THE AHARONOV-BOHM HAMILTONIAN WITH TWO VORTICES REVISITED

Abstract

<p>We consider an invariant quantum Hamiltonian <em>H</em> = −Δ<em>_LB</em> + <em>V</em> in the <em>L</em>² space based on a Riemannian manifold <em>˜M</em> with a discrete symmetry group Γ. To any unitary representation Λ of Γ one can relate another operator on <em>M</em> = <em>˜M</em> /Γ, called <em>H</em>_Λ, which formally corresponds to the same differential operator as <em>H</em> 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 <em>H</em>_Λ provided one knows the propagator associated with <em>H</em>. 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.</p>