QUANTUM RELATIVISTIC TODA CHAIN AT ROOT OF UNITY:

ISOSPECTRALITY, MODIFIED Q-OPERATOR AND FUNCTIONAL BETHE ANSATZ

S. PAKULIAK AND S. SERGEEV

Abstract. We investigate an N-state spin model called quantum relativistic Toda chain and based on the unitary finite dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter’s Q-operators. The classical counterpart of the modified Q-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q-operators.

Introduction

One of the indicative examples of integrable models of mathematical physics is the Toda chain. There are an enormous number of papers in the modern literature concerning the Toda chain, classical as well as quantum one. Let us mention only ref. [1] as an example of the application of the algebraic methods to the quantum Toda chain. One of the modifications of the Toda chain is known as a relativistic Toda chain, classical as well as quantum, see. e.g. [2, 3, 4, 5, 6].

1991 Mathematics Subject Classification. 82B23. Key words and phrases. Integrable models, Toda chain, Spin chains, Bilinear equations, Baxter equation,

Functional Bethe ansatz.

This work was supported in part by the grants INTAS OPEN 00-00055 and CGP CRDF RM1-2334-MO-02. 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

The algebra of observables in the quantum relativistic Toda chain is the local Weyl algebra. Due to an ambiguity of centers of the Weyl algebra with arbitrary Weyl’s q, one may talk about well defined quantum model only in three cases. The formal first one is the special limit q ↦ 1, corresponding to the usual quantum Toda chain [1]. The second case is the modular dualization of Weyl’s algebra, see [7, 8, 9] for details and [10] for the application of the modular concept to the quantum relativistic Toda chain. In this paper we are going to discuss the third known case – the case of the unitary finite dimensional representation of the Weyl algebra, arising when Weyl’s parameter q is the Nth primitive root of unity. Note, besides the finite dimensional representations at the root of unity, there exist the real infinite dimensional ones, but this case belongs to the modular double class.

In several integrable models with the unitary representations of the Weyl algebra at the Nth root of unity, the ℂ-valued Nth powers of Weyl’s elements form a classical discrete integrable system. So the parameters of the unitary representation of the Weyl algebra form a classical counterpart of the finite dimensional integrable system (i.e. spin integrable system), see [11, 12, 13, 14, 15] for examples. The most important finite dimensional operators, arising in the spin systems, have functional counterparts, defined as rational mappings in the space of functions of the Nth powers of Weyl’s elements. 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 have to be constructed in the terms of initial and final values of the parameters. Usual finite dimensional integrable models correspond to the case when the initial and final parameters coincide for all the operators involved. It means the consideration of the trivial classical dynamics. Algebraically, the conditions of the trivialization are the origin of e.g. Baxter’s curve for the chiral Potts model [16, 17], or of the spherical triangle parameterization for the Zamolodchikov-Bazhanov-Baxter model [18].

One may hope to achieve the progress in the spin integrable models by including into the consideration a nontrivial classical dynamics. This dynamics will complicate the situation, but sometimes more complicated systems in the mathematics and physics may be solved more easily – proverbial “a problem should be complicated until it becomes trivial” [19].

The quantum relativistic Toda chain at the root of unity is one of the simplest examples of such combined classical/spin system. This model was investigated in the preprints [20, 21, 22]. The present paper is an attempt to present the detailed and clarified exposition of this investigation.

We will use the language of the quantum inverse scattering method. Besides the formulation of the model in the terms of L-operators and transfer matrices, it implies the quantum intertwining relation, allowing one to construct Baxter’s Q-operator. Eigenvalues of the transfer matrix and Q-operator satisfy the Baxter, or T Q = Q′ + Q′′, equation. Baxter equation may hardly be solved explicitly for arbitrary N and for finite length of the chain M. It is the key relation for the investigation of the model in the thermodynamic limit. We will not investigate the Baxter equation in this paper even in the thermodynamic limit, we will restrict ourselves by a derivation of it.

Since any quantum Toda-type chain is the realm of the functional Bethe ansatz, the problem of eigenstates of the transfer matrix is related to the problem of the eigenstates of off-diagonal element of the monodromy matrix, see e.g. [1, 23, 24] for the details. This is true in our case as well. It is also known that Baxter’s Q-operators in the classical limit are related to the Bäcklund transformations of the classical chains, see e.g. [25, 26, 27].

In the following paragraph we describe in the shortest way the subject of the present paper.

The main feature of the quantum relativistic Toda chain at the root of unity with the nontrivial classical counterpart is that the quantum intertwining relation contains the Darboux transformation (see [25]) for the Nth powers of the Weyl elements. So the Bäcklund transformation is the classical counterpart of the finite dimensional Q-operator at the root of unity automatically. Such Q-operators we call the modified Q-operators, because they do not form the commutative family. Usual Q-operators are their particular cases. Modified Q-operators solve in general the isospectrality problem for the spin chain. Starting from a homogeneous quantum chain, one may create some number of solitons in parameterization of the inhomogeneities using the Bäcklund transformations. The finite dimensional counterpart of this procedure is a product of the modified Q-operators, giving a similarity transformation of the initial homogeneous transfer matrix. The term “soliton” is used here in Hirota’s sense: the system of the Nth powers of Weyl’s elements may be parameterized by τ-functions, obeying a system of discrete equations. A solitonic solution of this system is

4 S. PAKULIAK AND S. SERGEEV

given by the Hirota-type expressions. Solitonic τ-functions contain extra free complex parameters – the amplitudes of solitonic partial waves. Generally speaking, these parameters are a kind of “times” of the classical model conjugated to the commutative set of classical hamiltonians. In the other interpretation the amplitudes stand for a point on a jacobian of a classical spectral curve in its rational limit. Besides the complete solution of the quantum isospectrality problem, these degrees of freedom appeared to be useful in an unusual sense. The amplitudes may be chosen so that one component of a classical Backer-Akhiezer function becomes zero – this is related to the classical separation of variables. The corresponding finite dimensional similarity operator becomes a projector to an eigenvector of off-diagonal element of the monodromy matrix, i.e. the base of the quantum separation of variables. Moreover, since the similarity operator is the product of Q-operators, Sklyanin’s formula with the product of eigenfunctions Q appears in the direct way.

This paper is devoted to: formulation of the combined classical/spin model – the quantum relativistic Toda chain at the Nth root of unity; its classical Darboux transformation/spin intertwining relation; derivation and parameterization of Bäcklund transformation/modified Q-operator; derivation of the Baxter equation; explicit parameterization of the quantum separating operator.

1. Formulation of the model

In this section we formulate the model called the quantum relativistic Toda chain at the root of unity.

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

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

where λ is the spectral parameter and κ is an extra complex parameter, common for all sites, i.e. a modulus of eigenstates. 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 is an arbitrary positive integer greater then one and common for all sites. The Nth powers of the Weyl elements are centers of the algebra. We will deal with the finite dimensional unitary representation of the Weyl algebra 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 ⊗ ... ⊗ \underbrace{x}_{{m−th \atop place}} ⊗... ⊗ 1 , z_{m} = 1 ⊗ ... ⊗ \underbrace{z}_{{m−th \atop place}} ⊗... ⊗ 1 .

A convenient representation of x and z in 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 N × N dimensional matrices, normalized to the unity (x^{N} = z^{N} = 1), 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, such the chain is the inhomogeneous one.

