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Transactions of the American Mathematical Society

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



Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of the second order

Author: M. Faierman
Journal: Trans. Amer. Math. Soc. 168 (1972), 1-52
MSC: Primary 34B25
MathSciNet review: 0296390
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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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Keywords: Linear system, two parameters, real-valued functions, continuous functions, eigenvalues, absolute maximum, transition points, asymptotic integration, Weber equation, parabolic cylinder function, modified Bessel equation, modified Bessel function, matching of solutions, adjacent subintervals, main equation, inverse function theorem, perturbed equation, eigenfunctions, orthogonal properties of the eigenfunctions, equations in integers
Article copyright: © Copyright 1972 American Mathematical Society