Inverse problems for Jacobi operators: I. Interior mass–spring perturbations in finite systems

Abstract

We consider a linear finite spring–mass system which is perturbed by modifying one mass and adding one spring. We study when masses and springs can be recovered from the natural frequencies of the original and the perturbed systems. This is a problem about rank 2 or rank 3 perturbations of finite Jacobi matrices where we are able to describe quite explicitly the associated Green’s functions. We give necessary and sufficient conditions for two given sets of points to be eigenvalues of the original and modified systems, respectively.