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

Abstract

Let $\mathcal{H}(x,\xi )$ be a holomorphic Hamiltonian of quadratic growth on $R^{2n}$, $b$ a holomorphic exponentially localized observable, $H$, $B$ the corresponding operators on $L^2(R^n)$ generated by Weyl quantization, and $U(t)=\exp iHt/\hslash$. It is proved that the $L^2$ norm of the difference between the Heisenberg observable $B_t=U(t)BU(-t)$ and its semiclassical approximation of order $N-1$ is majorized by $K{N}^{(6n+1)N}(-\hslash ln\hslash {)}^N$ for $t\in [0,T_N(\hslash )]$ where $T_N(\hslash )=-\frac{2ln\hslash }{N-1}$. Choosing a suitable $N(\hslash )$ the error is majorized by $C{\hslash }^{ln|ln\hslash |}$, $0\le t\le |ln\hslash |/ln|ln\hslash |$. (Here $K,C$ are constants independent of $N,\hslash$).

14 pages, plain Tex