Spectral asymptotics of semiclassical unitary operators

TL;DR

This paper demonstrates that the high-frequency limit of quantum spectra for commuting semiclassical unitary operators converges to the classical spectrum, illustrating a form of Bohr's correspondence principle.

Contribution

It proves the convergence of convex hulls of quantum spectra to classical spectra for commuting semiclassical unitary operators under minimal assumptions.

Findings

Quantum spectra convex hulls converge to classical spectra

High-frequency limit aligns quantum and classical spectra

Results support Bohr's correspondence principle in this setting

Abstract

This paper establishes an aspect of Bohr's correspondence principle, i.e. that quantum mechanics converges in the high frequency limit to classical mechanics, for commuting semiclassical unitary operators. We prove, under minimal assumptions, that the semiclassical limit of the convex hulls of the quantum spectrum of a collection of commuting semiclassical unitary operators converges to the convex hull of the classical spectrum of the principal symbols of the operators.