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 $y” + \lambda y = 0$.
E. T. Copson, Theory of functions of a complex variable, Oxford University Press, London, 1935
- George Fix, Asymptotic eigenvalues of Sturm-Liouville systems, J. Math. Anal. Appl. 19 (1967), 519–525. MR 212256, DOI https://doi.org/10.1016/0022-247X%2867%2990009-1
- Harry Hochstadt, Asymptotic estimates for the Sturm-Liouville spectrum, Comm. Pure Appl. Math. 14 (1961), 749–764. MR 132863, DOI https://doi.org/10.1002/cpa.3160140408
- B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, American Mathematical Society, Providence, R.I., 1975. Translated from the Russian by Amiel Feinstein; Translations of Mathematical Monographs, Vol. 39. MR 0369797
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
E. T. Copson, Theory of functions of a complex variable, Oxford University Press, London, 1935
G. Fix, Asymptotic eigenvalues of Sturm-Liouville systems, J. Math. Anal. Appl. 19 519–525 (1967)
H. Hochstadt, Asymptotic estimates for the Sturm-Liouville spectrum, Comm. Pure Appl. Math. 14 749–764 (1961)
B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, Translations of Mathematical Monographs, vol. 39, American Mathematical Society, Providence, R.I., 1975 (Original Russian edition 1970)
E. C. Titchmarch, Eigenfunction expansions associated with second-order differential equations, Part I, Clarendon Press, Oxford, 1962
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Article copyright:
© Copyright 1984
American Mathematical Society