Propagators associated to periodic Hamiltonians: an example of the Aharonov-Bohm Hamiltonian with two vortices
Abstract
Description
We consider an invariant quantum Hamiltonian $H=-Δ_{LB}+V$ in the $L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a discrete symmetry group $Γ$. Typically, $\tilde{M}$ is the universal covering space of a multiply connected manifold $M$ and $Γ$ is the fundamental group of $M$. To any unitary representation $Λ$ of $Γ$ one can relate another operator on $M=\tilde{M}/Γ$, called $H_Λ$, which formally corresponds to the same differential operator as $H$ but which is determined by quasi-periodic boundary conditions. We give a brief review of the Bloch decomposition of $H$ and of a formula relating the propagators associated to the Hamiltonians $H_Λ$ and $H$. Then we concentrate on the example of the Aharonov-Bohm effect with two vortices. We explain in detail the construction of the propagator in this case and indicate all essential intermediate steps.