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 ${\chi }_k(A)$ 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.

References in corpus (2)

• Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
• Szegő kernel asymptotic expansion on CR manifolds with $S^1$ action