Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 182, pp. 1-16.

Title: Reconstructing the potential function for indefinite
       Sturm-Liouville problems using infinite product forms

Authors: Mohammad Dehghan    (Univ. of Tabriz, Tabriz, Iran)
         Ali Asghar Jodayree (Univ. of Tabriz, Tabriz, Iran)

Abstract:
 In this article we consider the linear second-order equation of
 Sturm-Liouville type
 $$
 y''+(\lambda\phi^2(t)-q(t))y=0, \quad 0\leq t\leq 1,
 $$
 where $\lambda$ is a real parameter, $q(t)$ is the potential function
 and $\phi^2(t)$ is the weight function. We use the infinite product
 representation of the derivative of the solution to the differential equation
 with Dirichlet-Neumann conditions, and for the system of dual equations
 which is needed for expressing inverse problem and for retrieving potential.
 It must be mentioned that the weight function has a zero whose order
 is an integer called a turning point.

Submitted July 15, 2013. Published August 07, 2013.
Math Subject Classifications: 34B24, 34A55, 34E20, 34E05.
Key Words: Indefinite Sturm-Liouville problem; turning point; dual equations;
           infinite product form.