On the Determination of the Singular Sturm-Liouville Operator from Two Spectra

Abstract

In this paper an inverse problem by two given spectrum for a second-order differential operator with coulomb singularity of the type A/x in zero point (here A is constant), is studied. It is well known that two spectrum {lambda(n)} and {mu(n)} uniquely determine the potential function q(x) in the singular Sturm-Liouville equation defined on interval (0,pi]. The aim of this paper is to prove the generalized degeneracy of the kernel K(x,t). In particular, we obtain a new proof of the Hochstadt's theorem concerning the structure of the difference (q) over tilde (x) - q(x).