Variables u_{m}^{N} and w_{m}^{N} form the classical counterpart of the quantum relativistic Toda chain. Define the classical L-operator as

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 matrices and integrability. Ordered product of the quantum L-operators

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 quantum model.

The integrability of the quantum chain is provided by the intertwining relation in the auxiliary two-dimensional spaces

R(λ, μ) ℓ(λ) ⊗ ℓ(μ) = (1 ⊗ ℓ(μ)) (ℓ(λ) ⊗ 1) R(λ, μ) . (11)

The tensor product of two 2 × 2 matrices with the identical Weyl algebra entries u, w, κ but different spectral parameters λ and μ is implied in (11), and

R(λ, μ) = ( \begin{matrix}λ − ω μ & 0 & 0 & 0 \\ 0 & λ − μ & μ (1 − ω) & 0 \\ 0 & λ (1 − ω) & ω (λ − μ) & 0 \\ 0 & 0 & 0 & λ − ω μ \end{matrix} ) (12)

is the slightly twisted six-vertex trigonometric R-matrix, used e.g. in [28] . Equation (11) may be verified directly. The repeated use of the intertwining relation (11) gives the analogous relation for the monodromy matrices (9), leading to the commutativity of the transfer matrices (10) with different spectral parameters but with the identical elements u_{m}, w_{m} and κ

[ t(λ, κ; {u_{m}, w_{m}}) , t(μ, κ; {u_{m}, w_{m}}) ] = 0 , (13)

where we emphasize the dependence of t(λ) = t(λ, κ; {u_{m}, w_{m}}_{m=1}^{M}) on the given set of ℂ-valued parameters u_{m}, w_{m}, m = 1...M.

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 , (14)

where

Y = ∏_{m=1}^{M} (−ω^{−1/2}x_{m}z_{m}) , Y^{N} = 1 . (15)

Transfer matrix (10) is invariant with respect to a gauge transformation of L-operators (1). Let G be 2 × 2 matrix with ℂ-valued coefficients, then the transformation

ℓ_{m} ↦ G ℓ_{m} G^{−1} ⇒ t̂ ↦ G t̂G^{−1} (16)

does not change the transfer matrix. For example,

G = ( \begin{matrix}1 & 0 \\ 0 & g \end{matrix} ) (17)

produces the transformation of the parameters

u_{m} ↦ g^{−1}u_{m} , w_{m} ↦ g w_{m} . (18)

Taking into account this freedom and also the possibility of redefinition of the spectral parameter λ, entering L-operator (1) always as λ^{−1}u_{m}, we may impose, without loss of generality, the following conditions for the parameters {u_{m}, w_{m}}_{m=1}^{M}

∏_{m=1}^{M} (−ω^{1/2}u_{m}) = ∏_{m=1}^{M}(−w_{m}) = 1 . (19)

1.3. Classical monodromy matrix. Define the monodromy of the classical Lax matrices (8) analogously to (9):

T̂ =^{def} L_{1} L_{2} ⋯ L_{M} = ( \begin{matrix}A(λ^{N}) & B(λ^{N}) \\ C(λ^{N}) & D(λ^{N}) \end{matrix} ) . (20)

For our purposes we have to mention the relation between the elements of the quantum monodromy matrix (9) and the classical ones. Note at first, relation (11) provides the commutativity

(21) [a(λ), a(μ)] = [b(λ), b(μ)] = [c(λ), c(μ)] = [d(λ), d(μ)] = 0 .

The spectra of a(λ), b(λ), c(λ) and d(λ) may be calculated by means of the following Proposition 1.

\begin{matrix}∏_{n∈ℤ_{N}} a(ω^{n}λ) = A(λ^{N}) , ∏_{n∈ℤ_{N}} b(ω^{n}λ) = B(λ^{N}) , \\ ∏_{n∈ℤ_{N}} c(ω^{n}λ) = C(λ^{N}) , ∏_{n∈ℤ_{N}} d(ω^{n}λ) = D(λ^{N}) . \end{matrix} (22)

Theorems of such kind were proved in [28] for the general case of cyclic representations like (6) of the quantum affine algebras. An alternative proof follows from the results of ref. [15].

Let us consider the homogeneous chain with u_{m} = u and w_{m} = w. Due to (18,19), we may fix the parameters of the homogeneous chain as

u_{m} = − ω^{−1/2} and w_{m} = − 1 . (23)

Then

T̂ = L(λ^{N})^{M} , L(λ^{N}) = ( \begin{matrix}1 − \frac{κ^{N}}{λ^{N}} & −\frac{ϵ}{λ^{N}} \\ ϵ & 0 \end{matrix} ) , (24)

where ϵ = (−1)^{N}, and the matrix elements of T̂ may be calculated with the help of projector decomposition of L(λ^{N}). Namely, let x_{1} and x_{2} are two eigenvalues of L(λ^{N}):

x_{1}x_{2} = \frac{1}{λ^{N}} , x_{1} + x_{2} = 1 − \frac{κ^{N}}{λ^{N}} . (25)

Then L(λ^{N}) may be presented in the form L(λ^{N}) = D diag(x_{1}, x_{2}) D^{−1}, therefore T̂ = D diag(x_{1}^{M}, x_{2}^{M}) D^{−1}, and the final answer reads

T̂ = ( \begin{matrix}\frac{x_{1}^{M+1} − x_{2}^{M+1}}{x_{1} − x_{2}} & , & −ϵx_{1}x_{2} \frac{x_{1}^{M} − x_{2}^{M}}{x_{1} − x_{2}} \\ ϵ \frac{x_{1}^{M} − x_{2}^{M}}{x_{1} − x_{2}} & , & \frac{x_{1}x_{2}^{M} − x_{2}x_{1}^{M}}{x_{1} − x_{2}} \end{matrix} ) . (26)

Now one may calculate \frac{x_{1}^{M} − x_{2}^{M}}{x_{1} − x_{2}} explicitly as a function of λ^{N}. First of all, it is an M −1-th

power polynomial with respect to \frac{1}{λ^{N}}, so

\frac{x_{1}^{M} − x_{2}^{M}}{x_{1} − x_{2}} = C_{0} ∏_{k=1}^{M−1} (1 − \frac{λ_{k}^{N}}{λ^{N}} ) . (27)

In the limit when 1/λ^{N} ↦ 0, one has x_{1} = 1 and x_{2} = 0, see (25). Therefore C_{0} = 1. The roots λ^{N} = λ_{k}^{N} of (27) correspond to the case when

x_{1}^{M} = x_{2}^{M} , x_{1} ≠ x_{2} ⇒ \frac{x_{2}}{x_{1}} = e^{2 i ϕ_{k}} , (28)

where ϕ_{k}, k = 1...M − 1, belong to the following set 𝔉_{M}:

ϕ_{k} ∈ 𝔉_{M} =^{def} \{\frac{π}{M} , \frac{2π}{M} , ... , \frac{(M − 1)π}{M} \} . (29)

The ordering of ϕ_{k} ∈ 𝔉_{M} is not essential. Introduce now three useful functions Δ_{ϕ}, Δ_{ϕ}^{∗} and Λ_{ϕ}:

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}^{∗} .

