Spectral Properties of the Symmetry Generators of Conformal Quantum Mechanics: A Path-Integral Approach

TL;DR

This paper employs a path-integral approach to analyze the spectral properties of symmetry generators in conformal quantum mechanics, especially focusing on the hyperbolic operators with continuous spectra and their quantum interpretation.

Contribution

It introduces a general framework for characterizing symmetry generators in CQM and derives spectral decompositions for elliptic, parabolic, and hyperbolic operators, including novel results for the hyperbolic case.

Findings

Developed a path-integral method for spectral analysis of symmetry generators.

Provided spectral decompositions for different classes of operators in CQM.

Discovered new insights into the continuous spectrum of hyperbolic operators.

Abstract

A path-integral approach is used to study the spectral properties of the generators of the SO(2,1) symmetry of conformal quantum mechanics (CQM). In particular, we consider the CQM version that corresponds to the weak-coupling regime of the inverse square potential. We develop a general framework to characterize a generic symmetry generator $G$ (linear combinations of the Hamiltonian $H$, special conformal operator $K$, and dilation operator $D$), from which the path-integral propagators follow, leading to a complete spectral decomposition. This is done for the three classes of operators: elliptic, parabolic, and hyperbolic. We also highlight novel results for the hyperbolic operators, with a continuous spectrum, and their quantum-mechanical interpretation. The spectral technique developed for the eigensystem of continuous-spectrum operators can be generalized to other operator problems.