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

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.