Stability of the inverse resonance problem for Jacobi operators

TL;DR

This paper investigates the stability of the inverse resonance problem for Jacobi operators, showing that small changes in resonance data lead to small changes in the operator coefficients, under certain conditions.

Contribution

It establishes a stability result for the inverse resonance problem of Jacobi operators, linking closeness of resonance data to closeness of the operators' coefficients.

Findings

Stability of the inverse resonance problem proven for Jacobi operators.

Closeness of zeros and poles of reflection coefficients implies closeness of coefficients.

Results hold under the condition of at most one half-bound state.

Abstract

When the coefficients of a Jacobi operator are finitely supported perturbations of the 1 and 0 sequences, respectively, the left reflection coefficient is a rational function whose poles inside, respectively outside, the unit disk correspond to eigenvalues and resonances. By including the zeros of the reflection coefficient, we have a set of data that determines the Jacobi coefficients up to a translation as long as there is at most one half-bound state. We prove that the coefficients of two Jacobi operators are pointwise close assuming that the zeros and poles of their left reflection coefficients are $\eps$-close in some disk centered at the origin.