On the second coefficient in the semi-classical expansion of Toeplitz Operators

TL;DR

This paper computes the second coefficient in the semi-classical expansion of Toeplitz operators on CR manifolds, which is crucial for advancing understanding in CR geometry.

Contribution

It provides an explicit calculation of the second coefficient in the asymptotic expansion of Toeplitz operators on CR manifolds, a previously uncomputed term.

Findings

Explicit formula for the second coefficient derived

Enhances understanding of semi-classical asymptotics in CR geometry

Facilitates further geometric analysis using Toeplitz operators

Abstract

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $A$ be the Toeplitz operator on $X$ associated with a Reeb vector field $\mathcal{T}\in\mathscr{C}^\infty(X,TX)$. Consider the operator $\chi_k(A)$ defined by functional calculus of $A$, where $\chi$ is a smooth function with compact support in the positive real line and $\chi_k(\lambda):=\chi(k^{-1}\lambda)$. It was established recently that $\chi_k(A)(x,y)$ admits a full asymptotic expansion in $k$. The second coefficient of the expansion plays an important role in the further study of CR geometry. In this work, we calculate the second coefficient of the expansion.