THE AHARONOV-BOHM HAMILTONIAN WITH TWO VORTICES REVISITED

Abstract

We consider an invariant quantum Hamiltonian H = −ΔLB + 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 HΛ, 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 HΛ 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.