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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Quantum Mechanics (General and Nonrelativistic)

J. Math. Phys. 53, 022101 (2012); http://dx.doi.org/10.1063/1.3679763 (12 pages)

Convergence of expansions in Schrödinger and Dirac eigenfunctions, with an application to the R-matrix theory

Julia Stasińska

Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, PL 80-233 Gdańsk, Poland and Grup de Física Teòrica: Informació i Fenómens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

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(Received 12 January 2011; accepted 6 January 2012; published online 2 February 2012)

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 13, 491 (1987)] and Szmytkowski and Hinze [J. Phys. B 29, 761 (1996); Szmytkowski and Hinze J. Phys. A 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 limit claimed by other authors.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. EXPANSIONS APPEARING IN THE R -MATRIX THEORIES
    1. Nonrelativistic R -matrix theory
    2. Relativistic R -matrix theory
  3. CONVERGENCE THEOREMS
  4. CONCLUSION

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KEYWORDS and PACS

Keywords

convergence, eigenvalues and eigenfunctions, Schrodinger equation, wave functions

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3679763

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947).

    A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30, 252 (1958).

    G. Goertzel, Phys. Rev. 73, 1463 (1948).

    R. Szmytkowski, J. Math. Phys. 39, 5231 (1998)JMAPAQ000039000010005231000001;, 40, 4181 (1999)JMAPAQ000040000008004181000001 (Erratum).

    R. Szmytkowski, J. Math. Phys. 42, 4606 (2001)JMAPAQ000042000009004606000001.

    R. Barkhudaryan, AIP Conf. Proc. 936, 86 (2007)APCPCS000936000001000086000001.


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