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 $\mathcal{T}\in {\mathcal{C}}^{\mathrm{\infty }}(X,TX)$. 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.