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.
Introduction
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 main result of [18] is the formulae that help to build a modified n × n matrix Hamiltonian for a given n × n matrix (non-Hermitian, in general) Hamiltonian. 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.
Introduction
In the one-dimensional QM with scalar Hamiltonians the isospectral transformations generally lead to a Nonlinear (Polynomial [36, 37] ) SUSY algebra. In respect to the SUSY partners there might be an infinite number of intertwining operators which provide the same pair of potentials but a different SUSY algebras [38, 39] . These intertwining operators differ in factors which are functions of the Hamiltonians themselves. Thus for intertwining operators and the SUSY algebra itself the problem arises to minimize the order of their differential representation -minimizability problem. For some classes of potentials minimized algebras may exist which contain the same partner Hamiltonians but different sets of intertwining operators. In this case the hidden symmetry operators appear [38] being built of products of intertwining operators from different algebras. 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] .
Introduction
In the present paper the following new results are elaborated:
Introduction
• The notions of weak and strong minimizability of a matrix intertwining operator are introduced.
Introduction
• Criterion of strong minimizability of a matrix intertwining operator is revealed.
Introduction
• Criterion and sufficient condition of existence of a constant symmetry matrix for a matrix Hamiltonian are presented.
Introduction
• 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.
Introduction
• 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.
Introduction
The basic notations for intertwining operator algebra are defined in Sec.2. In Sec.3 we explain the motivation for introduction of the notions of weak and strong minimizabilities and give their definitions. 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. This condition provide us with the opportunity to receive the useful in practice formula for constructing of a matrix Hamiltonian with a given constant symmetry matrix. The two examples of constructing of arbitrary 2 × 2 matrix Hamiltonian with a given symmetry matrix A are presented. In the first of these examples there are two different eigenvalues for the matrix A and in the second of these examples normal (Jordan) form of A is a Jordan block. In Conclusion we outline the perspectives for further studies of the criteria of minimizabilities and hidden symmetries induced by extended SUSY algebras.
Basic definitions and notation
Let us consider two n × n matrix Hamiltonians defined on the entire axis,
Basic definitions and notation
where I n is the identity matrix and V + (x) and V − (x) are square matrices, all elements of which are sufficiently smooth and, in general, complex-valued functions. We suppose that these Hamiltonians are intertwined by a matrix linear differential operator Q − N , so that
Basic definitions and notation
where X − j (x), j = 0, . . . , N are also square matrices of n-th order, all elements of which are sufficiently smooth and, in general, complex-valued functions.
Basic definitions and notation
It is not hard to check that intertwining (1) leads to the following consequences:
Basic definitions and notation
where and in what follows we restrict ourselves by the case det X − N = 0. Let's elucidate the structure of intertwining operator kernel and define the transformation vector-functions. By virtue of the intertwining the kernel of Q − N is an invariant subspace for H + :
Basic definitions and notation
A basis in the kernel of an intertwining operator Q − N in which the matrix T + has a normal (Jordan) form is called a canonical basis. Elements of a canonical basis are called a transformation vector-functions.
Basic definitions and notation
If a Jordan form of the matrix T + has block(s) of order higher than one, then the corresponding canonical basis contains not only formal vector-eigenfunctions of H + but also formal associated vector-function(s) of H + which are defined as follows [41] .
Basic definitions and notation
A finite or infinite set of vector-functions Φ m,i (x), i = 0, 1, 2, . . . is called a chain of formal associated vector-functions of H + for a spectral value λ m if
Weak and strong minimizability of a matrix intertwining operator
Let us introduce the notions of weak and strong minimizabilities and give their definitions.
Weak and strong minimizability of a matrix intertwining operator
It is evident that if to multiply Q − N by a polynomial of the Hamiltonian,
Weak and strong minimizability of a matrix intertwining operator
where A l , l = 0, . . . , L are either a complex numbers or constant symmetry matrices for H + (H − ), then such products are again an intertwining operators for the same Hamiltonians:
Weak and strong minimizability of a matrix intertwining operator
Thus, the question arises about possibility to simplify an intertwining operator by separation from it a superfluous factor which is a polynomial of the corresponding Hamiltonian.
Weak and strong minimizability of a matrix intertwining operator
Let us now present the formal definitions of weak and strong minimizability of a matrix intertwining operator. 
Weak and strong minimizability of a matrix intertwining operator
where P − M is an n×n matrix linear differential operator of M -th order, M = N −2L that intertwines the Hamiltonians H + and H − , so that P 
Weak and strong minimizability of a matrix intertwining operator
where Remark 1. In the scalar case n = 1 weak and strong minimizabilities are equivalent obviously. Thus. we shall speak below in the scalar case n = 1 about minimizability without additional specification "weak" or "strong".
