A note on parasupersymmetric quantum mechanics of arbitrary order

Abstract

We revisit parasupersymmetric quantum mechanics of arbitrary order and present a set of nontrivial relations, which characterizes the most general multilinear part of the associated parasupersymmetric algebra. We then show that the formulation of multilinear relations leads immediately to a polynomial of parasupersymmetric Hamiltonian in terms of the corresponding parasupercharges. The deduction of higher derivative supersymmetric quantum mechanics directly via this parasupersymmetric formulation is discussed. The complete degenerate structure of the energy spectrum for parasupersymmetric quantum mechanics of order p is systematically analyzed. Finally, the notion of cyclic symmetry is introduced and the algebra of cyclic charge operators of arbitrary order is developed, based on the parasupersymmetric formalism.

Parasupersymmetry in quantum graphs

“…where Id is the identity operator on H. We should emphasize that, as noted in Ref. [4], the parasupercharges . We also emphasize that the case N = 2 reduces to the ordinary N = 2 supersymmetry algebra, Q + Q − + Q − Q + = H + M 2 (α 1 ), where the origin of energy is shifted by M 2 (α 1 ).…”

“…We then impose a generic Z N cyclic symmetry in this system and obtain a Z N -graded Hilbert space. In Section 3.1 we show that the Huang-Su parasupersymmetry algebra of order p = N − 1 [4] is hidden behind the spectral problem with N-fold degeneracy. We classify boundary conditions invariant under parasupersymmetry transformations in Section 3.2 and then explicitly derive the parasuperspectrum in Section 4.…”

“…The above constructed hierarchy of Hamiltonians based on the factorization −d 2 /dx 2 = A ∓ α A ± α − M 2 (α) is well-described by the Huang-Su parasupersymmetry algebra of order p = N − 1 [4]. To see this, let us first work in the basis { e 0 , e 1 , • • • , e N−1 } in which both H and Z become diagonal,…”

“…Historically, parasupersymmetry was first introduced in non-relativistic quantum mechanics by Rubakov and Spiridonov [1], and nowadays there are two different formulations of parasupersymmetric quantum mechanics. One is parasupersymmetric quantum mechanics of order 2 proposed by Rubakov and Spiridonov [1], which is extended to arbitrary order by Khare [2], Tomiya [3] and Huang and Su [4], 1 and the other is that proposed by Beckers and Debergh [6], which is extended to arbitrary order by Chenaghlou and Fakhri [7]. Common ingredients of these two parasupersymmetric quantum mechanics are Hamiltonian operator H, parafermion number operator N PF , nilpotent parafermionic charge (parasupercharge) Q + and its adjoint Q − that satisfy (Q ± ) p+1 = 0, where p ∈ N is the order of parafermion.…”

Shape Invariant Potentials in Second-Order Supersymmetric Quantum Mechanics

“…Various aspects of nonlinear SUSY QM have been elaborated, including, for instance, the works on the classification of differential realizations of the nonlinear SUSY QM algebra [2] and on the intrinsic links between supersymmetric isospectrality and hidden symmetries in certain quantum systems [25,26,28,55]. In fact, the nth order nonlinear SUSY QM is closely related to the parasupersymmetric QM of order n [6,48,54,60]. It is known that the former quantum theory can be deduced from the latter one if the redundant information, namely the intermediate Hamiltonians, of the latter theory is completely truncated.…”