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

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 $\mathcal{N}=2$ supersymmetric extension of the same kind as that introduced by Freedman and Mende for the Calogero problem and based on an $\mathrm{osp}(2/2,\mathbb{R})\sim \mathrm{su}(1,1/1)$ 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.

18 pages, no figure

References in corpus (10)

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• N=4 superconformal Calogero models
• Superintegrability and higher order constants for classical and quantum systems
• Periodic orbits for an infinite family of classical superintegrable systems
• Calogero models and nonlocal conformal transformations
• Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$
• Superintegrability and higher order constants for quantum systems
• N=4 mechanics, WDVV equations and roots
• Families of classical subgroup separable superintegrable systems
• Exchange operator formalism for an infinite family of solvable and integrable quantum systems on a plane

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• Analytical Solution of Two-Dimensional Scarf II Model by Means of SUSY Methods
• The quantum $H_4$ integrable system
• The quantum $H_3$ integrable system
• Chiral super-Tremblay-Turbiner-Winternitz Hamiltonians and their dynamical superalgebra
• Comments on dihedral and supersymmetric extensions of a family of Hamiltonians on a plane
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• Superintegrability of the Post-Winternitz system on spherical and hyperbolic spaces