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

Source abstract

Let $\mathcal{H}$ be a holomorphic Hamiltonian of quadratic growth on $\mathbb{R}^{2n}$ , $b$ a holomorphic exponentially localized observable, $H,B$ the corresponding operators on $L^2(\mathbb{R}^n)$ generated by Weyl quantization, and $U(t)=\exp{\mathrm{i}Ht/\hbar}$ . 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 $K^N N^{(6n+1)N}\hbar^{-4/9}(-\hbar\log\hbar)^N$ for $t\in [0,T_n(\hbar)]$ , where $T_n(\hbar):=-2\log\hbar/[\alpha(6n+3)(N-1)]$ and $\alpha:=\Vert\mathrm{Hess}_{(x,\xi)}\,\mathcal{H}\Vert$ . Choosing a suitable $N(\hbar)$ the error is majorized by $C\hbar^{\log\vert\log\hbar\vert}$ , $0\leq t\leq \vert\log\hbar\vert/\log\vert\log\hbar\vert$ (here $K$ and $C$ are explicit constants independent of $N,\hbar$ ).