Quantum SU$(2|1)$ supersymmetric $\mathbb{C}^N$ Smorodinsky--Winternitz system

TL;DR

This paper explores the quantum characteristics of a supersymmetric extension of the Smorodinsky--Winternitz system on complex space, revealing energy level splitting due to supersymmetry and establishing connections with superconformal quantum mechanics.

Contribution

It provides a detailed construction of wave functions, energy spectra, and hidden symmetry operators for the SU(2|1) supersymmetric system, and introduces an equivalent superconformal framework.

Findings

Bosonic and fermionic states occupy separate energy levels.

Explicit wave functions and energy spectra are derived.

Hidden symmetry operators are characterized and related to superconformal generators.

Abstract

We study quantum properties of SU$(2|1)$ supersymmetric (deformed ${\cal N}=4$, $d=1$ supersymmetric) extension of the superintegrable Smorodinsky--Winternitz system on a complex Euclidian space $\mathbb{C}^N$. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU$(2|1)$ supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the "even-odd" splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU$(2|1)$ multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU$(2|1,1)$ quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU$(2|1,1)$ ones is found.