Eigenvalues of the discrete Schr\"odinger operator in the large coupling constant limit

Let $(\lambda_-,\lambda_+)$ be a spectral gap of a periodic Schr\"odinger 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 $\alpha$.