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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 ${\rm osp}(2/2, {{\mathbb{R}}}) \sim {\rm su}(1,1/1)$ 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.'
author:
- |
    C Quesne\
    [Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles,]{}\
    [Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium]{}
date:  
title: ' ${\cal N}=2$ supersymmetric extension of the Tremblay-Turbiner-Winternitz Hamiltonians on a plane'
---

plus 1pt minus 1pt

Keywords: quantum Hamiltonians, supersymmetric quantum mechanics, superalgebras

PACS numbers: 03.65.Fd, 11.30.Pb

Introduction
============

Recently an infinite family of exactly solvable quantum Hamiltonians on a plane $$\begin{split}
  & H_k = - \partial_r^2 - \frac{1}{r} \partial_r - \frac{1}{r^2} \partial_{\varphi}^2 + \omega^2 r^2 +
       \frac{k^2}{r^2} [a(a-1) \sec^2 k\varphi + b(b-1) \csc^2 k\varphi], \\
  & 