Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells

TL;DR

This paper investigates the asymptotic behavior of low-lying eigenvalues and eigenfunctions of Toeplitz operators on symplectic manifolds, focusing on cases with discrete wells, and provides bounds for eigenvalues of the Bochner-Laplacian.

Contribution

It introduces a semiclassical spectral analysis framework for Toeplitz operators with non-degenerate minima, extending understanding of eigenvalue asymptotics in symplectic geometry.

Findings

Asymptotic behavior of eigenvalues and eigenfunctions characterized.

Upper bounds established for low-lying eigenvalues of the Bochner-Laplacian.

Analysis applicable to operators with discrete wells.

Abstract

We consider Toeplitz operators associated with the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigenfunctions of a self-adjoint Toeplitz operator under assumption that its principal symbol has a non-degenerate minimum with discrete wells. As an application, we prove upper bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical limit.