Dirac-like Hamiltonians associated to Schr\"odinger factorizations

TL;DR

This paper extends the factorization method for scalar shape-invariant Schrödinger Hamiltonians to Dirac-like matrix Hamiltonians, including examples from spherical and hyperbolic geometries, revealing new algebraic structures and properties.

Contribution

It introduces a novel extension of the factorization method to Dirac-like Hamiltonians, incorporating intertwining and anti-intertwining operators, and provides explicit examples from curved spaces.

Findings

Extended factorization method to Dirac-like Hamiltonians

Constructed examples from spherical and hyperbolic geometries

Identified properties of non-Hermitian Dirac-like systems

Abstract

In this work, we have extended the factorization method of scalar shape-invariant Schr\"o\-din\-ger Hamiltonians to a class of Dirac-like matrix Hamiltonians. The intertwining operators of the Schr\"odinger equations have been implemented in the Dirac-like shape invariant equations. We have considered also another kind of anti-intertwining operators changing the sign of energy. The Dirac-like Hamiltonians can be obtained from reduction of higher dimensional spin systems. Two examples have been worked out, one obtained from the sphere ${\cal S}^2$ and a second one, having a non-Hermitian character, from the hyperbolic space ${\cal H}^2$.