Equivalence of resolvent and scattering resonances on quantum graphs

TL;DR

This paper proves that for certain quantum graphs with specific vertex couplings, the resolvent and scattering resonances are equivalent, using exterior complex scaling techniques.

Contribution

It demonstrates the equivalence of resolvent and scattering resonances on quantum graphs with delta-type couplings, a novel result in this context.

Findings

Resonances coincide for graphs with delta-type vertex coupling.

Exterior complex scaling effectively analyzes resonances on quantum graphs.

The result applies to graphs with finite compact parts and halflines.

Abstract

We discuss resonances for Schr\"odinger operators on metric graphs which consists of a finite compact part and a finite number of halflines attached to it; the vertex coupling is assumed to be of the $\delta$-type or certain modifications of it. Using exterior complex scaling on the graph we show that the resolvent and scattering resonances coincide in this case.