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We consider an invariant quantum Hamiltonian $H=-\Delta_{LB}+V$ in the$L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a discretesymmetry group $\Gamma$. Typically, $\tilde{M}$ is the universal covering spaceof a multiply connected manifold $M$ and $\Gamma$ is the fundamental group of$M$. To any unitary representation $\Lambda$ of $\Gamma$ one can relate anotheroperator on $M=\tilde{M}/\Gamma$, called $H_\Lambda$, which formallycorresponds to the same differential operator as $H$ but which is determined byquasi-periodic boundary conditions. We give a brief review of the Blochdecomposition of $H$ and of a formula relating the propagators associated tothe Hamiltonians $H_\Lambda$ and $H$. Then we concentrate on the example of theAharonov-Bohm effect with two vortices. We explain in detail the constructionof the propagator in this case and indicate all essential intermediate steps.

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