Scaling asymptotics for Szegő kernels on Grauert tubes

Abstract

Let $M_τ$ be the Grauert tube of radius $τ$ of a closed, real analytic manifold $M$. Associated to the Grauert tube boundary is the orthogonal projection $Π_τ\colon L^2(\partial M_τ) \to H^2(\partial M_τ)$, called the Szegő projector. Let $D_{\sqrtρ}$ denote the Hamilton vector field of the Grauert tube function $\sqrtρ$ acting as a differential operator. We prove scaling asymptotics for the spectral localization kernel of the Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$. We also prove scaling asymptotics for the tempered spectral projections kernel $P_{χ, λ}(z,w) = \sum_{λ_j \le λ} e^{-2τλ_j} ϕ_{λ_j}^\mathbb{C}(z) \overline{ϕ_{λ_j}^\mathbb{C}(w)}$, where $ϕ_{λ_j}^\mathbb{C}$ are analytic extensions to the Grauert tube of Laplace eigenfunctions on $M$.