(30)

Expressions for Δ and Δ^{∗} uniformize the rational curve

(31) ΔΔ^{∗} = Δ + Δ^{∗} + κ^{N} in the terms of \frac{Δ}{Δ^{∗}} = e^{2iϕ} . Comparing (25) and (31), we conclude λ_{k}^{N} = Λ_{ϕ_{k}} , ϕ_{k} ∈ 𝔉_{M} . (32) The application of Proposition 1 on page 7 to the the homogeneous chain gives in particular

∏_{n∈ℤ_{N}} b(ω^{n}λ) = −\frac{ϵ}{λ^{N}} ∏_{k=1}^{M−1} (1 − \frac{Λ_{ϕ_{k}}}{λ^{N}} ) , ϕ_{k} ∈ 𝔉_{M} . (33)

1.4. Eigenvalues and eigenstates. The natural aim of the investigation of any integrable quantum chain is to calculate the spectrum of the transfer matrix and to construct its eigenstates at least for the homogeneous chain. In this subsection we will fix several notations necessary for the spectral problems and for the functional Bethe ansatz.

For given transfer matrix t(λ) let t(λ) be its eigenvalue for the right eigenvector |Ψ_{t}⟩ and for the left eigenvector ⟨Ψ_{t}|: t(λ) |Ψ_{t}⟩ = |Ψ_{t}⟩ t(λ) , ⟨Ψ_{t}| t(λ) = t(λ) ⟨Ψ_{t}| . (34)

10 S. PAKULIAK AND S. SERGEEV

Here t(λ) = ∑_{k=0}^{M} λ^{−k} t_{k}, t_{k} are eigenvalues of the complete commutative set of t_{k} (10). Due to (14,15), t_{0} ≡ 1 and t_{M} = (−κ)^{M} ω^{γ}, where ω^{γ} is the eigenvalue of Y. The subscript ‘t’ in the notations of the eigenvectors stands for the M-components vector {t_{k}}, k = 1, ..., M. The eigenvectors do not depend on λ, but depend on κ and u_{m}, w_{m} for the inhomogeneous chain. In general t_{k} are not hermitian, so left and right eigenvectors are not conjugated. Nevertheless, let us imply that |Ψ_{t}⟩ and ⟨Ψ_{t}| are dual complete basises,

(35) t(λ) = ∑_{t} |Ψ_{t}⟩ t(λ) ⟨Ψ_{t}| and 1 = ∑_{t} |Ψ_{t}⟩ ⟨Ψ_{t}| ,

where the summations over t is by definition the summation over all N^{M} possible sets of the eigenvalues {t_{k}}_{k=1}^{M}.

All the observables are defined in the local basis initially (5):

|α⟩ = |α_{1}⟩ ⊗ |α_{2}⟩ ⊗ ⋯ ⊗ |α_{M}⟩ . (36)

The dual basis is defined via

⟨α|β⟩ = ∏_{m=1}^{M} δ_{α_{m},β_{m}} , 1 = ∑_{{α_{m}}} |α⟩ ⟨α| . (37)

Calculation of the eigenvector |Ψ_{t}⟩ means the calculation of the matrix elements ⟨α|Ψ_{t}⟩

|Ψ_{t}⟩ = ∑_{{α_{m}}} |α⟩ ⟨α|Ψ_{t}⟩ . (38)

Let us formulate now the isospectrality problem. Let {u_{m}, w_{m}}_{m=1}^{M} and {u′_{m}, w′_{m}}_{m=1}^{M} be two sets of parameters of Weyl elements for the quantum relativistic Toda chain at the root of unity. These sets are isospectral if

t(λ; {u_{m}, w_{m}}_{m=1}^{M}) = t(λ; {u′_{m}, w′_{m}}_{m=1}^{M}) . (39)

Let |Ψ_{t}⟩ and |Ψ′_{t}⟩ be the eigenvectors of two isospectral transfer matrices t(λ) and t′(λ) with the same eigenvalue t(λ). Two isospectral transfer matrices must be related by a similarity operator K

t(λ; {u_{m}, w_{m}}_{m=1}^{M}) K = K t(λ; {u′_{m}, w′_{m}}_{m=1}^{M}) , (40)

such that

K = ∑_{t} |Ψ_{t}⟩ K_{t} ⟨Ψ′_{t}| . (41)

Here K_{t} are arbitrary nonzero values. We will prove further that two transfer matrices are isospectral iff the traces of their classical monodromy matrices coincide.

The eigenvalues of the transfer matrix obey several functional relations. Sometimes they may be solved in the thermodynamic limit or at least some physical properties of the chain may be derived from the functional equations in the thermodynamic limit.

The eigenstates for quantum Toda chain, relativistic as well as usual, and in general the eigenstates for models based on the local Weyl algebra at root of the unity may be constructed with the help of so-called functional Bethe ansatz, or quantum separation of variables [1, 23, 24]. Eigenstates of off-diagonal element of the monodromy matrix b(λ)

(9) play the important rôle in the method of the functional Bethe ansatz. It is useful to parameterize the spectrum of b(λ) by its zeros λ_{k}, k = 1...M − 1. It follows from (33) for the homogeneous chain

λ_{k}^{N} = Λ_{ϕ_{k}} , k = 1...M − 1, ϕ_{k} ∈ 𝔉_{M}. (42)

The monodromy matrix (9) has the structure

t̂(λ) = t̂′(λ) · ℓ_{M}(λ) , (43)

where t̂′(λ) is the monodromy of the first M − 1 sites. Then

b(λ) = a′(λ) (− \frac{ω^{1/2}}{λ} u_{M}) . (44)

It is more convenient to work with the eigenvectors of the operator

b̃(λ) =^{def} b(λ) w_{M} = a′(λ) (− \frac{ω^{1/2}}{λ} u_{M} w_{M}) . (45)

Note that operator Y (15) commute with b̃. We denote the eigenvectors of b̃ as |{λ_{k}}_{k=1}^{M−1}, γ⟩ and define them by the following relation

b(λ_{j}) |{λ_{k}}_{k=1}^{M−1}, γ⟩ = 0 , λ_{j} ∈ {λ_{k}}_{k=1}^{M−1} , (46)

and

Y |{λ_{k}}_{k=1}^{M−1}, γ⟩ = |{λ_{k}}_{k=1}^{M−1}, γ⟩ ω^{γ} . (47)

The functional Bethe ansatz method, being applied to the quantum relativistic Toda chain at root of unity, implies the following structure of the matrix element (see [23]):

⟨Ψ_{t}|{λ_{k}}_{k=1}^{M−1}, γ⟩ = const ∏_{k=1}^{M−1} q_{t}(λ_{k}) . (48)

q_{t}(λ) is a function, which depends on the spectrum of t(λ), and obeys a functional equation, called the Baxter T −Q equation. Operator Y, being the member of t (14), has the eigenvalue ω^{γ} in all the components of formula (48). The explicit form of Baxter’s equation and the definition of q_{t}(λ) will be given later.

One obtains usually formula (48) from equation (11), see [1, 23, 24]. In this paper we will obtain (48) in a different way: we will construct first so-called modified Q-operator, whose spectrum is related to q_{t}. Then a projector to |{λ_{k}}_{k=1}^{M−1}, γ⟩ will be obtained using these modified Q-operators.

