Polynomial Supersymmetry for Matrix Hamiltonians

TL;DR

This paper develops a polynomial supersymmetry framework for matrix Hamiltonians, establishing intertwining relations, polynomial algebra, and criteria for reducibility, revealing the existence of irreducible matrix intertwining operators unlike the scalar case.

Contribution

It introduces a polynomial supersymmetry algebra for matrix Hamiltonians, including criteria for minimization and reducibility of intertwining operators, and demonstrates the existence of irreducible matrix operators.

Findings

Constructed polynomial supersymmetry algebra for matrix Hamiltonians

Established criteria for minimization and reducibility of intertwining operators

Proved existence of absolutely irreducible matrix intertwining operators

Abstract

We study intertwining relations for matrix one-dimensional, in general, non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any matrix intertwining operator Q_N^- of minimal order N there is a matrix operator Q_{N'}^+ of different, in general, order N' that intertwines the same Hamiltonians as Q_N^- in the opposite direction and such that the products Q_{N'}^+Q_N^- and Q_N^-Q_{N'}^+ are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and of reducibility of a matrix intertwining operator are considered and the criteria of minimizability and of reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.