Abstract
Type A N-fold supersymmetry of one-dimensional quantum mechanics can be constructed by using sl(2) generators represented on a finite dimensional functional space. Using this sl(2) formalism we show a general method of constructing Type A N-fold supersymmetric models. We also present systematic generation of known models and several new models using this method.
Highlights
N -fold supersymmetry is an extension of the ordinary supersymmetry in one-dimensional quantum mechanics and reduces to it for N = 1 [1]–[6]
It has supercharges that are N -th polynomials of momentum. (Similar higher derivative generalizations of supercharges were investigated in different contexts [7]-[20].) It shares the nonrenormalization theorems of ordinary supersymmetry [21, 22]: Some of the energy eigenvalues can be obtained in a closed form when N -fold supersymmetry is not spontaneously broken, while perturbative parts of some energy eigenvalues are obtained otherwise
We have shown that the Type A variety of N -fold supersymmetry can be constructed using sl(2) generators, which are defined on a functional space formed by solvable energy eigenstates [6]
Summary
(Similar higher derivative generalizations of supercharges were investigated in different contexts [7]-[20].) It shares the nonrenormalization theorems of ordinary supersymmetry [21, 22]: Some of the energy eigenvalues can be obtained in a closed form when N -fold supersymmetry is not spontaneously broken, while perturbative parts of some energy eigenvalues are obtained otherwise This property is related to what has been known as “quasi-solvability” [23]–[28]. We have shown that the Type A variety of N -fold supersymmetry can be constructed using sl(2) generators, which are defined on a functional space formed by solvable energy eigenstates [6]. In Ref.[6] we used this method of construction for the Type A subclass of N -fold supersymmetry, which is defined by the following form of the supercharges [4]: PN =. It is possible to derive the conditions (2.18)–(2.20) by a direct calculation of the N -fold supersymmetry algebra (2.8) [31]
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