Low energy resolvent asymptotics of the multipole Aharonov--Bohm Hamiltonian

TL;DR

This paper derives low energy resolvent asymptotics for the Aharonov--Bohm Hamiltonian with multiple poles, revealing distinct scattering behaviors depending on the total flux being integer, half-integer, or other values.

Contribution

It extends resolvent asymptotics to multi-pole Aharonov--Bohm Hamiltonians for all flux types, linking flux values to Euclidean scattering dimensions.

Findings

Integer flux leads to even-dimensional Euclidean scattering behavior.

Half-integer flux results in odd-dimensional Euclidean scattering.

Other flux values interpolate between these scattering regimes.

Abstract

We compute low energy asymptotics for the resolvent of the Aharonov--Bohm Hamiltonian with multiple poles for both integer and non-integer total fluxes. For integral total flux we reduce to prior results in black-box scattering while for non-integral total flux we build on the corresponding techniques using an appropriately chosen model resolvent. The resolvent expansion can be used to obtain long-time wave asymptotics for the Aharonov--Bohm Hamiltonian with multiple poles. An interesting phenomenon is that if the total flux is an integer then the scattering resembles even-dimensional Euclidean scattering, while if it is half an odd integer then it resembles odd-dimensional Euclidean scattering. The behavior for other values of total flux thus provides an `interpolation' between these.