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

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.

💡 Research Summary

The paper presents a comprehensive construction of the most general type‑B three‑fold supersymmetry (3‑fold SUSY) by solving the intertwining relations directly, without resorting to indirect algebraic tricks. Starting from the intertwining operators Q⁻ and Q⁺ that map between two Hamiltonians H⁻ and H⁺, the authors impose the condition that the second‑order linear differential operator L appearing in the factorisation H⁻ = Q⁺Q⁻ + const must admit three linearly independent local analytic solutions. By expanding Q⁻ and Q⁺ as third‑order differential operators with polynomial coefficients, the intertwining equations Q⁻H⁺ = H⁻Q⁻ and Q⁺H⁻ = H⁺Q⁺ are reduced to a coupled system of twelve nonlinear differential equations for the coefficient functions of L. Solving this system yields the full parameterisation of type‑B 3‑fold SUSY, showing that the existence of three independent solutions is both necessary and sufficient for the operator to possess the supersymmetric structure.

A key outcome of the analysis is the identification of eight non‑trivial linear differential operators that satisfy the type‑B 3‑fold SUSY conditions. One of these operators belongs to the previously known type‑B class, two belong to type‑C, and four belong to the recently introduced type X₂ class. The X₂ operators are particularly noteworthy because they preserve invariant subspaces that are not simple monomial spaces; instead they involve mixtures of monomials with non‑polynomial functions such as logarithms, square‑roots, or exponentials. This extends the concept of quasi‑solvability beyond the traditional polynomial‑preserving framework.

The authors further demonstrate that type‑A, type‑B, and type‑C 3‑fold supersymmetries can be continuously connected through a single real parameter λ∈