Ladder relations for a class of matrix valued orthogonal polynomials
aa r X i v : . [ m a t h . C A ] J u l LADDER RELATIONS FOR A CLASS OF MATRIX VALUED ORTHOGONALPOLYNOMIALS
ALFREDO DEA ˜NO, BRUNO EIJSVOOGEL, AND PABLO ROM ´AN
Abstract.
In this paper we study algebraic and differential relations for matrix valued orthogo-nal polynomials (MVOPs) defined on R . Using recent results by Casper and Yakimov, weinvestigate MVOPs with respect to a matrix weight of the form W ( x ) = e − v ( x ) e xA T e xA ∗ ,where v is a scalar polynomial of even degree with positive leading coefficient and A and T are constant matrices. We obtain ladder operators, discrete string equations for the recurrencecoefficients and multi-time Toda equations for deformations with respect to parameters in theweight, and we show that the Lie algebra generated by the ladder operators is finite dimensional.Hermite-type matrix valued weights are studied in detail: in this case the weight is characterizedby the ladder operators, and the Lie algebra generated by them can be extended to a Lie algebrathat is isomorphic to the standard Harmonic oscillator algebra. Freud-type matrix weights arealso discussed. Finally, we establish the link between these ladder relations and those consideredpreviously by A. Dur´an and M. Ismail. Introduction
Matrix valued orthogonal polynomials (MVOPs) were introduced by Krein in the 1940’s andthey appear in different areas of mathematics and mathematical physics, including spectral the-ory [22], scattering theory [21], tiling problems [14], integrable systems [6, 2, 28] and stochasticprocesses [24, 12, 13]. There is also a fruitful interaction between harmonic analysis of matrix val-ued functions on compact symmetric pairs and matrix valued orthogonal polynomials. The firstexample of such an interaction is a family of matrix valued orthogonal polynomials related withthe spherical functions of the compact symmetric pair (SU(3) , S(U(2) × U(1)), which appeared in[23]. Inspired by [37], the case of (SU(2) × SU(2) , diag) gave a direct approach [34, 35] leading to ageneral set-up in the context of multiplicity free pairs [27, 41]. In this context, some properties ofthe orthogonal polynomials such as orthogonality, recurrence relations and differential equationsare understood in terms of the representation theory of the corresponding symmetric spaces, seealso [1] for the quantum group case and [36] for multivariable matrix orthogonal polynomials.The interpretation of matrix valued orthogonal polynomials in terms of the representationtheory of a certain symmetric pair is typically only for a limited (discrete) number of the parametersinvolved. It is then necessary to develop analytic tools to extend to a general set of parameters.In this context, shift operators for matrix valued orthogonal polynomials turned out to be veryuseful [7, 8, 33, 31, 28].In the last two decades, there has been significant progress in understanding how the differen-tial and algebraic properties of the classical scalar orthogonal polynomials can be extended to thematrix valued setting. A. Dur´an and M. Ismail [19] introduced first order lowering and raisingoperators for MVOPs, and these results were rederived later on using the Riemann-Hilbert formu-lation by Gr¨unbaum and coauthors [25]. There is also extensive work on orthogonal polynomialsolutions of matrix valued differential equations of second order from an analytic point of view,we refer the reader for instance to [15, 18, 16, 17]. Recently, Casper and Yakimov [9] proposed ageneral framework to investigate the matrix Bochner problem, that is, the classification of N × N weight matrices W ( x ) whose associated MVOPs are eigenfunctions of a second order differentialoperator.The purpose of this paper is to apply the setup of [9] to MVOPs defined on the real line, inparticular for exponential-type weights. Given N ∈ N , we denote by M N ( C ) the space of all N × N matrices with complex entries. Let W : R → M N ( C ) be a positive definite matrix weight supported in the (possibly infinite) interval[ a, b ]. For M N ( C )-valued functions H, G , we define the matrix valued inner product(1.1) h H, G i = Z ba H ( y ) W ( y ) G ( y ) ∗ dy ∈ M N ( C ) . Using standard arguments it can be shown that there exists a unique sequence ( P ( x, n )) n of monicmatrix valued orthogonal polynomials (MVOPs) with respect to W , in the following sense: h P ( x, n ) , P ( x, m ) i = H ( n ) δ n,m , where the squared norm H ( n ) is a positive definite matrix, see for instance [11, 26]. As a directconsequence of orthogonality, the polynomials P ( x, n ) satisfy the following three-term recurrencerelation(1.2) xP ( x, n ) = P ( x, n + 1) + B ( n ) P ( x, n ) + C ( n ) P ( x, n − , where B ( n ) , C ( n ) ∈ M N ( C ). Note that these matrix coefficients multiply the MVOPs from theleft. From the orthogonality relations, we also obtain that B ( n ) = X ( n ) − X ( n + 1) , C ( n ) = H ( n ) H ( n − − , where X ( n ) is the one-but-leading coefficient of P ( x, n ), i.e. P ( x, n ) = x n + x n − X ( n ) + · · · .The structure of this paper is the following: in Section 2, following the approach of Casperand Yakimov in [9], we discuss differential and difference operators for these MVOPs. In thisnoncommutative setting operators can act both from the right and from the left. We considertwo isomorphic algebras of operators acting on MVOPs, one algebra of matrix valued differentialoperators acting from the right, F R ( P ), and a second algebra of matrix valued discrete operatorsacting from the left, F L ( P ). In this construction, a differential operator D ∈ F R ( P ) acts naturallyon the variable of the MVOPs, whereas a difference operator M ∈ F L ( P ) acts on its degree.The approach proposed in [9] is particularly explicit in the case of exponential weights de-fined on the real line; these weights are studied in Section 3 and written in the form W ( x ) = e − v ( x ) e xA T e xA ∗ , with x ∈ R , where the potential v is an even polynomial with positive leadingcoefficient and A and T are constant matrices. In this case, the differential operator D = ∂ x + A has a simple adjoint D † = −D + v ′ ( x ), with respect to the matrix valued inner product given by W . The actions of D and D † on the MVOPs are( P · D )( x, n ) = P ′ ( x, n ) + P ( x, n ) A, ( P · D † )( x, n ) = − P ′ ( x, n ) − P ( x, n ) A + v ′ ( x ) P ( x, n ) , which