2. Intertwining relations

The quantum relativistic Toda chain L-operator (1) is 2×2 matrix, whose matrix elements are N × N matrices. According to the conventional terminology, the two-dimensional vector space is called the auxiliary space, while N-dimensional one is called the quantum space. Equation (11) is the intertwining relation in the auxiliary spaces. In this section we will investigate the other type of the intertwining relations – in the quantum spaces.

It is easy to check that two L-operators (1) can not be intertwined in the quantum spaces (i.e. their intertwiner is exactly zero). In the case of the usual quantum Toda chain one needs an extra auxiliary L-operator (in its simplest form it is known as the L-operator for the “dimer self-trapping model” [25]). The same situation holds in our case: we need an auxiliary L-operator that may be intertwined with L-operator (1) in the quantum spaces. In general, this auxiliary L-operator is Bazhanov-Stroganov’s L-operator for the six-vertex model at the root of unity [17]. This fact is the origin of a relationship between the relativistic Toda chain at the Nth root of unity and the N-state chiral Potts model. The Bazhanov-Stroganov L-operator may be simplified to a “relativistic dimer self-trapping” L-operator.

In this section we introduce this auxiliary L-operator and write down the quantum intertwining relation. Being written in an appropriate way, this will give us a key for all further investigations.

2.1. Auxiliary L-operator. Define the auxiliary quantum L-operator as follows:

ℓ̃_{ϕ}(λ, λ_{ϕ}) =^{def} ( \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} ) . (49)

Here λ and λ_{ϕ} are two spectral parameters (actually, up to a gauge transformation (16), ℓ̃ depends on their ratio). κ_{ϕ} is a module analogous to κ. u_{ϕ} and w_{ϕ} are additional Weyl elements, defined analogously to (2,4,5,6)

u_{ϕ} = u_{ϕ} x_{ϕ} , w_{ϕ} = w_{ϕ} z_{ϕ} . (50)

In these notations the subscript ϕ labels the additional Weyl pair in the tensor product (5). Classical counterpart of ℓ̃ is by definition

L̃_{ϕ}(λ^{N}, λ_{ϕ}^{N}) =^{def} ( \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} ) . (51)

L-operators (49) are intertwined in their two dimensional auxiliary vector spaces by the same six-vertex trigonometric R-matrix (12). There exists also the fundamental quantum intertwiner for (49). It is the R-matrix for the chiral Potts model such that two rapidities are fixed to special singular values. The details are useless in this paper. But the method described here may be applied directly to the model, defined by (49). This is the subject of forthcoming papers.

2.2. Quantum intertwining relation. We are going to write out some quantum intertwining relation for L-operators (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} and u_{ϕ}, w_{ϕ} as the arguments of ℓ_{m} and ℓ̃_{ϕ}.

Proposition 2. 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} (52)

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

L̃_{ϕ}(λ^{N}, λ_{ϕ}^{N}; u_{ϕ,m}^{N}, w_{ϕ,m}^{N})L_{m}(λ^{N}; u_{m}^{N}, w_{m}^{N}) = L_{m}(λ^{N}; u_{m}^{′N}, w_{m}^{′N})L̃_{ϕ}(λ^{N}, λ_{ϕ}^{N}; u_{ϕ,m+1}^{N}, w_{ϕ,m+1}^{N}) , (53)

valid in each order of λ^{N}.

Proof. Consider the following equation

(54)  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})

and equate its coefficients in each order of λ. In (54 we used the Weyl elements as the formal arguments of L-operators (1) and (49). This system of equations has the unique solution with respect to the “primed” operators:

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} .

(55)

Obtaining (55) from (54), we used the exchange relations for u_{m}, w_{m}, u_{ϕ}, w_{ϕ} without assuming any exchange relations for “primed” operators. One may check directly that the rational homomorphism

(56) u_{m}, w_{m}, u_{ϕ}, w_{ϕ} ↦ u′_{m}, w′_{m}, u′_{ϕ}, w′_{ϕ} ,

given by (55), is the automomorphism of the local Weyl algebra.

Since the finite dimensional representations at root of unity are considered (see eqs. (4) and (50)), we may use the normalization of the operators entering (54) as usual:

(57) u_{m} = u_{m}x_{m} , w_{m} = w_{m}z_{m} , u_{ϕ} = u_{ϕ,m}x_{ϕ} , w_{ϕ} = w_{ϕ,m}z_{ϕ} , and

(58) u′_{m} = u′_{m}x′_{m} , w′_{m} = w′_{m}z′_{m} , u′_{ϕ} = u_{ϕ,m+1}x′_{ϕ} , w′_{ϕ} = w_{ϕ,m+1}z′_{ϕ} , where all x, z matrices are normalized to unity. With this normalization, the Nth powers of (55) may be calculated directly with the help of the following identity (u + w)^{N} = u^{N} + w^{N} . (59) The answer reads

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}} , (60)

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}} . (61)

It is easy to check that (60) and (61) are the exact and the unique solution of (53).

Automorphism (56) with normalizations (57) and (58) taken into account, gives the automorphism

(62) x_{m}, z_{m}, x_{ϕ}, z_{ϕ} ↦ x′_{m}, z′_{m}, x′_{ϕ}, z′_{ϕ} , where, recall, all x, z are normalized to unity finite dimensional operators. This is provided by (60,61) or by (53). Therefore this automorphism must be the internal one, i.e. due to the Shur lemma there must exist an unique (up to a multiplier) N^{2} × N^{2} matrix R_{m,ϕ}:

(63) x′_{m} = R_{m,ϕ}x_{m}R_{m,ϕ}^{−1} , z′_{m} = R_{m,ϕ}z_{m}R_{m,ϕ}^{−1} , and

(64) x′_{ϕ} = R_{m,ϕ}x_{ϕ}R_{m,ϕ}^{−1} , z′_{ϕ} = R_{m,ϕ}z_{ϕ}R_{m,ϕ}^{−1} . The last two equations are equivalent to (52). □

The local transformation u_{m}^{N}, w_{m}^{N}, u_{ϕ,m}^{N}, w_{ϕ,m}^{N} ↦ u_{m}^{′N}, w_{m}^{′N}, u_{ϕ,m+1}^{N}, w_{ϕ,m+1}^{N} , (65)

16 S. PAKULIAK AND S. SERGEEV

given by eqs. (60) and (61), is called the Darboux transformation for the classical relativistic Toda chain, see e.g. [25] for the analogous transformation for the usual Toda chain. (60) and (61) define the mapping (65) up to Nth roots of unity. These phases are the additional discrete parameters appeared when one takes the Nth roots in (65). Note, the matrix R_{m,ϕ} is unique if all these roots are fixed.

In the next section we will give matrix elements of R_{m,ϕ} in the basis (6).

2.3. Q-transformation and the isospectrality problem. Relations (52) and (53) may be iterated for the whole chain. The functional counterpart, eq. (53), gives

L̃_{ϕ}(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_{ϕ}(u_{ϕ,M+1}^{N}, w_{ϕ,M+1}^{N}) , (66)

where T̂ is the classical monodromy matrix (20) and the spectral parameters are implied. Values of u′_{m}, w′_{m}, m = 1, ..., M, and u_{ϕ,M+1}, w_{ϕ,M+1} in the terms of λ_{ϕ}, κ_{ϕ}, u_{m}, w_{m} and u_{ϕ,1}, w_{ϕ,1} must be obtained as the recursion iterating (60,61) for m = 1, ..., M.

