Quantum mechanical Liouville model with attractive potential

TL;DR

This paper investigates a quantum Liouville model with an attractive potential derived from symmetry reduction, revealing a family of quantum states characterized by an angle parameter and demonstrating quantum connectivity between subsystems.

Contribution

It introduces a quantum Liouville model with a novel attractive potential and analyzes its self-adjoint extensions and quantum connectivity, extending classical reduction methods.

Findings

Quantum theory labeled by an angle parameter $ heta$

Existence of probability flow between subsystems

Subsystems are quantum mechanically connected

Abstract

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.