Spectral Asymptotics at Thresholds for a Dirac-type Operator on $\mathbb{Z}^2$

TL;DR

This paper analyzes the spectral behavior of a Dirac-type operator on the two-dimensional integer lattice, revealing how the spectral shift function behaves near thresholds and how it encodes band interactions.

Contribution

It provides the first detailed spectral asymptotics at thresholds for a Dirac-type operator on ^2, linking the spectral shift function to band interactions via pseudo-differential operators.

Findings

Spectral shift function remains bounded outside a single threshold.

Main asymptotic term of the spectral shift function is characterized.

Interaction between flat and non-constant bands is encoded in the asymptotics.

Abstract

In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main asymptotic term in the unbounded case. Interestingly, we show that the constant in the main asymptotic term encodes the interaction between a flat band and whole non-constant bands. The strategy used is the reduction of the spectral shift function to the eigenvalue counting function of some compact operator which can be studied as a toroidal pseudo-differential operator.