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

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 {λn} and {µn} uniquely determine the potential function q(x) in the singular Sturm-Liouville equation defined on interval (0,π]. 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~(x) - q(x).