Type A N-fold Supersymmetry and Generalized Bender-Dunne Polynomials

TL;DR

This paper establishes the theoretical framework for Type A N-fold supersymmetry, linking algebraic and analytic methods, and generalizes Bender-Dunne polynomials to all such models, revealing their algebraic structure and invariants.

Contribution

It provides the necessary and sufficient conditions for Type A N-fold supersymmetry and generalizes Bender-Dunne polynomials without requiring normalizability, connecting algebraic and analytic approaches.

Findings

Derived conditions for Type A N-fold supersymmetry.

Established equivalence between analytic and sl(2) constructions.

Generalized Bender-Dunne polynomials for all models.

Abstract

We derive the necessary and sufficient condition for Type A N-fold supersymmetry by direct calculation of the intertwining relation and show the complete equivalence between this analytic construction and the sl(2) construction based on quasi-solvability. An intimate relation between the pair of algebraic Hamiltonians is found. The classification problem on Type A N-fold supersymmetric models is investigated by considering the invariance of both the Hamiltonians and N-fold supercharge under the GL(2,K) transformation. We generalize the Bender-Dunne polynomials to all the Type A N-fold supersymmetric models without requiring the normalizability of the solvable sector. Although there is a case where weak orthogonality of them is not guaranteed, this fact does not cause any difficulty on the generalization. It is shown that the anti-commutator of the Type A N-fold supercharges is expressed as the critical polynomial of them in the original Hamiltonian, from which we establish the complete Type A N-fold superalgebra. A novel interpretation of the critical polynomials in view of polynomial invariants is given.