Geometric condition for the observability of electromagnetic Schr\"odinger operators on $\mathbb{T}^2$

TL;DR

This paper establishes a geometric condition that determines when electromagnetic Schr"odinger operators on a 2D torus are observable, highlighting the impact of magnetic potentials on observability.

Contribution

It introduces a new geometric criterion that accounts for magnetic potentials, extending previous results on electric potentials alone.

Findings

The presence of magnetic potential can obstruct observability.

A sufficient and almost necessary geometric condition is identified.

The condition involves the magnetic field's geometric control.

Abstract

In this article we revisit the observability of the Schr\"odinger equation on the two-dimensional torus. In contrast to the Schr\"odinger operator with a purely electric potential, for which any non-empty open set guarantees observability, the presence of a magnetic potential introduces an additional obstruction. We establish a sufficient and almost necessary geometric condition for the observability of electromagnetic Schr\"odinger operators. This condition incorporates the magnetic potential, which can also be characterized by a geometric control condition for the corresponding magnetic field.