B\"acklund Transformation for the BC-Type Toda Lattice

We study an integrable case of n-particle Toda lattice: open chain with boundary terms containing 4 parameters. For this model we construct a B\"acklund transformation and prove its basic properties: canonicity, commutativity and spectrality. The B\"acklund transformation can be also viewed as a discretized time dynamics. Two Lax matrices are used: of order 2 and of order 2n+2, which are mutually dual, sharing the same spectral curve.


Introduction
In the present paper we study the Hamiltonian system of n one-dimensional particles with coordinates x j and canonical momenta X j , j = 1, . . . , n: e x j+1 −x j + α 1 e x 1 + 1 2 β 1 e 2x 1 + α n e −xn + 1 2 β n e −2xn (1.2) containing 4 arbitrary parameters: α 1 , β 1 , α n , β n . The model was missing from the early lists of integrable cases of the Toda lattice [1,2] based on Dynkin diagrams for simple affine Lie algebras. Its integrability was proved first in [3,4,5]. As for the more recent classifications, in [6] the model is enlisted as the case (i). In [7,8] particular cases of the Hamiltonian (1.2) are assigned to the C In Section 3 we describe the main result of our paper: construction of a Bäcklund transformation (BT) for our model as a one-parametric family of maps B λ : (Xx) → (Y y) from the variables (Xx) to the variables (Y y). We construct the BT choosing an appropriate gauge (or Darboux) transformation of the local Lax matrices. In Section 4, adopting the Hamiltonian point of view developed in [9,10], we prove the basic properties of the BT: 1. Preservation of the commuting Hamiltonians B λ : H j (X, x) → H j (Y, y). We also prove the following expansion of B λ in λ −1 ( 1.5) which allows to interpret the BT as a discrete time dynamics approximating the continuous-time dynamics generated by the Hamiltonian (1.2). In Section 5 we construct for our system an alternative Lax matrix L(v). The new Lax matrix of order 2n + 2 is dual to the matrix L(u) of order 2 in the sense that they share the same spectral curve with the parameters u and v having been swapped: (1. 6) In the same section we provide an interpretation of the BT in terms of the 'big' Lax matrix L(v) and establish a remarkable factorization formula for λ 2 − L 2 (v).
The concluding Section 6 contains a summary and a discussion. All the technical proofs and tedious calculations are removed to the Appendices.

Integrability of the model
In demonstrating the integrability of the model we follow the approach to the integrable chains with boundary conditions developed in [3,4] and use the notation of [9,10].
The Lax matrix L(u) for the BC-Toda lattice is constructed as the product of the following matrices (T t stands for the matrix transposition). The monodromy matrix T (u) is itself the product of the local Lax matrices each containing only the variables X j , x j describing a single particle. Note that tr T (u) is the generating function for the Hamiltonians of the periodic Toda lattice.
The matrices K ± (u) containing the information about the boundary interactions are defined as [3,4] The significance of the Lax matrix L(u) is that its spectrum is invariant under the dynamics generated by the Hamiltonian (1.2), the corresponding equations of motion dG/dt ≡Ġ = {H, G} for an observable G beinġ To prove the invariance of the spectrum of L(u) we introduce the matrices A j (u) which satisfy the easily verified identitieṡ From (2.2) and (2.9) it follows immediately thaṫ Then, using (2.1) and (2.10), we obtain the equalitẏ implying that the spectrum of L(u) is preserved by the dynamics. There are only two spectral invariants of a 2 × 2 matrix: the trace and the determinant. From (2.3) it follows that det ℓ(u) = 1 and, respectively, det T (u) = 1, so, by (2.1), the determinant of L(u) contains no dynamical variables Xx. The trace however, does contain dynamical variables and therefore can be used as a generating function of the integrals of motion, which can be chosen as the coefficients of the polynomial t(u) of degree 2n + 2 in u. Note that t(−u) = t(u) due to the symmetry The leading coefficient of t(u) at u 2n+2 is a constant (−1) n . Same is true for its free term which holds for any matrix M .
We are left then with n nontrivial coefficients H j which are integrals of motionḢ j = 0 sinceṫ(u) = 0 due to (2.12).
The conserved quantities H j are obviously polynomial in X, e ±x . Their independence can easily be established by setting e ±x = 0 in (2.3) and analysing the resulting polynomials in X. It is also easy to verify that the physical Hamiltonian (1.2) is expressed as The quantities H j are also in involution is proved in the standard way using the r-matrix technique [3,4]. Let 1 be the unit matrix of order 2 and for any matrix L define We have then the quadratic Poisson brackets [10,11] and, as a consequence, with the r-matrix Let r(u) = r t 1 (u) = r t 2 (u), (2.26) t 1 and t 2 being, respectively, transposition with respect to the first and second component of the tensor product C 2 ⊗ C 2 . Then for both T (u) = T (u)K − (u)T t (−u) and T (u) = T t (−u)K + (u)T (u) we obtain the same Poisson algebra [3,4] which ensures the commutativity (2.21) of t(u).

