Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

TL;DR

This paper develops semi-classical spectral asymptotics for Toeplitz operators on CR manifolds, providing asymptotic expansions and applications analogous to complex geometry results without requiring group actions.

Contribution

It introduces a full asymptotic expansion for functional calculus of Toeplitz operators on CR manifolds and derives several geometric embedding theorems.

Findings

Asymptotic expansion of $ ext{chi}_k(T_P)$ as $k o + $

CR analogues of high power line bundle results in complex geometry

Establishment of Kodaira type and Tian's convergence theorems

Abstract

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with some first order pseudodifferential operator $P$. We consider $\chi_k(T_P)$ the functional calculus of $T_P$ by any rescaled cut-off function $\chi$ with compact support in the positive real line. In this work, we show that $\chi_k(T_P)$ admits a full asymptotic expansion as $k\to+\infty$. As applications, we obtain several CR analogous of results concerning high power of line bundles in complex geometry but without any group action assumptions on the CR manifold. In particular, we establish a Kodaira type embedding theorem, Tian's convergence theorem and a perturbed spherical embedding theorem for strictly pseudoconvex CR manifolds.