Oscillation theorems for Sturm-Liouville problems with distribution potentials

Abstract

The Sturm-Liouville problem $$ \begin{array}{*{20}c} { - y'' + q(x)y = \lambda y,} \\ {y(0) = y(1) = 0} \\ \end{array} $$ is considered with a singular potential q(x) representing the derivative of a real function from the space L2[0, 1] in the distributional sense. Two approaches are developed for the study of oscillation properties of eigenfunctions of this problem. The first approach is based on generalization of methods of the Sturm theory. The second one is based on development of variational principles.