Geometric condition for the observability of electromagnetic Schrödinger operators on T2

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 nonempty 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.