Resonances on hedgehog manifolds

TL;DR

This paper investigates the resonance phenomena of quantum particles on hedgehog manifolds with attached leads, revealing unique high-energy asymptotics and the influence of magnetic fields, including cases with no true resonances.

Contribution

It provides new insights into the resonance behavior on hedgehog manifolds, especially regarding high-energy asymptotics and magnetic field effects, which differ from quantum graph models.

Findings

Resonances are confined to a strip in the momentum plane for single-point attachments.

Magnetic fields can eliminate true resonances via Aharonov-Bohm flux.

Resonance asymptotics differ from those in metric quantum graphs.

Abstract

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate it on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a `hedgehog' manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis.