Saint-Petersburg State University Physical Faculty Russia, Saint-Petertsburg, 198504, Petrodvorets, Ulianovskaya ul., 1 Andrei V. Sokolov Linear and Non-linear Supersymmetry for Non-Hermitian Matrix Hamiltonians Report at the Workshop \PHHQP XI: Non-Hermitian Operators in Quantum Physics" August 27-31, 2012, APC, Paris, France Main points of the report • Basic definitions and notation • Brief review of the literature and the purpose of this report • A method of constructing of any matrix differential intertwining operator of any order in terms of transformation vector-functions • Theorem on singular factorization of a matrix intertwining operator into a prod- uct of matrix intertwining operators of lower orders • The notion of minimizability of a matrix intertwining operator and criterion of minimizability • Sufficient condition of existence for an intertwining operator a \conjugate" inter- twining operator of the same order and of polynomial SUSY with these operators • Theorem on existence for any intertwining operator a \conjugate" intertwining operator and polynomial SUSY with these operators • Sufficient condition of reducibility of an intertwining operator and existence of absolutely irreducible intertwining operators 2 1 Basic definitions and notation 1.1 Intertwining relation Let us consider two defined on the entire axis matrix n × n Hamiltonians 2 2 H+ = −In@ + V+(x);H− = −In@ + V−(x);@ ≡ d=dx; where In is the identity matrix and V+(x) and V−(x) are square matrices, all ele- ments of which are sufficiently smooth and, in general, complex-valued functions. We suppose that these Hamiltonians are intertwined by a matrix linear differential − operator QN , so that N − − − X − j Q H+ = H−Q ;Q = X (x)@ ; (1) N N N j=0 j − where Xj (x), j = 0, . , N are as well square matrices of n-th order, elements of which are sufficiently smooth and, in general, complex-valued functions. It is not hard to check that intertwining (1) leads to the following consequences: − − − −1 − 0 − −1 XN (x) = Const;V−(x) = XN V+(x)(XN ) + 2XN−1(x)(XN ) ; − where and in the what follows we restrict ourselves by the case det XN 6= 0. 3 1.2 About \conjugate" intertwining operator + Existence of a \conjugate" matrix intertwining operator QM for a given matrix − intertwining operator QN such that M + + + X + j H+Q = Q H−;Q = X (x)@ ; (2) M M m j=0 j is not evident in general but is evident in the following cases: y y + + + − y PN j − y (1) H+ =H+;H− =H− ) H+QN =QN H−;QN =QN = j=0(−@) Xj (x); where y denotes Hermitian conjugation; t t + + + − t PN j − t (2) H+ =H+;H− =H− ) H+QN =QN H−;QN =QN = j=0(−@) Xj (x); where t denotes transposition; ∗ + + + − ∗ PN − ∗ j (3) H+ = H− ) H+QN = QN H−;QN = QN = j=0 Xj (x)@ ; where ∗ denotes complex conjugation. Existence of a \conjugate" matrix intertwining operator of the type (2) for any − matrix intertwining operator QN is guaranteed by one of theorems presented below. 4 1.3 Structure of intertwining operator kernel, transformation functions − By virtue of the intertwining the kernel of QN is an invariant subspace for H+: − − H+ ker QN ⊂ ker QN : − − − − Hence, for any basis Φ1 (x), . , Φd (x) in ker QN , d = dim ker QN = nN there is a + constant square matrix T ≡ kTijk of d-th order such that d − X + − H+Φ = T Φ ; i = 1; : : : ; d: i j=1 ij j − + A basis in the kernel of an intertwining operator QN 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. 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. A finite or infinite set of vector-functions Φm;i(x), i = 0, 1, 2, . is called a chain of formal associated vector-functions of H+ for a spectral value λm if H+Φm;0 =λmΦm;0; Φm;0(x)6≡0; (H+−λmIn)Φm;i =Φm;i−1; i=1; 2; 3;:::: 5 2 Brief review of the literature [1] R.D. Amado, F. Cannata, J.