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

TL;DR

This paper investigates how to reconstruct masses and springs in a finite spring-mass system from natural frequencies, focusing on rank two or three perturbations of Jacobi matrices, and provides explicit conditions for eigenvalue correspondence.

Contribution

It offers explicit descriptions of Green's functions and necessary and sufficient conditions for eigenvalue reconstruction in perturbed finite spring-mass systems.

Findings

Explicit formulas for Green's functions under perturbations

Necessary and sufficient conditions for eigenvalue correspondence

Characterization of reconstructibility of system parameters

Abstract

We consider a linear finite spring mass system which is perturbed by modifying one mass and adding one spring. From knowledge of the natural frequencies of the original and the perturbed systems we study when masses and springs can be reconstructed. This is a problem about rank two or rank three type 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 system respectively.