Asymptotics of Eigenvalues for Sturm-Liouville Problems with an Interior Singularity

We consider the asymptotic form of the eigenvalues of the linear differential equation −y″(x) + q(x) y(x) = λ y(x), −∞ < a < x < b < ∞, where a < 0 < b, q(x) is singular at x = 0, and y satisfies appropriate conditions at a, 0, and b. This extends previous work of Atkinson and of Harris. In particular, when q(x) = x−K, Atkinson derived asymptotic formulae which cover the case 1 ≤ K < ; Harris′s results cover the cases 1 ≤ K < . We now cover all of the cases 1 ≤ K < 2. Since the methods employed by both of these authors and ourselves apply only to limit circle, non-oscillatory expressions, our results now seem to take problems of this type to their conclusion.