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

TL;DR

This paper establishes scaling asymptotics for Szeg\

Contribution

It introduces new asymptotic formulas for spectral kernels on Grauert tubes, advancing understanding of their spectral and geometric properties.

Findings

Derived scaling asymptotics for the spectral localization kernel of a Toeplitz operator.

Established scaling asymptotics for tempered spectral projections kernels involving Laplace eigenfunctions.

Provided detailed asymptotic behavior near the boundary of 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 $\Pi_\tau \colon L^2(\partial M_\tau) \to H^2(\partial M_\tau)$, 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 $M$.