SUSY Intertwining Relations of Third Order in Derivatives

TL;DR

This paper derives the general solution for third-order intertwining relations between Schrödinger Hamiltonians, revealing how supercharges can add multiple energy levels to the spectrum and analyzing their factorization properties.

Contribution

It provides the first general solution expressed via an arbitrary function for third-order supercharges in supersymmetric quantum mechanics.

Findings

Spectrum can be extended by three levels using third-order supercharges

Explicit wave functions for three energy levels are constructed

Analysis of different factorization types of third-order supercharges

Abstract

The general solution of the intertwining relations between a pair of Schr\"odinger Hamiltonians by the supercharges of third order in derivatives is obtained. The solution is expressed in terms of one arbitrary function. Some properties of the spectrum of the Hamiltonian are derived, and wave functions for three energy levels are constructed. This construction can be interpreted as addition of three new levels to the spectrum of partner potential: a ground state and a pair of levels between successive excited states. Possible types of factorization of the third order supercharges are analysed, the connection with earlier known results is discussed.