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 = λru 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 Calderón 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 = λRu where P and R are diagonal n × n matrices and Q a symmetric n × n matrix with L∞( ) 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.