Quantum-mechanical Liouville model with attractive potential

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, R). 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 is an element of [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.