A Phase Space Criterion for Dynamical Amrein-Berthier Uncertainty

Abstract

We prove a phase space criterion for dynamical Amrein-Berthier uncertainty principles. The abstract result says that, for a Fourier integral operator $A\in FIO(\ddot{I}‡)$ associated with a tame canonical transformation $\ddot{I}‡$, the localized operator $1_{E}A{1}_F$ is compact on $L^2({\mathbb{R}}^d)$ whenever $\ddot{I}‡$ satisfies a vertical non refocusing condition: high frequency covectors issued from a spatially localized region cannot return to a vertical direction over the observation region. In the linear symplectic case this condition is equivalent to the familiar nondegeneracy $\det B\ne 0$ of the upper right block of the symplectic matrix. We apply this compactness theorem to Schrödinger propagators for Yajima--type Hamiltonians, including quadratic electric and linear magnetic growth, and obtain two--time Amrein--Berthier inequalities for compact localization sets at all nonrefocusing times. The result extends the compactness mechanism behind the dynamical Amrein-Berthier principle to a genuinely microlocal setting.

Submitted