Jitendriya Swain, M Krishna
Szego limit theorem on the lattice
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ABSTRACT.  In this paper, we prove a Szeg\"{o} type limit theorem 
on $\ell^2(\ZZ^d)$. We consider 
operators of the form $H=
ho\Delta+|\xi|^k, 0\leq
ho,~0<k<2$ 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 $A$ be a $0$th order self adjoint pseudo difference 
operator with symbol 
$a(\xi,x)$ satisfying $[A, H](H + 1)^{-\sigma}$ bounded for some 
$0 < \sigma < rac{1}{2}.$ Then for 
$f\in \mathcal{C}(\mathbb{R})$ and 
$(\xi,x)\in \mathbb{Z}^d	imes\mathbb{T}^d,$ 
$$\lim_{\lambda	o \infty}\dfrac{
m{tr}~ f(\pi_\lambda A \pi_\lambda)}{
m{rank}~ \pi_\lambda}=\lim_{\lambda	o\infty}\dfrac{1}{(2\pi)^d}\dfrac{1}{vol(h(\xi,x)\leq\lambda)}\sum_{(\xi,x): h(\xi,x)\leq\lambda} \int f(a(\xi,x))dx 
$$ 
assuming one of the limits exists. The limits are invariant under compact 
perturbation of $A$.