Discrete Spectrum of the Bilayer Graphene Operator

We consider the graphene operator $D_m$ perturbed by a decaying potential $\alpha V$, where $\alpha$ is a coupling constant. We study the number $N(\lambda ,\alpha )$ of eigenvalues of the operator $D(t)=D_m-tV$ passing through a regular point $\lambda \in \rho (D_m)$ as $t$ changes from $0$ to $\alpha$. We obtain asymptotic formulas for $N(\lambda ,\alpha )$ as $\alpha \to \mathrm{\infty }$.