Spectral design for matrix Hamiltonians: different methods of constructing of a matrix intertwining operator

Abstract

We study intertwining relations for n × n matrix non-Hermitian, in general, one-dimensional Hamiltonians 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 of general form are proposed and their interrelation is examined. As example we construct 2×2 matrix Hamiltonian of general form intertwined by operator of the first order with the Hamiltonian with zero matrix potential. It is shown that one can add for the final 2×2 matrix Hamiltonian with respect to the initial matrix Hamiltonian with the help of intertwining operator of the first order either up to two bound states for different energy values or up to two bound states described by vector-eigenfunctions for the same energy value or up to two bound states described by vector-eigenfunction and associated vector-function for the same energy value.

Confinement of an electron in a non-homogeneous magnetic field: Integrable vs superintegrable quantum systems

“…The problem of Darboux transformations (or intertwining transformations) of Dirac Hamiltonians has been previously considered in [10] but restricted to two-component wave functions, or in [13] (from a very different point of view than in our approach). The shapeinvariance for matrix Hamiltonians of Schrödinger-Pauli type has been studied in [14,15] and their Darboux properties in [16,17] and references therein. However, in our case we have characterized some interesting features not found before, which can be summarized in the following points: i) the Hamiltonians in the hierarchy have degenerate energy levels; ii) this implies that the intertwining operators are non-unique, the freedom depending on the dimension of the degeneracy; iii) the non-trivial symmetry operators of each Hamiltonian have been obtained; iv) anti-intertwining operators are natural in the context of Dirac Hamiltonians.…”

Polynomial supersymmetry for matrix Hamiltonians

“…. , nj m is different from identical zero and the determinant representation (see (41), (42) in [3]) for the operator Q − Nm,m in terms of these vector-functions takes place, m = 1, . .…”

“…The present paper is devoted to continuation of investigation of supersymmetry with matrix Hamiltonians from [1][2][3][4]. The papers [1,2] contain brief results without proofs on constructing of a matrix intertwining operator in terms of transformation vector-functions (including the case with formal associated vector-functions of initial Hamiltonian), on weak and strong minimizability and on reducibility of a matrix intertwining operator, on possibility to construct polynomial supersymmetry in some partial case and on constructing of matrix Hamiltonians with a given symmetry matrix.…”

“…The papers [1,2] contain brief results without proofs on constructing of a matrix intertwining operator in terms of transformation vector-functions (including the case with formal associated vector-functions of initial Hamiltonian), on weak and strong minimizability and on reducibility of a matrix intertwining operator, on possibility to construct polynomial supersymmetry in some partial case and on constructing of matrix Hamiltonians with a given symmetry matrix. In [3] general constructions of first-order and higher-order matrix n × n differential operators that intertwine matrix non-Hermitian, in general, Hamiltonians were described and founded. We consider in this paper the problems of weak minimization and of regular reducibility of matrix intertwining operators with proofs and show that polynomial algebra of supersymmetry can be constructed for arbitrary matrix intertwining operator (see review of papers on these problems in [1,3]).…”

“…In [3] general constructions of first-order and higher-order matrix n × n differential operators that intertwine matrix non-Hermitian, in general, Hamiltonians were described and founded. We consider in this paper the problems of weak minimization and of regular reducibility of matrix intertwining operators with proofs and show that polynomial algebra of supersymmetry can be constructed for arbitrary matrix intertwining operator (see review of papers on these problems in [1,3]). Some results of this paper were briefly presented earlier without proofs in [1,45].…”