QUANTUM-MECHANICAL LIOUVILLE MODEL WITH ATTRACTIVE POTENTIAL

We study the quantum-mechanical Liouville model with attractive potent ial, which is obtained by Hamiltonian symmetry reduction from the syst em of a free particle on SL(2, R). The classical reduced system consis ts of a pair of Liouville subsystems which are 'glued together' in suc h a way that the singularity of the Hamiltonian flow is regularized. I t 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 sp ectrum. There exists a probability flow between the two Liouville subs ystems, demonstrating that the two subsystems are also 'connected' qua ntum mechanically, even though all the wave functions in the Hilbert s pace vanish at the junction.