We consider scattering theory for Schrödinger-type operators with long-range perturbations. It is well-known that it is necessary to introduce a modified free propagation to define wave operators/scattering operator. We show the scattering matrix is a Fourier integral operator with the phase function corresponding to the (modified) classical scattering map. We also show the spectrum of the scattering matrix can be dense point and absolutely continuous, whereas for the short-range case it is well-known that the spectrum is always discrete with the essential spectrum only at 1.

We will first review some standard facts about the Uncertainty Principle in Quantum Mechanics, together with some of the main mathematical evidences of this phenomenon. After sketching some versions of the principle in Fourier Analysis, we will make a link with unique continuation properties for dispersive evolutions at distinct times. We will finally present some recent result obtained in collaboration with L. Cossetti and F. Linares.