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 = \lambda\rho u on a finite interval (0; h). We give some uniqueness results on q and \rho from the Gelfand spectral data, when the coefficients p and \rho 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 \rho and in the case where the coefficients p; q and \rho are smooth we give a uniqueness results on both q and \rho