Scaling asymptotics for Szegő kernels on Grauert tubes

Abstract

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 ${\Pi }_{\tau }:L^2(\partial M_{\tau })\to H^2(\partial M_{\tau })$, called the Szegő 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 $P_{\chi ,\lambda }(z,w)={\sum }_{{\lambda }_j\le \lambda }{e}^{-2\tau {\lambda }_j}{\varphi }_{{\lambda }_j}^{\mathbb{C}}(z)\overline{{\varphi }_{{\lambda }_j}^{\mathbb{C}}(w)}$, where ${\varphi }_{{\lambda }_j}^{\mathbb{C}}$ are analytic extensions to the Grauert tube of Laplace eigenfunctions on $M$.

Version to appear in The Journal of Geometric Analysis

References in corpus (1)

• A direct approach to the analytic Bergman projection