Growth of Sobolev norms for completely resonant quantum harmonic oscillators on $\mathbb{R}^2$

TL;DR

This paper demonstrates that for certain time-periodic quantum harmonic oscillators in two dimensions, solutions can have Sobolev norms that grow unbounded over time, with conditions on the potential's symbol ensuring this behavior.

Contribution

The authors identify generic conditions on the potential's principal symbol that lead to unbounded Sobolev norm growth in resonant quantum harmonic oscillators, using microlocal analysis and dynamical systems techniques.

Findings

Existence of solutions with unbounded Sobolev norm growth.

Conditions on the potential's symbol are generic in a functional space.

Explicit construction of the conjugate operator for resonant averaging.

Abstract

We consider time dependently perturbed quantum harmonic oscillators in $\mathbb{R}^2$: $$ {\rm i} \partial_t u=\frac12(-\partial_{x_1}^2-\partial_{x_2}^2 + x_1^2+x_2^2)u +V(t, x, D)u, \qquad \ x \in \mathbb{R}^2, $$ where $V(t, x, D)$ is a selfadjoint pseudodifferential operator of degree zero, $2\pi$ periodic in time. We identify sufficient conditions on the principal symbol of the potential $V(t, x, D)$ that ensure existence of solutions exhibiting unbounded growth in time of their positive Sobolev norms and we show that the class of symbols satisfying such conditions is generic in the Fr\'echet space of classical $2\pi$- time periodic symbols of order zero. To prove our result we apply the abstract Theorem of arXiv:2101.09055v1 : the main difficulty is to find a conjugate operator $A$ for the resonant average of $V(t,x, D)$. We construct explicitly the symbol of the conjugate operator $A$, called escape function, combining techniques from microlocal analysis, dynamical systems and contact topology.