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Asymptotic analysis of a singular Sturm-Liouville boundary value problem

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Abstract

Asymptotic expansions are given for the eigenvalues λn and eigenfunctions un of the following singular Sturm-Liouville problem with indefinite weight:

$$\begin{gathered} - ((1 - x^2 )u'(x))' = \lambda xu(x) on ( - 1,1), \hfill \\ lim_{| x | \to 1} u(x) finite \hfill \\ \end{gathered} $$

This eigenvalue problem arises if one separates variables in a partial differential equation which describes electron scattering in a one-dimensional slab configuration.

Asymptotic expansions of the normalization constants of the eigenfunctions are also given. The constants in these asymptotic expansions involve complete elliptic integrals. The asymptotic results are compared with the results of numerical calculations.

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Authors and Affiliations

  1. National Institute of Public Health and Environmental Hygiene, P. O. Box 150, 2260 AD, Leidschendam, The Netherlands

    E. J. M. Veling

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  1. E. J. M. Veling
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This paper was written while the author was a member of the Department of Applied Mathematics of the Mathematical Centre, Amsterdam, The Netherlands.

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Veling, E.J.M. Asymptotic analysis of a singular Sturm-Liouville boundary value problem. Integr equ oper theory 7, 561–587 (1984). https://doi.org/10.1007/BF01238866

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