Szeg\H{o} kernel asymptotics and concentration of Husimi Distributions of eigenfunctions

TL;DR

This paper derives asymptotic formulas for spectral projections and Husimi distributions of eigenfunctions on real analytic Riemannian manifolds, revealing their concentration behavior near geodesic flows.

Contribution

It provides explicit asymptotic formulas for spectral projection kernels and Husimi distributions using metaplectic representation, advancing understanding of eigenfunction concentration.

Findings

Asymptotic formulas expressed via metaplectic representation.

Sharp $L^p$ estimates for spectral projections and Husimi distributions.

Analysis near geodesic flow using parabolic rescaling.

Abstract

We work on the boundary $\partial M_\tau$ of a Grauert tube of a closed, real analytic Riemannian manifold $M$. The Toeplitz operator $\Pi_\tau D_{\sqrt{\rho}} \Pi_\tau$ associated to the Reeb vector field is a positive, self-adjoint, elliptic operator on $H^2(\partial M_\tau)$. We compute $\lambda \to \infty$ asymptotics under parabolic rescaling in a neighborhood of the geodesic (Reeb) flow $G^{t}_{\tau} = \exp t\Xi_{\sqrt{\rho}}$ for the spectral projection kernel $\Pi_{\chi, \lambda}$ associated to $\Pi_\tau D_{\sqrt{\rho}} \Pi_\tau$. We also compute scaling asymptotics for tempered sums of Husimi distributions (analytic continuations) on $\partial M_\tau$ of Laplace eigenfunctions on $M$. Both asymptotic formulae can be expressed in terms of the metaplectic representation of the linearization of the geodesic flow $G^{t}_\tau$ on Bargmann--Fock space. As a corollary, we obtain sharp $L^p \to L^{q}$ norm estimates for $\Pi_{\chi, \lambda}$ and sharp $L^p$ estimates for Husimi distributions.