Describing Bäcklund transformation
In this section we shall construct a Bäcklund transformation (BT) for our model. We shall stay in the framework of the Hamiltonian approach proposed in [9] and follow closely our previous treatment of the periodic Toda lattice [9,10], with the necessary modifications taking into account the boundary conditions. We are looking thus for a one-parametric family of maps B λ : (Xx) → (Y y) from the variables (Xx) to the variables (Y y) characterised by the properties enlisted in the Introduction: Invariance of Hamiltonians, Canonicity, Commutativity and Spectrality.
The invariance of the commuting Hamiltonians H j , or of their generating polynomial t(u) = tr L(u) will be ensured if we find an invertible matrix M 1 (u, λ) intertwining the matrices L(u) depending on the variables Xx and Y y: M 1 (u, λ)L(u; Y, y) = L(u; X, x)M 1 (u, λ). (3.1) To find M 1 (u, λ) let us look for a gauge transformation implying that det M j does not depend on j. From (3.2) and (2.2) we obtain Let J be the the standard skew-symmetric matrix of order 2 and define the antipode M a as for any matrix M of order 2. It is easy to see that Transposing (3.3) and using (3.6) together with the the fact that det M j is independent of j we obtain the relation We shall be able to obtain (3.1) if we impose two additional relations Then, starting with the right-hand side L(u; X, x)M 1 (u, λ) of (3.1) and using (2.1) and (3.3) we obtain Using then (3.8b) to move M n+1 (u, λ) through K + (u), then using (3.7) and finally (3.8a) we get, step by step, arriving finally at (3.1).
We have thus to find a set of matrices M j (u, λ), j = 1, . . . , n + 1 compatible with the conditions (3.2) and (3.8). A quick calculation shows that the so called DST-ansatz for M j used in [9,10] for the periodic Toda lattice contradicts the conditions (3.8).
The philosophy advocated in [10] requires that the ansatz for the gauge matrix M j (u) be chosen in the form of a Lax matrix satisfying the r-matrix Poisson bracket (2.23) with the same r-matrix (2.25) as the Lax operator ℓ(u). It was shown in [10] that the so-called DST-ansatz serves well for the periodic Toda case. The above ansatz is however not compatible with the boundary conditions (3.8) and we have to use a more complicated ansatz for M j in the form of the Lax matrix for the isotropic Heisenberg magnet (XXX-model): The same gauge transformation was used in [12] for constructing a Q-operator for the quantum XXX-magnet.
Substituting (3.12) into (3.2) we obtain the relations for j = 1, . . . , n, and from (3.8), respectively, . (3.14) Eliminating the variables S j , we arrive to the equations defining the BT (j = 1, . . . , n): The variables s j , j = 1, . . . , n + 1 in (3.15) are implicitly defined as functions of x, y and λ from the quadratic equations Like in the periodic case [9,10], the BT map B λ : (Xx) → (Y y) is described implicitly by the equations (3.15). Unlike the periodic case, we have extra variables s j . It is more convenient not to express s j from equations (3.16) and to substitute them into (3.15) but rather define the BT by the whole set of equations (3.15) and (3.16).
Equations (3.15) and (3.16) are algebraic equations and therefore define (Y y) as multivalued functions of (Xx), which is a common situation with integrable maps [13].
In this paper, to avoid the complications of the real algebraic geometry we allow all our variables to be complex.

Properties of the Bäcklund transformation
Having defined the map B λ : (Xx) → (Y y) in the previous section, we proceed to establish its properties from the list given in the Introduction.

Preservation of Hamiltonians
The equality H j (X, x) = H j (Y, y) ∀ λ, or, equivalently, t(u; X, x) = t(u; Y, y) holds by construction, being a direct consequence of (3.1).

