AbstractThe asymptotic conjugation relation limt→±∞ ‖g(χt)M(eitp)ƒ − M(g(∇ρ))M(eitp)ƒ‖2 = 0 is established for all ƒ∈L2(Rn) under mild assumptions on ϱ and g, where M(h)ƒ = F−1(hf̌) denotes Fourier multiplication. The asymptotic estimate limt→±∞ χj|χ|∂u∂t±∂u∂χj2 = 0 for finite energy solutions u of the wave equation is deduced from (∗), along with generalizations to a class of first-order symmetric hyperbolic systems of partial differential equations that are homogeneous and constant coefficient, and a weakened version for the Klein-Gordon equation. Also deduced from (∗) is the fact that for a free Schrödinger particle the probability of being in the set tA at time t tends to the probability that the velocity is in A as t → ±∞.