Applications of resonance theory without analyticity assumption

TL;DR

This paper extends the applicability of resonance theory in scattering for smooth, non-analytic potentials, demonstrating that key formulas remain valid without the analyticity assumption, using a novel resolvent estimate.

Contribution

It proves that resonance-related results in scattering theory hold for smooth, non-analytic potentials, introducing a new resolvent estimate technique.

Findings

Resonance formulas remain valid without analyticity.

A new resolvent estimate relates full and cut-off operators.

Results apply to smooth, decreasing potentials at infinity.

Abstract

We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schr\"odinger operator is supposed to be smooth and to decrease at infinity, the usual formulas relating scattering quantities and resonances still hold. The main ingredient for the proofs is a resolvent estimate of a new type, relating the resolvent of an operator with the resolvent of its cut-off counterpart.