Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains

TL;DR

This paper studies the semi-classical spectral asymptotics of Toeplitz operators on strictly pseudoconvex domains, revealing their kernel's structure as a Fourier integral operator with complex phase and deriving asymptotic expansions related to boundary geometry.

Contribution

It establishes that spectral projections of Toeplitz operators with Reeb-like symbols are semi-classical Fourier integral operators with detailed boundary asymptotics.

Findings

Kernel decays rapidly in the interior

Boundary asymptotic expansion of the kernel

Leading term expressed via Levi form and contact pairing

Abstract

On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension $n$ we consider a Toeplitz operator $T_R$ with symbol a Reeb-like vector field $R$ near the boundary. We show that the kernel of a weighted spectral projection $\chi(k^{-1}T_R)$, where $\chi$ is a cut-off function with compact support in the positive real line, is a semi-classical Fourier integral operator with complex phase, hence admits a full asymptotic expansion as $k\to+\infty$. More precisely, the restriction to the diagonal $\chi(k^{-1}T_R)(x,x)$ decays at the rate $O(k^{-\infty})$ in the interior and has an asymptotic expansion on the boundary with leading term of order $k^{n+1}$ expressed in terms of the Levi form and the pairing of the contact form with the vector field $R$.