Inequalities among eigenvalues of Sturm–Liouville problems with distribution potentials

Abstract

This paper deals with the eigenvalue problems for the Sturm–Liouville operators generated by the differential expression L(y)=−(p(x)y′)′+q(x)y with singular coefficients q(x) in the sense of distributions. We obtain the inequalities among the eigenvalues corresponding to different self-adjoint boundary conditions. The continuity region, the differentiability and the monotonicity of the nth eigenvalue corresponding to the separated boundary conditions are given. Oscillation properties of the eigenfunctions of all the coupled Sturm–Liouville problems are characterized. The main results of this paper can also be applied to solve a class of transmission problems.