On inverse problem for singular Sturm-Liouville operator from two spectra

Abstract

We study an inverse problem with two given spectra for a second-order differential operator with singularity of the type 2/r + ?(? + 1)/r 2 (here, l is a positive integer or zero) at zero point. It is well known that two spectra {? n} and {? n} uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, ?]. One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular, we obtain a new proof of the Hochstadt theorem concerning the structure of the difference q?(r) - q(r). © 2006 Springer Science+Business Media, Inc.