ITEP-TH-39/01

RELATIVISTIC TODA CHAIN AT ROOT OF UNITY II. MODIFIED Q-OPERATOR

S. PAKULIAK AND S. SERGEEV

Abstract. Matrix elements of quantum intertwiner as well as the modified Q-operator for the quantum relativistic Toda chain at root of unity are constructed explicitly. Modified Q-operators make isospectrality transformations of quantum transfer matrices so that the classical counterparts of Q-operators correspond to the Bäcklund isospectrality transformations of the classical transfer matrices. Separated vectors for the Functional Bethe Ansatz are constructed with the help of modified Q-operators.

Contents

Introduction 2 1. Formulation of the model 4 1.1. L-operators 4 1.2. Transfer matrix and integrability 5 1.3. Quantum separation of variables 7 2. Intertwining relations 8 2.1. Auxiliary L-operator 8 2.2. Intertwining 9 2.3. Q-transformation 10 3. Matrix R 12 3.1. w-function 12

1991 Mathematics Subject Classification. 82B23. Key words and phrases. Integrable models, Toda chain, Spin chain, Bilinear equations, Functional Bethe

Ansatz.

This work was supported in part by the grant INTAS OPEN 00-00055. S.P.’s work was supported by the grant RFBR 00-02-16477 and grant for support of scientific schools RFBR 00-15-96557, S.S’s work was supported by the grant RFBR 01-01-00201.

2 S. PAKULIAK AND S. SERGEEV

3.2. Matrix elements of R_{m,ϕ} 13 4. Rational Θ-function 14 5. Parameterization of K 16 5.1. Sequence of the Bäcklund transformations 16 5.2. Parameterization of modified Q-operators 17 5.3. Arbitrary value of λ_{ϕ} 18 5.4. Baxter equation 19 6. Separating operator 21 6.1. Spectrum of b 21 6.2. Limiting procedure 22 7. Discussion 25 References 26

Introduction

This paper is the continuation of Ref. [1] where the quantum relativistic Toda chain at a root of unity was investigated. In the previous paper we have established several features of the chain, concerning the relationship between its finite dimensional (spin) counterpart and its nontrivial classical counterpart.

For several integrable models, based on the simple Weyl algebra at N-th root of unity, the ℂ-numerical N-th powers of Weyl generators form a classical discrete integrable system. So the parameters of the unitary representation of Weyl algebra form the classical counterpart of the finite dimensional integrable system (i.e. spin integrable system), see [2, 3, 4, 5]. Most important finite dimensional operators, arising in the spin systems, have functional counterparts, defined as rational mappings in the space of Nth powers of the parameters. So the finite dimensional operators are the secondary objects: one has to define first the mapping of the parameters, and then the finite dimensional operators are to be constructed in the terms of initial and final values of the parameters. Usual finite dimensional integrable models correspond to the case when for all the operators involved the initial and final parameters coincide. Algebraically these conditions are the origin e.g. of Baxter’s curve in the case of the chiral Potts model [6, 7], or of the spherical triangle parameterization for the Zamolodchikov-Bazhanov-Baxter model.

The simplest case of such combined system is the so-called quantum Toda chain at root of unity. It differs significantly from the usual Toda chain [8] as well as from the relativistic Toda chain with arbitrary q [9]. For RTC at root of unity L-operators, quantum as well as classical, and the quantum intertwiners have rather simple form [9]. In [1] it was shown that the trace of monodromy of quantum intertwiners, giving usually Baxter’s Q-operator, has the nontrivial classical counterpart. Under several special additional conditions, the classical counterpart of Q operator is the Bäcklund transformation [10, 11] for the classical counterpart of relativistic Toda chain. Spin counterpart of the corresponding Bäcklund transformation makes an isospectral transformation of the spin system. Parameters of Weyl algebra are described in the terms of τ-function^{1}, and the transformation mentioned maps (n − 1)-solitonic state into n-solitonic state. Initial homogeneous state is supposed to be 0-solitonic, while the maximal number of solitons is M − 1 for the chain of the length M. The isospectral transformation of the spin system depends on the (n − 1)-solitonic origin and n-solitonic image of the classical states. Finite dimensional similarity operators, called the modified Q-operators, of course, do not form a commutative family. Nevertheless these isospectrality deformations play the important rôle in the Functional Bethe Ansatz [1] (see [12, 8, 13] for the details concerning the method of FBA). Namely, for a special set of the solitonic amplitudes the similarity operator becomes a projector into an eigenstate of nondiagonal element of the quantum Toda chain monodromy matrix. It was shown in [1], but we did not give explicit formulas for matrix elements of all finite dimensional operators (quantum intertwiner, similarity operators as well as the separating operator). All these is the subject of the current paper.

This paper is organized as follows. First, we recall the formulation of the quantum relativistic Toda chain at root of unity and its classical counterpart. Next, we recall the form of the auxiliary L-operators (which are the particular case of Bazhanov-Stroganov’s L-operator [7]), its classical counterpart, the intertwining relations and the isospectrality transformations of the Toda transfer matrix. In the third section the matrix elements of

1in this paper we use the term “τ-function” in the sense of bilinear discrete equation with constant coefficients, i.e. as a trigonometric limit of the θ-function on a high genus algebraic curve

4 S. PAKULIAK AND S. SERGEEV

quantum R-matrix and modified Q-operator are given in an appropriate basis. In the fourth section we recall the parameterization of τ-function in application to the relativistic Toda chain. In the fifth section we parameterize arguments of the modified Q-operators for the whole set of Bäcklund transformations and so construct explicitly the similarity operator transforming the homogeneous chain into the general (M − 1)-parametric inhomogeneous chain. In the sixth section, we describe an appropriate limiting procedure for the degenerative final state and construct the quantum separating operator. Finally, in conclusion, we discuss several further applications of the method described.

1. Formulation of the model

In this section we recall briefly the subject of the model called the quantum relativistic Toda chain at root of unity [1, 9].

1.1. L-operators. Let the chain is formed by M sites with the periodical boundary conditions. m-th site of the Toda chain is described by the following local Lax matrix:

ℓ_{m}(λ) = ( \begin{matrix}1 + \frac{κ}{λ} u_{m} w_{m} & , & − \frac{ω^{1/2}}{λ} u_{m} \\ w_{m} & , & 0 \end{matrix} ) . (1)

Here λ is the spectral parameter, κ is an extra complex parameter, common for all sites, i.e. the modulus. Elements u_{m} and w_{m} form the ultra-local Weyl algebra,

u_{m} · w_{m} = ω w_{m} · u_{m} , (2)

and u_{m}, w_{m} for different sites commute. Weyl’s parameter ω is the primitive root of unity,

ω = e^{2 π i/N} , ω^{1/2} = e^{i π/N} . (3)

N-th powers of the Weyl elements are centers of the algebra. We will deal with the finite dimensional unitary representation of the Weyl algebra, i.e.

u_{m} = u_{m} x_{m} , w_{m} = w_{m} z_{m} , (4)

where u_{m} and w_{m} are ℂ-numbers, and

(5) x_{m} = 1 ⊗ 1 ⊗ ... ⊗ \underbrace{x}_{{m−th \atop place}} ⊗... , z_{m} = 1 ⊗ 1 ⊗ ... ⊗ \underbrace{z}_{{m−th \atop place}} ⊗...

Convenient representation of x and z on the N-dimensional vector space |α⟩ = |α mod N⟩ is x |α⟩ = |α⟩ ω^{α} , z |α⟩ = |α + 1 ⟩ , ⟨α|β⟩ = δ_{α,β} . (6) Thus x and z are normalized to the unity (x^{N} = z^{N} = 1) N × N dimensional matrices, and the Nth powers of the local Weyl elements are ℂ-numbers u_{m}^{N} = u_{m}^{N} , w_{m}^{N} = w_{m}^{N} . (7) In general, all u_{m} and w_{m} are different, so we deal with the inhomogeneous chain.

