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Transactions of the American Mathematical Society

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

 

 

Distribution of eigenvalues of a two-parameter system of differential equations


Author: M. Faierman
Journal: Trans. Amer. Math. Soc. 247 (1979), 45-86
MSC: Primary 34B25; Secondary 34E05
DOI: https://doi.org/10.1090/S0002-9947-1979-0517686-7
MathSciNet review: 517686
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Abstract: In this paper two simultaneous Sturm-Liouville systems are considered, the first defined for the interval $ 0\, \leqslant \,{x_1}\, \leqslant \,1$, the second for the interval $ 0\, \leqslant \,{x_{2\,}}\, \leqslant \,1$, and each containing the parameters $ \lambda $ and $ \mu $. Denoting the eigenvalues and eigenfunctions of the simultaneous systems by $ ({\lambda _{j,k}},{\mu _{j,k}})$ and $ {\psi _{j,k}}({x_{1,}}{x_2})$, respectively, $ j,\,k\, = \,0,\,1,\, \ldots \,$, asymptotic methods are employed to derive asymptotic formulae for these expressions, as $ j + k \to \infty $ when $ (j,\,k)$ is restricted to lie in a certain sector of the $ (x,\,y)$ -plane. These results constitute a further stage in the development of the theory related to the behaviour of the eigenvalues and eigenfunctions of multiparameter Sturm-Liouville systems and answer an open question concerning the uniform boundedness of the $ {\psi _{j,k}}\,({x_1},\,{x_2})$.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0517686-7
Keywords: Two-parameter systems, simultaneous Sturm-Liouville systems, eigenvalues, eigenfunctions, asymptotic formulae, transition point, Bessel functions
Article copyright: © Copyright 1979 American Mathematical Society