On the second coefficient in the semi-classical expansion of 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 $\mathcal{T}\in {\mathcal{C}}^{\infty }(X,TX)$. Consider the operator $\ddot{I}{‡}_k(A)$ defined by functional calculus of $A$, where $\ddot{I}‡$ is a smooth function with compact support in the positive real line and $\ddot{I}{‡}_k(\hat{I}»):=\ddot{I}‡(k^{-1}\hat{I}»)$. It was established recently that $\ddot{I}{‡}_k(A)(x,y)$ 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.