Convergent Disfocality and Nondisfocality Criteria for Second-Order Linear Differential Equations

Abstract

This paper presents a method to determine whether the second-order linear differential equation $y^{\prime \prime }+q(x)y=0$ is either disfocal or nondisfocal in a fixed interval. The method is based on the recursive application of a linear operator to certain functions and yields upper and lower bounds for the distances between a zero and its adjacent critical points, which will be shown to converge to the exact values of such distances as the recursivity index grows.