Szegö limit theorem on the lattice

In this paper, we prove a Szego type limit theorem on \(\ell ^2(\mathbb {Z}^d)\). We take the self-adjoint operator \(H=-\Delta +V\) on \(\ell ^2(\mathbb {Z}^d)\), where \(\displaystyle (\Delta u)(\mathbf{n}) = \sum \nolimits _{|\mathbf{n}-\mathbf{k}|=1} (u(\mathbf{k}) - u(\mathbf{n}))\) and the operator V is the multiplication by a positive sequence \(\{V(\mathbf{n}), \mathbf{n}\in \mathbb {Z}^d\}\) with \(V(\mathbf{n}) \rightarrow \infty \) as \(|\mathbf{n}| \rightarrow \infty . \) We take the orthogonal projection \(\pi _{\lambda }\) onto the subspace, in \(\ell ^2(\mathbb {Z}^d)\), spanned by eigenfunctions of H with eigenvalues \(\le \lambda \). Let B be a zeroth order self-adjoint pseudo-difference operator associated with symbol \(b \in S_{1,0, \infty }(\mathbb {T}^d\times \mathbb {Z}^d)\). We then show for “nice functions” f, that $$\begin{aligned} \lim _{\lambda \rightarrow \infty } \frac{Tr(f(\pi _\lambda B\pi _\lambda ))}{Tr(\pi _\lambda )} = \lim _{\lambda \rightarrow \infty } \frac{1}{(2\pi )^d} \frac{\sum _{V(\mathbf{n}) \le \lambda } \int _{\mathbb {T}^d} f(b(\mathbf{x},\mathbf{n})) ~ d\mathbf{x}}{\sum _{V(\mathbf{n})\le \lambda } 1}. \end{aligned}$$