A novel multi-parameter family of quantum systems with partially broken -fold supersymmetry

Abstract

We develop a systematic algorithm for constructing an N -fold supersymmetric system from a given vector space invariant under one of the supercharges. Applying this algorithm to spaces of monomials, we construct a new multi-parameter family of N -fold supersymmetric models, which shall be referred to as "type C". We investigate various aspects of these type C models in detail. It turns out that in certain cases these systems exhibit a novel phenomenon, namely, partial breaking of N -fold supersymmetry.

-fold supersymmetry in quantum systems with position-dependent mass

“…Type A N -fold supersymmetry [33] is characterized by the so-called type A monomial space [30] Applying the algorithm with this type A monomial space, we can construct the most general form of type A N -fold supersymmetric PDM quantum systems. Fortunately, we can omit the process of constructing gauged Hamiltonians.…”

“…Later, a significant breakthrough was achieved by the proof of the equivalence between N -fold supersymmetry and (weak) quasi-solvability [29]. Based on the equivalence, a systematic algorithm for constructing an N -fold supersymmetric system was developed [30]. Owing to the fact that the framework of N -fold supersymmetry includes all the ordinary supersymmetric methods as its special cases and to its equivalence to weak quasi-solvability, which is less restrictive concept than quasi-exact solvability [31,32], it provides one of the most powerful methods for investigating one-body quantum mechanical systems which admit analytic solutions in so far the least restrictive sense.…”

Quasi-exact solvability beyond the sl(2) algebraization

“…Further research along this line suggests to regard a Darboux or SUSY transformation not just as a transformation on the potential and eigenfunctions, but as an algebraic deformation of an infinite polynomial flag [17]. Non Lie-algebraic potentials appear also in the recent work of González-López and Tanaka in the context of supersymmetry [18].…”

Simultaneous ordinary and type A N-fold supersymmetries in Schrödinger, Pauli, and Dirac equations

“…Up to now, three different families of N -fold supersymmetric systems have been found for arbitrary finite integer N , namely, type A [19,20], type B [21], and type C [22], which have correspondence with the classification of second-order linear differential operators preserving a monomial-type vector space [23]. Of the three types, type A and type C under a specific condition have been completely classified [20,22].…”