Hochstadt’s results for inverse Sturm–Liouville problems with finite number of transmission and parameter dependent boundary conditions

Abstract

This paper deals with the boundary value problem involving the differential equation ?y ?? + qy = ?y, subject to the parameter dependent boundary conditions with finite number of transmission conditions. The potential function q ? L ² (0, ?) is real and ? is a spectral parameter. We develop the Hochstadt’s results based on the transformation operator for inverse Sturm–Liouville problem when there are finite number of transmission and parameter dependent boundary conditions. Furthermore, we establish a formula for q(x) ? q˜(x) in the finite interval [0, ?], where q(x) and ˜q(x) are analogous functions.