will imply that D , D † ∈ F R ( P ). Our first result states that D , D † induce ladder relations: P · D ( x, n ) = X j = − k +1 A j ( n ) P ( x, n + j ) , P · D † ( x, n ) = k − X j =0 e A j ( n ) P ( x, n + j ) , where k = deg v , with some matrix coefficients A j ( n ) and e A j ( n ). These operators are closelyrelated to the creation and annihilation operators given in [19], with the advantage that D and D † are each other’s adjoint. This property is crucial to show that the Lie algebra generated bythe operators D and D † is finite dimensional and it is isomorphic to the algebra generated by theladder operators for the scalar weight w ( x ) = e − v ( x ) , see for instance [10], [29, Chapter 3]. Fromthe ladder relations, we obtain nonlinear algebraic equations for the coefficients of the recurrencerelation (1.2). In the literature these identities are often called discrete (or Freud) string equations.We include two examples: Hermite-type weights with v ( x ) = x + tx and t ∈ R , and Freud-typeweights with v ( x ) = x + tx , and in this last case the discrete string equations are in fact amatrix analogue of the discrete Painlev´e I equation [40]. We remark that this kind of identity,which is very relevant in integrable systems, is obtained here as a result of the relation betweenthe two Fourier algebras of operators and in particular from the fact that F L ( P ) and F R ( P ) areisomorphic.Section 4 is devoted to the detailed study of Hermite-type matrix valued weights. In this setting,we show that the ladder relations in fact characterize the matrix valued weight. We establish the link between the Lie algebra generated by D and D † and the standard Harmonic oscillator algebra.An important feature of this case is that the matrix valued Hermite polynomials are related toa degenerate Hamiltonian which is a composition of the quantum harmonic oscillator with anon-relativistic spin Hamiltonian. This generalizes the well known property of scalar Hermitepolynomials being related to the Schr¨odinger Hamiltonian.Complementing the previous results, in Section 5, we investigate similar identities of differentialand algebraic type for a deformation of the matrix weight with respect to extra parameters.Examples include the non-Abelian Toda and Langmuir lattice equations. For the particular caseof a multi-time Toda deformation, we give a Lax pair formulation, analogous to [29, (2.8.5)] forthe scalar case.In the appendix we establish the link between the ladder relations obtained with this metho-dology and the ladder operators previously considered by A. Dur´an and M. Ismail in [19]. Acknowledgements.
The authors would like to thank Erik Koelink and Mourad E. H. Ismailfor fruitful discussions about the content and scope of this paper. Bruno Eijsvoogel thanks KoenReijnders, John van de Wetering and Walter Van Assche for useful discussions as well. The supportof an Erasmus+ travel grant and EPSRC grant “Painlev´e equations: analytical properties andnumerical computation”, reference EP/P026532/1 is gratefully acknowledged. The work of PabloRom´an was supported a FONCyT grant PICT 2014-3452 and by SeCyTUNC and the work ofBruno Eijsvoogel is supported by FWO research project G0C9819N.2.
Preliminaries
In this section we introduce the left and right Fourier algebras related to the sequence of monicMVOPs, following a recent work of Casper and Yakimov [9]. For this, we view the sequence P ( x, n ) as a function P : C × N → M N ( C ). It is, therefore, natural to consider the space offunctions P = { Q : C × N → M N ( C ) : Q ( x, n ) is rational in x for fixed n } . A differential operator of the form(2.1) D = n X j =0 ∂ jx F j ( x ) , where F j : C → M N ( C ) is a rational function of x , acts on an element Q ∈ P from the right by( Q · D )( x, n ) = n X j =0 ( ∂ jx Q )( x, n ) F j ( x ) = n X j =0 d j Qdx j ( x, n ) F j ( x ) . We denote the algebra of all differential operators of the form (2.1) by M N . Now we consider aleft action on P by discrete operators. For j ∈ Z , let δ j be the discrete operator which acts on asequence A : N → M N ( C ) by ( δ j · A )( n ) = A ( n + j ) , where we take the value of a sequence at a negative integer to be equal to 0. A discrete operator(2.2) M = k X j = − ℓ A j ( n ) δ j , where A − ℓ , . . . , A k are sequences, acts on elements of P from the left by( M · Q )( x, n ) = k X j = − ℓ A j ( n ) ( δ j · Q )( x, n ) = k X j = − ℓ A j ( n ) Q ( x, n + j ) . We denote the algebra of all discrete operators of the form (2.2) by N N , and we adapt theconstruction given in [9, Definition 2.20] to our setting: ALFREDO DEA˜NO, BRUNO EIJSVOOGEL, AND PABLO ROM´AN
Definition 2.1.
For the sequence ( P ( x, n )) n of MVOPs we define: F L ( P ) = { M ∈ N N : ∃D ∈ M N , M · P = P · D} ⊂ N N , F R ( P ) = {D ∈ M N : ∃ M ∈ N N , M · P = P · D} ⊂ M N . (2.3)Using these Fourier algebras, we prove the following uniqueness result: Lemma 2.2.
Given
D ∈ F R ( P ) , there exists a unique M ∈ F L ( P ) such that M · P = P · D .Conversely, given M ∈ F L ( P ) , there exists a unique D ∈ F R ( P ) such that M · P = P · D .Proof. Let us assume that there exist M , M ∈ F L ( P ) such that( M · P )( x, n ) = ( P · D )( x, n ) , ( M · P )( x, n ) = ( P · D )( x, n ) , then (( M − M ) · P )( x, n ) = 0. Suppose that M − M has the following expression(2.4) (( M − M ) · P )( x, n ) = k X j = − ℓ A j ( n ) P ( x, n + j ) . By taking the leading coefficient of (2.4) we obtain that A k ( n ) = 0. Proceeding recursively weconclude that A j ( n ) = 0 for all j = − ℓ, . . . , k . The converse is proven in a similar way. (cid:3) It follows directly from the definition that the elements of F L ( P ) are related to the elements of F R ( P ). Lemma 2.2 shows that the map ϕ : F L ( P ) → F R ( P ) , defined by M · P = P · ϕ ( M ) , is in fact a bijection. Remark 2.3.
For M , M ∈ F L ( P ) we have that(2.5) M M · P = M · P · ϕ ( M ) = P · ϕ ( M ) ϕ ( M ) , which implies that M M ∈ F L ( P ). Therefore the linear space F L ( P ) is a subalgebra of N N . Asimilar computation shows that F R ( P ) is an algebra. We shall refer to F L ( P ) and F R ( P ) as theleft and right Fourier algebras respectively.Now it follows from (2.5) that M M · P = P · ϕ ( M ) ϕ ( M ) for all M , M ∈ F L ( P ). On theother hand, by the definition of ϕ , we have that M M · P = P · ϕ ( M M ) and, since ϕ is bijective,we conclude that ϕ is an isomorphism of algebras. Remark 2.4.
