薛定谔和狄拉克本征函数展开式的收敛性及其在R-矩阵理论中的应用
Convergence of expansions in Schrödinger and Dirac eigenfunctions, with an application to the R-matrix theory
摘要：
在微分本征值问题的本征函数基中展开波函数是薛定谔和狄拉克粒子的R矩阵方法的核心。在应用函数级数时，应该仔细分析的一个中心问题是它们的收敛性。本文研究了非相对论和相对论$R$矩阵理论中本征函数展开式的性质。特别地，我们证实了Rosenthal [J. Phys. G：Nucl. Phys. 13，491（1987）]和Szmytkowski和Hinze [J. Phys. B：At.摩尔Opt.Phys.29，761（1996）; J.Phys.A：Math.Gen.29，6125（1996）]，在狄拉克粒子的R-矩阵理论的最流行的公式中，泛函级数不能收敛到所要求的极限。
英文摘要：
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 Schrödinger 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.
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