  This paper is a continuation of a paper by the authors: arXiv:0706.0415,
where short range perturbations of the flat Euclidian metric where considered.
Here, we generalize the results of the paper to long-range perturbations (in
particular, we can allow potentials growing like $<x>^{2-\varepsilon}$ at
infinity). More precisely, we construct a modified quantum free evolution
$G_0(-s, hD_z)$ acting on Sj\"ostrand's spaces, and we characterize the
analytic wave front set of the solution $e^{-itH}u_0$ of the Schr\"odinger
equation, in terms of the semiclassical exponential decay of $G_0(-th^{-1},
hD_z)T u_0$, where $T$ stands for the Bargmann-transform. The result is valid
for $t<0$ near the forward non trapping points, and for $t>0$ near the backward
non trapping points. It is an extension of a paper by Nakamura
(arXiv:math/0605742) to the analytic framework.