Weak and strong minimizability of a matrix intertwining operator
For comparison with our new results on strong minimizability of a matrix intertwining operator we present the previously formulated criterion of complete minimizability for a scalar intertwining operator [38, 39] and the criterion of complete weak minimizability for a matrix intertwining operator [25, 40] . The word "complete" means here that we consider such a minimization of an intertwining operator Q − N that the corresponding residual intertwining operator P − M in (3) is (weakly) non-minimizable.
Weak and strong minimizability of a matrix intertwining operator
Criterion of complete minimizability of a scalar intertwining operator. In the scalar case n = 1 an intertwining operator Q − N can be represented in the form
Weak and strong minimizability of a matrix intertwining operator
where P − M is a non-minimizable linear differential operator of the M -th order that intertwines the Hamiltonians H + and H − , so that P
and only if
(1) all numbers λ l , l = 1, . . . , s belong to the spectrum of the matrix T + and there are no equal numbers between them;
and only if
(2) there are 2 Jordan blocks in a normal (Jordan) form of the matrix T + for any eigenvalue from the set λ l , l = 1, . . . , s; (3) there are no 2 Jordan blocks in a normal (Jordan) form of T + for any eigenvalue of this matrix that does not belong to the set λ l , l = 1, . . . , s; (4) k l is the minimal of the orders of Jordan blocks corresponding to the eigenvalue λ l in a normal (Jordan) form of the matrix T + , l = 1, . . . , s.
Criterion of complete weak minimizability of a matrix intertwining operator.
A matrix intertwining operator Q − N can be represented in the form
Criterion of complete weak minimizability of a matrix intertwining operator.
where P − M is a non-minimizable matrix linear differential operator of the M -th order that intertwines the Hamiltonians H + and H − , so that P
and only if
(1) all numbers λ l , l = 1, . . . , s belong to the spectrum of the matrix T + and there are no equal numbers between them; (2) there are 2n Jordan blocks in a normal (Jordan) form of the matrix T + for any eigenvalue from the set λ l , l = 1, . . . , s; (3) there are no 2n Jordan blocks in a normal (Jordan) form of T + for any eigenvalue of this matrix that does not belong to the set λ l , l = 1, . . . , s; (4) k l is the minimal of the orders of Jordan blocks corresponding to the eigenvalue λ l in a normal (Jordan) form of the matrix T + , l = 1, . . . , s.
and only if
Now let us give the criterion of partial strong minimizability from the right for a matrix intertwining operator. The word "partial" means here that we consider such minimization of a matrix intertwining operator Q − N that the corresponding residual intertwining operator P − M in (4) is not necessarily strongly non-minimizable from the right. As well we present in this section the interesting corollary of the mentioned above criterion that contains the criterion for a constant n × n matrix to be a symmetry matrix for an n × n matrix Hamiltonian.
and only if
Criterion of partial strong minimizability from the right of a matrix intertwining operator. A matrix intertwining operator Q − N can be represented in the form
and only if
where A is a constant n × n matrix and P − N −2 is a matrix linear differential operator of the order N − 2 that intertwines H + and H − , so that P 
and only if
where (a) X ial , i = 1, . . . , m, a = 1, . . . , g i , l = 0, . . . , ν ia − 1 is a complete set of constant eigenand associated vectors of the matrix A, so that
and only if
(b) g il is the total number of vectors X ial with fixed i = 1, . . . , m and l = 0, . . . , ν i1 , so that
and only if
is the algebraic multiplicity of the eigenvalue λ i of the matrix A, i = 1, . . . , m; (c) ϕ ialb (x) and ψ ialb (x) for any i, a, l and b are a scalar functions; (3) the Wronskian of all vector-functions Φ ial (x) and Ψ ial (x) does not vanish on the entire axis.
and only if
In view of the latter criterion, relation (2) and the result of [25, 40] on constructing of a matrix intertwining operator from a set of associated vector-functions the following corollary is valid.
and only if
Corollary 1 (criterion of existence of a symmetry matrix for a matrix Hamiltonian). A constant n× n matrix A is a symmetry matrix for an n× n matrix Hamiltonian H of Schrödinger form, [H, A] = 0, if and only if there is a double set of associated vector-functions
and only if
of the Hamiltonian H such that this set satisfies conditions 1 -3 of the Criterion of partial strong minimizability from the right of a matrix intertwining operator.
and only if
The following Section 4 contains more useful in practice sufficient condition of existence of a constant symmetry matrix for a matrix Hamiltonian.
Existence of a constant symmetry matrix for a matrix Hamiltonian
Let us formulate the sufficient condition of existence of a constant symmetry matrix for a matrix Hamiltonian. This condition provide us with the opportunity to receive the formula for constructing of a matrix Hamiltonian with a given constant symmetry matrix which is useful in practice.
