Szego limit theorem on the lattice (1102.4131v2)
Abstract: In this paper, we prove a Szeg\"{o} type limit theorem on $\ell2(\ZZd)$. We consider operators of the form $H=\Delta+V$, $V$ multiplication by a positive sequence ${V(n), n \in \ZZd}$ with $V(n) \rightarrow \infty, |n| \rightarrow \infty $ on $\ell2(\ZZd)$ and $\pi_{\lambda}$ the orthogonal projection of $\ell2(\mathbb{Z}d)$ 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 $b(x,n), (x,n) \in \TTd\times \ZZd$ and show that for nice functions $f$ $$ \lim_{\lambda \rightarrow \infty} Tr(f(\pi_\lambda B\pi_\lambda))/Tr(\pi_\lambda) = \lim_{\lambda \rightarrow \infty} \frac{1}{(2\pi)d} \frac{\sum_{V(n) \leq \lambda} \int_{\TTd} f(b(x,n)) ~ dx}{\sum_{V(n)\leq\lambda} 1}. $$