Hidden Supersymmetry in Wigner-Yang Quantum Mechanics

Abstract:We consider a quantum system on the real line obeying a deformed Heisenberg algebra originally proposed by Wigner in 1950. Its explicit coordinate representation was provided by Yang in 1951 and in essence is identical in form with Dunkl's difference-differential operator introduced in 1989 in connection with roots systems of refection groups. Under certain conditions such quantum systems exhibit a supersymmetric (SUSY) structure where the reflection operator acts as the grading operator. We present a generalisation of Yang's representation by first considering only of one the two equations of motion in phase space. The corresponding non-interacting system is found to represent Witten's model of SUSY quantum mechanics. Imposing also the second equation of motion the original result of Wigner and Yang is reconsidered by extending their discussion to general symmetric potentials on the real line. As explicit example we discuss the harmonic oscillator and an attractive Coulomb-like potential $V(x)=-\gamma/|x|$. We also establish a Hooke-Newton duality between this Coulomb-like system and the original Wigner-Yang harmonic oscillator system.

Dedicated to Asim Orhan Barut (1926-1994) on the occasion of his $100^{\rm th}$ birthday. Talk presented at the GROUP36, Valladolid, Spain, July 13-17, 2026. 14 pages

arXiv:2608.04503 [math-ph]

(or arXiv:2608.04503v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2608.04503 arXiv-issued DOI via DataCite (pending registration)