Resonances of third order differential operators

TL;DR

This paper investigates the resonances of third order differential operators with compactly supported coefficients, establishing bounds on their number and deriving a trace formula based solely on resonances.

Contribution

It provides new upper bounds for the number of resonances and expresses the trace formula explicitly in terms of resonances for third order differential operators.

Findings

Upper bounds on resonance counts at large radii

Resonances characterized as zeros of a Fredholm determinant

Trace formula expressed solely through resonances

Abstract

We consider resonances for third order ordinary differential operator with compactly supported coefficients on the real line. Resonance are defined as zeros of a Fredholm determinant on a non-physical sheet of three sheeted Riemann surface. We determine upper bounds of the number of resonances in complex discs at large radius. We express the trace formula in terms of resonances only.