Canonicity
The canonicity of the BT means that the variables Y (X, x; λ) and y(X, x; λ) have the same canonical Poisson brackets (1.1) as (Xx). An equivalent formulation can be given in terms of symplectic spaces and Lagrangian manifolds. Consider the 4n-dimensional symplectic space V 4n with coordinates XxY y and symplectic 2-form Equations (3.15) and (3.16) define a 2n-dimensional submanifold Γ 2n ⊂ V 4n which can be considered as the graph Y = Y (X, x; λ), y = y(X, x; λ) of the BT (the parameter λ is assumed here to be a constant). The canonicity of the BT is then equivalent to the fact that the manifold Γ 2n is Lagrangian, meaning that: (a) it is isotropic, that is nullifies the form Ω 4n and (b) it has maximal possible dimension for an isotropic manifold: One way of proving the canonicity is to present explicitly the generating function Φ λ (y; x) of the canonical transformation, such that The required function is given by the expression

Commutativity
The commutativity B λ 1 •B λ 2 = B λ 2 •B λ 1 of the BT follows from the preservation of the complete set of Hamiltonians and the canonicity by the standard argument [9,10] based on Veselov's theorem [13] about the action-angle representation of integrable maps.

Spectrality
The spectrality property formulated first in [9] generalises the canonicity by allowing the parameter λ of the BT to be a dynamical variable like x and y.
An amazing fact is that e µ is proportional to an eigenvalue of the matrix L(λ), see (1.4). In fact, the two eigenvalues of L(λ) can be found explicitly to be Λ = (α 2 n + β n λ 2 ) see Appendix B for the proof.

Bäcklund transformation as discrete time dynamics
One of applications of a BT is that it might provide a discrete-time approximation of a continuoustime integrable system [14,15]. Indeed, iterations of the canonical map B λ generate a discrete time dynamics. Furthermore, if we find a point λ = λ 0 that (a) the map B λ 0 becomes the identity map, and (b) in a neighbourhood of λ 0 the infinitesimal map B λ 0 +ε ∼ ε{H, ·} reproduces the Hamiltonian flow with the Hamiltonian (1.2), we can claim that B λ is a discrete time approximation of the BC-Toda lattice. An attractive feature of this approximation is that, unlike some others [14], the discrete-time system and the continuous-time one share the same integrals of motion. In our case λ 0 = ∞. Letting ε = λ −1 and assuming the ansatz and substituting expansion (4.11) for s 1 into formula (3.14) for S 1 we obtain where A j coincides with the matrix (given by (2.7) and (2.8)) which describes the continuoustime dynamics of the Lax matrix. From (3.2) we obtain then ℓ(u; Y j , y j ) = ℓ(u; X j , x j ) − 2ε A j+1 ℓ(u; X j , x j ) − ℓ(u; X j , x j )A j + O(ε 2 ), (4.14) for j = 1, . . . , n + 1. Comparing the result to (2.9) we get the expansion (1.5).

Dual Lax matrix
Many integrable systems possess a pair of Lax matrices sharing the same spectral curve with the parameters u and v swapped like in (1.6), see [16] for a list of examples and a discussion. In particular, the periodic n-particle Toda lattice has two Lax matrices: the 'small' one, of order 2 [11], and the 'big' one, of order n [17]. For various degenerate cases of the BC-Toda lattice 'big' Lax matrices are also known [2,7,8,17]. In this section we present a new Lax matrix of order 2n + 2 for the most general, 4-parametric BC-Toda lattice. Here we describe the result, removing the detailed derivation to Appendix C.
Let E jk be the square matrix of order 2n + 2 with the only nonzero entry (E jk ) jk = 1. The Lax matrix L(v) is then described for the generic case n ≥ 3 as and consists of a bulk 'Jacobian' strip (the main diagonal and two adjacent diagonals) which reproduces the Lax matrix for the open Toda lattice together with boundary blocks containing parameters α 1 β 1 α n β n . We do not consider here the special case of small dimensions n = 1, 2 when the two boundary blocks interfere with each other and the structure of the Lax matrices becomes more complicated To help visualise the matrix L(v) we present an illustration for the case n = 3, using the shorthand notation ξ j ≡ e x j , η j ≡ e y j : The matrix L(v) possesses the symmetry . The matrix L(v) shares the same spectral curve with the 'small' Lax operator L(u) satisfying the determinantal identity (1.6) and thus generates the same commuting Hamiltonians H j .
The Lax matrix L(v) of order 2n + 2 seems to be new. When one or more of the constants α 1 β 1 α n β n vanish it degenerates (with a drop of dimension) into known Lax matrices for simple affine Lie algebras [2,7,8,17]. For the general 4-parametric case a Lie-algebraic interpretation of L(v) is still unknown. In particular, it is an interesting question whether L(v) satisfies a kind of r-matrices Poisson algebra.
Inozemtsev [5] presented a different Lax matrix for the BC-Toda lattice, of order 2n instead of 2n + 2 and with a more complicated dependence on the spectral parameter. The relation of these two Lax matrices is yet to be investigated.
For the dynamics (2.5), (2.6) we have an analog of the Lax equation (2.12): and satisfying The analog of the formula (3.1) for the Bäcklund transformation is where M(v) is given by (using again the notation ξ j ≡ e x j , η j ≡ e y j ) and M(v) is defined as One of common ways to obtain a Bäcklund transformation is from factorising a Lax matrix in two different ways, see [18] for Toda lattices and [13] for other integrable models. For our model we also have a remarkable factorisation, only instead of L(v) we have to take its square:

