Semi-classical Spectral Asymptotics of Toeplitz Operators on Strictly Pseudodonvex Domains

On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension 'n we consider a Toeplitz operator T-R with symbol a Reeb-like vector field R near the boundary. We show that the kernel of a weighted spectral projection chi(k(-1)T(R)), where chi is a cut-off function with compact support in the positive real line, is a semi-classical Fourier integral operator with complex phase, hence admits a full asymptotic expansion as k -> +infinity. More precisely, the restriction to the diagonal chi(k(-1)T(R))(x, x) decays at the rate O(k(-infinity)) in the interior and has an asymptotic expansion on the boundary with leading term of order k(n+1) expressed in terms of the Levi form and the pairing of the contact form with the vector field R.