We can write the three-term recurrence relation (1.2) as xP = P · x = L · P, where L = δ + α ( n ) + β ( n ) δ − . Therefore x ∈ F R , L ∈ F L and ϕ ( L ) = x . Moreover, for every polynomial v ∈ C [ x ], we have P · v ( x ) = P · v ( ϕ ( L )) = v ( L ) · P. The main result from [9] that we use in this paper is the existence of an adjoint operation † inthe Fourier algebras F L ( P ) and F R ( P ), see [9, § F L ( P ),we first note that the algebra of discrete operators N N has a ∗ -operation given by(2.6) k X j = − ℓ A j ( n ) δ j ∗ = k X j = − ℓ A j ( n − j ) ∗ δ − j , where A j ( n − j ) ∗ denotes the conjugate transpose of A j ( n − j ). The adjoint of M ∈ N N is(2.7) M † = H ( n ) M ∗ H ( n ) − , where the squared norm H ( n ) is viewed as a sequence. The following relation holds: h ( M · P )( x, n ) , P ( x, m ) i = h P ( x, n ) , ( M † · P )( x, m ) i . On the other hand, we say that D † is the adjoint of an operator D ∈ F R ( P ) if h P · D , Q i = h P, Q · D † i , for all P, Q ∈ M N ( C )[ x ]. It follows from [9, Theorem 3.7] that ϕ ( M † ) = ϕ ( M ) † for all M ∈ F L ( P ),so that the Fourier algebras F L ( P ) and F R ( P ) are closed under the the adjoint operation † . Definition 2.5.
Given a pair ( M, D ) with M ∈ F L ( P ) and D ∈ F R ( P ) , a relation of the form M · P = P · D , where M = k X j = − ℓ A j ( n ) δ j . is called a ladder relation. If the operator M only contains nonpositive (nonnegative) powers of δ ,we say that it is a lowering (raising) relation. Observe that if a pair ( M, D ) gives a raising relation, then it follows from (2.7) and (2.6) that( M † , D † ) gives a lowering relation and viceversa.3. Ladder relations for exponential weights
In this section we investigate the existence of lowering and raising relations for a class of matrixvalued weights. Assume that ( a, b ) = ( −∞ , ∞ ) and that(3.1) W ( x ) = e − v ( x ) e xA T e xA ∗ , v ( x ) = x k + v k − x k − + · · · + v , where k is even, A and T are constant matrices and T is invertible and self-adjoint. In the approachof [19, 10], the lowering and raising operators depend on both the degree of the polynomials andthe variable x . In our case, we split the dependence on the variables x and n . We start with thematrix valued differential operator(3.2) D = ∂ x + A, and we compute the following explicit form of the adjoint: Proposition 3.1.
The adjoint of the differential operator (3.2) with respect to W is D † = − ∂ x − A + v ′ ( x ) . Proof.
For polynomials
P, Q ∈ M N ( C )[ x ], we have from (1.1) and (3.2) that h P · D , Q i = h P ′ , Q i + h P A, Q i . Integrating h P ′ , Q i by parts and using that W is invertible and self-adjoint, we obtain h P ′ , Q i = −h P, Q ′ i − h P, QW ′ W − i = h P, − Q ′ − QW ′ W − i . (3.3)Observe that the boundary terms on the right hand side of (3.3) vanish because of the exponentialdecay of the matrix weight at ±∞ . Since h P A, Q i = h P, QW A ∗ W − i , we have h P · D , Q i = h P, − Q ′ − QW ′ W − + QW A ∗ W − i . Therefore D † = − ∂ x − W ′ ( x ) W ( x ) − + W ( x ) A ∗ W ( x ) − . Using the explicit expression for W wecomplete the proof of the proposition. (cid:3) Remark 3.2.
For a given polynomial q , we denote by ( q ( L )) j ( n ) the coefficient of the differenceoperator q ( L ) of order j in δ . In other words, we have(3.4) q ( L ) = deg q X j = − deg q ( q ( L )) j ( n ) δ j . The calculation of ( q ( L )) j ( n ) can be carried out following the scheme shown in Figure 1: ( q ( L )) j ( n )is equal to the sum over all possible paths from P ( x, n ) to P ( x, n + j ) in deg q steps, where in eachpath we multiply the coefficients corresponding to each arrow.For example if q ( x ) = x we have q ( L ) = L , and in order to compute (cid:0) L (cid:1) − ( n ) we have atotal of six paths from P ( x, n ) to P ( x, n −
1) in three steps: (cid:0) L (cid:1) − ( n ) = C ( n ) B ( n − + B ( n ) C ( n ) B ( n −
1) + B ( n ) C ( n )+ C ( n + 1) C ( n ) + C ( n ) C ( n −
1) + C ( n ) . ALFREDO DEA˜NO, BRUNO EIJSVOOGEL, AND PABLO ROM´AN · · · P ( x, n + 1) P ( x, n ) P ( x, n − · · · C ( n +2) C ( n +1) I B ( n +1) I C ( n ) B ( n ) I C ( n − B ( n − I Figure 1.
Scheme for the calculation of ( q ( L )) j ( n ). Proposition 3.3.
Let W be a matrix weight as in (3.1) . Then the monic polynomials P ( x, n ) satisfy the lowering relation P · D = M · P, D = ∂ x + A, M = X j = − k +1 A j ( n ) δ j , where k = deg v ( x ) and, using the notation (3.4) , we have A ( n ) = A, A j ( n ) = ( v ′ ( L )) j ( n ) . Proof.
Notice that ( P · ∂ x )( x, n ) is a polynomial of degree n −
1. Furthermore ( P · D )( x, n ) is apolynomial of degree n with A as its leading coefficient. Therefore( P · D )( x, n ) = X j = − n A j ( n ) (cid:0) δ j · P (cid:1) ( x, n ) , A ( n ) = A. Moreover for j <
0, we have A j ( n ) = h P · D , δ j · P iH ( n − j ) − = h P, δ j · P · D † iH ( n − j ) − = h P, δ j · P · v ′ ( x ) iH ( n − j ) − = h P · v ′ ( x ) , δ j · P iH ( n − j ) − = h v ′ ( L ) · P, δ j · P iH ( n − j ) − . In the third equality we have used that D † = −D + v ′ ( x ) and that, for j < (cid:0) δ j · P · D (cid:1) has alower degree than P . In the fourth equality we use the fact that v ′ ( x ) is a scalar function.Using (3.4) we get A j ( n ) = ( v ′ ( L )) − j ( n ) . In order to complete the proof we note that ( v ′ ( L )) − j ( n ) = 0 for all j ≥ k . (cid:3) Corollary 3.4.
