Analysis of the R-matrix method on Lagrange meshes

The R-matrix method on a Lagrange mesh is a very simple approximation of the R-matrix method with a basis. By analysing an exactly solvable example, we observe that the mesh approximation does not reduce the accuracy of the R-matrix bound-state energies and phase shifts. This property is obtained with two different meshes, the shifted Legendre and shifted Jacobi meshes, which correspond to equivalent polynomial bases. Their comparison shows that the orthogonality of the Lagrange basis functions is not as crucial as was previously assumed: the Legendre mesh, which corresponds to a nonorthogonal Lagrange basis, is at least as accurate as the Jacobi mesh based on an orthogonal basis. We also emphasize the surprising origin of a known property of the R-matrix method: the results are much more accurate with basis functions without uniform boundary conditions because the quality of the matching is realized by a few highly excited eigenfunctions, with weak physical content, of the sum of the Hamiltonian and Bloch operators.