For the periodic chain the cyclic boundary conditions for the recursion have to be imposed,

u_{ϕ,M+1} = u_{ϕ,1} , w_{ϕ,M+1} = w_{ϕ,1} . (67)

Now suppose that (60) and (67) are solved, i.e. u_{ϕ,m}, w_{ϕ,m} are parameterized (at least implicitly) in the terms of u_{m}, w_{m}, m = 1, ..., M, and some extra parameters, possible degrees of freedom of (60) and (67) (e.g. λ_{ϕ}, κ_{ϕ}, etc.) Then (61) defines in general the transformation Q_{ϕ}

Q_{ϕ} : {u_{m}, w_{m}}_{m=1}^{M} ↦ {u′_{m}, w′_{m}}_{m=1}^{M} . (68)

The transfer matrices for two sets {u_{m}, w_{m}} and {u′_{m}, w′_{m}} have the same spectrum, because of the existence of N^{M} × N^{M} matrix Q_{ϕ}, nondegenerative in general,

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

such that

t(λ; {u_{m}, w_{m}}) · Q_{ϕ} = Q_{ϕ} · t(λ; {u′_{m}, w′_{m}}) . (70)

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}, so Q_{ϕ} means in particular Q(λ_{ϕ}). It is important to point out ones more that Q_{ϕ} “remembers” about its initial {u_{m}, w_{m}}_{m=1}^{M} and its final {u′_{m}, w′_{m}}_{m=1}^{M} sets of the parameters. In terms of the eigenvectors of the transfer matrix t(λ) (34), operator Q_{ϕ} can be written as

Q_{ϕ} = ∑_{t} |Ψ_{t}⟩ q_{t,ϕ} ⟨Ψ′_{t}| . (71)

Consider now the repeated application of the transformations Q_{ϕ}, eq. (68),

{u_{m}, w_{m}} ↦^{Q_{ϕ_{1}}} {u′_{m}, w′_{m}} ↦ ...{u_{m}^{(n−1)}, w_{m}^{(n−1}} ↦^{Q_{ϕn}} {u_{m}^{(n)}, w_{m}^{(n)}} ↦ ...{u_{m}^{(g)}, w_{m}^{(g)}} , (72)

such that the set of isospectral quantum transfer matrices t^{(n)}(λ) = t(λ, κ; {u_{m}^{(n)}, w_{m}^{(n)}}_{m=1}^{M}) , n = 0, ..., g (73) has arisen. Sequence (72) defines the transformation K, K^{(g)} : {u_{m} ≡ u_{m}^{(0)}, w_{m} ≡ w_{m}^{(0)}}_{m=1}^{M} ↦ {u_{m}^{(g)}, w_{m}^{(g)}}_{m=1}^{M} , (74) with the finite dimensional counterpart K^{(g)} = Q_{ϕ_{1}}^{(1)} Q_{ϕ_{2}}^{(2)} ... Q_{ϕ_{n}}^{(n)} ... Q_{ϕ_{g}}^{(g)} , (75) where Q_{ϕ_{n}}^{(n)} = tr_{ϕ_{n}} (R_{1,ϕ_{n}}^{(n)} R_{2,ϕ_{n}}^{(n)} … R_{M,ϕ_{n}}^{(n)}) , (76) such that

(77) t^{(n−1)}(λ) Q_{ϕ_{n}}^{(n)} = Q_{ϕ_{n}}^{(n)} t^{(n)}(λ) and t^{(0)}(λ) K^{(g)} = K^{(g)} t^{(g)}(λ) . Thus in general the transformations Q_{ϕ} and their iterations K^{(g)} give the isospectral transformations of the initial quantum chain in the space of the parameters, so that their finite dimensional counterparts Q_{ϕ} and K^{(g)} respectfully describe the change of the eigenvector basis (see eq. (41).

3. Matrix R

In this section we construct explicitly the finite dimensional matrix R_{m,ϕ}, obeying (52). Since a basis invariant formula for R_{m,ϕ} is useless and rather complicated, we will find matrix elements of R_{m,ϕ} in the basis (6). But first we have to introduce several notations concerning the functions on the Fermat curve.

18 S. PAKULIAK AND S. SERGEEV

3.1. w-function. Let p be a point on the Fermat curve F

p =^{def} (x, y) ∈ F ⇔ x^{N} + y^{N} = 1 . (78)

Actually, the identity (59) is the origin of the Fermat curve. 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 . (79)

Function w_{p}(n) has a lot of remarkable properties, see the Appendix of ref. [18] 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) . (80) Then w_{p}(n) w_{Op}(−n) Φ(n) = 1 , where Φ(n) = (−)^{n} ω^{n^{2}/2} . (81) In the subsequent sections we will use also two simple properties of w-function:

w_{(x,ωy)}(n) = ω^{n}w_{(x,y)}(n) , w_{(x,y)}(n + 1) = \frac{y}{1 − ωx}w_{(ωx,y)}(n) . (82)

Define also three special points on the Fermat curve: q_{0} = (0, 1) , q_{∞} = Oq_{0} , q_{1} = (ω^{−1}, 0) . (83) Then

w_{q_{0}}(n) = 1 , w_{q_{∞}}(n) = \frac{1}{Φ(n)} , \frac{1}{w_{q_{1}}(n)} = δ_{n,0} . (84)

The inversion relation may be mentioned for the completeness

∑_{n∈ℤ_{M}} \frac{w_{(ωx,ωy)}(n + a)}{w_{(x,y)}(n + b)} = N \frac{1 − ωx}{ωx} \frac{x^{N}}{1 − x^{N}} δ_{a,b} . (85)

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} (86)

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

x_{1} x_{2} = x_{3} . (87)

Eq. (87) and the spin structure of (86) provide the dependence of (86) 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 3. Matrix R_{m,ϕ}(p_{1}, p_{2}, p_{3}), whose matrix elements (86) 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} .

(88)

Proof. Each relation of (88) should be rewritten in the form

⟨α_{m}, α_{ϕ}|R_{m,ϕ} f = f′ R_{m,ϕ}|β_{m}, β_{ϕ}⟩ . (89)

Such identities may be verified directly with the help of (82). □ Compare (88) with (55). R_{m,ϕ} solves (52) if

x_{1} = ω^{−1/2} \frac{u_{ϕ,m}}{κ u_{m}} , \frac{x_{3} y_{1}}{x_{1} y_{3}} = ω^{−1/2} \frac{λ_{ϕ}}{u_{ϕ,m}w_{m}} , (90)

and

\begin{matrix}u′_{m} = \frac{κ_{ϕ}}{κ} u_{ϕ,m} , & w′_{m} = w_{m}w_{ϕ,m} \frac{y_{2}}{y_{3}} , \\ u_{ϕ,m+1} = ω x_{3} u_{m} , & w_{ϕ,m+1} = \frac{κ}{κ_{ϕ}} \frac{u_{m}}{u_{ϕ,m}} \frac{y_{3}}{y_{2}} . \end{matrix} (91)

(90) and (91) are the complete set of the relations following from the identification of (88) with (55). Using (90), (87) and (78), one has to fix first the parameters p_{1}, p_{2}, p_{3} and thus to define R_{m,ϕ}. Then (91) parameterize u′_{m}, w′_{m}, u_{ϕ,m+1}, w_{ϕ,m+1} in the terms of u_{m}, w_{m}, u_{ϕ,m}, w_{ϕ,m} and the phases of x_{3} and y_{2}/y_{3}.