Variables u_{m}^{N} and w_{m}^{N} form the classical counterpart of the quantum relativistic Toda chain, and the classical Lax matrix is

L_{m}(λ^{N}) =^{def} ( \begin{matrix}1 + \frac{κ^{N}}{λ^{N}} u_{m}^{N}w_{m}^{N} & , & \frac{u_{m}^{N}}{λ^{N}} \\ w_{m}^{N} & , & 0 \end{matrix} ) . (8)

1.2. Transfer matrix and integrability. Ordered product of the quantum L-operators (1)

t̂(λ) =^{def} ℓ_{1}(λ) ℓ_{2}(λ) ⋯ ℓ_{M}(λ) = ( \begin{matrix}a(λ) & b(λ) \\ c(λ) & d(λ) \end{matrix} ) , (9)

and its trace

t(λ) = a(λ) + d(λ) = ∑_{k=0}^{M} λ^{−k} t_{k} . (10)

are the monodromy and the transfer matrices of the model.

Transfer matrices, defined with the given κ and with the given set of u_{m}, w_{m}, m = 1...M, t(λ) = t(λ, κ; {u_{m}, w_{m}}_{m=1}^{M}) . (11) form the commutative set: ∀ λ, μ [ t(λ, κ; {u_{m}, w_{m}}) , t(μ, κ; {u_{m}, w_{m}}) ] = 0 . (12)

6 S. PAKULIAK AND S. SERGEEV

Eq. (12) is provided by the intertwining relation for L-operators (1) in the auxiliary twodimensional vector spaces with the help of well known six-vertex trigonometric R-matrix, see [1] for the details.

In the spectral decomposition of the transfer matrix (10) the utmost operators are

t_{0} = 1 , t_{M} = κ^{M} ∏_{m=1}^{M} (−ω^{1/2}u_{m}w_{m}) Y , (13)

where

Y = ∏_{m=1}^{M} (−ω^{−1/2}x_{m}z_{m}) , Y^{N} = 1 . (14)

Further we will need also the monodromy of the classical Lax matrices (8),

T̂ = L_{1} L_{2} ⋯ L_{M} = ( \begin{matrix}A(λ^{N}) & B(λ^{N}) \\ C(λ^{N}) & D(λ^{N}) \end{matrix} ) (15)

and its trace

T (λ^{N}) = A(λ^{N}) + D(λ^{N}) . (16)

M nontrivial coefficients in the decomposition of T (λ^{N}) are involutive with respect to the Poisson brackets, associated with the following symplectic form:

Ω = ∑_{m=1}^{M} \frac{d u_{m}}{u_{m}} ∧ \frac{d w_{m}}{w_{m}} . (17)

It is the consequence of (2) and (12).

The utmost coefficient in the decomposition of the classical transfer matrix T (λ^{N}), analogous to (14), generates the simple gauge transformation of L-operators (1). Thus we may fix without lost of generality

∏_{m=1}^{M} (−ω^{1/2}u_{m}) = ∏_{m=1}^{M}(−w_{m}) = 1 . (18)

1.3. Quantum separation of variables. Eigenstates of off-diagonal element of the monodromy matrix (9) b(λ) play the important rôle in the constructing of the spectrum of the quantum transfer matrix. This method is known as the method of the Functional Bethe Ansatz, or the method of the quantum separation of variables [12, 8, 13]. It is useful to parameterize the spectrum of b(λ) by its zeros λ_{k}, k = 1...M − 1. To deal with the eigenstates of b(λ) explicitly is not useful. We should better define the set of vectors |{λ_{k}}_{k=1}^{M−1}, γ⟩, such that

b(λ_{j}) |{λ_{k}}_{k=1}^{M−1}, γ⟩ = 0 , λ_{j} ∈ {λ_{k}}_{k=1}^{M−1} , (19)

and

Y |{λ_{k}}_{k=1}^{M−1}, γ⟩ = |{λ_{k}}_{k=1}^{M−1}, γ⟩ ω^{γ} , (20)

where the integral of motion Y is given by (14).

The matrix element between an eigenstate of t_{k} and |{λ}, γ⟩ is a product of (M − 1) Baxter’s functions Q(λ_{k}), taken for the same λ_{k}, k = 1...M − 1, as in (19), see e.g. [12, 13]. For the details concerning the Baxter equation for the quantum relativistic Toda chain at root of unity see [1], and later we will recall it. Note here, this Q is a meromorphic function on Baxter’s curve. In our parameterization it is given by

λ^{N} = δ^{N} + \frac{λ^{N}}{δ^{N}} + κ^{N} . (21)

Convenient Baxter’s form x^{N} + y^{N} = 1 + k^{2}x^{N}y^{N} may be obtained by the substitution x = 1/δ, y = δ/λ, k^{2} = −κ^{N}.

Spectra of a, ..., d are described by A, ..., D, see [14]. For example,

∏_{n∈ℤ_{N}} b(ω^{n}λ) = B(λ^{N}) . (22)

Eq. (22) allows one to find the set of λ_{k}. We will do it later for the homogeneous model.

The real reason why we are considering the classical counterpart is that the quantum separation of variables is associated with the Bäcklund transformation of the classical model.

8 S. PAKULIAK AND S. SERGEEV

2. Intertwining relations

In this section we recall the auxiliary L-operators, quantum R-matrix and the Darboux relation for the classical counterpart, providing the solution of the isospectrality problem, construction of modified Q-operators and Bäcklund transformation.

2.1. Auxiliary L-operator. Toda L-operators (1) are intertwined in the quantum spaces with the Bazhanov-Stroganov L-operators [7]. This fact is the origin of the relationship between the relativistic Toda chain at the Nth root of unity and the N-state chiral Potts model. In general the intertwiner is rather complicated, but there exists a special limit of Bazhanov-Stroganov’s L-operator such that the intertwiner simplifies. This limit is the analogue of the so-called dimer self-trapping L-operators for usual Toda chain, see [10, 8, 9] for the details. We define the auxiliary L-operator as follows:

\begin{matrix}ℓ̃_{ϕ}(λ, λ_{ϕ}) = \\ = ( \begin{matrix}1 − ω^{1/2}κ_{ϕ} \frac{λ_{ϕ}}{λ} w_{ϕ} & , & −\frac{ω^{1/2}}{λ} (1 − ω^{1/2}κ_{ϕ}w_{ϕ})u_{ϕ} \\ −ω^{1/2}λ_{ϕ}u_{ϕ}^{−1}w_{ϕ} & , & w_{ϕ} \end{matrix} ) . \end{matrix} (23)

Here λ and λ_{ϕ} are two spectral parameters (actually, up to a gauge transformation, ℓ̃depends on their ratio). κ_{ϕ} is the module as well as κ. u_{ϕ} and w_{ϕ} form the same Weyl algebra at the root of unity (4),

u_{ϕ} = u_{ϕ} x_{ϕ} , w_{ϕ} = w_{ϕ} z_{ϕ} . (24)

In all these notations the subscript ϕ stands as the “number” of additional Weyl algebra. Classical counterpart of ℓ̃ is

L̃_{ϕ} = ( \begin{matrix}1 + κ_{ϕ}^{N} \frac{λ_{ϕ}^{N}}{λ^{N}} w_{ϕ}^{N} & , & \frac{u_{ϕ}^{N}}{λ^{N}} (1 + κ_{ϕ}^{N}w_{ϕ}^{N}) \\ λ_{ϕ}^{N} \frac{w_{ϕ}^{N}}{u_{ϕ}^{N}} & , & w_{ϕ}^{N} \end{matrix} ) . (25)

L-operators (23) are intertwined in their two dimensional auxiliary vector spaces by the same six-vertex trigonometric R-matrix as (1). Also there exists the fundamental quantum intertwiner for (23): it is the R-matrix for the chiral Potts model such that two rapidities are fixed to a special singular value. The method describing in this paper may be applied directly to the model, defined by (23), this is the subject of the forthcoming paper.

