Construction of Quasi-solvable Quantum Mechanical Matrix Models: Lie Superalgebra v.s. N-fold Supersymmetry

TL;DR

This paper develops quasi-solvable quantum matrix models using Lie superalgebra and N-fold supersymmetry, revealing their equivalences and differences, and highlighting the role of superalgebra constraints and conjugation.

Contribution

It compares Lie superalgebraic and N-fold supersymmetric methods for constructing quasi-solvable matrix models, uncovering new features and constraints in the osp(2/2) case.

Findings

Lie-superalgebraic and N-fold supersymmetric constructions coincide for q(2) models.

Distinct features arise in osp(2/2) models due to invariant subspace dimensions.

N-fold superalgebra closure imposes stronger constraints on the models.

Abstract

We construct quasi-solvable quantum mechanical matrix models by employing two different methods, the one is universal enveloping algebra of Lie superalgebra and the other is N-fold supersymmetry. For the former we examine the q(2) and osp(2/2) Lie-superalgebraic quasi-solvable matrix operators in the literature, and then compare them with the corresponding N-fold supersymmetric matrix systems. In the q(2) case, Lie-superalgebraic construction and the intertwining relation lead to the identical result. In the osp(2/2) case, however, some novel features emerge due to the difference in dimension of linear spaces which consist of the two-component invariant subspace. In both cases, the closure of N-fold superalgebra imposes stronger constraint on the admissible form of the systems and the concept of conjugation plays a key role in the formulation.