Existence of a constant symmetry matrix for a matrix Hamiltonian
Sufficient condition of existence of a constant symmetry matrix for a matrix Hamiltonian. A constant n× n matrix A is a symmetry matrix for an n× n matrix Hamiltonian H of Schrödinger form, [H, A] = 0, if there is a set of associated vector-functions
Existence of a constant symmetry matrix for a matrix Hamiltonian
of the Hamiltonian H such that
Existence of a constant symmetry matrix for a matrix Hamiltonian
(1) this set satisfies conditions 1 and 2 of the Criterion of partial strong minimizability from the right of a matrix intertwining operator;
Existence of a constant symmetry matrix for a matrix Hamiltonian
(2) the Wronskian of all vector-functions Φ ial (x) does not vanish on the entire axis.
Existence of a constant symmetry matrix for a matrix Hamiltonian
This condition allows us to use the following method of constructing of a matrix Hamiltonian with a given constant symmetry matrix.
Existence of a constant symmetry matrix for a matrix Hamiltonian
Arbitrary n × n matrix Hamiltonian H of Schrödinger form with a given constant symmetry n × n matrix A can be found with the help of the following formula,
Existence of a constant symmetry matrix for a matrix Hamiltonian
where Φ(x) is an n × n matrix-valued function constructed from n vector-functions
Existence of a constant symmetry matrix for a matrix Hamiltonian
as from columns. In this case a scalar functions ϕ ialb (x) from decompositions
Existence of a constant symmetry matrix for a matrix Hamiltonian
ϕ ia,l−s,t (x)X its (see above) are arbitrary parameterizing functions such that the Wronskian of all vector-functions Φ ial (x), i.e. det Φ(x), does not vanish on the entire axis. Let us elucidate the advantages of the above construction with two examples of constructing of arbitrary 2 × 2 matrix Hamiltonian with a given symmetry matrix A. In the first of these examples there are two different eigenvalues for the matrix A and in the second of these examples normal (Jordan) form of A is a Jordan block.
Existence of a constant symmetry matrix for a matrix Hamiltonian
Example 1: Constructing of a matrix 2 × 2 Hamiltonian with a given symmetry matrix A that has two different eigenvalues.
Existence of a constant symmetry matrix for a matrix Hamiltonian
If A is a given 2 × 2 matrix that has two different eigenvalues,
Existence of a constant symmetry matrix for a matrix Hamiltonian
then arbitrary 2 × 2 matrix Hamiltonian H, for which A is a symmetry matrix, can be found as follows:
Existence of a constant symmetry matrix for a matrix Hamiltonian
where ϕ 1 (x) and ϕ 2 (x) are arbitrary smooth scalar functions without zeroes. The fact that [H, A] = 0 follows from evident commutativity of all matrices from the right-hand parts of (5) and (6).
Existence of a constant symmetry matrix for a matrix Hamiltonian
Example 2: Constructing of a matrix 2 × 2 Hamiltonian with a given symmetry matrix A that has eigen-and associated vectors. If A is a given 2 × 2 matrix that has eigen-and associated vectors,
Existence of a constant symmetry matrix for a matrix Hamiltonian
then arbitrary 2 × 2 matrix Hamiltonian H, for which A is a symmetry matrix, can be found as follows: 
Existence of a constant symmetry matrix for a matrix Hamiltonian
where ϕ 0 (x) and ϕ 1 (x) are arbitrary smooth scalar functions and ϕ 0 is without zeroes. The fact that A is a symmetry matrix for H is obvious in view of (7) and (8).
Conclusions
Let us outline some possible problems for future studies.
Conclusions
(1) To find a criterion of complete strong minimizability for a matrix intertwining operator.
Conclusions
(2) To elaborate methods of spectral design for matrix Hamiltonians with the help of matrix intertwining operators of arbitrary order and, in particular, to reveal a criterion for transformation vector-functions that provides required changes for the spectrum of the corresponding final matrix Hamiltonian with respect to the spectrum of an initial matrix Hamiltonian. For this purpose one could try to generalize Index Theorem and Lemma 4 of [39, 42] for the matrix case.
Conclusions
(3) To study (in)dependence of matrix differential intertwining operators in the way analogous to one of [38] and, in particular, to define the notions of dependence and independence for these operators, to find a criterion of dependence for them and to solve the questions on maximal number of independent matrix differential intertwining operators and on a basis of such operators.
Conclusions
(4) By analogy with [38, 43] to investigate properties of a minimal matrix differential hidden symmetry operator.
Conclusions
(5) To study (ir)reducibility of matrix differential intertwining operators and, in particular, to classify irreducible and absolutely irreducible [25] matrix differential intertwining operators in the way analogous to one of [37, [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] .

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