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

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