  We consider semiclassical Schroedinger operators on R^n, with C^\infty
potentials decaying polynomially at infinity. The usual theories of resonances
do not apply in such a non-analytic framework. Here, under some additional
conditions, we show that resonances are invariantly defined up to any power of
their imaginary part. The theory is based on resolvent estimates for families
of approximating distorted operators with potentials that are holomorphic in
narrow complex sectors around R^n.
