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We prove controllability of the Schr\"odinger equation in $\mathbb{R}^d$ inany time $T > 0$ with internal control supported on nonempty, periodic, opensets. This demonstrates in particular that controllability of the Schr\"odingerequation in full space holds for a strictly larger class of control supportsthan for the wave equation and suggests that the control theory ofSchr\"odinger equation in full space might be closer to the diffusive nature ofthe heat equation than to the ballistic nature of the wave equation. Ourresults are based on a combination of Floquet-Bloch theory with Ingham-typeestimates on lacunary Fourier series.

Controllability of the Schr\"odinger equation on unbounded domains without geometric control condition