-P. Dedonder, Coupled-channel supersymmetric quantum mechanics, Phys. Rev. A 38 (1988) 3797. + − +: the simplest formulae for H+, H−, Q1 and Q1 in the case where N = 1, n is arbitrary, all transformation vector-functions are formal vector-eigenfunctions of H+ and belong to the only its spectral value; −: very partial case. [2] A.A. Andrianov, F.Cannata, D.N.Nishnianidze, M.V.Ioffe, Matrix Hamiltonians: SUSY approach to hidden symmetries, JPA 30 (1997) 5037; quant-ph/9707004. + −† y y +: systematic studying of the system where QN = QN , H+ = H+ and H− = H− in the case N = 1 for n = 2 mostly and in the case N = 2 for n = 2; −: rather restricted case. 6 [3] V.M. Goncharenko, A.P. Veselov, Monodromy of the matrix Schr¨odingerequa- tions and Darboux transformations, JPA 31 (1998) 5315. − − +: there are formulae in [3] that allow us to construct any QN with XN = In and the corresponding H− for any H+ (n and N are arbitrary); −: the formulae of [3] are rather complex since they contain quasideterminants and they are in terms of a basis in an H+-invariant subspace; there is no any − condition that provides implementability of the described procedure of QN con- − structing and absence of pole(s) for QN coefficients and for the potential of H−. [4] B.F. Samsonov, A.A. Pecheritsin, Chains of Darboux transformations for the matrix Schr¨odingerequation, JPA 37 (2004) 239; quant-ph/0307145. − +: there are more simple than in [3] formulae for QN and H− in terms of usual determinants and of formal vector-eigenfunctions of H+; −: very restricted case since only vector-eigenfunctions of H+ are used; there is − no condition that provides implementability of the procedure of QN constructing − and absence of pole(s) for QN and H−. 7 [5] T. Tanaka, N -fold Supersymmetry in Quantum Mechanical Matrix Models, Mod. Phys. Lett. A 27 (2012) 1250051; arXiv:1108.0480. +: the author proposes to consider SUSY generated by two intertwining opera- − + + − − + tors of the same order QN and QN such that QN QN and QN QN are the identical polynomials with matrix coefficients of H+ and H− respectively; −: there is no any general method of construction of such SUSY and there is no proof of existence of such SUSY except the case n = N = 2. The purpose of this report is to generalize the results of [6] A.A. Andrianov, A.V. Sokolov, Nonlinear supersymmetry in quantum mechan- ics: algebraic properties and differential representation, Nucl. Phys. B 660 (2003) 25; hep-th/0301062. to the matrix case. 8 3 A method of constructing of any matrix differential intertwining operator of any order in terms of transformation vector-functions Let us consider a set of formal associated vector-functions − − − − t Φl (x) ≡ 'l1(x);'l2(x);:::;'ln(x) ; l = 1; : : : ; nN; n; N 2 N of a matrix n × n Hamiltonian H+ such that this set can be divided into a chains of formal associated vector-functions of H+ for different, in general, spectral values − of H+ and the Wronskian W (x) of all Φl (x), l = 1, . , nN does not vanish on − the real axis. Then a matrix n × n differential operator QN of N-th order with − N arbitrary nondegenerate matrix coefficient XN at @ that intertwines H+ with some Hamiltonian H− and such that the considered set form a basis in its kernel can be 9 uniquely found as follows, 1 Q− = X− N W (x) N '− :::'− '−0 :::'−0 ::: ('− )(N−1) ::: ('− )(N−1) (Φ−)(N) 11 1n 11 1n 11 1n 1 − − −0 −0 − (N−1) − (N−1) − (N) '21 :::'2n '21 :::'2n ::: ('21) ::: ('2n) (Φ2 ) . .. .. .. .. × . ; '− :::'− '−0 :::'−0 ::: ('− )(N−1) ::: ('− )(N−1) (Φ− )(N) nN;1 nN;n nN;1 nN;n nN;1 nN;n nN N−1 N−1 N P1 :::Pn P1@:::Pn@:::P1@ :::Pn@ In@ PlΦ = 'l; 8 Φ(x) ≡ '1(x);'2(x);:::;'n(x) ; l = 1; : : : ; n; where during calculation of the determinant in each of its terms the corresponding N−1 N−1 N of the operators P1,..., Pn, P1@,..., Pn@, P1@ ,..., Pn@ , In@ must be placed on the last position. The potential of the Hamiltonian H− intertwined with − H+ by QN can be calculated with the help of the formula from the section 1. We emphasize that the condition that the Wronskian W (x) does not vanish on the − real axis provides existence for QN and H− and smoothness (absence of pole(s)) for − − − the potential of H− and all matrix-valued coefficients X0 (x), . , XN−1(x) of QN . 10 4 Theorem on singular factorization of a matrix intertwining operator into a product of matrix intertwining operators of lower orders The following theorem takes an important part in studying of polynomial supersym- metry and minnimizability and reducibility of matrix intertwining operators. Theorem 1. Suppose that − − (1) Φl (x), l = 1,..., nN are elements of a canonical basis in ker QN renumbered − so that the set Φl (x); l = 1,..., nj for any j = 1,..., N − 1 can be divided into a chains of formal associated vector-functions of H+; − (2) Wj(x) is the Wronskian of Φl (x), l = 1,..., nj, where j = 1,..., N; (3) jm, m = 1,..., M is monotonically increasing sequence of natural numbers such that jM = N and Wjm (x) 6≡ 0, m = 1,..., M − 1; (4) Nm = jm − jm−1, m = 1,..., M, j0 ≡ 0.