Periodic Jacobi operator with finitely supported perturbations

TL;DR

This paper analyzes the spectral properties of a periodic Jacobi operator with finitely supported perturbations, providing asymptotic behavior of the spectrum under small perturbations.

Contribution

It characterizes the spectral changes of a periodic Jacobi operator when finitely supported perturbations are introduced, including asymptotic spectrum analysis for small perturbations.

Findings

Spectral properties are described for the perturbed Jacobi operator.

Asymptotics of the spectrum are obtained for small perturbations.

The impact of finitely supported perturbations on the spectrum is quantified.

Abstract

We describe the spectral properties of the Jacobi operator $(Hy)_n= a_{n-1} y_{n-1}+a_{n}y_{n+1}+b_ny_n,$ $n\in\Z,$ with $a_n=a_n^0+ u_n,$ $b_n= b_n^0+ v_n,$ where sequences $a_n^0>0,$ $b_n^0\in\R$ are periodic with period $q$, and sequences $ u_n,$ $ v_n$ have compact support. In the case $ u_n\equiv 0$ we obtain the asymptotics of the spectrum in the limit of small perturbations $ v_n.$