Let W and D be as in Proposition 3.3, then D ∈ F R ( P ) .Proof. This is an immediate consequence of Definition 2.1 and Proposition 3.3. (cid:3)
This is a refinement for exponential weights of the general results in [25] where the authorsuse a Riemann-Hilbert formulation for matrix orthogonal polynomials. In particular, we give anelementary proof for the lowering relation and obtain the exact degree of the lowering operator M .It follows from the discussion in Section 2 that there exists a unique M † ∈ F L ( P ) such that(3.5) M · P = P · D and M † · P = P · D † . Using the explicit expressions of D and D † , we find that(3.6) (cid:2) D † , D (cid:3) = v ′′ ( x ) , D + D † = v ′ ( x ) . The second equation of (3.6) and Remark 2.4 imply that we can write(3.7) ϕ − ( D + D † ) = M + M † = v ′ ( L ) . explicitly in terms of the difference operator coming from the three-term recurrence relation. Usingthe explicit formula for D in Proposition 3.3 and the definition of M † in (2.7), we verify(3.8) ( v ′ ( L )) ( n ) = (cid:0) ϕ − ( D + D † ) (cid:1) ( n ) = (cid:0) M + M † (cid:1) ( n ) = A + H ( n ) A ∗ H ( n ) − , using the notation (3.4) again. Moreover we can use (3.5) to arrive at (cid:2) M † , M (cid:3) · P = P · (cid:2) D † , D (cid:3) = P · v ′′ ( x ) = v ′′ ( L ) · P. as well as similar formulas for higher order commutators. Note that each element in a Fourieralgebra corresponds to a discrete-differential relation for the monic orthogonal polynomials. Inparticular, there is a distinguished subalgebra of F R ( P ), namely the Lie algebra g generated by D and D † . In the following theorem we prove that g is independent of the matrix A and isisomorphic to the Lie algebra corresponding to the scalar case A = 0, N = 1 which was studiedin [10, Theorem 3.1]. Theorem 3.5.
The differential operators D and D † generate a Lie algebra g of dimension k + 1 .Proof. Let v ( j ) be the j -th derivative of v . Using that v ( j ) is scalar, and so it commutes with thematrix A , we first observe that [ D , v ( j ) ] = − v ( j +1) . Then for any M N ( C )-valued smooth function F we have F · [ D , v ( j ) ] = F · D v ( j ) − ( F v ( j ) ) · D = − v ( j +1) F. Since D † = −D + v ′ ( x ), we obtain that the Lie algebra generated by D and D † is generated by {D , v ′ ( x ) , . . . , v ( k ) } , and is, therefore, ( k + 1)-dimensional. (cid:3) From the previous results, we obtain nonlinear relations for the coefficients of the three-termrecurrence relation. These identities can be seen as a non-Abelian analogue of the discrete stringor Freud equations, see for instance [4, § Theorem 3.6.
Let W be a matrix weight with monic orthogonal polynomials P ( x, n ) such that D = ∂ x + A ∈ F R ( P ) , and D † = −D + v ′ ( x ) , for some polynomial v ( x ) of degree k . Then the coefficients of the three term recurrence relationfor P ( x, n ) satisfy the following commutation relations (3.9) [ B ( n ) , A ] = I + ( v ′ ( L )) − ( n ) − ( v ′ ( L )) − ( n + 1) , [ C ( n ) , A ] = C ( n ) ( v ′ ( L )) ( n − − ( v ′ ( L )) ( n ) C ( n ) . Proof.
Consider ( P · D ) ( x, n ) = ( M · P ) ( x, n ) . In particular the coefficient of x n − gives(3.10) nI + X ( n ) A = AX ( n ) + A − ( n ) . Taking the difference of (3.10) for n + 1 and n we obtain[ B ( n ) , A ] − I = A − ( n ) − A − ( n + 1) , which together with Proposition 3.3 gives the first desired result.For the second commutation relation we take (3.8) with n replaced by n − C ( n ) from the left, and we subtract from it (3.8) with parameter n and multiplied by C ( n ) fromthe right. The result follows after cancellation of H ( n ) A ∗ H ( n − − terms. (cid:3) Remark 3.7.
Theorem 3.6 does not require the weight to be of exponential type as in (3.1). Inthe next section, however, we prove that for a polynomial v of degree two, the ladder relations(3.9), together with the moment of order zero, determine the weight to be of Hermite type. Example 3.8. (Hermite-type weight)
In the case of a Hermite-type weight with a Toda deforma-tion, namely a matrix weight of the form (3.1) with v ( x ) = x + tx , we have the operators(3.11) D = ∂ x + A, D † = − ∂ x − A + 2 x + t. The Lie algebra generated by D and D † is 3-dimensional and we have the following relations(3.12) D + D † = 2 x + t, (cid:2) D † , D (cid:3) = 2 . Using Proposition 3.3 with v ′ ( x ) = 2 x + t , we obtain A − ( n ) = 2 C ( n ) = 2 H ( n ) H ( n − − .Moreover(3.13) M = A + 2 C ( n ) δ − , M † = 2 δ + H ( n ) A ∗ H ( n ) − = 2 δ + 2 B ( n ) − A + tI. ALFREDO DEA˜NO, BRUNO EIJSVOOGEL, AND PABLO ROM´AN
The discrete string equations from Theorem 3.6 are(3.14) [ B ( n ) , A ] = 2 ( C ( n ) − C ( n + 1)) + I, [ C ( n ) , A ] = 2 ( C ( n ) B ( n − − B ( n ) C ( n )) . Example 3.9. (Freud type weight)
