Eigenvalue Asymptotics near a flat band in presence of a slowly decaying potential

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues near a flat band for Dirac-type and Laplace operators on periodic structures, revealing accumulation rates influenced by potential decay and flat band structure.

Contribution

It provides new eigenvalue asymptotics for Dirac-type operators with slowly decaying potentials and extends results to Laplace operators on periodic graphs.

Findings

Eigenvalues accumulate near the flat band at a semiclassical rate.

The accumulation rate constant encodes the flat band's structure.

Similar behavior observed for Laplace operators on periodic graphs.

Abstract

We provide eigenvalue asymptotics for a Dirac-type operator on $\mathbb Z^n$, $n\geq 2$, perturbed by multiplication operators that decay as $|\mu|^{-\gamma}$ with $\gamma<n$. We show that the eigenvalues accumulate near the value of the flat band at a ''semiclassical'' rate with a constant that encodes the structure of the flat band. Similarly, we show that this behaviour can be obtained also for a Laplace operator on a periodic graph.