We consider the semiclassical Schroedinger operator with a well-in-an-island potential, on which we assume C-infinity 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 Sj"ostrand 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