Discussion
The method for constructing a Bäcklund transformation presented in this paper seems to be quite general and applicable as well to other integrable sl(2)-type chains with the boundary conditions treatable within the framework developed in [3,4]. There is little doubt that a similar BT can be constructed for the D-type Toda lattice and a more general Inozemtsev's Toda lattice [5] with the boundary terms like b n sinh 2 x n since those, as shown in [20], can also be described in the formalism based on the boundary K matrices (2.1) and the Poisson algebra (2.27).
The 'big' Lax matrix L(v) still awaits a proper Lie-algebraic interpretation. Obtaining a BT from the factorisation of λ 2 − L 2 like in (5.11) might prove to be useful for other integrable systems related to classical Lie algebras.
It is well known that the quantum analog of a BT is the so-called Q-operator [21], see also [9]. Examples of Q-operators for quantum integrable chains with a boundary have been constructed recently for the XXX magnet [12] and for the Toda lattices of B, C and D types [22]. Our results for the BC-Toda lattice agree with those of [22], the generating function of the BT being a classical limit of the kernel of the Q-operator. Hopefully, our results will help to construct the Q-operator for the general 4-parametric quantum BC-Toda lattice.

A Proof of canonicity
Here we adapt to the BC-Toda case the argument from [10] developed originally for the periodic case. The trick is to obtain the graph Γ 2n of the BT as a projection of another manifold in a bigger symplectic space, the mentioned manifold being Lagrangian for trivial reason.
Consider the 8-dimensional symplectic space W 8 with coordinates XxY ySsT t and the symplectic form The matrix relation M (u, λ; T, t)ℓ(u; Y, y) = ℓ(u; X, x)M (u, λ; S, s) is equivalent to 4 relations defining a 4-dimensional submanifold G 4 ⊂ W 8 . The fact that G 4 is Lagrangian, that is ω 8 | G 4 = 0, is proved by presenting explicitly the generating function An alternative proof [10] is based on the fact that ℓ(u) and M (u, λ) are symplectic leaves of the same Poisson algebra (2.23).
Relation (A.2) defines thus a canonical transformation from XxSs to Y yT t. Let us take n copies W f λ (y j , t j ; x j , s j ).
After the elimination of the variables T t from equations (A.3) and (A.6), the resulting set of equations defining the submanifold G 2n = G 4n+2 ∩ W 6n+2 ⊂ W 6n+2 coincides with equations (3.13) and (3.14) defining the BT.
As we have seen in Section 3, the variables S j s j can also be eliminated leaving a 2n dimensional submanifold Γ 2n ⊂ V 4n coinciding with the graph of the BT discussed in Section 4.2. By construction, Γ 2n is the projection of G 2n from W 6n+2 onto V 4n parallel to W 2n+2 . Furthermore, Γ 2n is Lagrangian since ω 8n+4 vanishes on G 4n+2 , therefore on G 2n = G 4n+2 ∩ W 6n+2 , and therefore on Γ 2n . The canonicity of the BT is thus established geometrically, without tedious calculations.
The same argument as in [10] shows that the generating function Φ λ of the Lagrangian submanifold Γ 2n is obtained by setting t j = s j+1 in (A.8), which produces formula (4.4).

B Proof of spectrality
Here we provide the proof of formulae (4.8) for the eigenvalues of L(λ). For the proof we use an observation from [10] and show that the eigenvectors of L(λ) are given by null-vectors of M 1 (±λ, λ).
After setting u = −λ in (3.12) the matrix M j becomes a projector with the null-vector Let us set u = −λ in the matrix equality (3.1) and apply it to the vector σ 1 . By (B.2), the right-hand side gives 0. Therefore, L(−λ)σ 1 should be proportional to the same null-vector σ 1 of M j (−λ, λ), and σ 1 is an eigenvector of L(−λ).
The rest of the formulae of Section 5 are obtained by a straitforward calculation not much different from the periodic case [10,19].