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 \(k\rightarrow +\infty \) . More precisely, the restriction to the diagonal \(\chi (k^{-1}T_R)(x,x)\) decays at the rate \(O(k^{-\infty })\) 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.