On the instability of the essential spectrum for block Jacobi matrices

TL;DR

This paper investigates the conditions under which unbounded block Jacobi matrices exhibit instability in their essential spectrum, providing criteria for the emergence of a discrete spectrum on a half-axis and illustrating these with numerous examples.

Contribution

It introduces new sufficient conditions for essential spectrum instability in unbounded block Jacobi matrices and demonstrates their sharpness through extensive examples.

Findings

Identified conditions leading to essential spectrum instability.

Established criteria for the appearance of discrete spectrum on a half-axis.

Provided numerous examples confirming the sharpness of results.

Abstract

We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a half-axis of a real line. An extensive list of examples showing the sharpness of obtained results is provided.