We study the small-time approximate controllability of bilinear Schrödinger equations, where the drift is a magnetic Schrödinger operator and the control is an electric potential. We prove this property in two circumstances: (i) in Rᵈ, with a quadratic and an additional generic bounded electric potential in the control, and with a uniform magnetic field in the drift; (ii) in Rᵈ or Tᵈ, with control electric potentials supported on a finite number of Hermite or Fourier eigenfunctions, and with any differentiable magnetic potential in the drift.
Eugenio Pozzoli (2026) studied this question.