Localization and delocalization of eigenmodes of Harmonic Oscillators

TL;DR

This paper investigates how the eigenfunctions of coupled quantum harmonic oscillators behave in the semi-classical limit, revealing that the structure of their measures depends on the arithmetic relations between oscillator frequencies.

Contribution

It provides a detailed characterization of semi-classical measures for coupled quantum harmonic oscillators with arbitrary frequencies, highlighting the impact of frequency relations on measure convexity.

Findings

Semi-classical measures depend on arithmetic relations between frequencies.

Non-rational frequency ratios lead to non-convex sets of measures.

Many invariant measures are not semi-classical when frequencies are incommensurate.

Abstract

We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems of coupled quantum harmonic oscillators with arbitrary frequencies. The structure of the set of semi-classical measures turns out to depend strongly on the arithmetic relations between frequencies of each decoupled oscillator. In particular, we show that as soon as these frequencies are not rational multiples of a fixed fundamental frequency, the set of semi-classical measures is not convex and therefore, infinitely many measures that are invariant under the classical harmonic oscillator are not semi-classical measures.