When one considers the complicated operator K^{(g)}, eq. (75), its matrix elements must be calculated using matrices R_{m,ϕ_{n}}^{(n)}, entering (76). The matrix elements of all R_{m,ϕ_{n}}^{(n)} are the same functions,

R_{m,ϕ_{n}}^{(n)} = R_{m,ϕ_{n}}(p_{1,m}^{(n)}, p_{2,m}^{(n)}, p_{3,m}^{(n)}) , (92)

given by (86), but with different and rather complicated parameters. Eqs. (90) and (91) are written for p_{1,m}^{(1)}, p_{2,m}^{(1)}, p_{3,m}^{(1)} in fact. Nevertheless, for given K^{(g)}, i.e. for given sets of {u_{m}^{(n)}, w_{m}^{(n)}}, m = 1, ..., M and n = 0, ..., g, see (72), one may parameterize the corresponding K^{(g)} explicitly via parameterizations (90,91), the form of the matrix elements of R_{m,ϕ} (86) and formulas (76,75).

3.3. Normalization of the matrix R. The matrix elements of R_{m,ϕ} are defined in general up to a constant multiplier. Definition (79) and formula (86) are the most simple expressions in the terms of the Fermat curve coordinates p_{1}, p_{2}, p_{3}. But this R_{m,ϕ} is not normalized. For several applications we need the normalization of R_{m,ϕ} connected with the determinant of R_{m,ϕ}.

To calculate the determinant, we need several definitions. Let

V (x) =^{def} ∏_{n=1}^{N−1} (1 − ω^{n+1}x)^{n} . (93)

V (x) obeys the following relations:

\frac{V (ω^{−1}x)}{V (x)} = \frac{1 − x^{N}}{(1 − x)^{N}} , (94)

and

V (ω^{−1}x^{−1}) = \frac{(1 − x^{N})^{N−1}ω^{N(N−1)(2N−1)/6}}{(−x)^{N(N−1)/2} V (x)} . (95)

Besides, one may calculate the particular value:

V (ω^{−1}) ≡ N^{N/2} e^{iπ(N−1)(N−2)/12} . (96)

Function V (x) appears in the following expressions, where p = (x, y):

\begin{matrix}∏_{n} ∑_{σ} \frac{ω^{nσ}}{w_{p}(σ)} = (ωx)^{N(N−1)/2} \frac{V (ω^{−1})}{V (x)} , \\ ∏_{n} ∑_{σ} \frac{ω^{nσ}}{Φ(σ)w_{p}(σ)} = \frac{V (ω^{−1})}{V (x)} , \\ ∏_{n} w_{p}(n) = \frac{V (x)}{y^{N(N−1)/2}} . \end{matrix} (97)

The third expression is rather trivial. One may prove the first two formulas considering the poles and zeros of the left and right hand sides. Details may be found in the appendix of [18].

Proposition 4. The determinant of the matrix R, defined by (86), is

det R = (−)^{N(N−1)/2} (∏_{n} \frac{w_{p_{2}}(n)}{w_{Op_{1}}(n)})^{N} (∏_{n} ∑_{σ} \frac{ω^{nσ}}{Φ(σ)w_{p_{3}}(σ)})^{N} . (98)

Proof. To obtain (98), one has to use O-automorphism (81) for w_{p_{1}} in (86). The factors w_{p_{2}} and w_{Op_{1}} correspond to the diagonal matrices D and D′ in the matrix decomposition of R_{m,ϕ},

R_{m,ϕ} = D(p_{1})R′(p_{3})D′(p_{2}) . (99)

The determinants of D and D′ give the term with p_{1} and p_{2} in (98). The Fourieur transform applied to the matrix R_{m,ϕ} in the m-th space yields F_{m}R′(p_{3})F_{m}^{−1} = D′′(p_{3})P_{m,ϕ}, where ⟨α_{m}|F |β_{m}⟩ = N^{−1/2}ω^{α_{m}β_{m}}, D′′ is a diagonal matrix and P_{m,ϕ} is the permutation. det D′′ gives the term, depending on p_{3} in (98), and det P is the sign factor in (98). □

Let the normalization factor for R_{m,ϕ} be ρ_{R}:

ρ_{R}^{N} = (ω^{−1/2} \frac{y_{1}}{x_{1}y_{2}} )^{N(N−1)/2} \frac{V (x_{1})}{V (ω^{−1}x_{2}^{−1})V (x_{3})} , (100)

22 S. PAKULIAK AND S. SERGEEV

so that

(101) det R = const · ρ_{R}^{N^{2}} .

Analogously, the normalization factor for the monodromy of R_{m,ϕ} and so of Q_{ϕ} is

ρ_{Q}^{N} = ∏_{m=1}^{M} (ω^{−1/2} \frac{y_{1,m}}{x_{1,m}y_{2,m}} )^{N(N−1)/2} \frac{V (x_{1,m})}{V (ω^{−1}x_{2,m}^{−1})V (x_{3,m})} . (102)

4. Parameterization of the recursion

The main object of the present investigations is the system of the recursion relations (60). Our goal is the construction of operator K^{(g)}, (75), for the homogeneous initial state (103). In this case, the fist step transformation Q_{ϕ_{1}} has a remarkably simple structure, clarifying nevertheless the structure of all subsequent Q_{ϕ_{n}}.

4.1. The first step. Let the initial parameters u_{m}, w_{m} are homogeneous

u_{m} = −ω^{−1/2} , w_{m} = −1 , (103)

see (23). The main recursion relation is the first equation in (60). For the homogeneous initial state it contains only one unknown u_{ϕ,m}. Without loss of generality let us introduce complex numbers δ_{ϕ} and δ_{ϕ}^{∗} (∗ does not stand for the complex conjugation!) and a function τ′_{m}, m ∈ ℤ, such that

u_{ϕ,m} = −ω^{−1/2} δ_{ϕ} \frac{τ′_{m−1}}{τ′_{m}} , δ_{ϕ}^{∗} = \frac{λ_{ϕ}}{δ_{ϕ}} . (104)

The first relation in (60) may be rewritten as the second order linear recursion for (τ′_{m})^{N}:

δ_{ϕ}^{N}δ_{ϕ}^{∗N} (τ′_{m})^{N} = δ_{ϕ}^{N} (τ′_{m−1})^{N} + δ_{ϕ}^{∗N} (τ′_{m+1})^{N} + κ^{N} (τ′_{m})^{N} . (105)

Parameterization (104) and linear recursion (105) are considered now in the general position, without mentioning ℤ_{M}-invariance. It is known that the general solution of a second order linear recursion with constant coefficients is a linear combination of two fundamental solutions, (τ′_{m})^{N} = αz_{1}^{m} + βz_{2}^{m}, where z_{1} and z_{2} are two roots of the characteristics equation (δ_{ϕ}δ_{ϕ}^{∗})^{N} = δ_{ϕ}^{N}z^{−1} + δ_{ϕ}^{∗N}z + κ^{N}. One may always redefine (τ′_{m})^{N} ↦ α^{−1}z_{1}^{−m}(τ′_{m})^{N}. This corresponds to a redefinition of δ_{ϕ}. One may choose δ_{ϕ}

δ_{ϕ}^{N}δ_{ϕ}^{∗N} = δ_{ϕ}^{N} + δ_{ϕ}^{∗N} + κ^{N} , λ_{ϕ} = δ_{ϕ}δ_{ϕ}^{∗} , (106) such that z_{1} = 1 and τ′_{m} = 1 solves (105). Compare (106) with equations (30), (31). One may identify δ_{ϕ}^{N} = Δ_{ϕ} δ_{ϕ}^{∗N} = Δ_{ϕ}^{∗N} , λ_{ϕ}^{N} = Λ_{ϕ} , (107) so the index ϕ may be understood conveniently as the spectral parameter ϕ.

