Quantum resonances without Analyticity

We propose a definition for the resonances of Schr"o-dinger operators with slowly decaying $CC^infty$ potentials without any analyticity assumption. Our definition is based on almost analytic extensions for these potentials, and we describe a systematic way to build such an extension that coincide with the function itself whenever it is analytic. That way, if the potential is dilation analytic, our resonances are the usual ones. We show that our resonances with negative real part are exactly the eigenvalues of the operator. We also prove that our definition coincides with the usual ones in the case of smooth exponentially decaying potentials. Then we consider semiclassical results. We show that, if the trapped set for some energy $E$ is empty, there is no resonance in any complex vicinity of $E$ of size $O(hlog(1/h))$. Finally, we investigate the semiclassical shape resonances and generalize some results of Helffer and Sj"ostrand