2.2. Intertwining. We are going to write out some quantum intertwining relation for Loperators (1) as well as for the whole monodromy matrix (9). So we use notations, applicable for the recursion in m. Also in this section we will point out parameters u_{m}, w_{m}, (4), as the arguments of L-operators.

Proposition 1. There exists unique (up to a constant multiplier) N^{2}×N^{2} matrix R_{m,ϕ}(λ_{ϕ}), such that R_{m,ϕ}, ℓ_{m} and ℓ̃ obey the modified intertwining relation

\begin{matrix}ℓ̃_{ϕ}(λ, λ_{ϕ}; u_{ϕ,m}, w_{ϕ,m}) · ℓ_{m}(λ; u_{m}, w_{m}) R_{m,ϕ}(λ_{ϕ}) = \\ = R_{m,ϕ}(λ_{ϕ}) ℓ_{m}(λ; u′_{m}, w′_{m}) · ℓ̃_{ϕ}(λ, λ_{ϕ}; u_{ϕ,m+1}, w_{ϕ,m+1}) , \end{matrix} (26)

if and only if their classical counterparts of L_{m} and L̃ obey the following Darboux relation:

L̃(u_{ϕ,m}^{N}, w_{ϕ,m}^{N})L(u_{m}^{N}, w_{m}^{N}) = L(u_{m}^{′N}, w_{m}^{′N})L̃(u_{ϕ,m+1}^{N}, w_{ϕ,m+1}^{N}) , (27)

Proof: the direct verification. Eq. (26) may be rewritten as

(28)  ℓ̃_{ϕ}(λ; u_{ϕ}, w_{ϕ}) · ℓ_{m}(λ; u_{m}, w_{m}) = ℓ_{m}(λ; u′_{m}, w′_{m}) · ℓ̃_{ϕ}(λ; u′_{ϕ}, w′_{ϕ}) ,

where

(29) u′_{m} = u′_{m} R_{m,ϕ}x_{m}R_{m,ϕ}^{−1} , w′_{m} = w′_{m} R_{m,ϕ}z_{m}R_{m,ϕ}^{−1} ,

and

(30) u′_{ϕ} = u_{ϕ,m+1} R_{m,ϕ}x_{ϕ}R_{m,ϕ}^{−1} , w′_{ϕ} = w_{ϕ,m+1} R_{m,ϕ}z_{ϕ}R_{m,ϕ}^{−1} ,

10 S. PAKULIAK AND S. SERGEEV

Solution of (28) with respect to the primed operators is unique and given by

u′_{m} = \frac{κ_{ϕ}}{κ} u_{ϕ} ,

w′_{m} = w_{m}w_{ϕ} − ω^{1/2}λ_{ϕ}u_{ϕ}^{−1}w_{ϕ} ,

u′_{ϕ} = (λ_{ϕ} + κu_{m}w_{m} − ω^{1/2}w_{m}u_{ϕ})^{−1} λ_{ϕ}u_{m} ,

w′_{ϕ} = \frac{κ}{κ_{ϕ}} u_{m}w_{m} (w_{m}u_{ϕ} − ω^{1/2}λ_{ϕ})^{−1}

(31)

Nth powers of all primed operators in (31) give

u_{ϕ,m+1}^{N} = \frac{λ_{ϕ}^{N} u_{m}^{N}}{λ_{ϕ}^{N} + κ^{N}u_{m}^{N}w_{m}^{N} + w_{m}^{N}u_{ϕ,m}^{N}} , w_{ϕ,m+1}^{N} = \frac{κ^{N}}{κ_{ϕ}^{N}} \frac{u_{m}^{N}w_{m}^{N}}{λ_{ϕ}^{N} + w_{m}^{N}u_{ϕ,m}^{N}} , (32)

and

u_{m}^{′N} = \frac{κ_{ϕ}^{N}}{κ^{N}} u_{ϕ,m}^{N} , w_{m}^{′N} = \frac{κ^{N}}{κ_{ϕ}^{N}} \frac{u_{m}^{N}w_{m}^{N}}{u_{ϕ,m}^{N}} \frac{w_{ϕ,m}^{N}}{w_{ϕ,m+1}^{N}} . (33)

Eqs. (32) and (33) are the exact and unique solution of (27). Therefore eq. (27) is the consistency condition for (26). Next, eq. (26) is a set of linear equations for the matrix elements of R_{m,ϕ}, this set is defined appropriately, and this provides the uniqueness of R_{m,ϕ}.

The local transformation

u_{m}, w_{m}, u_{ϕ,m}, w_{ϕ,m} ↦ u′_{m}, w′_{m}, u_{ϕ,m+1}, w_{ϕ,m+1} , (34)

given by eqs. (32) and (33), is called usually as the Darboux transformation for the classical relativistic Toda chain. Eqs. (32) and (33) define the mapping (34) up to N-th roots of unity. These roots are the additional discrete parameters of the transformation (34). Note, matrix R_{m,ϕ} is unique if all these roots are fixed.

2.3. Q-transformation. Relations (26) and (27) may be iterated for the whole chain. The functional counterpart gives

L̃(u_{ϕ,1}^{N}, w_{ϕ,1}^{N})T̂({u_{m}^{N}, w_{m}^{N}}_{m=1}^{M}) = T̂({u_{m}^{′N}, w_{m}^{′N}}_{m=1}^{M})L̃(u_{ϕ,M+1}^{N}, w_{ϕ,M+1}^{N}) , (35)

where the mapping u_{ϕ,1}, w_{ϕ,1} ↦ u_{ϕ,M+1}, w_{ϕ,M+1} is M-th iteration of (34).

For the periodic chain the cyclic boundary conditions for the recursion (32) are to be imposed, u_{ϕ,M+1} = u_{ϕ,1} , w_{ϕ,M+1} = w_{ϕ,1} . (36) Now suppose (32) and (36) are solved, i.e. u_{ϕ,m}, w_{ϕ,m} are parameterized in the terms of u_{m}, w_{m}, m = 1...M, and some extra parameters, possible degrees of freedom of (32) and (36) (e.g. λ_{ϕ}). Then (33) defines in general the transformation Q_{ϕ} Q_{ϕ} : {u_{m}, w_{m}}_{m=1}^{M} ↦ {u′_{m}, w′_{m}}_{m=1}^{M} (37) The transfer matrices of two sets, {u_{m}, w_{m}} and {u′_{m}, w′_{m}}, have the same spectrum, because of there exists N^{M} × N^{M} matrix Q_{ϕ}, nondegenerative in general,

Q_{ϕ} =^{def} tr_{ϕ} (R_{1,ϕ} R_{2,ϕ} … R_{M,ϕ}) , (38)

such that t(λ; {u_{m}, w_{m}}) · Q_{ϕ} = Q_{ϕ} · t(λ; {u′_{m}, w′_{m}}) . (39) Subscript ϕ of Q_{ϕ} stands as the reminder for the parameters, arising in the solution of recursion, including at least the spectral parameter λ_{ϕ}.

