Polynomial supersymmetry for matrix Hamiltonians

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

Spectrally isomorphic Dirac systems: Graphene in an electromagnetic field

“…As the operatorH 2 represents a Schrödinger operator with matrix potential (the antidiagonal terms with first derivative can be transformed out be an additional unitary transformation), the presented procedure can be beneficial for analysis of these nonrelativistic systems that are of growing interest recently [32], [33], [28].…”

Minimal realizations of supersymmetry for matrix Hamiltonians

“…The matrix models with supersymmetry appear in Quantum Mechanics in several areas: in particular, for spectral design of potentials describing multichannel scattering and the motion of spin particles in external fields. The different cases of such models are considered in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and their systematic study was undertaken in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] (see also the recent reviews [34,35]). In [17] intertwining of matrix Hermitian Hamiltonians by n × n first-order and 2 × 2 second-order matrix differential operators was investigated and the corresponding supersymmetric algebras were constructed.…”

“…The intertwining operator for this purpose is given by a n × n matrix linear differential operator of arbitrary order with the identity matrix coefficient at derivative d/dx in the highest degree that intertwines these Hamiltonians. The systematic study of intertwining relations for n × n matrix non-Hermitian, in general, one-dimensional Hamiltonians has been performed in [25,33] with intertwining realized by n×n matrix linear differential operators with nondegenerate coefficients at d/dx in the highest degree. Some methods of constructing of n×n matrix intertwining operator of the first order in derivative and of general form were proposed and their interrelations were examined.…”

“…In the case of scalar Hamiltonians the number of independent minimized algebras cannot exceed two. The similar program has not been realized for matrix SUSY partner Hamiltonians although the important steps in this direction were done in [25].…”

“…The previous results on minimizability in scalar case are briefly formulated: in particular, the criterion of complete minimizability for a scalar intertwining operator from [38,39]. At the end of this section the criterion of complete weak minimizability is given for a matrix intertwining operator from [25,40] and the criterion of partial strong minimizability from the right for a matrix intertwining operator is presented. In Sec.4 the sufficient condition of existence of a constant symmetry matrix for a matrix Hamiltonian is found.…”