On the one dimensional Gelfand and Borg-Levinson spectral problems for discontinuous coefficients.

In this paper, we deal with the inverse spectral problem for the equation −(pu′)′+qu = λρu on a finite interval (0, h). We give some uniqueness results on q and ρ from the Gelfand spectral data, when the coefficients p and ρ are piecewise Lipschitz and q is bounded. We also prove an equivalence result between the Gelfand spectral data and the Borg-Levinson spectral data. As a consequence, we have similar uniqueness results if we consider the Borg-Levinson spectral data. Finally, we consider the inverse problem from the nodes and give uniqueness results on ρ and in the case where the coefficients p, q and ρ are smooth we give a uniqueness results on both q and ρ