Polynomial supersymmetry for matrix Hamiltonians: proofs

Abstract

In this paper we continue studying of matrix $n\times n$ linear differential intertwining operators. The problems of minimization and of reducibility of matrix intertwining operators are considered and criterions of weak minimizability and of regular reducibility are proven. It is shown that in contrast to the scalar case $n=1$ there are for any $n⩾2$ regularly absolutely irreducible matrix intertwining operators of any order $N⩾2$, {\it i.e.} operators which cannot be factorized into a product of a matrix intertwining operators of lower orders even with a pole singularity(-ies) into coefficients. The theorem is proven on existence for any matrix intertwining operator $Q_N^{-}$ of the order $N$ a matrix differential operator $Q_{N^{\prime }}^{+}$ of other, in general, order $N^{\prime }$ that intertwines the same Hamiltonians $H_{+}$ and $H_{-}$ as $Q_N^{-}$ in the opposite direction and such that the products $Q_{N^{\prime }}^{+}{Q}_N^{-}$ and $Q_N^{-}{Q}_{N^{\prime }}^{+}$ are identical polynomials of $H_{+}$ and $H_{-}$ respectively. Polynomial algebra of supersymmetry corresponding to this case is constructed.

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