Inverse Sturm-Liouville problem with discontinuity conditions

Abstract

This paper deals with the boundary value problem involving the di erential equation ly := -y''+qy = λy, subject to the standard boundary conditions along with the following discontinuity conditions at a point a ε(0,π) y(a + 0) = a1y(a - 0), y'(a + 0) = a1-1y'(a - 0) + a2y(a - 0), where q(x), a1,a2 are real, q ε L2(0,π ) and λ is a parameter independent of x. We develop the Hochestadt's result based on the transformation operator for inverse Sturm-Liouville problem when there are discontinuous conditions. Furthermore, we establish a formula for q(x) - q~(x) in the nite interval where q(x) and q~(x) are analogous functions.

Keywords

• Inverse problem
• Sturm-Liouville problems
• Discontinuous conditions
• Green's function