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 \(\frac{2}{r} + \frac{{\ell (\ell + 1)}}{{r^2 }}\) (here, l is a positive integer or zero) at zero point. It is well known that two spectra {λ n } and {λ n } uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, π]. 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 \(\tilde q(r) - q(r)\).