Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A uniformly convergent series for Sturm-Liouville eigenvalues

Author: Davis Cope
Journal: Quart. Appl. Math. 42 (1984), 373-380
MSC: Primary 34B25
DOI: https://doi.org/10.1090/qam/757175
MathSciNet review: 757175
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Abstract: For the regular Sturm-Liouville problem with equation $ y'' + (\lambda - q(x))y = 0$ on $ 0 \le x \le \pi $, there are well-known asymptotic expansions for the eigenvalues and eigenfunctions. We show that these asymptotic expansions can be replaced by convergent series for sufficiently large eigenvalues. Convergence is uniform on the interval $ 0 \le x \le \pi $ and uniform with respect to the eigenvalues, in the sense that a single majorant bounds all series. The basic idea is to replace the asymptotic results, which use an expansion of powers of $ {n^{ - 1}}or{(n + 1/2)^{ - 1}}$ for integers $ n$, by a series in powers of $ {\mu ^{ - 1}}$, where $ {\mu ^2}$ is an eigenvalue for the corresponding constant coefficient Sturm-Liouville problem with equation $ y'' + \lambda y = 0$.

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DOI: https://doi.org/10.1090/qam/757175
Article copyright: © Copyright 1984 American Mathematical Society

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