Analysis of theR-matrix method on Lagrange meshes

Abstract

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.

Integrals of Lagrange functions and sum rules

“…This mesh can be useful in the R-matrix method since the basis functions can simulate r l+1 behaviours at the origin and do not vanish at r = 1 [14]. Equation (59) becomes for β > 0,…”

“…by using Lagrange functions modified by an appropriate multiplicative factor. The same problem has been encountered with shifted Jacobi polynomials over the (0, 1) interval [14]. The corresponding Lagrange basis can be useful for the study of collisions within the R-matrix method [15].…”

Study of 14Be with core excitation

“…To calculate the positions of the 5/2 + resonances and the energies of the bound or virtual 1/2 + states we used the R-matrix method on a Lagrange mesh (see [31], and references wherein), which allows bound and continuum states to be found from the same small basis set. The energies of the 1 2 + virtual states are determined using second-order effective range theory.…”

CalculableR-matrix method for the Dirac equation

“…An important clarification of this problem was made possible by the introduction of the Bloch operator [11]. The convergence and accuracy become much better when a basis with a variety of boundary conditions is used [12] (see Ref. [4] for a discussion).…”

“…In the Schrödinger case, a further simplification keeping a good convergence is obtained by combining the Lagrange-mesh method [17,18] with the R matrix [19,12]. The resulting method does not require any analytical or numerical calculation of matrix elements.…”