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 nsupersymmetry 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. *

P T symmetric models with nonlinear pseudosupersymmetry

“…It can be shown that for |µ| ≤ λ + 1 4 , the wavefunctions ψ are also PT invariant, and can be normalized following (7). The new potential V 0,2 (x) has real bound state spectrum given by…”

“…While nonlinear SUSY for N = 2, has been investigated widely for Hermitian Hamiltonians [5][6][7][8][9][10], such studies have not been carried out as yet for non Hermitian Hamiltonians. Motivated by the importance of such systems in the recent development of quantum mechanics, our aim in the present work is to generalise the concept of nonlinear SUSY to include non Hermitian quantum systems.…”

Dolan–Grady relations and noncommutative quasi-exactly solvable systems

“…This observation revealed the intimate connection of the relations with the QES systems. The algebraic origin of the construction allowed us to apply it to the two-dimensional systems [10,11]. In the present paper we demonstrated that…”

“…The Dolan-Grady relations [1] can be represented in the form [10,9] Z, Z, Z,Z = ω 2 Z,Z , Z , Z , Z,Z =ω 2 Z,Z . (2.1)…”

“…In accordance with general algebraic construction of the n-HSUSY [9], the Hamiltonian depending on a natural parameter n is a quadratic form in two mutually conjugate operators subjected to DGR, whereas the supercharges are (anti) holomorphic polynomials of the order n in the same operators. It was shown that the n-HSUSY Hamiltonian belongs to an infinite set of mutually commuting even operators being quadratic in the Onsager algebra generators, and that these even integrals together with the odd supercharges form a polynomial superalgebra of the order n. Realization of this construction in various physical systems [8,10,11] revealed an intimate relation of the n-HSUSY with the quasi-exactly solvable (QES) systems [12].…”