For a quartic even potential v ( x ) = x + tx we have theoperators: D = ∂ x + A, D † = − ∂ x − A + 4 x + 2 tx. The following relations hold true: (cid:2) D † , D (cid:3) = 12 x + 2 t, (cid:2)(cid:2) D † , D (cid:3) , D (cid:3) = 24 x, (cid:2)(cid:2)(cid:2) D † , D (cid:3) , D (cid:3) , D (cid:3) = 24 . The relation P · D = M · P in Proposition 3.3 is written explicitly as follows: P ′ ( x, n ) + P ( x, n ) A = (cid:0) A + A − ( n ) δ − + A − ( n ) δ − + A − ( n ) δ − (cid:1) P ( x, n ) . where the coefficients are computed using that A − j ( n ) = ( v ′ ( L )) j ( n ) and the scheme in Figure 1: A − ( n ) = 4 (cid:16) C ( n ) C ( n −
1) + C ( n ) + C ( n + 1) C ( n ) + B ( n ) C ( n )+ B ( n ) C ( n ) C ( n −
1) + C ( n ) B ( n − (cid:17) + 2 tC ( n ) A − ( n ) = 4 ( B ( n ) C ( n ) C ( n −
1) + C ( n ) B ( n − C ( n −
1) + C ( n ) C ( n − B ( n − ,A − ( n ) = 4 C ( n ) C ( n − C ( n − . Furthermore, we use Theorem 3.6 to compute the discrete string equations[ C ( n ) , A ] = C ( n ) ( v ′ ( L )) ( n − − ( v ′ ( L )) ( n ) C ( n ) , [ B ( n ) , A ] = I + ( v ′ ( L )) − ( n ) − ( v ′ ( L )) − ( n + 1) . If we replace ( v ′ ( L )) ( n ) = A + H ( n ) A ∗ H ( n ) − in the first equation, we obtain a trivial identity,however in terms of the coefficients of the recurrence relation we have( v ′ ( L )) ( n ) = B ( n )( C ( n ) + C ( n + 1)) + ( C ( n ) + C ( n + 1) + B ( n ) + 2 t ) B ( n ) , which implies the identity[ C ( n ) , A ] = C ( n ) B ( n − C ( n −
1) + C ( n ))+ C ( n )( C ( n −
1) + C ( n ) + B ( n − + 2 t ) B ( n − − B ( n )( C ( n ) + C ( n + 1)) C ( n ) − ( C ( n ) + C ( n + 1) + B ( n ) + 2 t ) B ( n ) C ( n ) . On the other hand, we can sum the second identity from 0 to n −
1, to obtain n − X k =0 [ B ( k ) , A ] = n + ( v ′ ( L )) − (0) − ( v ′ ( L )) − ( n ) = n − ( v ′ ( L )) − ( n ) , since C (0) = 0. Also, because B ( n ) = X ( n ) − X ( n + 1), in terms of the subleading coefficient of P ( x, n ), we obtain (3.10). If A = 0 then the weight is even and therefore X ( n ) = 0. In this case(3.10) reduces to the discrete Painlev´e I equation, see e.g. [40, § Hermite-type weights
In this section we investigate further properties of Hermite-type matrix weights. First we willshow that an arbitrary matrix weight W having operators D and D † as in Theorem 3.6 and witha specific moment of order zero is equivalent to a Hermite-type weight. Later we investigate aparticular case of the matrix A that leads to the Harmonic oscillator algebra and a link with aquantum mechanical composite system. Characterization of Hermite-type weights with a ladder relation.
The proof of thischaracterization follows the lines of the main result in [5], where the authors discuss a scalar Freudweight.
Theorem 4.1.
Let f W be a matrix weight, supported on R , and let ( P ( x, n )) n be the sequence ofmonic orthogonal polynomials . Let A be a matrix such that D = ∂ x + A ∈ F R ( P ) , D † = −D + 2 x + t, and H (0) = Z ∞−∞ e − x − xt e xA T e xA ∗ dx. Then f W is equivalent to the Hermite-type weight (4.1) W ( x ) = e − x − xt e xA T e xA ∗ . Proof.
In this proof, we first show that the string equations determine uniquely, up to the zerothmoment, the coefficients of the recurrence relation for the monic MVOPs with respect to f W . Thenthe theorem follows by showing that the matrix weight (4.1) corresponds to a determinate momentproblem, in the sense of [3].We recall from Remark 3.7 that the string equations in Theorem 3.6 hold for the case of thematrix weight f W . We need to write the string equations in terms of the squared norms. Usingthe isomorphism ϕ − and (3.12) we verify that 2 L = M + M † − t and using (3.13) we obtain(4.2) 2 B ( n ) = A + H ( n ) A ∗ H ( n ) − − tI. If we replace (4.2) in the first identity of (3.14), we obtain C ( n ) − C ( n + 1) = 14 (cid:2) H ( n ) A ∗ H ( n ) − , A (cid:3) − I. Since C ( n ) = H ( n ) H ( n − − for all n >
0, we get H ( n + 1) = 12 H ( n ) + H ( n ) H ( n − − H ( n ) − H ( n ) A ∗ H ( n ) − A H ( n ) + 14 A H ( n ) A ∗ , n > . Moreover, C (0) = 0 gives the initial condition H (1) = 12 H (0) − H (0) A ∗ H (0) − A H (0) + 14 A H (0) A ∗ . Therefore, the squared norms H ( n ) for n > H (0).Furthermore, using the identities C ( n ) = H ( n ) H ( n − − and 2 B ( n ) = A + H ( n ) A ∗ H ( n ) − , wefind that the coefficients of the recurrence relation and, thus, the monic orthogonal polynomialsare completely determined as well.Finally we need to prove that the moment problem for the Hermite weight W has a uniquesolution. By [3, Theorem 3.6], it suffices to show that the diagonal entries of the matrix valuedmeasure W ( x ) dx are determinate. We follow the approach given by Freud in [20, Theorem 5.1.and 5.2]: We observe that (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ e β | x | W ( x ) i,i dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z ∞−∞ e β | x | e − x − xt e xA T e xA ∗ dx (cid:13)(cid:13)(cid:13)(cid:13) ≤ Z ∞−∞ e β | x | e − x − xt e x k A k k T k e x k A ∗ k dx ≤ M < ∞ , for any β >
0. Therefore by [20, Theorem 5.2] the diagonal measures W ( x ) i,i dx are determinateand so is W . (cid:3) The harmonic oscillator algebra.
