Nonlinear supersymmetry on the plane in magnetic field and quasi-exactly solvable systems

Abstract

The nonlinear $n$-supersymmetry with holomorphic supercharges is investigated for the 2D system describing the motion of a charged spin-1/2 particle in an external magnetic field. The universal algebraic structure underlying the holomorphic $n$-supersymmetry is found. It is shown that the essential difference of the 2D realization of the holomorphic $n$-supersymmetry from the 1D case recently analysed by us consists in appearance of the central charge entering non-trivially into the superalgebra. The relation of the 2D holomorphic $n$-supersymmetry to the 1D quasi-exactly solvable (QES) problems is demonstrated by means of the reduction of the systems with hyperbolic or trigonometric form of the magnetic field. The reduction of the $n$-supersymmetric system with the polynomial magnetic field results in the family of the one- dimensional QES systems with the sextic potential. Unlike the original 2D holomorphic supersymmetry, the reduced 1D supersymmetry associated with $x^6+...$ family is characterized by the non-holomorphic supercharges of the special form found by Aoyama et al.

17 pages; references updated, to appear in Nucl. Phys. B

References in corpus (6)

• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Supersymmetry and the spontaneous breakdown of PT symmetry
• General Forms of a N-fold Supersymmetric Family
• Nonlinear Supersymmetry, Quantum Anomaly and Quasi-Exactly Solvable Systems
• Quantum Complex Henon-Heiles Potentials
• PT - symmetrized supersymmetric quantum mechanics

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