Szego limit theorem on the lattice

TL;DR

This paper establishes a Szeg"{o} limit theorem for operators on the lattice , describing the asymptotic behavior of traces of functions of certain projections and operators as the spectral parameter tends to infinity.

Contribution

It extends Szeg"{o} limit theorems to lattice operators with unbounded potentials, providing a new asymptotic trace formula for pseudo difference operators.

Findings

Proves the limit formula for trace ratios involving projections and pseudo difference operators.

Shows the asymptotic behavior of spectral projections for lattice operators with growing potentials.

Provides a framework for analyzing spectral asymptotics on  lattices.

Abstract

In this paper, we prove a Szeg\"{o} type limit theorem on $\ell^2(\ZZ^d)$. We consider operators of the form $H=\Delta+V$, $V$ multiplication by a positive sequence $\{V(n), n \in \ZZ^d\}$ with $V(n) \rightarrow \infty, |n| \rightarrow \infty $ on $\ell^2(\ZZ^d)$ and $\pi_{\lambda}$ the orthogonal projection of $\ell^2(\mathbb{Z}^d)$ on to the space of eigenfunctions of $H$ with eigenvalues $\leq \lambda$. We take $B$ to be a pseudo difference operator of order zero with symbol $b(x,n), (x,n) \in \TT^d\times \ZZ^d$ and show that for nice functions $f$ $$ \lim_{\lambda \rightarrow \infty} Tr(f(\pi_\lambda B\pi_\lambda))/Tr(\pi_\lambda) = \lim_{\lambda \rightarrow \infty} \frac{1}{(2\pi)^d} \frac{\sum_{V(n) \leq \lambda} \int_{\TT^d} f(b(x,n)) ~ dx}{\sum_{V(n)\leq\lambda} 1}. $$