Resonances for nonanalytic potentials

Abstract

We consider semiclassical Schrödinger operators on ${ℝ}^n$, with $C^{\infty }$ potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a nonanalytic 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 ${ℝ}^n$.