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. 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.Mathematics Subject Classification(2010): 47B36, 15A29, 47A75, 15A18, 70J50.

Spectral analysis for linear semi-infinite mass-spring systems

“…Note that the validity of our results includes the case of finite dimensional Jacobi matrices. In this particular case, the results of this work coincide with the corresponding ones in [6].…”

“…3.1, 3.2]). We point out that the techniques and ideas developed in this work allow to tackle the corresponding generalizations of the inverse spectral analysis carried out in [6] and [4]. This is the subject of a forthcoming paper.…”

Ambarzumian-type Problems for Discrete Schrödinger Operators

“…[8,12,13,15,20] and references therein) and discrete (see e.g. [2][3][4][5][6][7][8][9][10][11]16,18] and references therein) settings.…”

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

“…Inverse spectral problems for Jacobi operators have been amply studied (see for instance [6,7,8,13,15,18,19,20,25,26,28] for the finite case and [9,12,13,16,17,32,33] for the infinite case). However, inverse spectral problems that involve the kind of perturbation producing J n from J have been treated, to the best of our knowledge, only in the finite case [8,25,26]. Yet, this sort of perturbation arises in a natural way from the view point of physics: it corresponds to the modification of one mass and spring constant at any place in the chain.…”

“…Inverse spectral problems for Jacobi operators have been amply studied (see for instance [6,7,8,13,15,18,19,20,25,26,28] for the finite case and [9,12,13,16,17,32,33] for the infinite case). However, inverse spectral problems that involve the kind of perturbation producing J n from J have been treated, to the best of our knowledge, only in the finite case [8,25,26].…”