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# Distribution of Eigenvalues of A Two-Parameter System of Differential Equations

M. Faierman
Transactions of the American Mathematical Society
Vol. 247 (Jan., 1979), pp. 45-86
DOI: 10.2307/1998775
Stable URL: http://www.jstor.org/stable/1998775
Page Count: 42
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## Abstract

In this paper two simultaneous Sturm-Liouville systems are considered, the first defined for the interval $0 \geqslant x_1 \geqslant 1$, the second for the interval $0 \geqslant x_2 \geqslant 1$, 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 $j + k \rightarrow \infty$, 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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