Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds

We consider any compact CR manifold whose Levi form is non-degenerate of constant signature $(n_-,n_+)$, $n_-+n_+=n$. For $\lambda>0$ and $q\in\{0,\cdots,n\}$, we let $\Pi_\lambda^{(q)}$ be the spectral projection of the Kohn Laplacian of $(0,q)$-forms corresponding to the interval $[0,\lambda]$. For certain classical pseudodifferential operators $P$, we study a class of generalized elliptic Toeplitz operators $T_{P,\lambda}^{(q)}:=\Pi_\lambda^{(q)}\circ P\circ \Pi_\lambda^{(q)}$. For any cut-off $\chi\in\mathscr C^\infty_c(\mathbb R\setminus\{0\})$, we establish the full asymptotics of the semi-classical spectral projector $\chi(k^{-1}T_{P,\lambda}^{(q)})$ as $k\to+\infty$. Our main result conclude that the smooth Schwartz kernel $\chi(k^{-1}T_{P,\lambda}^{(n_-)})(x,y)$ is the sum of two semi-classical oscillatory integrals with complex-valued phase functions.