Spectral Asymptotics at Thresholds for a Dirac-Type Operator on z2

In this article, we provide the spectral analysis of a Dirac-type operator on Z2 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. © 2022 Elsevier Inc.