Resonances for non-analytic potentials

TL;DR

This paper develops a new framework for defining and analyzing resonances in semiclassical Schrödinger operators with smooth, polynomially decaying, non-analytic potentials, extending resonance theory beyond analytic cases.

Contribution

It introduces a novel approach to define and study resonances for non-analytic potentials using resolvent estimates and approximating distorted operators.

Findings

Resonances are invariantly defined up to any power of their imaginary part.

The theory applies to smooth, polynomially decaying potentials that are not necessarily analytic.

Resonance invariance is established under certain additional conditions.

Abstract

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.