Consider now the repeated application of the transformations Q_{ϕ}, eq. (37),

{u_{m}, w_{m}} ↦^{Q_{ϕ_{1}}} {u′_{m}, w′_{m}} ↦^{Q_{ϕ_{2}}} ... ↦^{Q_{ϕn}} {u_{m}^{(n)}, w_{m}^{(n)}} , (40)

such that the set of isospectral quantum transfer matrices t^{(n)}(λ) = t(λ, κ; {u_{m}^{(n)}, w_{m}^{(n)}}_{m=1}^{M}) (41) has arisen. Sequence (40) defines the transformation K, K^{(n)} : {u_{m} ≡ u_{m}^{(0)}, w_{m} ≡ w_{m}^{(0)}}_{m=1}^{M} ↦ {u_{m}^{(n)}, w_{m}^{(n)}}_{m=1}^{M} , (42) with the finite dimensional counterpart K^{(n)} = Q_{ϕ_{1}}^{(1)} Q_{ϕ_{2}}^{(2)} ... Q_{ϕ_{n}}^{(n)} , (43) where Q_{ϕ_{n}}^{(n)} = tr_{ϕ_{n}} (R_{1,ϕ_{n}}^{(n)} R_{2,ϕ_{n}}^{(n)} … R_{M,ϕ_{n}}^{(n)}) (44)

12 S. PAKULIAK AND S. SERGEEV

makes t^{(n−1)}(λ) Q_{ϕ_{n}}^{(n)} = Q_{ϕ_{n}}^{(n)} t^{(n)}(λ) , (45) and t^{(0)}(λ) K^{(n)} = K^{(n)} t^{(n)}(λ) . (46)

3. Matrix R

In this section we construct explicitly the finite dimensional matrix R_{m,ϕ}, obeying (26), in the basis (6). But first we have to introduce several notations. 3.1. w-function. Let p be a point on the Fermat curve F

p =^{def} (x, y) ∈ F ⇔ x^{N} + y^{N} = 1 . (47)

Very useful function on the Fermat curve is w_{p}(n), p ∈ F, n ∈ ℤ_{N}, defined as follows:

\frac{w_{p}(n)}{w_{p}(n − 1)} = \frac{y}{1 − x ω^{n}} , w_{p}(0) = 1 . (48)

Function w_{p}(n) has a lot of remarkable properties, see the appendix of ref. [15] for an introduction into ω-hypergeometry. In this paper it is necessary to mention just a couple of properties of w-function. Let O be the following automorphism of the Fermat curve: p = (x, y) ⇔ Op = (ω^{−1}x^{−1}, ω^{−1/2}x^{−1}y) . (49) Then

w_{Op}(n) = \frac{1}{Φ(n)w_{p}(−n)} , (50)

where Φ(n) = (−)^{n} ω^{n^{2}/2} . (51) In the subsequent sections we will use implicitly two simple automorphisms else:

w_{(x,ωy)}(n) = ω^{n}w_{(x,y)}(n) , w_{(x,y)}(n + 1) = \frac{y}{1 − ωx}w_{(ωx,y)}(n) . (52)

Define also three special points on the Fermat curve: q_{0} = (0, 1) , q_{∞} = Oq_{0} , q_{1} = (ω^{−1}, 0) . (53)

Then

w_{q_{0}}(n) = 1 , w_{q_{∞}}(n) = \frac{1}{Φ(n)} , \frac{1}{w_{q_{1}}(n)} = δ_{n,0} . (54)

3.2. Matrix elements of R_{m,ϕ}. Consider the N^{2} × N^{2} matrix R_{m,ϕ}(p_{1}, p_{2}, p_{3}) with the following matrix elements:

\begin{matrix}⟨α_{m}, α_{ϕ}|R_{m,ϕ}|β_{m}, β_{ϕ}⟩ = \\ = ω^{(α_{m}−β_{m})β_{ϕ}} \frac{w_{p_{1}}(α_{ϕ} − α_{m}) w_{p_{2}}(β_{ϕ} − β_{m})}{w_{p_{3}}(β_{ϕ} − α_{m})} δ_{α_{ϕ},β_{m}} . \end{matrix} (55)

Here p_{1}, p_{2}, p_{3} are three points on the Fermat curve, such that

x_{1} x_{2} = x_{3} . (56)

Eq. (56) and the spin structure of (55) provide the dependence of (55) on two continuous parameters, say x_{1} and x_{3}, and on two discrete parameters, say the phase of y_{1} and the phase of y_{3}.

Proposition 2. Matrix R_{m,ϕ}(p_{1}, p_{2}, p_{3}), whose matrix elements (55) are given in the basis (6), makes the following mapping:

R_{m,ϕ}x_{m}R_{m,ϕ}^{−1} = x_{ϕ} ,

R_{m,ϕ}z_{m}R_{m,ϕ}^{−1} = \frac{y_{3}}{y_{2}} z_{m}z_{ϕ} − ω \frac{x_{3}y_{1}}{x_{1}y_{2}} x_{ϕ}^{−1}z_{ϕ} ,

R_{m,ϕ}x_{ϕ}^{−1}R_{m,ϕ}^{−1} = ωx_{3}x_{m}^{−1} − ω \frac{x_{1}y_{3}}{y_{1}} x_{m}^{−1}z_{m}x_{ϕ} + \frac{y_{3}}{y_{1}} z_{m} ,

R_{m,ϕ}z_{ϕ}^{−1}R_{m,ϕ}^{−1} = \frac{y_{3}}{y_{2}} x_{ϕ}x_{m}^{−1} − ω \frac{x_{3}y_{1}}{x_{1}y_{2}} z_{m}^{−1}x_{m}^{−1} .

(57)

Proof: direct verification with the help of (52).

Compare (57) with (31). Obviously, R_{m,ϕ} solves (26) if

x_{1,m} = ω^{−1/2} \frac{u_{ϕ,m}}{κ u_{m}} , \frac{x_{3,m} y_{1,m}}{x_{1,m} y_{3,m}} = ω^{−1/2} \frac{λ_{ϕ}}{u_{ϕ,m}w_{m}} , (58)

and

\begin{matrix}u′_{m} = \frac{κ_{ϕ}}{κ} u_{ϕ,m} , & w′_{m} = w_{m}w_{ϕ,m} \frac{y_{2,m}}{y_{3,m}} , \\ u_{ϕ,m+1} = ω x_{3,m} u_{m} , & w_{ϕ,m+1} = \frac{κ}{κ_{ϕ}} \frac{u_{m}}{u_{ϕ,m}} \frac{y_{3,m}}{y_{2,m}} . \end{matrix} (59)

Eqs. (58) and (59) allow one to parameterize R_{m,ϕ_{n}}^{(n)} in (44) in terms of u_{m}, w_{m}, .... So for any appropriate set of {u_{m}^{(k)}, w_{m}^{(k)}}, m = 1...M and k = 0...n, see formula (40), we may construct the corresponding K^{(n)} explicitly via the parameterization (58,59), form of the matrix elements (55), and the formulas (44,43).

Further we will consider the sequences like (40) for the homogeneous initial state {u_{m} = u_{0}, w_{m} = w_{0}}_{m=1}^{M}, and parameterize {u_{m}^{(n)}, w_{m}^{(n)}} in the terms of solitonic Hirotatype expressions. To do it, several notations are to be introduced.

4. Rational Θ-function

In this section we introduce several useful functions and notations. The main purpose is to introduce the form of solitonic τ-function for the classical relativistic Toda chain. The reader may find the details concerning the corresponding classical integrable model, reduction from 2DToda lattice hierarchy and so on in Ref. [17].

Fix any sequence of ϕ_{k} from the set 𝔉_{M}:

ϕ_{k} ∈ 𝔉_{M} =^{def} \{\frac{π}{M} , \frac{2π}{M} , ... , \frac{(M − 1)π}{M} \} , (60)

Introduce the functions

Δ_{ϕ} = e^{i ϕ} (\sqrt{cos^{2} ϕ + κ^{N}} + cos ϕ) ,
Δ_{ϕ}^{∗} = e^{−i ϕ} (\sqrt{cos^{2} ϕ + κ^{N}} + cos ϕ) ,
Λ_{ϕ} = Δ_{ϕ} Δ_{ϕ}^{∗} .

