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

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.