Szego limit theorem on the lattice

Abstract

In this paper, we prove a Szegö type limit theorem on $\ell^2(\ZZ^d)$. We consider operators of the form $H=Δ+V$, $V$ 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 $π_λ$ the orthogonal projection of $\ell^2(\mathbb{Z}^d)$ on to the space of eigenfunctions of $H$ with eigenvalues $\leq λ$. 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 $f$ $$ \lim_{λ\rightarrow \infty} Tr(f(π_λBπ_λ))/Tr(π_λ) = \lim_{λ\rightarrow \infty} \frac{1}{(2π)^d} \frac{\sum_{V(n) \leq λ} \int_{\TT^d} f(b(x,n)) ~ dx}{\sum_{V(n)\leqλ} 1}. $$