(61)

Expressions for Δ and Δ^{∗} parameterize the curve

(62) ΔΔ^{∗} = Δ + Δ^{∗} + κ^{N} in the terms of e^{2iϕ} = \frac{Δ}{Δ^{∗}} .

Rational Θ-functions are to be defined recursively as

\begin{matrix}Θ_{m}^{(0)} = 1 , \\ Θ_{m}^{(1)} = 1 − f_{1} e^{2imϕ_{1}} , \\ Θ_{m}^{(2)} = 1 − f_{1}e^{2imϕ_{1}} − f_{2}e^{2imϕ_{2}} + d_{1,2}f_{1}f_{2}e^{2im(ϕ_{1}+ϕ_{2})} , \end{matrix} (63)

and so on, Θ_{m}^{(n)} = Θ_{m}^{(n−1)}({f_{k}, ϕ_{k}}_{k=1}^{n−1}) − f_{n}e^{2imϕ_{n}}Θ_{m}^{(n−1)}({d_{k,n}f_{k}, ϕ_{k}}_{k=1}^{n−1}) , (64) where all ϕ_{k} are different and belong to the set (60). The phase shift d_{j,k} = d_{ϕ_{j},ϕ_{k}} is given by (see (61))

(65) d_{ϕ,ϕ′} = (Δ_{ϕ} − Δ_{ϕ′}) (Δ_{ϕ}^{∗} − Δ_{ϕ′}^{∗}) ; (Δ_{ϕ} − Δ_{ϕ′}^{∗}) (Δ_{ϕ}^{∗} − Δ_{ϕ′}) .

Arguments of Θ_{m}^{(n)} are the set of ϕ_{k} ∈ 𝔉_{M} and corresponding amplitudes f_{k}, k = 1...n. Maximal Θ-function is

Θ_{m} =^{def} Θ_{m}^{(M−1)}({f_{k}}_{k=1}^{M−1}) . (66)

𝔉_{M}-set, (60), provides the M-periodicity of Θ-functions, Θ_{m+M} ≡ Θ_{m}.

Introduce next a couple of useful functions

s_{ϕ}(ξ) = \frac{Δ_{ϕ}^{∗}}{Δ_{ϕ}} \frac{Δ_{ϕ} − ξ}{Δ_{ϕ}^{∗} − ξ} , (67)

and

s_{ϕ,ϕ′} = s_{ϕ}(Δ_{ϕ′}^{∗}) = \frac{Δ_{ϕ}^{∗}}{Δ_{ϕ}} \frac{Δ_{ϕ′}^{∗} − Δ_{ϕ}}{Δ_{ϕ′}^{∗} − Δ_{ϕ}^{∗}} . (68)

Necessary is to consider the re-parameterization of the amplitudes f_{k} in the special form: for any n, k, 0 ≤ n < M, 1 ≤ k ≤ n let

f_{k}^{(n)}(\{g\}) = g_{k} ∏_{{j=1 \atop j≠k}}^{n} s_{k,j} , (69)

where s_{k,j} is given by (68). Denote the corresponding Θ-functions as follows: Θ_{m}^{(n)}({g}) = Θ_{m}^{(n)}({f_{k}^{(n)}({g}), ϕ_{k}}_{k=1}^{n}) (70)

16 S. PAKULIAK AND S. SERGEEV

In advance, we will always imply the free amplitudes g_{k}, k = 1, ..., as the arguments of Θ-function, (69,70).

Particular case when all g_{k} = s_{k}(ξ) is also important: let Θ_{m}^{(n)}(ξ) = Θ_{m}^{(n)}({g_{k} = s_{k}(ξ)}_{k=1}^{n}) (71) This particular case has the following property: Proposition 3. For any n

(72) Θ_{n}^{(n)}(ξ) = ∏_{k=1}^{n} \frac{Δ_{k}^{∗} − Δ_{k}}{Δ_{k}^{∗} − ξ} ∏_{1≤j<k≤n} \frac{Λ_{j} − Λ_{k}}{Δ_{j}^{∗} − Δ_{k}^{∗}} ,

and for m = 0...n

Θ_{m}^{(n)}(ξ) = (− κ^{N})^{(n−m)(n−m−1)/2} (\frac{ξ}{Δ_{1}...Δ_{n}} )^{n−m} Θ_{n}^{(n)}(ξ) (73)

If ξ = 1, eq. (73) is valid also for m = −1.

5. Parameterization of K

5.1. Sequence of the Bäcklund transformations. Turn now to the original quantum chain. Let the initial parameters u_{m}, w_{m} are homogeneous: u_{m} = −ω^{−1/2} , w_{m} = −1 . (74) It corresponds to (18).

For the sequence of ϕ_{k} ∈ 𝔉_{M} fixed, and for generic set of corresponding g_{k}, k = 1...M − 1, let (see eqs. (69) and (70) and the note right after (70)) (τ_{m}^{(n)})^{N} = Θ_{m}^{(n)}({g_{k}}_{k=1}^{n}) , (θ_{m}^{(n)})^{N} = Θ_{m}^{(n)}({g_{k}s_{k}(1)}_{k=1}^{n}) . (75) The phases of τ_{m}^{(n)} and θ_{m}^{(n)} are arbitrary. Proposition 4. Consider the set of (M −1) transformations (37), such that the initial state is the homogeneous one (74), and the mapping Q_{ϕ_{n}} is derived for λ_{ϕ_{n}}^{N} = Λ_{ϕ_{n}} , (76) where sequence of ϕ_{n} ∈ 𝔉_{M} is fixed, and function Λ_{ϕ} is given by the last formula of (61). Let besides

κ_{ϕ_{n}}^{N} = \frac{κ^{N}}{Δ_{ϕ_{n}}} . (77)

Then the sequence (40) of {u_{m}^{(n)}, w_{m}^{(n)}} may be parameterized as follows:

u_{m}^{(n)} = −ω^{−1/2} \frac{τ_{m−1}^{(n)}}{τ_{m}^{(n)}} , w_{m}^{(n)} = − \frac{θ_{m}^{(n)}}{θ_{m−1}^{(n)}} , ; m ∈ ℤ_{M} , (78)

where τ_{m}^{(n)}, θ_{m}^{(n)} are given by (75) with arbitrary g_{1}...g_{M−1}.

See [1] for the sketch of the proof of this proposition. Eq. (77) needs a comment. Arbitrary value of κ_{ϕ_{n}} corresponds to a gauge transformation of {u_{m}^{(n)}, w_{m}^{(n)}}_{m=1}^{M} ↦ {cu_{m}^{(n)}, c^{−1}w_{m}^{(n)}}_{m=1}^{M} (79) with some c proportional to κ_{ϕ_{n}}, while the structure of τ and θ is not changed. So we impose (77) and obtain (78) without lost of generality. 5.2. Parameterization of modified Q-operators. Let now for the shortness λ_{ϕ_{n}} = λ_{n} are given by (76), and let us fix further the roots for δ_{n} and c_{n}, 1 ≤ n < M :

δ_{n}^{N} = Δ_{ϕ_{n}} , κ_{ϕ_{n}} = \frac{κ}{δ_{n}} , c_{n}^{N} = \frac{Δ_{ϕ_{n}}^{∗} − 1}{Δ_{ϕ_{n}}^{∗}} . (80)

Auxiliary values in (58,59) are given by

u_{ϕ_{n},m} = − ω^{−1/2} δ_{n} \frac{τ_{m−1}^{(n)}}{τ_{m}^{(n)}} , w_{ϕ_{n},m+1} = ω^{−1/2} \frac{δ_{n}}{λ_{n}c_{n}} \frac{θ_{m}^{(n−1)}τ_{m}^{(n)}}{τ_{m}^{(n−1)}θ_{m}^{(n)}} . (81)

