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

In this article we revisit the observability of the Schrödinger equation on the two-dimensional torus.

In contrast to the Schrödinger 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 Schrödinger operators.

This condition incorporates the magnetic potential, which can also be characterized by a geometric control condition for the corresponding magnetic field.