Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry

TL;DR

This paper investigates the behavior of small eigenvalues of Toeplitz operators on complex manifolds, revealing exponential decay and linking eigenvalue distribution to geometric structures like Mabuchi geodesics.

Contribution

It establishes a novel connection between eigenvalue decay, geometric envelopes, and Mabuchi geometry, extending results to Toeplitz matrices.

Findings

Eigenvalues decay exponentially with the semiclassical parameter.

Distribution of eigenvalues relates to Mabuchi geodesics.

Results apply to Toeplitz matrices as well.

Abstract

We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices.