Propagators associated to periodic Hamiltonians: an example of the Aharonov-Bohm Hamiltonian with two vortices

We consider an invariant quantum Hamiltonian $H=-\Delta_{LB}+V$ in the $L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a discrete symmetry group $\Gamma$. Typically, $\tilde{M}$ is the universal covering space of a multiply connected manifold $M$ and $\Gamma$ is the fundamental group of $M$. To any unitary representation $\Lambda$ of $\Gamma$ one can relate another operator on $M=\tilde{M}/\Gamma$, called $H_\Lambda$, 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 $H_\Lambda$ 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.