Some uniqueness results of discontinuous coefficients for the one-dimensional inverse spectral problem

In this paper, we deal with the inverse spectral problem for the equation -(pu')'+qu = lambdaru on a finite interval (0, h). The above equation subjected to appropriate boundary conditions on zero and h gives the vibrations of a string of length h. Using the Dirichlet-to-Neumann map type approach known in the multidimensional Calderon problem, we prove some uniqueness results of one or two discontinuous coefficients among p, q and r, and the length h from the vibrations of the end point zero. We also consider Sturm-Liouville systems of the form -(Pu')' + Qu = lambdaRu where P and R are diagonal n x n matrices and Q a symmetric n x n matrix with Linfinity (Omega) entries. In the case n = 2, this problem models the small vibrations of two connected beams. We prove the uniqueness of the matrix of rigidity P or matrix density R when its entries are piecewise constant functions.