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

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

1998-05-19

arxiv.org/abs/math-ph/9805018v1

Physics

Mathematical Physics

14 pages, plain Tex

Scientific paper

Let ${\cal H}(x,\xi)$ 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 $U(t)=\exp{iHt/\hbar}$. 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 $K N^{(6n+1)N}(-\hbar ln\hbar)^N$ for $t\in [0,T_N(\hbar)]$ where $T_N(\hbar)=-{2 ln\hbar\over {N-1}}$. Choosing a suitable $N(\hbar)$ the error is majorized by $C\hbar^{ln|ln\hbar|}$, $0\leq t\leq |ln\hbar|/ln|ln\hbar|$. (Here $K,C$ are constants independent of $N,\hbar$).

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