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