2021/07/11 by Chang, Robert, Rabinowitz, Abraham · 1 citation
#Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2107.05105
Let Mτ be the Grauert tube of radius τ of a closed, real analytic manifold M. Associated to the Grauert tube boundary is the orthogonal projection Πτ\colon L2(∂ Mτ) → H2(∂ Mτ), called the Szegő projector. Let D√ρ denote the Hamilton vector field of the Grauert tube function √ρ acting as a differential operator. We prove scaling asymptotics for the spectral localization kernel of the Toeplitz operator ΠτD√ρ Πτ. We also prove scaling asymptotics for the tempered spectral projections kernel Pχ, λ(z,w) = ∑λj ≤ λ e-2τλj ϕλj^ℂ(z) ϕλj^ℂ(w), where ϕλj^ℂ are analytic extensions to the Grauert tube of Laplace eigenfunctions on M.