R_{m,ϕ_{n}}^{(n)}-matrices, entering to (44),have the arguments R_{m,ϕ_{n}}^{(n)} : p_{1,m}^{(n)}, p_{2,m}^{(n)}, p_{3,m}^{(n)}, (82) given by

\begin{matrix}x_{1,m}^{(n)} = ω^{−1/2} \frac{δ_{n}}{κ} \frac{τ_{m}^{(n−1)}τ_{m−1}^{(n)}}{τ_{m−1}^{(n−1)}τ_{m}^{(n)}} , & x_{3,m}^{(n)} = ω^{−1}δ_{n} \frac{τ_{m}^{(n−1)}τ_{m}^{(n)}}{τ_{m−1}^{(n−1)}τ_{m+1}^{(n)}} , \\ \frac{y_{3,m}^{(n)}}{y_{2,m}^{(n)}} = ω^{−1/2} \frac{δ_{n}}{λ_{n}c_{n}} \frac{θ_{m}^{(n−1)}τ_{m−1}^{(n)}}{τ_{m−1}^{(n−1)}θ_{m}^{(n)}} , & \frac{y_{1,m}^{(n)}}{y_{3,m}^{(n)}} = ω^{1/2} \frac{λ_{n}}{κδ_{n}} \frac{θ_{m−1}^{(n−1)}τ_{m+1}^{(n)}}{θ_{m}^{(n−1)}τ_{m}^{(n)}} , \end{matrix} (83)

and with (56) implied,

x_{2,m}^{(n)} = ω^{−1/2}κ \frac{(τ_{m}^{(n)})^{2}}{τ_{m−1}^{(n)}τ_{m+1}^{(n)}} . (84)

The trace of quantum monodromy (44) may be calculated explicitly in the basis (6), and the answer is

⟨α|Q_{ϕ_{n}}^{(n)}|β⟩ = ∏_{m∈ℤ_{M}} ω^{(α_{m}−β_{m})β_{m+1}} \frac{w_{p_{1,m}^{(n)}}(β_{m} − α_{m}) w_{p_{2,m}^{(n)}}(β_{m+1} − β_{m})}{w_{p_{3,m}^{(n)}}(β_{m+1} − α_{m})} . (85)

Operator K, (43), calculated as the product of all (M − 1) Q^{(n)}-operators, is thus defined explicitly.

Proposition 5. Operator K({g}), given by (43), does not depend on the ordering of (g_{k}, ϕ_{k}) (up to a multiplier, arising in general when one takes arbitrary phases for τ_{m}^{(n)} and θ_{m}^{(n)}). Thus without lost of generality one may regard ϕ_{n} = \frac{πn}{M}.

Thus for the generic set of amplitudes g_{k} constructed is the similarity operator K({g}), transforming the homogeneous initial transfer matrix into most generic isospectral one.

5.3. Arbitrary value of λ_{ϕ}. Previously we considered the set of ϕ_{k} ∈ 𝔉_{M}. Actually, all the calculations in the classical relativistic Toda chain are based on (62). Therefore, in general, we may consider a generic sequence of complex numbers ϕ_{k}, k ∈ ℤ_{+}. For any generic complex ϕ one has to restore λ_{ϕ} and δ_{ϕ} via (61), and take into account (80). It corresponds to the Q operator for the generic value of λ_{ϕ}. Then all the formulas for the parameterization of transfer matrix in the terms of τ-functions (78,75,70) as well as the parameterization of Qoperator (83,85) are valid if the amplitudes g_{ϕ}, corresponding to those ϕ ∉ 𝔉_{M} and −ϕ ∉ 𝔉_{M} are zeros, and in the case when ϕ ∈ 𝔉_{M} or −ϕ ∈ 𝔉_{M} the amplitudes are arbitrary. Important is that in the calculation of the amplitudes (69) all ϕs are to be taken into account. Also, if in the generic sequence of ϕ_{k} it happens ϕ_{k} = ϕ_{m} or ϕ_{k} = −ϕ_{m}, then Θ-function is to be understood as the corresponding residue of formal singular expression (70).

For example, usual Q-operators for the homogeneous initial state corresponds to ϕ_{1} = ϕ ∉ 𝔉_{M}, g_{1} = 0. The cases when ϕ_{k} = ±ϕ_{m} correspond to the annihilation of the solitons.

Nevertheless, considering operator K, (43), we will always imply ϕ_{k} ∈ 𝔉_{M}.

5.4. Baxter equation. Consider initial homogeneous transfer matrix t(λ) and corresponding Q operators with arbitrary value of λ_{ϕ}. As it was mentioned in the previous subsection, corresponding functional counterpart of Q-operator is trivial, i.e. such Q-operators commute with the transfer matrices and form the commutative family. Such Q-operators are usual Baxter’s Q-operators, obeying Baxter’s equation.

Explicit form of simple Q-operator is given by (85) with the homogeneous parameterization

p_{1,m} = p_{1} , p_{2,m} = p_{2} , p_{3,m} = p_{3} (86)

with

\begin{matrix}x_{1} = ω^{−1/2} \frac{δ_{ϕ}}{κ} , x_{2} = ω^{−1/2}κ , x_{3} = ω^{−1}δ_{ϕ} , \\ \frac{y_{3}}{y_{2}} = \frac{ω^{−1/2}δ_{ϕ}}{λ_{ϕ}c_{ϕ}} , \frac{y_{1}}{y_{3}} = \frac{ω^{1/2}λ_{ϕ}}{κδ_{ϕ}} , \end{matrix} (87)

where δ_{ϕ}, λ_{ϕ} and c_{ϕ} are given by (80,76) and (61), but with generic value of ϕ. Using (87), we may regard

Q = Q(λ_{ϕ}, δ_{ϕ}) . (88)

Several simple properties of simple Q-operator may be derived with the help of the matrix elements of (85) and (87), and with the help of (52):

XQ(λ_{ϕ}, δ_{ϕ})X^{−1} = Q(ω^{−1}λ_{ϕ}, δ_{ϕ}) ,

ZQ(λ_{ϕ}, δ_{ϕ})X = (ω^{1/2} \frac{λ_{ϕ}}{δ_{ϕ}} \frac{1 − δ_{ϕ}}{κ − ω^{1/2}δ_{ϕ}} )^{M} Q(ωλ_{ϕ}, ωδ_{ϕ}) ,

YQ(λ_{ϕ}, δ_{ϕ}) = Q(λ_{ϕ}, δ_{ϕ})Y = (−\frac{λ_{ϕ}}{δ_{ϕ}} \frac{1 − δ_{ϕ}}{κ − ω^{1/2}δ_{ϕ}} )^{M} Q(λ_{ϕ}, ωδ_{ϕ}) ,

(89)

where

X = ∏_{m=1}^{M} x_{m} , Z = ∏_{m=1}^{M} z_{m} , (90)

and Y is given by (14).

