Linear and Non-Linear Supersymmetry for Non-Hermitian Matrix Hamiltonians

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 product 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” intertwining 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 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 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:

† † + + + − † PN j − † (1) H+ =H+,H− =H− ⇒ H+QN =QN H−,QN =QN = j=0(−∂) Xj (x), where † 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.

+ −† † † +: 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 mechanics: 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) ≡ ϕl1(x), ϕl2(x), . . . , ϕln(x) , l = 1, . . . , nN, n, N ∈ 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

ϕ− . . . ϕ− ϕ−0 . . . ϕ−0 ... (ϕ− )(N−1) ... (ϕ− )(N−1) (Φ−)(N) 11 1n 11 1n 11 1n 1 − − −0 −0 − (N−1) − (N−1) − (N) ϕ21 . . . ϕ2n ϕ21 . . . ϕ2n ... (ϕ21) ... (ϕ2n) (Φ2 )

...... × ...... ,

ϕ− . . . ϕ− ϕ−0 . . . ϕ−0 ... (ϕ− )(N−1) ... (ϕ− )(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 = ϕl, ∀ Φ(x) ≡ ϕ1(x), ϕ2(x), . . . , ϕn(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 supersymmetry 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.

Then there exist a matrix n × n intertwining operators Q− and intermediate Nm,m

11 Hamiltonians Hm of Schr¨odingerform, m = 1,..., M such that:

(1) Q− is an operator of N -th order with the coeﬃcient I at ∂Nm and all other Nm,m m n its coeﬃcients have, in general, a pole singularities, m = 1,..., M;

(2) the potential of Hm has, in general, a pole singularities, m = 1,..., M;

− (3) the following factorization of QN is valid,

Q− = X−Q− · ... · Q− ; N N NM ,M N1,1

(4) the following chain relations hold,

Q− H =H Q− , m=1,...,M,H ≡H ,X−H =H X−. Nm,m m−1 m Nm,m 0 + N M − N

Remark 1. In the conditions of Theorem 1 in the case where

jm = m, m = 1,...,M = N ⇒ Nm = 1, m = 1,...,M = N

− all intertwining operators Q1,m, m = 1, . . . , N are intertwining operators of the ﬁrst order and it is possible to present the following additional chain relations as well as

12 to simplify some formulae of Theorem 1:

− − Q1,jHj−1 = HjQ1,j, j = 1,...,N,H0 ≡ H+,

+ − − + Hj = Q1,j+1Q1,j+1 + U0,j+1(x) = Q1,jQ1,j + U0,j(x), j = 1,...,N − 1, + − − + H0 = Q1,1Q1,1 + U0,1(x),HN = Q1N Q1N + U0,N (x), − [U0,j(x),Q1,j] = 0, j = 1,...,N, − − + def − 2 Q1,j ≡In∂+X0,j(x),Q1,j =−In∂+X0,j(x),Hj ≡ −In∂ + Vj(x), j =1,...,N,

− 2 −0 − 2 −0 U0,j(x) = Vj−1(x) − (X0,j(x)) + X0,j(x),Vj(x) = (X0,j(x)) + X0j (x) + U0,j(x)

−0 ≡ Vj−1(x) + 2X0,j(x), j = 1,...,N,V0(x) ≡ V+(x),

− 2 −0 − 2 −0 U0,j(x) = Vj(x) − (X0,j(x)) − X0,j(x),Vj−1(x) = (X0,j(x)) − X0j (x) + U0,j(x)

−0 − −1 − ≡ Vj(x) − 2X0,j(x), j = N,..., 1,VN (x) ≡ (XN ) V−(x)XN , where the matrices U0,j(x), j = 1, . . . , N have, in general, a pole singularities.

13 5 The notion of minimizability of a matrix intertwining operator and criterion of minimizability

− It is evident that if to multiply QN by a polynomial of the Hamiltonian, L L − h X l i h X l i − Q alH ≡ alH Q , al ∈ , l = 0, . . . , L, N l=0 + l=0 − N C then such product is again an intertwining operator for the same Hamiltonians: L L L n − h X l io − h X l i n − h X l io Q alH H+ = Q H+ alH = H− Q alH . N l=0 + N l=0 + N l=0 + Thus, the question arises about possibility to simplify an intertwining operator by separation from it a superﬂuous polynomial factor.

− Deﬁnition 1. An intertwining operator QN is called minimizable if this operator can be represented in the form L − − h X l i Q =P alH , al ∈ , l =1, . . . , L, aL 6=0, 1 L N/2, N M l=0 + C 6 6 − where PM is a matrix n × n linear diﬀerential operator of M-th order, M = N − 2L − − that intertwines the Hamiltonians H+ and H−, so that PM H+ = H−PM . Otherwise, − the operator QN is called non-nminimizable.

