Remarks on the Space-Time Behavior of Scattering Solutions to the Schrödinger Equations

Abstract

We consider the space-time behavior of scattering solutions to the Schrödinger equation
i∂_tu=Hu   on   (x, t)∈Rⁿ×R   (n{≥}1),
u(x, 0)=φ(x),   x∈Rⁿ,
where H denotes a self-adjoint operator in the Hilbert space \mathscr{H}:=L²(Rⁿ). We prove the fact that every scattering solution which has the estimate
| (e^−itHφ)(x) | {≤}C | t | ^−α(1+ | x | )^−β   (| t | {≥}1, x∈Rⁿ)
for some α, β∈R with α+β>n/2 vanishes identically.
Futhermore, we show, every non-trivial scattering solution has the estimate
lim inf\limits_R→+∞ R⁻¹∫_ΓR^± | (e^−itHφ) (x) | ²dxdt>0,
where
Γ_R⁺={(x, t)∈Rⁿ×R; R<(| t | ²+ | x | ²)^1/2<2R, t>0}
and
Γ_R⁻={(x, t)∈Rⁿ×R; R<(| t | ²+ | x | ²)^1/2<2R, t<0}.