In the special case of a Hermite-type weight of theform (3.1) with v ( x ) = x and A = P Nj =2 µ j E j,j − , for some coefficients µ j , there exists a secondorder differential operator D which is symmetric with respect to W and so it has the orthogonalpolynomials as eigenfunctions, see for instance [18, 31]. The nilpotent matrix considered in [18] isupper triangular instead of lower triangular as in this paper so, in the rest of the section we will refer to [31]. It follows from [31, Proposition 3.5] that the monic orthogonal polynomials P ( x, n )with respect to W satisfy(4.3) ( P · D )( x, n ) = Γ( n ) · P ( x, n ) , where D and Γ( n ) are given by D = − ∂ x + ∂ x ( xI − A ) −
12 ( A + I ) + J, Γ( n ) = nI + J −
12 ( A + I ) , and J is the diagonal matrix defined by J i,i = i . The expressions that we have are not identicalto those in [31] because our weight W is related to theirs by a lower triangular matrix L (0): f W = L (0) W L (0) ∗ , ( L (0)) k,ℓ = ( − ( k − ℓ ) / ( k − ℓ )!!( k − ℓ )!when k − ℓ is non-negative and even. Note that the matrices J and A satisfy the commutationrelation [ J, A ] = A . By Definition 2.3, we have that D ∈ F R ( P ), Γ ∈ F L ( P ) and ϕ (Γ) = D . Weconsider the operators D and D † from (3.11) with t = 0 and normalized in the following way: e D = 2 − ( ∂ x + A ) , e D † = 2 − ( − ∂ x − A + 2 xI ) . Using the explicit expressions of the differential operators, we easily verify that(4.4) [ D, e D ] = e D , [ D, e D † ] = − e D † , [ e D , e D † ] = − . Therefore, D , e D , e D † and the identity operator generate a four dimensional Lie algebra calledthe harmonic oscillator algebra which we denote by h . Note that since D , e D , e D † are elementsof F R ( P ), we have that h ⊂ F R ( P ). Moreover the Lie algebra g in Theorem 3.5 is a threedimensional ideal in the Lie algebra h . The isomorphism ϕ immediately gives an isomorphicsubalgebra ϕ − ( h ) ⊂ F L ( P ).With the identification e D ←→ J + , e D † ←→ J − , D ←→ J , I ←→ E , we find that h is isomorphic to the four dimensional Lie algebra G ( a, b ) given in [38, § a = 0 and b = 1, see also [39, Chapter 10, (1.1)]. The Casimir operator is given by C = D − e D † e D + I, and commutes with e D , e D † and D . This follows directly from the commutation relations (4.4).Moreover, C ∈ F R ( P ) and C is self-adjoint, i.e. C † = C . Using the explicit expressions of theoperators, we obtain that C is the zeroth order operator C = J − I − xA. Furthermore, if we denote f M = 2 − M and f M † = 2 − M † , using (3.13) we obtain ϕ − ( C ) = Γ( n ) − f M † f M + I = − Aδ + J + (cid:18) n + 12 (cid:19) I − C ( n + 1) − B ( n ) A + ( AC ( n ) − B ( n ) C ( n )) δ − , which gives the following relation for the monic orthogonal polynomials: − A P ( x, n + 1) + (cid:18) J + (cid:18) n + 12 (cid:19) I − C ( n + 1) − B ( n ) A (cid:19) P ( x, n )+ ( AC ( n ) − B ( n ) C ( n )) P ( x, n −
1) = P ( x, n ) (cid:18) J − − xA (cid:19) . Observe that the Casimir operator ϕ − ( C ) of the Lie algebra ϕ − ( h ) is a second order differenceoperator having the sequence of monic MVOPs as eigenfunctions with a non-diagonal eigenvalueacting on P ( x, n ) from the right. The fact that the operator C commutes with e D , e D † and D is translated via the isomorphism ϕ into the following relations[ ϕ − ( C ) , f M ] = 0 , [ ϕ − ( C ) , f M † ] = 0 , [ ϕ − ( C ) , Γ( n )] = 0 . Writing the last equation explicitly, we obtain two new commutation relations for the coefficientsof the recurrence relation: A + (cid:20) C ( n + 1) + B ( n ) A, J − A (cid:21) = 0 , B ( n ) C ( n ) − AC ( n ) − (cid:20) B ( n ) C ( n ) − AC ( n ) , J − A (cid:21) = 0 . Note that these commutation relations involve the diagonal matrix J which does not appear inthe matrix valued string equations.4.2.1. Vector-valued eigenfunctions of D . For every λ ∈ C we define V λ to be the vector space ofall vector-valued eigenfunctions of the differential operator D which are in L ( W ), i.e. V λ = (cid:26) F : R → C N : F · D = λF, Z ∞−∞ F ( x ) W ( x ) F ( x ) ∗ dx < ∞ (cid:27) . Note that from (4.3), the k -th row of P ( x, n ) is an eigenfunction of D with eigenvalue n + k and,thus, it is in V n + k . Since C commutes with D , we have that F · C ℓ ∈ V λ for all F ∈ V λ and ℓ ∈ N .From the first commutation relation in (4.4), we obtain that( λ − F · e D = F · e D D, for F ∈ V λ , in such a way that e D : V λ → V λ − . Similarly we verify that e D † : V λ → V λ +1 . The operators D , D , D † and C act on the vector spaces V λ in the following way: D : V λ → V λ , e D : V λ → V λ − , e D † : V λ → V λ +1 , C : V λ → V λ . In this sense, e D and e D † are lowering and raising operators respectively.If we conjugate the differential operators D , e D and e D † by Φ( x ) = e − x e xA , we obtain E = Φ( x ) − e D Φ( x ) = 1 √ ∂ x + x ) I, E † = Φ( x ) − e D † Φ( x ) = 1 √ − ∂ x + x ) I,H = Φ( x ) − D Φ( x ) = hI + ∆ , h = − ∂ xx + 12 x , ∆ = Φ( x ) − C Φ( x ) = J − I, In this way the second order differential operator H splits into a scalar x -dependent part h plusa constant diagonal matrix ∆. Moreover, note that for N = 1, the H reduces to the operator h which is precisely the Hamiltonian for a spinless particle subject to a harmonic potential as in [39,Chapter 10, (1.1)] and the operators E and E † are, respectively, the operators J − and J + of [39,Chapter 10].For β ∈ C , we consider the following vector space of eigenfunctions of H . e V β = (cid:26) G : R → C N : G · H = βG, Z ∞−∞ G ( x ) T G ( x ) ∗ dx < ∞ (cid:27) . Observe that for every β ∈ C , the function η : e V β → V β , G ( x ) G ( x )Φ( x ) − , is an isomorphism of vector spaces. The operators H , E , E † and ∆ act on the vector spaces e V β in the following way: H : e V β → e V β , E : e V β → e V β − , E † : e V β → e V β +1 , ∆ : e V β → e V β . In order to find the eigenfunctions of H , we treat the operators h and ∆ separately. Thescalar operator h is the well-known Schr¨odinger operator corresponding to the quantum harmonic oscillator. Its square-integrable eigenfunctions are the Hermite functions ψ n ( x ) = e − x / p n n ! √ π H n ( x ) , ψ n ( x ) h = (cid:18) n + 12 (cid:19) ψ n ( x ) , where H n ( x ) = ( − n e x d n dx n e − x are the scalar Hermite polynomials. On the other hand, the lefteigenvectors of the matrix ∆ are exactly the canonical vectors e k , k = 1 , . . . , N . Therefore(4.5) { ψ λ − k ( x ) e k } min( λ,N ) k =1 is an orthogonal basis for V λ for λ ∈ N > . We can however construct a different orthogonal basisusing the MVOPs. The row vectors(4.6) q ( k ) λ − k ( x ) = e k · Q ( x, λ − k ) , with Q ( x, n ) := P ( x, n )Φ( x ) , is also an orthogonal basis of V λ . Remark 4.2.
