Eigenvalues of the discrete Schrödinger operator in the large coupling constant limit

Let $({\lambda }_{-},{\lambda }_{+})$ be a spectral gap of a periodic Schrödinger operator $A$ on the lattice ${\mathbb{Z}}^d$. Consider the operator $A(\alpha )=A-\alpha V$ where $V$ is a decaying positive potential on ${\mathbb{Z}}^d$. We study the asymptotic behavior of the number of eigenvalues of $A(t)$ passing through a point $\lambda \in ({\lambda }_{-},{\lambda }_{+})$ as $t$ grows from $0$ to $\alpha$.