Constant-step Lagrange meshes for central potentials

Abstract

The existence conditions of Lagrange meshes, i.e. meshes leading to very accurate results without analytical calculation of potential matrix elements, are studied in relation with sine and cosine bases on a finite interval. For each one of three different types of boundary conditions, three Lagrange meshes are found. A numerical example shows that (i) the Lagrange-mesh calculations nearly reach the same accuracy as a variational calculation involving the same basis in spite of the approximation on the potential matrix elements and (ii) they are much more accurate than mesh calculations which do not satisfy the existence conditions. The striking accuracy of the Gauss quadrature employed in the mesh approximation for the significant eigenvalues remains unexplained in view of the limited accuracy of the same quadrature for individual potential matrix elements.