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

Abstract

Let $(\hat{I}{»}_{-},\hat{I}{»}_{+})$ be a spectral gap of a periodic Schrödinger operator $A$ on the lattice ${\mathbb{Z}}^d$. Consider the operator $A(\hat{I}\pm )=A-\hat{I}\pm V$ 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 $\hat{I}»\in (\hat{I}{»}_{-},\hat{I}{»}_{+})$ as $t$ grows from $0$ to $\hat{I}\pm$.