Scaling asymptotics for Szeg\H{o} kernels on Grauert tubes

Let $M_\tau$ be the Grauert tube of radius $\tau$ of a closed, real analytic manifold $M$. Associated to the Grauert tube boundary is the orthogonal projection $\Pi_\tau \colon L^2(\partial M_\tau) \to H^2(\partial M_\tau)$, called the Szeg\H{o} projector. Let $D_{\sqrt{\rho}}$ denote the Hamilton vector field of the Grauert tube function $\sqrt{\rho}$ acting as a differential operator. We prove scaling asymptotics for the spectral localization kernel of the Toeplitz operator $\Pi_\tau D_{\sqrt{\rho}} \Pi_\tau$. We also prove scaling asymptotics for the tempered spectral projections kernel $P_{\chi, \lambda}(z,w) = \sum_{\lambda_j \le \lambda} e^{-2\tau\lambda_j} \phi_{\lambda_j}^\mathbb{C}(z) \overline{\phi_{\lambda_j}^\mathbb{C}(w)}$, where $\phi_{\lambda_j}^\mathbb{C}$ are analytic extensions to the Grauert tube of Laplace eigenfunctions on $M$.