Szegő kernel asymptotics and concentration of Husimi Distributions of eigenfunctions

2022/02/28 by Chang, Robert, Rabinowitz, Abraham
#Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2202.14013

Abstract

We work on the boundary ∂ M_τ of a Grauert tube of a closed, real analytic Riemannian manifold M. The Toeplitz operator Π_τD_√ρ Π_τ associated to the Reeb vector field is a positive, self-adjoint, elliptic operator on H²(∂ M_τ). We compute λ→ ∞ asymptotics under parabolic rescaling in a neighborhood of the geodesic (Reeb) flow G^t_τ = exp tΞ_√ρ for the spectral projection kernel Π_{χ, λ} associated to Π_τD_√ρ Π_τ. We also compute scaling asymptotics for tempered sums of Husimi distributions (analytic continuations) on ∂ M_τ 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_τ on Bargmann--Fock space. As a corollary, we obtain sharp L^p → L^q norm estimates for Π_{χ, λ} and sharp L^p estimates for Husimi distributions.

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