On inverse problem for singular Sturm-Liouville operator from two spectra - Panakhov, E. S.; Yilmazer, R. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
In the paper, an inverse problem with two given spectra for second order differential operator with singularityof type $\cfrac{2}{r} + \cfrac{l(l+1)}{r^2}$(here, $l$ is a positive integer or zero) at zero point is studied. It is well known thattwo spectra $\{λ_n\}$ and $\{μ_n\}$ uniquely determine the potential function $q(r)$ in a singular Sturm-Liouville equation defined on 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 Hochstadt's theorem concerning the structure of the difference $\widetilde{q}(r) - q(r)$.