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

Faierman, M.

doi:10.1090/S0002-9947-1979-0517686-7

Distribution of eigenvalues of a two-parameter system of differential equations
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by M. Faierman

Trans. Amer. Math. Soc. 247 (1979), 45-86

DOI: https://doi.org/10.1090/S0002-9947-1979-0517686-7

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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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