98-351 Bambusi D., Graffi S., Paul T.

Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time. (49K, Plain TeX) May 18, 98

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Abstract. Let $ \h$ be a holomorphic Hamiltonian of quadratic growth on $ \Re^{2n}$, $b$ a holomorphic exponentially localized observable, $H$, $B$ the corresponding operators on $L^2(\Re^n)$ 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 $K N^{(6n+1)N}(-\hbar{\rm log}\hbar)^N$ for $t\in [0,T_N(\hbar)]$ where $ \ds T_N(\hbar)=-{2{\rm log}\hbar\over {N-1}}$. Choosing a suitable $N(\hbar)$ the error is majorized by $C\hbar^{\log|\log\hbar|}$, $0\leq t\leq |\log\hbar|/\log|\log\hbar|$. (Here $K,C$ are constants independent of $N,\hbar$).

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