The second fundamental solution of the characteristics equation is z_{2} = (δ_{ϕ}/δ_{ϕ}^{∗})^{N}, i.e. (τ′_{m})^{N} = (δ_{ϕ}^{N}/δ_{ϕ}^{∗N})^{m}. The complete solution of recursion (105) is (τ′_{m})^{N} = 1 − f_{ϕ} e^{2imϕ} , (108) where f_{ϕ} is arbitrary complex number, and (see (30))

e^{2 i ϕ} = \frac{δ_{ϕ}^{N}}{δ_{ϕ}^{∗N}} ≡ \frac{Δ_{ϕ}}{Δ_{ϕ}^{∗}} . (109)

Turn now to the ℤ_{M}-invariance condition. It demands τ′_{M+m} = τ′_{m}, hence the possible values of ϕ become discrete, e^{2iϕM} = 1 and e^{2iϕ} ≠ 1. Since there exists the symmetry between δ_{ϕ} and δ_{ϕ}^{∗} in (105), we may regard ϕ ∈ 𝔉_{M} in (108), see (29) for the definition of 𝔉_{M}.

Turn further to the first relation in (61) (or to the first relation in (91)). It follows

u′_{m} = −ω^{−1/2} \frac{κ_{ϕ}}{κ} δ_{ϕ} \frac{τ′_{m−1}}{τ′_{m}} . (110)

Taking into account the gauge invariance (18), we may fix κ_{ϕ} as follows

κ_{ϕ} = \frac{κ}{δ_{ϕ}} (111)

and so

u′_{m} = −ω^{−1/2} \frac{τ′_{m−1}}{τ′_{m}} . (112)

It agrees with (19). The second equations in (60) and (61) give

w′_{m} = − \frac{θ′_{m}}{θ′_{m−1}} , (113)

where

(θ′_{m})^{N} = 1 − f_{ϕ} \frac{Δ_{ϕ}^{∗}}{Δ_{ϕ}} \frac{Δ_{ϕ} − 1}{Δ_{ϕ}^{∗} − 1} e^{2imϕ} . (114)

24 S. PAKULIAK AND S. SERGEEV

These simple calculations show that with the useful parameterizations (107), where the functions of ϕ are defined in (30), and with additional condition (111), the image of Q_{ϕ}- transformation (108,112,113,114), contains one arbitrary complex parameter f_{ϕ} if ϕ ∈ 𝔉_{M}. The case when −ϕ ∈ 𝔉_{M} is equivalent to the case when ϕ ∈ 𝔉_{M}. It follows from the symmetry between δ_{ϕ} and δ_{ϕ}^{∗}. The case when ϕ is arbitrary complex number, ℤ_{M}-invariance requirement yields f_{ϕ} ≡ 0, therefore τ′_{m} = 1 and the chain remains the homogeneous one.

Equations (108) and (114) define the Nth powers of τ′ and θ′. Their phases are arbitrary. The Nth rooting is the subject of the definition of p_{j,m} for R_{m,ϕ}.

Functions (τ′_{m})^{N} and (θ′_{m})^{N} look like solutions of a Hirota-type discrete equation. We call these expressions “one-soliton τ-functions”. Therefore Q_{ϕ}-transformation with ϕ ∈ 𝔉_{M} must be identified with the Bäcklund transformation for the classical relativistic Toda chain, see [25] for the application of this ideology to the usual Toda chain.

4.2. Nth step. Turn now to the sequence (72) of Q_{ϕ_{n}}-transformations

Q_{ϕ_{n}} : {u_{m}^{(n−1)}, w_{m}^{(n−1)}}_{m=1}^{M} ↦ {u_{m}^{(n)}, w_{m}^{(n)}}_{m=1}^{M} (115)

with generic λ_{ϕ_{n}}, parameterized according to (107,30) as follows

δ_{n}^{N} = Δ_{n} , δ_{n}^{∗N} = Δ_{n}^{∗} , λ_{ϕ_{n}} = δ_{n}δ_{n}^{∗} ,

Δ_{n}Δ_{n}^{∗} = Δ_{n} + Δ_{n}^{∗} + κ^{N} , e^{2iϕ_{n}} = \frac{Δ_{n}}{Δ_{n}^{∗}} .

(116)

Taking into account (19), we may parameterize the n-th state in the sequence (72), n = 0, ..., g, as

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} . (117)

The homogeneous initial state (23) corresponds to τ_{m}^{(0)} = θ_{m}^{(0)} = 1. Parameterizations (117) with the first relation of (61) taken into account implies that

κ_{ϕ_{n}} = \frac{κ}{δ_{n}} (118)

and

u_{ϕ_{n},m} = −ω^{−1/2} δ_{n} \frac{τ_{m−1}^{(n)}}{τ_{m}^{(n)}} (119)

are imposed for Q_{ϕ_{n}}-transformation. The key recursion relation, the first one from (60), looks now as follows

\begin{matrix}(δ_{ϕ_{n}}δ_{ϕ_{n}}^{∗} τ_{m}^{(n)} τ_{m}^{(n−1)} θ_{m−1}^{(n−1)})^{N} = (δ_{ϕ_{n}} τ_{m−1}^{(n)} τ_{m}^{(n−1)} θ_{m}^{(n−1)})^{N} \\ + (δ_{ϕ_{n}}^{∗} τ_{m+1}^{(n)} τ_{m−1}^{(n−1)} θ_{m−1}^{(n−1)})^{N} + (κ τ_{m}^{(n)} τ_{m−1}^{(n−1)} θ_{m}^{(n−1)})^{N} . \end{matrix} (120)

The second equations in (60) and (61) provide

w_{ϕ_{n},m+1} = w_{ϕ_{n}} \frac{θ_{m}^{(n−1)} τ_{m}^{(n)}}{τ_{m}^{(n−1)} θ_{m}^{(n)}} , w_{ϕ_{n}}^{N} = \frac{1}{1 − Δ_{ϕ_{n}}^{∗}} , (121)

and

(θ_{m}^{(n)})^{N} = (\frac{w_{ϕ_{n}}}{τ_{m−1}^{(n−1)}} )^{N} ((θ_{m}^{(n−1)}τ_{m−1}^{(n)})^{N} − (δ_{ϕ_{n}}^{∗}θ_{m−1}^{(n−1)}τ_{m}^{(n)})^{N}) . (122)

Consider (120) and (122) as the set of discrete equations with respect to m ∈ ℤ and n ∈ ℤ_{+} with the initial data τ_{m}^{(0)} = θ_{m}^{(0)} = 1.

