Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry
We consider an invariant quantum Hamiltonian $H=-Δ_{LB}+V$ in the $L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a countable discrete symmetry group $Γ$. Typically, $\tilde{M}$ is the universal covering space of a multiply connected Riemannian manifold $M$ and $Γ$ is the fundamental group of $M$. On the one hand, following the basic step of the Bloch analysis, one decomposes the $L^{2}$ space over $\tilde{M}$ into a direct integral of Hilbert spaces formed by equivariant functions on $\tilde{M}$. The Hamiltonian $H$ decomposes correspondingly, with each component $H_Λ$ being defined by a quasi-periodic boundary condition. The quasi-periodic boundary conditions are in turn determined by irreducible unitary representations $Λ$ of $Γ$. On the other hand, fixing a quasi-periodic boundary condition (i.e., a unitary representation $Λ$ of $Γ$) one can express the corresponding propagator in terms of the propagator associated to the Hamiltonian $H$. We discuss these procedures in detail and show that in a sense they are mutually inverse.