Imposing too strong simultaneous decay or localization constraints on a function and its Fourier transform causes the function to vanish identically (Uncertainty Principle). The same applies to a solution to some dispersive evolutionary PDE if similar constraints are imposed on its profile at two different time points.

In the first part of the talk, I will introduce a manifestation of the Uncertainty Principle, the Benedicks-Amrein-Berthier Theorem. The $L^2$ norm on $\mathbb{R}^n$ of a function is controlled by the sum of the $L^2$ norms of the function and its Fourier transform observed outside sets of finite Lebesgue measure (Amrein-Berthier). In particular, no non-zero function supported on a set of finite Lebesgue measure can have a Fourier transform with the same property (Benedicks).

In the second part I will show how the above statements translate into qualitative and even quantitative unique continuation results when the Fourier transform is replaced by the propagator, evaluated at a non-zero time T, associated to some dispersive PDE, in particular considering linear and non-linear instances of the Schr\"odinger equation.