Semiclassical quantization and spectral limits of h-pseudodifferential and Berezin-Toeplitz operators

TL;DR

This paper introduces a minimalistic semiclassical quantization framework to analyze spectral limits, showing the convex hull of quantum spectra converges to classical spectra, providing new insights into quantum-classical correspondence.

Contribution

It presents a novel minimalistic approach to semiclassical quantization and proves spectral convex hull convergence for both commuting and non-commuting operators.

Findings

Convex hull of quantum spectra converges to classical spectrum convex hull.

Provides an alternative solution to the isospectrality problem for quantum toric systems.

Convergence is uniform for bounded operators.

Abstract

We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system. This gives a quick alternative solution to the isospectrality problem for quantum toric systems. If the operators are uniformly bounded, the convergence is uniform. Analogous results hold for non-commuting operators.