Spectral asymptotics for Sturm-Liouville equations with indefinite weight

The Sturm-Liouville equation

is considered subject to the boundary conditions

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.