Concentration of Quasimodes for Perturbed Harmonic Oscillators

TL;DR

This paper investigates how small perturbations affect the concentration and distribution of quasimodes in semiclassical harmonic oscillators, revealing the influence of resonance, Diophantine conditions, and spectral clustering.

Contribution

It provides a detailed description of the set of semiclassical measures for perturbed harmonic oscillators, incorporating resonance, spectral, and dynamical properties, with new constructions and normal form techniques.

Findings

Characterization of semiclassical measures for perturbed 2D harmonic oscillators.

Identification of the impact of Diophantine properties on measure concentration.

Development of a novel quasimode construction using propagation of coherent states.

Abstract

In this work, concentration properties of quasimodes for perturbed semiclassical harmonic oscillators are studied. The starting point of this research comes from the fact that, in the presence of resonances between frequencies of the harmonic oscillator and for a generic bounded perturbation, the set of semiclassical measures for quasimodes of sufficiently small width is strictly smaller than the corresponding set of semiclassical measures for the unperturbed system. In this work we precise the description of this set taking into account the Diophantine properties of the vector of frequencies, the separation between clusters of eigenvalues in the spectrum produced by the presence of a perturbation, and the dynamical properties of the classical Hamiltonian flow generated by the average of the symbol of the perturbation by the harmonic oscillator flow. In particular, for the perturbed two-dimensional periodic harmonic oscillator, we characterize the set of semiclassical measures of quasimodes with a width that is critical for this problem. Two of the main ingredients in the proof of these results, which are of independent interest, are: (i) a novel construction of quasimodes based on propagation of coherent states by several commuting flows and (ii) a general quantum Birkhoff normal form for harmonic oscillators.