  We study the quantum mechanical Liouville model with attractive potential
which is obtained by Hamiltonian symmetry reduction from the system of a free
particle on $SL(2, \Real)$. The classical reduced system consists of a pair of
Liouville subsystems which are `glued together' in such a way that the
singularity of the Hamiltonian flow is regularized. It is shown that the
quantum theory of this reduced system is labelled by an angle parameter $\theta
\in [\,0,2\pi)$ characterizing the self-adjoint extensions of the Hamiltonian
and hence the energy spectrum. There exists a probability flow between the two
Liouville subsystems, demonstrating that the two subsystems are also
`connected' quantum mechanically, even though all the wave functions in the
Hilbert space vanish at the junction.
