Szeg\"o type limit theorems on the Heisenberg group

Let $\mathcal{H}=-{\mathrm{\Delta }}_{\mathbb{H}}+V$ be the Schr"odinger operator on the Heisenberg group ${\mathbb{H}}^n$, where ${\mathrm{\Delta }}_{\mathbb{H}}$ is the full laplacian on ${\mathbb{H}}^n$ and $V$ is a positive smooth potential, bounded below and grows like $\mathrm{\mid }g{\mathrm{\mid }}^{\kappa },\kappa >0$ for large $\mathrm{\mid }g\mathrm{\mid }$. Let ${\mathcal{P}}_r$ be the orthogonal projection of $L^2({\mathbb{H}}^n)$ onto the space of eigenfunctions of $\mathcal{H}$ with eigenvalue $\le r$; 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 $\mathbb{R}$^$withsymbol$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 $L^2({\mathbb{H}}^n)$.