A dynamical Amrein-Berthier uncertainty principle

2025/04/18 by D'Ancona, Piero, Fiorletta, Diego · 1 citation
#35J10 #35Q41 #42B37 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2504.13746

Abstract

Given a selfadjoint magnetic Schrödinger operator H = ( i ∂ + A(x) )² + V(x) on L²(ℝⁿ), with V(x) strictly subquadratic and A(x) strictly sublinear, we prove that the flow u(t)=e^-itHu(0) satisfies an Amrein--Berthier type inequality ‖u(t)‖_L²\lesssim_E,F,T,A,V ‖u(0)‖_L²(E^c) + ‖u(T)‖_L²(F^c), 0≤ t≤ T for all compact sets E,F ⊂ ℝⁿ. In particular, if both u(0) and u(T) are compactly supported, then u vanishes identically. Under different assumptions on the operator, which allow for time--dependent coefficients, the result extends to sets E,F of finite measure. We also consider a few variants for Schrödinger operators with singular coefficients, metaplectic operators, and we include applications to control theory.

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