 98351 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
${N1}$ 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 {N1}}$. 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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