MINIMAL ESCAPE VELOCITIES

We give a new derivation of the minimal velocity estimates (SiSo1) for unitary evolutions. LetH andA be selfadjoint operators on a Hilbert space H. The starting point is Mourre's inequality i(H,A) ≥ � > 0, which is supposed to hold in form sense on the spectral subspace Hof H for some interval � ⊂ R. The second assumption is that the multiple commutators ad (k) A (H) are well- behaved fork = 1...n (n ≥ 2) . Then we show that, for a dense set of 's in Hand allm < n−1, t = exp(−iHt) is contained in the spectral subspace A ≥ �t as t → ∞, up to an error of order t −m in norm. We apply this general result to the case where H is a Schrodinger operator on R n and A the dilation generator, proving that t(x) is asymptotically supported in the set |x| ≥ t √ � up to an error of order t −m in norm.