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

Abstract

We study quantum properties of SU$(2|1)$ supersymmetric (deformed $\mathcal{N}=4$, $d=1$ 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.

arXiv comment1+25 pages, 2 figures, a new affiliation added and Acknowledgments extended; published version

References in corpus (17)

• Rigid Supersymmetric Theories in Curved Superspace
• Exploring Curved Superspace
• Superconformal mechanics
• Representation theory of sl(2|1)
• Weak supersymmetry
• Deformed Supersymmetric Mechanics
• N=4 supersymmetric mechanics with nonlinear chiral supermultiplet
• Super Kähler oscillator from SU(2|1) superspace
• Superconformal mechanics in SU(2|1) superspace
• Quantum Oscillator on $\DC P^n$ in a constant magnetic field
• From worldline to quantum superconformal mechanics with/without oscillatorial terms: $D(2,1;\alpha )$ and $sl(2|1)$ models
• ${\mathbb{C}}^N$-Smorodinsky-Winternitz system in a constant magnetic field
• A surprise in mechanics with nonlinear chiral supermultiplet
• ${\mathbb{CP}}^N$-Rosochatius system, superintegrability, supersymmetry
• Deformed supersymmetric quantum mechanics with spin variables
• A three-dimensional superconformal quantum mechanics with $sl(2|1)$ dynamical symmetry
• Symmetries of deformed supersymmetric mechanics on Kähler manifolds

Cited by in corpus (2)

• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• A Note on Shape Invariant Potentials for Discretized Hamiltonians