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Annali di Matematica Pura ed Applicata

, Volume 135, Issue 1, pp 363–398 | Cite as

Asymptotic formulae for eigenvalues of limit circle problems on a half line

  • F. V. Atkinson
  • C. T. Fulton
Article

Summary

For the classical limit-circle eigenvalue problem for −y″+qy=λy on [α, ∞) asymptotic formulae for the eigenvalues are obtained. For the positive spectrum a Prüfer transformation and an iterative procedure of Atkinson are employed to obtain higher order terms in the asymptotic expansions of λn. For the negative spectrum a piecewise turningpoint analysis making use of a linear approximation to q in the neighborhood of the turning point and a WKB approximation away from the turning point is employed to obtain first approximations to both the eigenvalues and eigenfunctions. In both cases assumptions are placed on q, and the turning-point analysis gives rise to slightly stronger assumptions in the case of the negative spectrum. An example of Titchmarsh, q=−exp (2x (for which solutions are available for all values of the eigenvalue parameter in terms of Bessel functions), together with the limit-circle theory of Fulton, provides an independent verification of the general results for a specific case; in particular, Olver's uniform asymptotic expansion for Jv(vz) as v→∞ is used to double-check the asymptotic formula for the eigenfunctions in the case of the negative spectrum.

Keywords

Asymptotic Expansion Eigenvalue Problem Bessel Function Linear Approximation Iterative Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1983

Authors and Affiliations

  • F. V. Atkinson
    • 1
  • C. T. Fulton
    • 2
  1. 1.TorontoCanada
  2. 2.MelbourneUSA

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