Let us derive now Baxter’s equation for general inhomogeneous chain. To do it, one has to consider the degeneration point of ℓ̃, (23):

ℓ̃_{ϕ}(λ_{ϕ}, λ_{ϕ}) = \binom{1 − ω^{1/2}κ_{ϕ}w_{ϕ}}{−ω^{1/2}λ_{ϕ}u_{ϕ}^{−1}w_{ϕ}} · ( 1 , −ω^{1/2}λ_{ϕ}^{−1}u_{ϕ}) . (91)

Therefore eq. (26) in the degeneration point may be rewritten in the following forms:

\begin{matrix}( 1 , −ω^{1/2} \frac{u_{ϕ,m}}{λ_{ϕ}} x_{ϕ} ) · ℓ_{m}(λ_{ϕ}; u_{m}, w_{m}) · R_{m,ϕ} = \\ = R_{m,ϕ}^{(1)} · ( 1 , −ω^{1/2} \frac{u_{ϕ,m+1}}{λ_{ϕ}} x_{ϕ} ) , \end{matrix} (92)

and

\begin{matrix}ℓ_{m}(λ_{ϕ}; u′_{m}, w′_{m}) · R_{m,ϕ} · ( \begin{matrix}ω^{1/2} \frac{u_{ϕ,m+1}}{λ_{ϕ}} x_{ϕ} \\ 1 \end{matrix} ) = \\ = ( \begin{matrix}ω^{1/2} \frac{u_{ϕ,m}}{λ_{ϕ}} x_{ϕ} \\ 1 \end{matrix} ) · R_{m,ϕ}^{(2)} , \end{matrix} (93)

where

R_{m,ϕ}^{(1)} = \frac{u_{m}}{u_{ϕ,m+1}} x_{m} R_{m,ϕ} x_{ϕ}^{−1} ,

R_{m,ϕ}^{(2)} = ω^{1/2} \frac{u_{ϕ,m+1} w_{m}}{λ_{ϕ}} z_{m} R_{m,ϕ} x_{ϕ} .

(94)

Let further Q_{ϕ}^{(1)} and Q_{ϕ}^{(2)} are the traces of the monodromies of R_{m,ϕ}^{(1)} and R_{m,ϕ}^{(2)} (ℤ_{M} boundary condition for u_{ϕ,m} is taken into account again – the subscript ϕ reminds this). Then from (92) and (93) it follows the Baxter equation in its operator form:

t(λ_{ϕ}) Q_{ϕ} = Q_{ϕ} t′(λ_{ϕ}) = Q_{ϕ}^{(1)} + Q_{ϕ}^{(2)} . (95)

Using the first relation of (57), one may obtain

(96) Q_{ϕ}^{(1)} = ∏_{m} \frac{u_{m}}{u_{ϕ,m}} XQX^{−1} , Q^{(2)} = ∏_{m} ω^{1/2} \frac{u_{ϕ,m}w_{m}}{λ_{ϕ}} ZQX ,

where X and Z are given by (90).

Turn now to the homogeneous chain. Using (89), we obtain

t(λ_{ϕ})Q(λ_{ϕ}) = δ_{ϕ}^{−M}Q(ω^{−1}λ_{ϕ}) + (−ω^{−1/2} \frac{δ_{ϕ}}{λ_{ϕ}} )^{M} YQ(ωλ_{ϕ}) , (97)

where δ_{ϕ}-argument of all Q-s remains unchanged, and t and Q may be diagonalized simultaneously. Let t(λ) and q_{t}(λ) be the eigenvalues of t and Q for the same eigenvector, then (97) provides the functional equation

t(λ)q_{t}(λ) = δ^{−M}q_{t}(ω^{−1}λ) + (−ω^{−1/2} \frac{δ}{λ})^{M} ω^{γ}q_{t}(ωλ) , (98)

where ω^{γ} is the eigenvalue of Y, and q_{t} is the meromorphic function on the curve (21). Detailed investigations of (98) show that for generic λ any solution of (98) such that t(λ) is a polynomial and q_{t}(λ, δ) is a meromorphic function on the curve (21), gives the eigenvalue of t(λ) and Q(λ).

6. Separating operator

6.1. Spectrum of b. For the homogeneous chain (74) the value of B(λ^{N}), (15), may be calculated immediately:

B(λ^{N}) = (−)^{N−1} \frac{1}{λ^{N}} ∏_{k=1}^{M−1} (1 − \frac{Λ_{ϕ_{k}}}{λ^{N}} ) , (99)

where Λ_{ϕ} is given by the last equation of (61) and ϕ_{k} is exactly the set 𝔉_{M}, (60). This defines the parameterization of the vectors |{λ_{k}}_{k=1}^{M−1}, γ⟩, see (19).

In [1] it was established the following fact:

Proposition 6. Let λ_{k} = λ_{ϕ_{k}} are defined by (76) and the phases of λ_{k} are fixed. Operator K_{{λ}}, corresponding to the case when all g_{k} = 1, k = 1...M − 1 in eqs. (75,70), makes the quantum separation of variables:

b(λ_{k}) K_{{λ}} = 0 , k = 1...M − 1 . (100)

This means

K_{{λ}} = ∑_{γ} |{λ_{k}}_{k=1}^{M−1}, γ⟩ ⟨χ_{γ}| , (101)

22 S. PAKULIAK AND S. SERGEEV

where χ_{γ} are some vectors, such that ⟨χ_{γ}| Y = ω^{γ} ⟨χ_{γ}|, where Y is given by (14). Parameterization (τ_{m}^{(n)})^{N} = Θ_{m}^{(n)}(0) , (θ_{m}^{(n)})^{N} = Θ_{m}^{(n)}(1) (102) is not appropriate for (83), because of τ_{m}^{(n)} = 0 for m = 0...n − 1, and u_{m}^{(n)} are ambiguous. Thus one needs to define in addition a limiting procedure for (102). In general, the ambiguity in (78) corresponds to the arbitrariness of the right eigenstates χ_{γ} of K_{{λ}}. 6.2. Limiting procedure. In this section we describe an appropriate limiting procedure. Most simple K_{{λ}} appears when we choose u_{m}^{(M−1)} = 0 for m = 1...M − 1. Namely, consider the set of infinitely small numbers, ε_{k} ↦ 0 , k = 1...M − 1 , (103) such that any ratio ε_{n}/ε_{m} ≠ 1 is finite, and

u_{m}^{(M−1)} = −ω^{−1/2} \frac{ε_{M−m}}{δ_{1}....δ_{M−1}} , m = 1...M − 1 , (104)

so that

u_{M}^{(M−1)} = −ω^{−1/2} \frac{(δ_{1}...δ_{M−1})^{M−1}}{ε_{1}...ε_{M−1}} . (105)

Recursion (32,33) with

κ_{ϕ_{n}} = \frac{κ}{δ_{n}} (106)

implies

u_{m}^{(n)} = −ω^{−1/2} \frac{ε_{n−m+1}}{δ_{1}...δ_{n}} + o(ε_{1}) , m = 1...n (107)

while for m = n + 1...M − 1 all u_{m}^{(n)} are regular. This means

τ_{m}^{(n)} = τ_{n}^{(n)} \frac{ε_{1}...ε_{n−m}}{(δ_{1}...δ_{n})^{n−m}} + o(ε_{1}^{n−m}) , m = 0...n . (108)

w_{m}^{(n)} as explicit functions of ε_{k} are rather complicated, and speaking truly, we may say nothing about t^{(M−1)} in the meantime. But in the limit ε_{k} = 0 the following formulas may be chosen:

\frac{τ_{n}^{(n)}}{θ_{n}^{(n)}} = ∏_{k=1}^{n} c_{k} , (109)

where c_{k} are given by (80) and

θ_{m}^{(n)} = (ω^{1/2}κ)^{(n−m)(n−m−1)/2} \frac{θ_{n}^{(n)}}{(δ_{1}...δ_{n})^{n−m}} . (110)

On N-th powers these formulas follow from proposition (3). Eq. (110) implies in part

\frac{w_{m+1}^{(n)}}{w_{m}^{(n)}} = ω^{1/2}κ , m = 0...n − 1 . (111)

Now in the limit ε_{k} ↦ 0 the following parameterization of the ambiguous and singular points arises (notation for the singular points are described by (53,54)):

• \underline{m = 1...n − 1}: p_{3,m}^{(n)} = q_{1} , p_{2,m}^{(n)} = O(p_{1,m}^{(n)}) . (112)

where x_{1,m}^{(n)} = \frac{1}{ω^{1/2}κ} \frac{ε_{n−m+1}}{ε_{n−m}}.

