Non-Weyl resonance asymptotics for quantum graphs in a magnetic field

TL;DR

This paper investigates how magnetic fields influence the resonance asymptotics of quantum graphs, showing that magnetic fields do not alter non-Weyl behavior but can affect the effective size of certain graphs.

Contribution

It proves that magnetic fields cannot change non-Weyl asymptotics into Weyl ones, and provides examples where magnetic fields influence the effective size of the graph.

Findings

Magnetic fields do not convert non-Weyl to Weyl asymptotics.

Magnetic fields can affect the effective size of some non-Weyl graphs.

Resonance asymptotics are sensitive to magnetic fields in specific cases.

Abstract

We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the ``effective size'' of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field.