Regularization of singularities in Lagrange-mesh calculations

Abstract

The authors propose a regularization technique which allows Lagrange-mesh calculations to retain their accuracy, efficiency and simplicity when the Hamiltonian is singular at finite distance. Compact analytical expressions of kinetic-energy matrix elements are an essential ingredient of the method. The authors present general procedures for deriving them. Their approach is tested on the hydrogen atom and the squared orbital-momentum operator. They demonstrate its accuracy on the non-separable problem of the hydrogen atom in a magnetic field. This method opens the way to numerous new applications of Lagrange-mesh calculations in atomic and molecular physics.

Lagrange-mesh calculations in momentum space

“…The LMM relies on the existence of a N -point mesh {x i }, which is associated with an orthonormal set of N indefinitely derivable functionsf j (x), called the Lagrange function [2][3][4]. Each functionf j (x) satisfies the Lagrange conditions,f…”

“…The use of the LMM in configuration space is described in [2][3][4][5][6][7][8]13]. We will present here the formulation for the integral equation (11) within the LMM.…”

“…An exact expression can easily be obtained (see the Appendix in [3]). However, as shown in [4], it is preferable to use the approximation (27)- (28).…”

“…Among these techniques, the Lagrange-mesh method (LMM), which is especially easy to implement, can produce very accurate results. First created to compute eigenvalues and eigenfunctions of a two-body Schrödinger equation [2][3][4][5][6][7], it has been extended to treat semirelativistic Hamiltonian [8][9][10][11]. The trial eigenstates are developed in a basis of particular functions, the Lagrange functions, which vanish at all mesh points, except one.…”

Lagrange mesh, relativistic flux tube, and rotating string

“…The Lagrange mesh method is a very accurate and simple procedure to compute eigenvalues and eigenfunctions of a two-body Schrödinger equation [1,2,3]. The trial eigenstates are developed in a basis of well-chosen functions, the Lagrange functions, and the Hamiltonian matrix elements are obtained with a Gauss quadrature.…”