Low-energy spectrum of Toeplitz operators: the case of wells

TL;DR

This paper investigates the localization and spectral properties of Toeplitz operators with specific symbols, extending semiclassical analysis techniques to describe ground state concentration and eigenvalues in the presence of wells.

Contribution

It provides new results on the localization of the ground state and asymptotic eigenvalue descriptions for Toeplitz operators with finite non-degenerate critical points.

Findings

Ground state concentrates near critical points

First K eigenvalues are asymptotically described

Results extend semiclassical analysis to Toeplitz operators

Abstract

In the 1980s, Helffer and Sj\"ostrand examined in a series of articles the concentration of the ground state of a Schr\"odinger operator in the semiclassical limit. In a similar spirit, and using the asymptotics for the Szeg\"o kernel, we show a theorem about the localization properties of the ground state of a Toeplitz operator, when the minimal set of the symbol is a finite set of non-degenerate critical points. Under the same condition on the symbol, for any integer K we describe the first K eigenvalues of the operator.