Convergence of expansions in Schr\"odinger and Dirac eigenfunctions, with an application to the R-matrix theory

TL;DR

This paper investigates the convergence properties of eigenfunction expansions in Schrödinger and Dirac R-matrix theories, confirming that in the Dirac case, the series often do not converge as previously claimed.

Contribution

It provides a detailed analysis of the convergence issues in R-matrix eigenfunction expansions for relativistic particles, clarifying longstanding misconceptions.

Findings

Eigenfunction series in Dirac R-matrix theory often fail to converge.

Confirmed previous findings that the series do not converge to the claimed limit.

Analyzed convergence properties in both nonrelativistic and relativistic contexts.

Abstract

Expansion of a wave function in a basis of eigenfunctions of a differential eigenvalue problem lies at the heart of the R-matrix methods for both the Schr\"odinger and Dirac particles. A central issue that should be carefully analyzed when functional series are applied is their convergence. In the present paper, we study the properties of the eigenfunction expansions appearing in nonrelativistic and relativistic $R$-matrix theories. In particular, we confirm the findings of Rosenthal [J. Phys. G: Nucl. Phys. 13, 491 (1987)] and Szmytkowski and Hinze [J. Phys. B: At. Mol. Opt. Phys. 29, 761 (1996); J. Phys. A: Math. Gen. 29, 6125 (1996)] that in the most popular formulation of the R-matrix theory for Dirac particles, the functional series fails to converge to a claimed limit.