Controllability of the Schr\"odinger equation on unbounded domains without geometric control condition

TL;DR

This paper proves the controllability of the Schrödinger equation in unbounded domains with internal controls supported on periodic sets, showing it is more flexible than wave equation control and closer to heat equation behavior.

Contribution

It demonstrates controllability of the Schrödinger equation in full space without geometric control conditions, using Floquet-Bloch theory and Fourier series estimates.

Findings

Controllability holds for any positive time T.

Control supports can be nonempty, periodic, open sets.

Results suggest Schrödinger control is more diffusive-like than wave control.

Abstract

We prove controllability of the Schr\"odinger equation in $\mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schr\"odinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schr\"odinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.