Deformed Conformal and Supersymmetric Quantum Mechanics

TL;DR

This paper introduces a q-deformation of N=2 supersymmetry in quantum mechanics, leading to spectra related by q-scaling, and explores conformal symmetry in self-similar potentials with q-deformed harmonic oscillator algebra.

Contribution

It proposes a novel q-deformation of supersymmetry algebra and analyzes its implications for spectral properties and conformal symmetry in quantum systems.

Findings

Superpartner Hamiltonians are q-isospectral with spectra related by q^2 scaling.

Self-similar potentials obey the su_q(1,1) conformal algebra.

Raising and lowering operators satisfy the q-deformed harmonic oscillator algebra.

Abstract

Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e. the spectrum of one can be obtained from another by the q^2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra su_q(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane.