Minimal Realizations of Supersymmetry for Matrix Hamiltonians

Abstract

The notions of weak and strong minimizability of a matrix intertwining operator are introduced. Criterion of strong minimizability of a matrix intertwining operator is revealed. Criterion and sufficient condition of existence of a constant symmetry matrix for a matrix Hamiltonian are presented. A method of constructing of a matrix Hamiltonian with a given constant symmetry matrix in terms of a set of arbitrary scalar functions and eigen- and associated vectors of this matrix is offered. Examples of constructing of $2\times 2$ matrix Hamiltonians with given symmetry matrices for the cases of different structure of Jordan form of these matrices are elucidated.

arXiv comment9 pages

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• Spectrally isomorphic Dirac systems: graphene in electromagnetic field
• Confluent Crum-Darboux transformations in Dirac Hamiltonians with $PT$-symmetric Bragg gratings
• Polynomial supersymmetry for matrix Hamiltonians: proofs
• Symmetries and Supersymmetries of Generalized Schrödinger equations