Inverse problem having special singularity type from two spectra

Abstract

It is well known that the two spectra {?n} and {?n} uniquely determine the potential function q(x) in a Sturm-Liouville equation defined on the unit interval and having of the special singularity type q(x) =x?p+q0 (x) (where ? is an constant, 1<p<2) at the point zero. In this work, we give the solution of the inverse problem on two partially non-coinciding spectra for the Sturm-Liouville equation with the peculiarity at zero. In particular in this case we obtain Hochstadt's theorem concerning the structure of the difference q (x) - q? (x).