Polynomial SUSY in Quantum Mechanics and Second Derivative Darboux Transformation

TL;DR

This paper classifies second-order polynomial supersymmetric quantum mechanics in one and two dimensions, focusing on irreducible supercharges and revealing a dynamic symmetry in two dimensions linked to a central charge operator.

Contribution

It provides a classification of second-order polynomial SUSY quantum mechanics and introduces the concept of irreducible supercharges not constructed by repeated Darboux transformations.

Findings

Irreducible supercharges cannot be formed by repeated Darboux transformations.

In two dimensions, binomial superalgebra induces a dynamic symmetry with a central charge.

Classification of polynomial SUSY in 1D and 2D quantum systems.

Abstract

We give the classification of second-order polynomial SUSY Quantum Mechanics in one and two dimensions. The particular attention is paid to the irreducible supercharges which cannot be built by repetition of ordinary Darboux transformations. In two dimensions it is found that the binomial superalgebra leads to the dynamic symmetry generated by a central charge operator.