Multiple resonances in the semi-classical limit

We construct for the Schrödinger operator in the semi-classical limit compact perturbations of a radial symmetric potential which give rise to resonances associated to arbitrarily high order poles for the meromorphic extension of the resolvent. Our results concern the hamiltonianP ₀=–h ²–x ² in the 2-dimensional case, as well as a fairly large class of radial-symmetric potentials in the 3-dimensional case. We show that the poles of the resolvent for such a potential are necessarily simple, and subsequently the degeneracy is due to a lack of symmetry.