A note on discrete Borg-type theorems

TL;DR

This paper explores discrete Borg-type theorems for Schrödinger operators, showing that small spectral gaps imply the potential is nearly constant, with extensions to broader contexts and connections to the Ten Martini problem.

Contribution

It introduces new discrete Borg-type theorems using linear algebra techniques, linking spectral gap size to potential constancy, and extends results to more general settings.

Findings

Periodic potential is nearly constant if spectral gaps are small.

Extended results to broader operator settings.

Discussion of connections to the Ten Martini problem.

Abstract

We consider the discrete versions of the well known Borg theorem and use simple linear algebraic techniques to obtain new versions of the discrete Borg type theorems. To be precise, we prove that the periodic potential of a discrete Schrodinger operator is almost a constant if and only if the possible spectral gaps of the operator are of small width. This result is further extended to more general settings and the connection to the well known Ten Martini problem is also discussed.