Quantum SU(2|1) supersymmetric ℂ N Smorodinsky-Winternitz system

Abstract We study quantum properties of SU(2|1) supersymmetric (deformed N $$ \mathcal{N} $$ = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space 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.