N=2 supersymmetric extension of the Tremblay-Turbiner-Winternitz Hamiltonians on a plane

TL;DR

This paper extends the Tremblay-Turbiner-Winternitz Hamiltonians on a plane to include ${ m N}=2$ supersymmetry using an ${ m osp}(2/2, )$ superalgebra, providing explicit bases and analyzing ground states.

Contribution

It introduces an ${ m N}=2$ supersymmetric extension of the $H_k$ Hamiltonians on a plane, based on a specific superalgebra, with explicit representation bases and ground state analysis.

Findings

Supersymmetric extension exists for all positive $k$

Explicit construction of representation bases

Ground states are atypical lowest-weight states

Abstract

The family of Tremblay-Turbiner-Winternitz Hamiltonians $H_k$ on a plane, corresponding to any positive real value of $k$, is shown to admit a ${\cal N} = 2$ supersymmetric extension of the same kind as that introduced by Freedman and Mende for the Calogero problem and based on an ${\rm osp}(2/2, \R) \sim {\rm su}(1,1/1)$ superalgebra. The irreducible representations of the latter are characterized by the quantum number specifying the eigenvalues of the first integral of motion $X_k$ of $H_k$. Bases for them are explicitly constructed. The ground state of each supersymmetrized Hamiltonian is shown to belong to an atypical lowest-weight state irreducible representation.