Spectral asymptotics for Sturm-Liouville equations with indefinite weight

Abstract:

The Sturm-Liouville equation \begin{equation*} -(py’)’ + qy =\lambda ry \;\; \text {on}\;\; [0,l] \end{equation*} is considered subject to the boundary conditions \begin{align*} y(0)\cos \alpha &= (py’)(0)\sin \alpha ,\\ y(l)\cos \beta &= (py’)(l)\sin \beta . \end{align*} We assume that $p$ is positive and that $pr$ is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to $O(1/\sqrt {n})$ for $\sqrt {\lambda _n}$, or equivalently up to $O(\sqrt {n})$ for $\lambda _n$, the eigenvalues of the above boundary value problem.