• \underline{m = n}: p_{1,n}^{(n)} = q_{0} , p_{2,n}^{(n)} = q_{∞} ∀ n (113)

and p_{3,n}^{(n)} are regular, except

p_{3,M−1}^{(M−1)} = q_{∞} . (114)

• \underline{m = n + 1...M − 2}: All p_{j,m}^{(n)} in this region are regular.

• \underline{m = M − 1}: p_{1,M−1}^{(n)} are regular except p_{1,M−1}^{(M−1)} = q_{0} , (115)

and p_{2,M−1}^{(n)} = p_{3,M−1}^{(n)} = q_{∞} ∀ n . (116)

• \underline{m = M}: p_{1,M}^{(n)} = q_{∞} , p_{2,M}^{(n)} = p_{3,M}^{(n)} = q_{0} ∀ n. (117)

Substituting these expressions into (85), one obtains the following form of n-th modified Q-matrix, n ≠ M − 1:

\begin{matrix}⟨α|Q^{(n)}|β⟩ = \frac{1}{Φ(β_{1} + β_{M} − α_{M})} ∏_{m=1}^{n−1} δ_{α_{m},β_{m+1}}× \\ Φ(α_{n}) w_{Op_{3,n}^{(n)}}(α_{n} − β_{n+1})× \\ ∏_{m=n+1}^{M−2} ω^{(α_{m}−β_{m})β_{m+1}} \frac{w_{p_{1,m}^{(n)}}(β_{m} − α_{m}) w_{p_{2,m}^{(n)}}(β_{m+1} − β_{m})}{w_{p_{3,m}^{(n)}}(β_{m+1} − α_{m})}× \\ \frac{Φ(α_{M−1})}{Φ(β_{M−1})} w_{p_{1,M−1}^{(n)}}(β_{M−1} − α_{M−1}) , \end{matrix} (118)

and for the last (M − 1)th one

⟨α|Q^{(M−1)}|β⟩ = \frac{1}{Φ(β_{1} + β_{M} − α_{M})} ∏_{m=1}^{M−2} δ_{α_{m},β_{m+1}} Φ(α_{M−1}) . (119)

Here Φ is given by (51). Explicit form of the modified Q-operators (118) and (119) allows one to prove the following Proposition 7. K_{{λ}} (z_{m} − z_{m+1}) = 0 , m ∈ ℤ_{M} . (120)

So, in our limiting procedure χ is simple: ⟨χ_{γ}|α⟩ = χ_{γ}(\overline{α}) ≡ χ_{γ}(\overline{α} mod N), (121) where

\overline{α} =^{def} ∑_{m∈ ℤ_{M}} α_{m} (122)

and

\frac{χ_{γ}(\overline{α} + M)}{χ_{γ}(\overline{α})} = (−ω^{−1/2})^{M} ω^{γ−\overline{α}} . (123)

7. Discussion

In this paper we investigated the simplest integrable model, associated with the local Weyl algebra at the root of unity. All such models always contain the classical discrete dynamic of parameters. Nontrivial solution of the classical counterpart provides the solution of the isospectrality problem of the finite dimensional counterpart. Well known result by Sklyanin, Kuznetsov at al was that in the classical limit of the usual Toda chain (and many other models) Baxter’s quantum Q-operator corresponds to the Bäcklund transformation of the classical system, see e.g. [8, 10, 11, 12, 16] etc. In our case we have the Bäcklund transformation of the classical counterpart and modified Q-operator in the quantum space simultaneously. Unusual is that solving the quantum isospectrality problem, we miss the commutativity of the modified Q-operators.

Nevertheless, our main result, we think, is the explicit construction of (M −1)-parametric family of quantum inhomogeneous transfer matrices with the same spectrum as the initial homogeneous one, and the explicit construction of the corresponding similarity operator (43). We hope, the solution of the isospectrality problem will help to solve the model with arbitrary N explicitly.

As one particular application of it we have obtained the quantum separation of variables. Previously there was a hypothesis, formulated for the usual quantum Toda chain, that the product of operators Q, taken in the special points, is related to the quantum Functional Bethe Ansatz. In this paper we have established it explicitly, but for the product of modified operators Q. Below some explanations are given.

Let the solitonic amplitudes g_{k} are in general position. Let |Ψ_{t}^{(n)}⟩ be the complete set of eigenvectors of t^{(n)}(λ), and ⟨Ψ_{t}^{(n)}| be the corresponding set of co-vectors,

t^{(n)}(λ) = ∑_{t} |Ψ_{t}^{(n)}⟩ t(λ) ⟨Ψ_{t}^{(n)}| . (124)

Then

Q_{ϕ_{n}}^{(n)} = ∑_{t} |Ψ_{t}^{(n−1)}⟩ q_{t}(λ_{n}) ⟨Ψ_{t}^{(n)}| , (125)

where q_{t}(λ) is evidently the eigenvalues of simple Q-operators, as in (98). Note, the functional Baxter equation exists also and for arbitrary inhomogeneous chain, but to obtain it from (95) one has to use the decomposition like (125).

26 S. PAKULIAK AND S. SERGEEV

With (125) taken into account, K-operator for general set of g_{k} is

K = ∑_{t} |Ψ_{t}^{(0)}⟩ q_{t}(λ_{1})...q_{t}(λ_{M−1}) ⟨Ψ_{t}^{(M−1)}| , (126)

In the limit when all g_{k} = 1, the sets ⟨Ψ_{f}^{(n)}| become degenerative, e.g.

⟨Ψ_{t}^{(n)}| (z_{1} − z_{M}) = 0 ∀ t . (127)

For us it is still mysterious what happens with the set |Ψ_{t}^{(n)}⟩ and co-set ⟨Ψ_{t}^{(n)}| when g_{k} = 1 – this is the subject of separate investigation. But nevertheless, in [1] and in this paper it is proven that K is given by (126) and simultaneously by (101). Thus Sklyanin’s formula appears explicitly:

⟨Ψ_{t}^{(0)}|{λ_{k}}_{k=1}^{M−1}, γ⟩ = const ∏_{k=1}^{M−1} q_{t}(λ_{k}) . (128)

Well known is that to solve Baxter’s equation for N > 2 is quite hopeless (pure fermionic case N = 2 is trivial, (98) may be solved in one string). Only way is to avoid Baxter’s equation for high genus curve. So, we think, the attention is to be paid to the special cases of the isospectral transfer matrices like g_{k} = 1, when several simplifications for the eigenstates are expected.

Note in conclusion, this method may be applied to any model, based on the local Weyl algebra. Mentioned are to be the chiral Potts model [6, 7] and the Zamolodchikov-Bazhanov-Baxter model in the vertex formulation [15]. To be exact, all two dimensional integrable models with the local Weyl algebra are particular cases of the general three dimensional model, and their classical counterparts are known [5].

Acknowledgements The authors are grateful to R. Baxter, V. Bazhanov, V. Mangazeev, G. Pronko, E. Sklyanin, A. Belavin, Yu. Stroganov, A. Isaev, G. von Gehlen, V. Tarasov, P. Kulish, F. Smirnov and R. Kashaev for useful discussions and comments.

S.P. would also like to thank Max-Planck Instiut für Mathematik (Bonn) for support and hospitality and S.S. would thank the hospitality of MPIM during his short visit to Bonn supported by Heisenberg-Landau program.

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Stanislav Pakuliak, Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia and Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

   E-mail address: pakuliak@thsun1.jinr.ru

Sergei Sergeev, Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia

   E-mail address: sergeev@thsun1.jinr.ru