14 Theorem 2 (criterion of minimizability). − A matrix intertwining operator QN can be represented in the form s − − Y kl 0 QN =PM (λlIn −H+) , λl ∈C, kl ∈N, l =1, . . . , s, λl 6=λl0 , l 6=l , l=1 − where PM is a non-minimizable matrix linear diﬀerential operator of the M-th order − − that intertwines the Hamiltonians H+ and H−, so that PM H+ = H−PM , if and only if

+ (1) all numbers λl, 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 λl, l = 1,..., s; (3) there are no 2n Jordan blocks in a normal (Jordan) form of T + for any eigen-

value of this matrix that does not belong to the set λl, l = 1,..., s;

(4) kl is the minimal of the orders of Jordan blocks corresponding to the eigenvalue + λl in a normal (Jordan) form of the matrix T , l = 1,..., s.

15 6 Suﬃcient condition of existence for an intertwining operator a “conjugate” intertwining operator of the same order and of polynomial SUSY with these operators

Theorem 3. Suppose that (1) the conditions of Theorem 1 and Remark 1 takes place; (2) for any eigenvalue of T + there are n (and no more) Jordan blocks in a Jordan form of T + and all these blocks (for this eigenvalue) have identical sizes; − + (3) all Φl (x), l =n(m−1)+1,..., nm correspond to the same eigenvalue λm of T

and are associated vector-functions of H+ of the same order, m = 1,..., N; QN (4) the polynomial PN (λ) is deﬁned by the equality PN (λ) = m=1(λ − λm); + + + + − −1 (5) the operator QN is deﬁned by the the equality QN = Q1,1 · ... · Q1,N (XN ) . Then

+ N N − −1 (1) QN is an operator of the N-th order, its coeﬃcient at ∂ is equal to (−1) (XN ) + and all other its coeﬃcients are smooth (without pole(s)); moreover, QN does not

depend on the order of numbering of the numbers λm, m = 1,...,N;

16 + (2) the operator QN intertwines the Hamiltonians H+ and H−, so that

+ + H+QN = QN H−;

(3) the following equalities hold,

+ − − + QN QN = PN (H+),QN QN = PN (H−).

Corollary 1. In the conditions of Th. 3 with the help of the super-Hamiltonian ! H 0 H = + 0 H− and the nilpotent supercharges ! ! 0 Q+ 0 0 Q = N , Q¯ = , Q2 = Q¯ 2 = 0 − 0 0 QN 0 one can construct the following polynomial algebra of supersymmetry:

{Q, Q¯ } = PN (H), [H, Q] = [H, Q¯ ] = 0.

17 7 Theorem on existence for any intertwining operator a “conjugate” intertwining operator and polynomial SUSY with these operators

Theorem 4. Suppose that

+ (1) λl, l = 1,..., L is the set of all diﬀerent eigenvalues of T ;

− + (2) gl is the geometric multiplicity of λl in the spectrum of T , l = 1,..., L; − − (3) kl,j, j = 1,..., gl are the orders of Jordan blocks corresponding to λl in a Jordan form of T +, l = 1,..., L;

− (4) l = max − k , l = 1,. . . , . κ 16j6gl l,j + 0 Then there is a non-minimizable operator QN 0 of the order N = 2(κ1 +...+κL)−N with smooth coeﬃcients that intertwines H+ and H− as follows,

+ + H+QN 0 = QN 0 H− and such that: L + − Y κl Q 0 Q = (H+ − λlIn) . N N l=1

18 − Corollary 2. If in the conditions of Theorem 4 there is no a matrix operator PM of the order M, M < N such that the following intertwining holds,

− − PM H+ = H−PM , then L + − − + Y κl QN 0 QN =P(N+N 0)/2(H+),QN QN 0 =P(N+N 0)/2(H−), P(N+N 0)/2(λ)≡ (λ−λl) . l=1 In this case with the help of the super-Hamiltonian ! H 0 H = + 0 H− and the nilpotent supercharges + ! ! 0 Q 0 0 0 Q = N , Q¯ = , Q2 = Q¯ 2 = 0 − 0 0 QN 0 one can construct the following polynomial algebra of supersymmetry: ¯ ¯ {Q, Q} = P(N+N 0)/2(H), [H, Q] = [H, Q] = 0.

