Low-energy spectrum of Toeplitz operators with a miniwell

TL;DR

This paper refines the understanding of eigenvector concentration for Toeplitz operators in the semiclassical limit, especially when the minimal set of the symbol forms a submanifold, extending miniwell conditions.

Contribution

It provides a precise criterion for eigenvector concentration on submanifolds, generalizing the miniwell condition for Toeplitz operators.

Findings

Eigenvectors concentrate on the minimal set of the symbol.

A new criterion for concentration when the minimal set is a submanifold.

Extension of miniwell conditions to Toeplitz operators.

Abstract

In the semiclassical limit, it is well-known that the first eigenvector of a Toeplitz operator concentrates on the minimal set of the symbol. In this paper, we give a more precise criterion for concentration in the case where the minimal set of the symbol is a submanifold, in the spirit of the "miniwell condition" of Helffer-Sj{\"o}strand.