Szegö type limit theorems on the Heisenberg group

Abstract

Let $\mathcal{H}=-{\Delta }_{\mathbb{H}}+V$ be the Schrödinger 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 $\le r$; Let $A$ be a 0-th order self-adjoint pseudo-differential operator on $L^2({\mathbb{H}}^n)$ relative to the operator $1+|\lambda |H+V(g),g\in {\mathbb{H}}^n,\lambda \in {\mathbb{R}}^{\ast }$ 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, λ}(ξ, x)) \,dξ\,dx \,dg\,dμ(λ) }{\int_{G^{r}} \,dξ\,dx \,dg\,dμ(λ)}, \end{align*} (Assuming one limit exists) where $G^r=\{(g,\lambda ,\xi ,x)\in {\mathbb{H}}^n\times {\mathbb{R}}^{\ast }\times {\mathbb{R}}^n\times {\mathbb{R}}^n:|\lambda |(1+|\xi {|}^2+|x{|}^2)+V(g)\le r\}$, $a(g,\lambda )=O{p}^W(a_{g,\lambda })$, 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 Schrödinger operator $\mathcal{H}$ by bounded self-adjoint operators on $L^2({\mathbb{H}}^n)$.

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