Localization and delocalization of eigenmodes of Harmonic Oscillators

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.

12 pages. This version corrects a gap in the definition of the decomposition of the harmonic oscillator as a sum of hamiltonians with periodic flows. The statement of the main result remains unchanged, as well as the main lines of the proof