THE AHARONOV-BOHM HAMILTONIAN WITH TWO VORTICES REVISITED

Abstract

We consider an invariant quantum Hamiltonian H = −Δ_LB + V in the L² space based on a Riemannian manifold ˜M with a discrete symmetry group Γ. To any unitary representation Λ of Γ one can relate another operator on M = ˜M /Γ, called H_Λ, which formally corresponds to the same differential operator as H but which is determined by quasi-periodic boundary conditions. As originally observed by Schulman in theoretical physics and Sunada in mathematics, one can construct the propagator associated with H_Λ provided one knows the propagator associated with H. This approach is reviewed and demonstrated on a quantum model describing a charged particle on the plane with two Aharonov-Bohm vortices. The construction of the propagator is explained in full detail including all substantial intermediate steps.