Spectral analysis for linear semi-infinite mass-spring systems

Abstract

We study how the spectrum of a Jacobi operator changes when this operator is modified by a certain finite rank perturbation. The operator corresponds to an infinite mass‐spring system and the perturbation is obtained by modifying one interior mass and one spring of this system. In particular, there are detailed results of what happens in the spectral gaps and which eigenvalues do not move under the modifications considered. These results were obtained by a new tecnique of comparative spectral analysis and they generalize and include previous results for finite and infinite Jacobi matrices.

Inverse problems for Jacobi operators IV: interior mass-spring perturbations of semi-infinite systems

“…This paper is a continuation of recent work on the matter [9][10][11][12] and presents substantial generalizations of previous results. We are now able to manage the situation where the perturbation takes place at any arbitrary interior mass and spring of the system.…”

“…Within the regime of validity of the Hooke's law, the Jacobi operator J models the semiinfinite linear mass-spring system of figure 1 [10,12] with (1.2). See [15,25] for an explanation of the deduction of these formulae in the finite case.…”

“…To tackle the inverse problem, we use the characterization of the relative distribution of the spectra of J and J n given in [12]. Here a central role is played by the Green functions of the original and perturbed operators.…”