Type B 3-fold Supersymmetry and Non-polynomial Invariant Subspaces

TL;DR

This paper characterizes the most general type B 3-fold supersymmetry, linking it to differential operators with specific solution spaces, and introduces new quasi-solvable models and operators with previously overlooked properties.

Contribution

It provides the complete classification of type B 3-fold supersymmetry and discovers new quasi-solvable operators and models, connecting different supersymmetry types continuously.

Findings

Identified eight linearly independent non-trivial differential operators.

Discovered new quasi-solvable operators preserving specific subspaces.

Established continuous connection among types A, B, and C 3-fold supersymmetries.

Abstract

We obtain the most general type B 3-fold supersymmetry by solving directly the intertwining relation. We then show that it is a necessary and sufficient condition for a second-order linear differential operator to have three linearly independent local analytic solutions. We find that there are eight linearly independent non-trivial linear differential operators of this kind. As a by-product, we find new quasi-solvable second-order operators preserving a monomial or polynomial subspace, one in type B, two in type C, and four in type X_2, all of which have been missed in the existing literature. In addition, we show that type A, type B, and type C 3-fold supersymmetries are connected continuously via one parameter. A few new quasi-solvable models are also presented.