Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time

Let H be a holomorphic Hamiltonian of quadratic growth on ℝ2n, b a holomorphic exponentially localized observable, H, B the corresponding operators on L2(ℝn) generated by Weyl quantization, and U(t) = exp iHt/ℏ. It is proved that the L2 norm of the difference between the Heisenberg observable Bt = U(t)BU(-t) and its semiclassical approximation of order N - 1 is majorized by KN N(6n+1)N ℏ-4/9(-ℏlogℏ)N for t ∈ [0, Tn(ℏ)], where Tn(ℏ) := -2logℏ/[α(6n + 3)(N - 1)] and α := ∥Hess(x,ξ)H∥. Choosing a suitable N(ℏ) the error is majorized by Cℏlog| log ℏ|, 0 ≤ t ≤ | log ℏ|/log | log ℏ| (here K and C are explicit constants independent of N, ℏ).