  We consider the semiclassical Schroedinger operator with a "well in an
island" potential, on which we assume smoothness only, except near infinity. We
give the asymptotic expansion of the imaginary part of the shape resonance at
the bottom of the well. This is a generalization of a result by Helffer and
Sjoestrand in the globally analytic case. We use an almost analytic extension
in order to continue the WKB solution coming from the well beyond the caustic
set, and, for the justification of the accuracy of this approximation, we
develop some refined microlocal arguments in h-dependent small regions.
