An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra

Abstract

A classical inverse problem is “can you hear the density of a string clamped at both ends?” The mathematical model gives rise to an inverse Sturm--Liouville problem for the unknown density $\rho$, and it is well known that the answer is negative: the Dirichlet spectrum from the clamped end-point conditions is insufficient. There are many known ways to add additional information to gain a positive answer, and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode---the so-called norming constants. We make the assumption that neither of these changes are possible. Instead we will add known mass-densities to the string in a way we can prescribe and remeasure the Dirichlet spectrum. We will not be able to answer the uniqueness question in its most general form, but will give some insight to what “added masses” should be chosen and how this can lead to a reconstruction of the original string density.