The Eigenvalues and The Optimal Potential Functions of Sturm-Liouville Operators

Abstract

In this article, we study the eigenvalues and potential functions of Sturm-Liouville operators with ordinary separated-type boundary condition. When the gap of the first two eigenvalues reaches minimum, we give the specific form of potential function. Meanwhile, for step potential function, we establish an one-to-one relationship between the eigenvalues and the nonnegative real roots of a class of algebraic equation, which provide an effective method for the approximate calculation of eigenvalues.