**Abstract: **
We consider an invariant quantum Hamiltonian $H=-\Delta_{LB}+V$ in
the $L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a
countable discrete symmetry group $\Gamma$. Typically, $\tilde{M}$
is the universal covering space of a multiply connected Riemannian
manifold $M$ and $\Gamma$ 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
$H_{\Lambda}$ being defined by a quasi-periodic boundary condition.
The quasi-periodic boundary conditions are in turn determined by
irreducible unitary representations $\Lambda$ of $\Gamma$. On the
other hand, fixing a quasi-periodic boundary condition (i.e., a
unitary representation $\Lambda$ of $\Gamma$) one can express the
corresponding propagator in terms of the propagator associated to
the Hamiltonian $H$. We aim to provide a mathematically rigorous
basis for the two procedures and we show that in a sense these
procedures are mutually inverse.