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
$N,\hbar$
).
