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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of the second order
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by M. Faierman PDF
Trans. Amer. Math. Soc. 168 (1972), 1-52 Request permission

Abstract:

We consider a two-point boundary value problem associated with an ordinary differential equation defined over the unit interval and containing the two parameters $\lambda$ and $\mu$. If for each real $\mu$ we denote the $m$th eigenvalue of our system by ${\lambda _m}(\mu )$, then it is known that ${\lambda _m}(\mu )$ is real analytic in $- \infty < \mu < \infty$. In this paper we concern ourselves with the asymptotic development of ${\lambda _m}(\mu )$ as $\mu \to \infty$, and indeed obtain such a development to an accuracy determined by the coefficients of our differential equation. With suitable conditions on the coefficients of our differential equation, the asymptotic formula for ${\lambda _m}(\mu )$ may be further developed using the methods of this paper. These results may be modified so as to apply to ${\lambda _m}(\mu )$ as $\mu \to - \infty$ if the coefficients of our differential equation are also suitably modified.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 168 (1972), 1-52
  • MSC: Primary 34B25
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0296390-0
  • MathSciNet review: 0296390