19 8 Suﬃcient condition of reducibility of an intertwining operator and existence of absolutely irreducible intertwining operators

8.1 Deﬁnitions of reducible, irreducible and absolutely irreducible matrix intertwining operators

− − − Deﬁnition 2. QN is called reducible if there are a matrix operators KN−M and PM of the orders N − M and M, 0 < M < N respectively with smooth coeﬃcients and a matrix Hamiltonian HM with smooth potential such that the following relations hold, − − − − − − − QN = KN−M PM ,PM H+ = HM PM ,KN−M HM = H−KN−M . (3) − Otherwise the operator QN is called irreducible.

− Deﬁnition 3. QN is called absolutely irreducible if for any M, 0 < M < N there − − are no a matrix intertwining operators KN−M and PM of the orders N − M and M respectively and a matrix intermediate Hamiltonian HM , even with the potential − − of HM and the coeﬃcients of QN−M and PM possessing by a pole singularity(-ies), such that (3) hold.

20 8.2 Suﬃcient condition of reducibility of a matrix intertwining operator

Theorem 5 (suﬃcient condition of reducibility). Suppose that

− − (1) a vector-functions Φl (x), l = 1,..., nM, 1 6 M < N belong to ker QN and

can be divided into a chains of formal associated vector-functions of H+;

− (2) the Wronskian of Φl (x), l = 1,..., nM does not vanish on the real axis.

− − Then there exist a matrix operators KN−M and PM of the orders N − M and M respectively and a matrix Hamiltonian HM such that:

− − N−M M − (1) the coeﬃcients of KN−M and PM at ∂ and ∂ are equal to XN and In − − respectively and all other coeﬃcients of KN−M and PM and the potential of HM are smooth (without pole(s));

− − − − − − − (2) the relations QN =KN−M PM , PM H+ =HM PM and KN−M HM =H−KN−M hold;

− − (3) the vector-functions Φl (x), l = 1,..., nM form a canonical basis in ker PM

− and the operator QN is reducible.

21 8.3 Example of absolutely irreducible matrix intertwining operator

In contrast to the scalar case n = 1 where absolutely irreducible intertwining operators absent there are in the matrix case with any n > 2 absolutely irreducible intertwining operators of any order. Restrict ourselves to the case n = N = 2 and consider two chains of associated functions of two scalar Hamiltonians h1 and h2 for the same spectral value λ0 ∈ C:

h1ϕ1,0 = λ0ϕ1,0, (h1 − λ0)ϕ1,l = ϕ1,l−1, l = 1, 2, 3,

h2ϕ2,0 = λ0ϕ2,0, (h2 − λ0)ϕ2,1 = ϕ2,0 such that the Wronskians

0 0 0 0 W1(x) ≡ ϕ1,0ϕ1,1 − ϕ1,0ϕ1,1,W2(x) ≡ ϕ2,0ϕ2,1 − ϕ2,0ϕ2,1 do not vanish on the real axis. Then the vector-functions

− t − t − t − t Φ0 = (ϕ1,0, 0) , Φ1 = (ϕ1,1, 0) , Φ2 = (ϕ1,2, ϕ2,0) , Φ3 = (ϕ1,3, ϕ2,1) , form a chain of formal associated vector-functions of the matrix Hamiltonian

H+ = diag (h1, h2)

22 for the spectral value λ0,

− − − − H+Φ0 = λ0Φ0 , (H+ − λ0In)Φl = Φl−1, l = 1, 2, 3

− − and the following equalities for the Wronskian of Φ0 (x), . . . , Φ3 (x) hold, 0 0 ϕ1,0 0 ϕ1,0 0 ϕ1,0 ϕ1,0 0 0 0 0 ϕ1,1 0 ϕ1,1 0 ϕ1,1 ϕ1,1 0 0 0 0 = − 0 0 = −W1,N (x)W2,N (x). ϕ1,2 ϕ2,0 ϕ1,2 ϕ2,0 ϕ1,2 ϕ1,2 ϕ20 ϕ2,0 0 0 0 0 ϕ1,3 ϕ2,1 ϕ1,3 ϕ2,1 ϕ1,3 ϕ1,3 ϕ2,1 ϕ2,1 − − Thus, the Wronskian of Φ0 (x), . . . , Φ3 (x) does not vanish on the real axis and − there exist the matrix Hamiltonian H− and the matrix operator Q2 intertwining − − − H+ and H− such that Φ0 (x), . . . , Φ3 (x) form a canonical basis in ker Q2 . Absolute − irreducibility of Q2 takes place in view of the facts that any canonical basis in the − kernel of possible separated from Q2 intertwining operator can be constructed of − − Φ0 (x) and Φ1 (x) but their Wronskian

ϕ 0 1,0 ≡ 0. ϕ1,1 0 Generalization of this construction for arbitrary n and N is straightforward.