# Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

# Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with a first order pseudodifferential operator $P$. We consider the operator $\chi_k(T_P)$ defined by functional calculus of $T_P$, where $\chi$ is a smooth function with compact support in the positive real line and $\chi_k(\lambda):=\chi(k^{-1}\lambda)$. We show that $\chi_k(T_P)$ admits a full asymptotic expansion as $k\to+\infty$. As applications, we obtain several CR analogues of results concerning the high powers of line bundles in complex geometry. In particular, we establish a Kodaira type embedding theorem, Tian's convergence theorem and an embedding theorem of strictly pseudoconvex CR manifolds into perturbed spheres.