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

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.

💡 Research Summary

The paper studies the low‑energy behavior of the resolvent of the Aharonov–Bohm Hamiltonian in the plane when several magnetic flux tubes (poles) are present. The magnetic vector potential is singular at the pole locations (s_k) and carries a strength (\alpha_k); the total flux is (\beta=\sum_{k=1}^n\alpha_k). The authors consider the Friedrichs self‑adjoint extension of the operator \