For Schrödinger equations on tori with real-part controls, the maximal null-controllable subspace is the real orthogonal complement of the purely imaginary stationary modes.
Ammar-Khodja, A
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Controllable subspaces and real-part observability for Schr\"odinger equations on $\mathbb T^d$
For Schrödinger equations on tori with real-part controls, the maximal null-controllable subspace is the real orthogonal complement of the purely imaginary stationary modes.