We can interpret the differential operator H as a Hamiltonian of a quantum me-chanical composite system. See [32, § h as its Hamiltonian. Thesecond part would be a static spin s = ( N − / suchthat the spacing of energy levels of the two separate systems, is equal. The combined system couldthen be seen as a particle with spin s and without electrical charge, that is constrained to movein one direction subject to a quadratic potential with an additional (tuned) magnetic field in aperpendicular direction.Then the basis in eq. (4.5) corresponds to an orthogonal basis of separable states and the basisin eq. (4.6), to an orthogonal basis of non-separable (i.e. entangled) states.5. Deformation of the weight and multi-time Toda lattice
In this section, we consider an arbitrary matrix weight of the form(5.1) W ( x ) = e − v ( x,t ) f W ( x ) , where v ( x ; t ) is a polynomial of even degree with positive leading coefficient depending smoothlyon a parameter t ≥
0. In the following theorem we study the effect of differentiating the recurrencecoefficients with respect to t , an idea that is natural when one considers orthogonal polynomialsin the context of integrable systems. Theorem 5.1.
If we denote by . the derivative with respect to t , then the recurrence coefficientsin (1.2) satisfy the following deformation equations: (5.2) . B ( n ) = ( . v ( L )) − ( n ) − ( . v ( L )) − ( n + 1) . C ( n ) = ( . v ( L )) − ( n ) − ( . v ( L )) − ( n + 1) + ( . v ( L )) − ( n ) B ( n − − B ( n ) ( . v ( L )) − ( n ) , where we use the same notation as in Remark 3.4.Proof. Let P ( x, n ) be the monic orthogonal polynomials with W ( x ). Since h P ( x, n ) , P ( x, m ) i = 0for n > m , we have0 = ∂∂t h P ( x, n ) , P ( x, m ) i = h . P ( x, n ) , P ( x, m ) i − h P ( x, n ) · . v ( x ) , P ( x, m ) i , and then we can expand . P ( x, n ) = n − X m =0 h P ( x, n ) · . v ( x ) , P ( x, m ) i H ( m ) − P ( x, m ) = n − X m =0 ( . v ( L )) m − n ( n ) P ( x, m ) , Equivalently one could tune the quartic potential of the harmonic oscillator. where we use the notation in Remark 3.4 for ( . v ( L )) k ( n ). On the other hand, if we differentiatethe three-term recurrence relation (1.2) with respect to t , we obtain(5.3) x . P ( x, n ) = . P ( x, n + 1) + B ( n ) . P ( x, n ) + C ( n ) . P ( x, n − . B ( n ) P ( x, n ) + . C ( n ) P ( x, n − n X m =0 ( . v ( L )) m − n − ( n + 1) P ( x, m ) + B ( n ) n − X m =0 ( . v ( L )) m − n ( n ) P ( x, m )+ C ( n ) n − X m =0 ( . v ( L )) m − n +1 ( n − P ( x, m ) + . B ( n ) P ( x, n ) + . C ( n ) P ( x, n − , while on the left hand side we get(5.4) x . P ( x, n ) = n − X m =0 ( . v ( L )) m − n ( n ) P ( x, m + 1) + n − X m =0 ( . v ( L )) m − n ( n ) B ( m ) P ( x, m )+ n − X m =1 ( . v ( L )) m − n ( n ) C ( m ) P ( x, m − . Combining (5.3) and (5.4), isolating the derivatives of the recurrence coefficients, we get . B ( n ) P ( x, n ) + . C ( n ) P ( x, n −
1) = n − X m =0 ( . v ( L )) m − n ( n ) P ( x, m + 1)+ n − X m =0 ( . v ( L )) m − n ( n ) B ( m ) P ( x, m ) + n − X m =1 ( . v ( L )) m − n ( n ) C ( m ) P ( x, m − − n X m =0 ( . v ( L )) m − n − ( n + 1) P ( x, m ) − B ( n ) n − X m =0 ( . v ( L )) m − n ( n ) P ( x, m ) − C ( n ) n − X m =0 ( . v ( L )) m − n +1 ( n − P ( x, m ) . Comparing coefficients of P ( x, n ) and P ( x, n −
1) we obtain (5.2) for . B ( n ) and . C ( n ). (cid:3) Example 5.2. If . v ( x ) = x , we obtain the non-Abelian Toda lattice . B ( n ) = C ( n ) − C ( n + 1) , . C ( n ) = C ( n ) B ( n − − B ( n ) C ( n ) . Note that for v ( x ) = x + xt the relations (3.14) give2 . B ( n ) = [ B ( n ) , A ] − I, . C ( n ) = [ C ( n ) , A ] . Example 5.3. If . v ( x ) = x , we obtain the non-Abelian Langmuir lattice . B ( n ) = B ( n ) C ( n ) − B ( n + 1) C ( n + 1) + C ( n ) B ( n − − C ( n + 1) B ( n ) , . C ( n ) = C ( n ) C ( n − − C ( n + 1) C ( n ) + C ( n ) B ( n − − B ( n ) C ( n ) . Multi-time Toda lattice equations.
Let W be a weight of the form (5.1) with a multi-timeToda deformation, namely a polynomial v of the form v ( x, ~t ) = v ( x, t , . . . , t k ) = k X j =1 t j x j . If we denote by . the derivative with respect to t j , then we have . v ( L ) = L j . Theorem 5.1 gives theexpressions for the derivatives of the recurrence coefficients, but if j is large, then the coefficients( . v ( L )) k ( n ) in eq. (5.2) can be difficult to compute, and a much more convenient formulation isgiven as a Lax pair. In the spirit of [29, § L with the block tridiagonal matrix withblock entries ( L nm ), L n,n +1 = I , L n,n = B ( n ), L n,n − = C ( n ), and L n,m = 0 I if | n − m | ≥
2. Fora N × N -block semi-infinite matrix S = ( S nm ), we define S + as the matrix obtained by replacingall the N × N blocks of S below the main diagonal by zero. Analogously, we let S − to be thematrix obtained by replacing all the N × N blocks above the subdiagonal by zero.Then, we have the following result: Theorem 5.4.
For j = 1 , . . . , k − , we have . L = (cid:2) L, ( L j ) + (cid:3) = − (cid:2) L, ( L j ) − (cid:3) . Proof.
