Szeg\"o type limit theorems on the Heisenberg group

Abstract

Let $\mathcal{H}=-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and grows like $|g|^\kappa, \kappa>0$ for large $|g|$. Let $\mathcal{P}_{r}$ be the orthogonal projection of $L^2(\mathbb{H}^n)$ onto the space of eigenfunctions of $\mathcal{H}$ with eigenvalue $\leq r$; Let $A$ be a 0-th order self-adjoint pseudo-differential operator on $L^2(\mathbb{H}^n)$ relative to the operator $1+|\lambda|H+V(g), g\in \mathbb{H}^n, \lambda \in \mathbb{R}^*$ with symbol $a(g, {\lambda}),$ where $H$ is the Hermite operator on $L^2(\mathbb{R}^n)$ then \begin{align*} \lim_{r\to\infty} \frac{tr~{f(\mathcal{P}_rA\mathcal{P}_r)}}{tr~(\mathcal{P}_r)} &= \lim_{r\to\infty} \frac{\int_{G^{r}}f(a_{g, {\lambda}}(\xi, x)) \,d\xi\,dx \,dg\,d\mu(\lambda) }{\int_{G^{r}} \,d\xi\,dx \,dg\,d\mu(\lambda)}, \end{align*} (Assuming one limit exists) where $G^{r}=\{(g, \lambda, \xi, x)\in \mathbb{H}^n \times \mathbb{R}^*\times \mathbb{R}^n\times \mathbb{R}^n : |\lambda |(1+|\xi| ^2+|x|^2)+V(g)\leq r \}$, $a(g, {\lambda})=Op^W(a_{g, {\lambda}})$, and $\mu(\lambda)$ is the Plancherel measure on the Heisenberg group. Also we show that the above limit on the right hand side remains unaltered under a compact perturbation of the pseudo-differential operator $A$ or a perturbation of the Schr\"odinger operator $\mathcal{H}$ by bounded self-adjoint operators on $L^2(\mathbb{H}^n)$.