Covariant forms of Lax one-field operators: from Abelian to non-commutative

TL;DR

This paper explores the extension of Lax one-field operators from Abelian to non-commutative cases, linking factorization, supersymmetry, and Darboux transformations within quantum theory, and analyzing matrix operators like Pauli and Dirac types.

Contribution

It introduces covariant forms of Lax operators in non-commutative settings and derives infinite chain equations for factorizing operators in quantum spectral problems.

Findings

Derived infinite chain equations for factorizing operators

Identified symmetries through chain closure

Analyzed matrix operators of Pauli and Dirac types

Abstract

Links of factorization theory, supersymmetry and Darboux transformations as isospectral deformations are considered in the context of quantum theory. The infinite chain equations for factorizing operators for a spectral problem are derived. A closure of the chain defines a symmetry of the system. Examples of matrix-differential operators of Pauli and Dirac type are analyzed.