Inverse problems: recovery of BV coefficients from nodes

We consider the Sturm-Liouville problem on a finite interval with Dirichlet boundary conditions. Let the elastic modulus and the density be of bounded variation. Results for both the forward problem and the inverse problem are established. For the forward problem, new bounds are established for the eigenfrequencies. The bounds are sharp. For the inverse problem, it is shown that the elastic modulus is uniquely determined, up to one arbitrary constant, by a dense subset of the nodes of the eigenfunctions when the density is known. Similarly it is shown that the density is uniquely determined, up to one arbitrary constant, by a dense subset of the nodes of the eigenfunctions when the elastic modulus is known. Algorithms for finding piecewise constant approximations to the unknown elastic modulus or density are established and are shown to converge to the unknown function at every point of continuity. Results from numerical calculations are presented.