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

We provide eigenvalue asymptotics for a Dirac-type operator on ${\mathbb{Z}}^n$, $n\ge 2$, perturbed by multiplication operators that decay as $\mathrm{\mid }\mu {\mathrm{\mid }}^{-\gamma }$ 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.