Spectral properties of the symmetry generators of conformal quantum mechanics: A path-integral approach

Author:

Camblong H. E.1ORCID,Chakraborty A.234ORCID,Lopez Duque P.2ORCID,Ordóñez C. R.2ORCID

Affiliation:

1. Department of Physics and Astronomy, University of San Francisco 1 , San Francisco, California 94117-1080, USA

2. Department of Physics, University of Houston 2 , Houston, Texas 77024-5005, USA

3. 3 Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada

4. 4 Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario, Canada

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.

Funder

Air Force Office of Scientific Research

University of San Francisco Faculty Development Fund

Army Research Office

Publisher

AIP Publishing

Subject

Mathematical Physics,Statistical and Nonlinear Physics

Reference107 articles.

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