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 × 2 matrix Hamiltonians with given symmetry matrices for the cases of different structure of Jordan form of these matrices are elucidated.

Polynomial supersymmetry for matrix Hamiltonians

“…, N . The intertwining relations (13) are analogous to (20) from [3] and take place in view of Remark 1 by virtue of the validity of the conditions analogous to (21) from [3] which follow from the statement (2). The intertwining relation (14) holds due to (8) with M = N .…”

“…Remark 4. In the conditions of Theorem 3 the matrix n × n coefficient of P − M at ∂ in the highest degree is equal obviously to X − N (see (2)) and any element of any other matrix-valued coefficient of P − M is smooth (without pole(s)) in view of (18) and smoothness of all elements of all matrix-valued coefficients of Q − N .…”

“…that take place due to (2), is mapped by Q − N into a chain of formal associated vectorfunctions of the Hamiltonian H − for the same spectral value λ m with possible exception of some number of vector-functions Q − N Ψ − m,l with lower numbers which can be identical zeroes. It is clear in view of (5) that if Q − N Ψ − m,l 0 ≡ 0 for some l 0 then Q − N Ψ − m,l ≡ 0 for any l < l 0 and if Q − N Ψ − m,l 0 ≡ 0 for some l 0 then Q − N Ψ − m,l ≡ 0 for any l > l 0 .…”

“…Definition 7. The intertwining operator Q + N ′ that exists in accordance with Theorem 5 for any matrix n × n linear differential operator Q − N of the N -th order with nondegenerate matrix coefficient at ∂ N , which intertwines a matrix n × n Hamiltonians H + and H − of Schrödinger form according to (2), will be called by us complement for the operator Q − N with respect to the Hamiltonian H + . We shall denote the complement for a matrix linear differential intertwining operator Q − N as (Q − N ) C , so that the following equality takes place in the conditions of Theorem 5,…”

“…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. 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.…”