We first observe that (cid:2) L, ( L j ) + (cid:3) + (cid:2) L, ( L j ) − (cid:3) = (cid:2) L, ( L j ) + + ( L j ) − (cid:3) = (cid:2) L, L j (cid:3) = 0 , which proves the second equality. Using that . B ( n ) = ( . L ) n,n and . C ( n ) = ( . L ) n,n − , we willcomplete the proof by showing that (cid:2) L, ( L j ) + (cid:3) n,n equals the right hand side of the first equationin (5.2), that (cid:2) L, ( L j ) + (cid:3) n,n − equals the right hand side of the second equation of (5.2) and that (cid:2) L, ( L j ) + (cid:3) n,m = 0 otherwise.Note that ( . v ( L )) m ( n ) = ( L j ) n,n + m for any indices m, n so the first equation in (5.2) reads . B ( n ) = ( L j ) n,n − − ( L j ) n +1 ,n , n ≥ . On the other hand, bearing in mind that L is block tridiagonal and L j − is lower triangular withzeros on the diagonal, we have (cid:2) L, ( L j ) − (cid:3) n,n = L n,n +1 ( L j ) n +1 ,n − ( L j ) n,n − L n − ,n = ( L j ) n +1 ,n − ( L j ) n,n − , since L n,n +1 = I for any n ≥
0, which proves the result for the main diagonal. The secondequation in (5.2) is . C ( n ) = ( L j ) n,n − − ( L j ) n +1 ,n − + ( L j ) n,n − B ( n − − B ( n )( L j ) n,n − , n ≥ . We also have (cid:2) L, ( L j ) − (cid:3) n,n − = L n,n ( L j ) n,n − + L n,n +1 ( L j ) n +1 ,n − − ( L j ) n,n − L n − ,n − − ( L j ) n,n − L n − ,n − = B ( n )( L j ) n,n − + ( L j ) n +1 ,n − − ( L j ) n,n − B ( n − − ( L j ) n,n − , which proves the result for the first subdiagonal.Finally, if k ≥ n + 1 we repeat the calculation using (cid:2) L, ( L j ) − (cid:3) n,k , which gives 0, consistentlywith ( . L ) n,k , and if k ≤ n −
2, we compute (cid:2) L, ( L j ) + (cid:3) n,k , which gives 0 on both sides again. (cid:3) Appendix A. Comparison of ladder operators
In this article we have taken a different approach to ladder operators for matrix valued orthog-onal polynomials than for example the one in [19]. Their approach is inspired by [10] and [30] forthe scalar orthogonal polynomials. This appendix is meant to compare our approach with theirsfor our class of weight functions, i.e. the exponential weights in (3.1).The ladder relations for exponential weights are stated in Proposition 3.3:(A.1) P ′ ( x, n ) = k − X j =1 A − j ( n ) P ( x, n − j ) + [ A, P ( x, n )] , for monic polynomials P . For exponential weights on the real line, we have the following identity:(A.2) W ′ ( x ) = − W ( x ) V ( x ) , V ( x ) = v ′ ( x ) I − A ∗ − ρ ( x ) , where ρ ( x ) = W − ( x ) AW ( x ).The ladder relation given in [19] for exponential weights reads(A.3) P ′ ( x, n ) = F ( x, n ) P ( x, n ) − E ( x, n ) P ( x, n − , where the coefficients are E ( x, n ) H ( n −
1) = − Z R P ( y, n ) W ( y ) V ( x ) − V ( y ) x − y P ( y, n ) ∗ dy, (A.4) F ( x, n ) H ( n −
1) = − Z R P ( y, n ) W ( y ) V ( x ) − V ( y ) x − y P ( y, n − ∗ dy. These identities are obtained in the following way: we expand the derivative of P ( x, n ) in the basisof MVOPs, with coefficients multiplying on the left: P ′ ( x, n ) = n − X k =0 h P ′ ( x, n ) , P ( x, k ) iH ( k ) − P ( x, k )= n − X k =0 (cid:18)Z R P ′ ( y, n ) W ( y ) P ( y, k ) ∗ dy (cid:19) H ( k ) − P ( x, k )= Z R P ′ ( n, y ) W ( y ) n − X k =0 P ( y, k ) ∗ H ( k ) − P ( x, k ) ! dy. We integrate by parts, and the boundary terms vanish because of the decay of W ( x ) at ±∞ . This,together with (A.2), gives P ′ n ( x ) = − Z R P ( n, y ) W ( y )( − V ( y )) n − X k =0 P ( y, k ) ∗ H ( k ) − P ( x, k ) dy = − Z R P ( n, y ) W ( y )( V ( x ) − V ( y )) n − X k =0 P ( y, k ) ∗ H ( k ) − P ( x, k ) dy, where we have used the fact that the integral with − V ( x ) vanishes by orthogonality. If we nowapply the Christoffel-Darboux formula(A.5)( x − y ) n − X k =0 P ( y, k ) ∗ H ( k ) − P ( x, k ) = P ( y, n − ∗ H ( n − − P ( x, n ) − P ( y, n ) ∗ H ( n − − P ( x, n − , we obtain the formulas (A.4) for the coefficients E ( x, n ) and F ( x, n ). Furthermore, using theformula for V ( x ) in (A.2), we can write F ( x, n ) H ( n −
1) = − Z R P ( y, n ) W ( y ) (cid:20) v ′ ( x ) − v ′ ( y ) x − y − ρ ( x ) − ρ ( y ) x − y (cid:21) P ( y, n − ∗ dy (A.6) E ( x, n ) H ( n −
1) = − Z R P ( y, n ) W ( y ) (cid:20) v ′ ( x ) − v ′ ( y ) x − y − ρ ( x ) − ρ ( y ) x − y (cid:21) P ( y, n ) ∗ dy. On the other hand, by direct computation using the fact that P ( x, n ) is monic and (A.2), we have(A.7) P ( x, n ) A − AP ( x, n ) = n − X k =0 h P n A, P k iH ( k ) − P k ( x )= n − X k =0 (cid:18)Z R P ( y, n ) W ( y ) ( ρ ( y ) − ρ ( x )) P ( y, k ) ∗ dy (cid:19) H ( k ) − P ( x, k ) . Therefore, applying (A.5) again, we obtain P ( x, n ) A − AP ( x, n ) = (cid:18)Z R P ( y, n ) W ( y ) ρ ( y ) − ρ ( x ) x − y P ( y, n − ∗ dy (cid:19) H ( n − − P ( x, n )+ (cid:18)Z R P ( y, n ) W ( y ) ρ ( y ) − ρ ( x ) x − y P ( y, n ) ∗ dy (cid:19) H ( n − − P ( x, n − . Comparing this last equation with (A.6), we find a relation between the two ladder operators,since F ( x, n ) H ( n −
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