Decomposition of spectral asymptotics for Sturm-Liouville equations with a turning point

Abstract

The Sturm-Liouville equation $-y'' + qy = \lambda ry,$ on $ [0,l],$ is considered subject to the boundary conditions \noindent $y(0)\cos\alpha(0) = y'(0)\sin\alpha(0),$ $y(l)\cos\alpha(l) = y'(l)\sin\alpha(l).$ We assume that $r$ is piecewise continuous with a variety of behaviours at $0, l$ and an interior turning point. We give asymptotic approximations for the eigenvalues $\lambda_n$ of the above boundary value problem in forms equivalent to $\lambda_n = an^2+bn+O(n^\tau)$, where $\tau < 1$.