On weakly turbulent solutions to the perturbed linear Harmonic oscillator

TL;DR

This paper constructs special solutions called bubbles for the linear harmonic oscillator, demonstrating how certain time-dependent potentials can cause solutions to grow logarithmically in Sobolev norms, revealing complex resonance phenomena.

Contribution

It introduces the concept of bubbles as invariant tori in the linear harmonic oscillator and uses them to build potentials leading to Sobolev norm growth in solutions.

Findings

Existence of bubble solutions forming invariant tori.

Construction of potentials causing logarithmic Sobolev norm growth.

Explicit resonance mechanism leading to oscillatory solutions.

Abstract

We introduce specific solutions to the linear harmonic oscillator, named bubbles. They form resonant families of invariant tori of the linear dynamics, with arbitrarily large Sobolev norms. We use these modulated bubbles of energy to construct a class of potentials which are real, smooth, time dependent and uniformly decaying to zero with respect to time, such that the corresponding perturbed quantum harmonic oscillator admits solutions which exhibit a logarithmic growth of Sobolev norms. The resonance mechanism is explicit in space variables and produces highly oscillatory solutions. We then give several recipes to construct similar examples using more specific tools based on the continuous resonant (CR) equation in dimension two.