B. J. Harris and F. Marzano: Eigenvalue approximations for linear periodic differential equations with a singularity

Eigenvalue approximations for linear periodic differential equations with a singularity

B. J. Harris, Northern Illinois University, DeKalb, Illinois, U.S.A.
F. Marzano, Edinboro University of Pennsylvania, Edinboro, Pennsylvania, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 7. (1999), pp. 1-18.

Communicated by Xinzhi Liu.

Appeared on 1999-01-01

Abstract: We consider the second order, linear differential equation
$$y''(x) + (\lambda.q(x)) y(x) = 0 \eqno{(1)}$$
where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semiperiodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semiperiodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$.

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