Dynamical Amrein-Berthier Uncertainty for Fractional Schrödinger Flows

Abstract

We prove dynamical Amrein-Berthier uncertainty principles for fractional Schrödinger flows. For the free Hamiltonian $H=(-\Delta {)}^{\alpha }$ on $L^2({\mathbb{R}}^n)$, with $\alpha >\frac{1}{2}$, we show that two--time localization on finite measure sets $E,F$ forces the quantitative estimate \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,n,α} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad T\neq0,\ t\in \mathbb{R} \end{equation*} for $u(t)=e^{-itH}u(0)$ at every time. The threshold $\alpha >\frac{1}{2}$ is tied to the stationary phase structure of the fractional kernel. If $\alpha \ge 1$ the sets can be arbitrary finite measure sets; if $\frac{1}{2}<\alpha <1$ we impose the finiteness of a natural interaction energy \begin{equation*} \textstyle \mathcal{I}_γ(E,F) = \int_{\mathbb{R}^n \times \mathbb{R}^n} \mathbf{1}_{F}(x)|x-y|^{2γ}\mathbf{1}_{E}(y)\,dx\,dy<\infty, \qquad γ= \frac{n(1-α)}{2 α-1} \end{equation*} of the pair $(E,F)$, essentially equivalent to a sufficiently fast joint decay of the measure of the sets at infinity. In particular, compact support at two distinct times is impossible for a nonzero solution. We also prove corresponding results for one dimensional fractional Hamiltonians $(-{\partial }_x^2+V{)}^{\alpha }$ under weighted scattering assumptions, and for higher order Hamiltonians $(-\Delta {)}^m+V$ for suitable classes of decaying potentials $V$.

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