Resolvent estimates and resonance free domains for Schr\"odinger operators with matrix-valued potentials

TL;DR

This paper develops semiclassical resolvent estimates for matrix-valued Schr"odinger operators with long-range potentials, leading to new resonance free domain results applicable in both trapping and non-trapping scenarios.

Contribution

It generalizes existing scalar Schr"odinger operator results to matrix-valued potentials, establishing new resolvent estimates and resonance free domains.

Findings

Resonance free domains established for matrix-valued potentials

Semiclassical resolvent estimates derived for long-range interactions

Results extend scalar cases to more complex matrix-valued operators

Abstract

We establish semiclassical resolvent estimates for Schr\"odinger operators with long-range matrix-valued potentials. As an application we prove resonance free domains both in trapping and non-trapping situations. Our results generalize the well-known results of \cite{Burq1, Martinez1} in the case of scalar Schr\"odinger operators.