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

TL;DR

This paper investigates how the eigenvalues of a discrete Schr"odinger operator behave asymptotically as the coupling constant becomes large, especially focusing on eigenvalues passing through a specific spectral point within a gap.

Contribution

It provides an analysis of the asymptotic distribution of eigenvalues for large coupling constants in a discrete Schr"odinger operator with decaying potential.

Findings

Eigenvalues pass through a spectral point within the gap as the coupling increases.

Asymptotic formulas describe the number of eigenvalues crossing a point.

Results extend understanding of spectral flow in discrete operators.

Abstract

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 $\alpha$.