Annali di Matematica Pura ed Applicata

, Volume 135, Issue 1, pp 363–398

# 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

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