11-25 Jitendriya Swain, M Krishna

Szego limit theorem on the lattice (289K, pdf) Feb 20, 11

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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 $A$.

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