larger meshes for Yukawa potentials.

Abstract

The Lagrange-mesh method is a powerful method to solve eigenequations written in configuration space. It is very easy to implement and very accurate. Using a Gauss quadrature rule, the method requires only the evaluation of the potential at some mesh points. The eigenfunctions are expanded in terms of regularized Lagrange functions which vanish at all mesh points except one. It is shown that this method can be adapted to solve eigenequations written in momentum space, keeping the convenience and the accuracy of the original technique. In particular, the kinetic operator is a diagonal matrix. Observables in both configuration space and momentum space can also be easily computed with a good accuracy using only eigenfunctions computed in the momentum space. The method is tested with Gaussian and Yukawa potentials, requiring respectively a small or a great mesh to reach convergence.