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 the 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 when \(k\) becomes large. The second coefficient of the expansion plays an important role in the further studies of CR geometry. In this work, we calculate the second coefficient of the expansion.