Proposition 5. General solution of the infinite dimensional in m and in n system of discrete equations (120), (122) with generic values of ϕ_{n}, n = 0, 1, ... (see parameterizations (116) and with the initial data τ_{m}^{(0)} = θ_{m}^{(0)} = 1 is given by

\begin{matrix}(τ_{m}^{(n)})^{N} = h^{(n)}(\{f_{k}e^{2imϕ_{k}}\}_{k=1}^{n}) , \\ (θ_{m}^{(n)})^{N} = h^{(n)}(\{f_{k}s_{k}(1)e^{2imϕ_{k}}\}_{k=1}^{n}) , \end{matrix} (123)

where f_{1}, ..., f_{n} ≡ {f_{k}}_{k=1}^{n} is a set of arbitrary complex variables, h^{(n)} are defined by

h^{(1)}(f_{1}) = 1 − f_{1} , (124)

and recursively

\begin{aligned}h^{(n)}(\{f_{k}\}_{k=1}^{n}) & = & h^{(n−1)}(\{f_{k}s_{k}(Δ_{n}^{∗})\}_{k=1}^{n−1}) \\ & − & f_{n} ∏_{m=1}^{n−1} s_{n}(Δ_{m}^{∗}) h^{(n−1)}(\{f_{k}s_{k}(Δ_{n})\}_{k=1}^{n−1}) , \end{aligned} (125)

where functions s_{n} are

s_{n}(ξ) = \frac{Δ_{n}^{∗}}{Δ_{n}} \frac{Δ_{n} − ξ}{Δ_{n}^{∗} − ξ} . (126)

26 S. PAKULIAK AND S. SERGEEV

Proof. Equations (120) and (122) become the algebraic identities after the substitution (123) (see Proposition 14 on page 50 of the Appendix). The first step of this recursion with respect to n is described before, hence (τ′_{m})^{N} = h^{(1)}(fe^{2iϕm}) etc.

Suppose that the statement of the Proposition is true for n = 1, ..., g − 1. h^{(g)} depends on f_{1}, ..., f_{g}, while h^{(g−1)} depends only on f_{1}, ..., f_{g−1}, therefore τ_{m}^{(g)} contains one extra degree of freedom f_{g} ∈ ℂ with respect to τ_{m}^{(g−1)}, θ_{m}^{(g−1)}. On the other side, (120) with n = g is the second order linear recursion with respect to (τ_{m}^{(g)})^{N}, so it must have exactly two fundamental solutions. Hence, due to the homogeneity, general normalized solution of (120) with n = g with respect to (τ_{m}^{(g)})^{N} for given τ_{m}^{(g−1)}, θ_{m}^{(g−1)} must have exactly one extra degree of freedom. This degree of the freedom is parameter f_{g}. □

Proposition (5) is formulated for the generic sequence of ϕ_{n}. One may verify, due to definition (125), h^{(g)}({f_{n}}_{n=1}^{g}) is the symmetrical function with respect to any permutation of the pairs (f_{n}, ϕ_{n}), n = 1, ..., g.

Two special cases should be discussed. The fist one is the case when ϕ_{n} = ϕ_{n′} mod π for some n and n′, and the second one is ϕ_{n} = −ϕ′_{n} mod π. Both these cases are the singular ones for definition (125). Nevertheless, since the “observables” u_{m}^{(g)} and w_{m}^{(g)} are the ratios of τ-functions, corresponding τ_{m}^{(g)} and θ_{m}^{(g)} for these cases may be obtained as the residues of formula (123). Technically, both cases are equivalent (up to some re-parameterizations of κ_{ϕ_{n}} or κ_{ϕ_{n′}}). The equivalence is connected again with the symmetry between δ_{ϕ} and δ_{ϕ}^{∗}, so we may consider only the case

ϕ_{n} = − ϕ_{n′} mod π . (127)

In this case condition (118) is convenient for any n, and the residues of τ_{m}^{(g)} and θ_{m}^{(g)} are given by the same formula (123), but with

f_{n} = f_{n′} ≡ 0 . (128)

Turn now the the finite chain with the periodic boundary conditions

τ_{M+m}^{(g)} = τ_{m}^{(g)} , θ_{M+m}^{(g)} = θ_{m}^{(g)} . (129)

This boundary conditions, being applied to (123), give the following rule: in the sequence of f_{1}, ..., f_{g}

f_{n} = 0 ⇔ ϕ_{n} ∉ 𝔉_{M} mod π , (130)

where the set 𝔉_{M} is given by (29). It means that any Q_{ϕ} with ϕ ∉ 𝔉_{M} just changes the values of f_{n}. Therefore the maximal number of the possible exponents e^{2imϕ_{n}} is M − 1. Without loss of generality, let us define the minimal complete sequence of Q_{ϕ_{n}} described by

{(f_{n}, ϕ_{n})}_{n=1}^{M−1} : ϕ_{n} ∈ 𝔉_{M} , ϕ_{n} ≠ ϕ_{n′} , (131)

such that the corresponding operator K^{(M−1)} creates the state of the quantum chain with (M − 1) arbitrary complex parameters.

Usually the expressions for (τ_{m}^{(g)})^{N}, (θ_{m}^{(g)})^{N}, given by (123), are called the solitonic solutions of discrete Hirota-type equations. g is the number of solitons, e^{2imϕ_{n}} is nth solitonic wave (or exponent) and f_{n} is the amplitude of nth partial wave. Q_{ϕ_{n}}-mapping, ϕ_{n} ∈ 𝔉_{M}, increases the number of the solitons, hence it is the Bäcklund transformation.

4.3. Parameterization of modified Q-operators. Turn now to the parameterizations of the matrix elements of the finite dimensional counterpart of the transformation Q_{ϕ_{n}} – the matrix Q_{ϕ_{n}}. By definition (76), it is constructed with the help of R-matrices, and to parameterize it, one has to point out the Fermat curve parameters (92).

All the ingredients are already prepared: one has just substitute the parameterizations (117,118,119,121) into (90) and (91). As the result, R_{m,ϕ_{n}}^{(n)}-matrices, entering (76), have the arguments (see (92)):

\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)}} = w_{ϕ_{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} (132)

and due to (87)

x_{2,m}^{(n)} = ω^{−1/2}κ \frac{(τ_{m}^{(n)})^{2}}{τ_{m−1}^{(n)}τ_{m+1}^{(n)}} . (133)

[figure: {u_{m}, w_{m}}_{m=1}^{M}; {u′′_{m}, w′′_{m}}_{m=1}^{M}; {ũ′_{m}, w̃′_{m}}_{m=1}^{M}; {u′_{m}, w′_{m}}_{m=1}^{M}; Q_{ϕ_{1}}; Q_{ϕ_{2}}; Q_{ϕ_{2}}; Q_{ϕ_{1}}]
Figure 1. Two different ordering of Q_{ϕ_{1}} and Q_{ϕ_{2}}.

The trace of quantum monodromy (76) may be calculated explicitly in the basis (6)

⟨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})} . (134)

Thus, the operator K^{(g)}, (75), calculated as the product of g Q^{(n)}-operators, is defined explicitly.

4.4. Permutation of the modified Q-operators. For a given state {u_{m}, w_{m}}_{m=1}^{M} let us consider two successive Q-transformations (68) with different ϕ-parameters ϕ_{1} and ϕ_{2}. For the simplicity let {u_{m}, w_{m}}_{m=1}^{M} be a solitonic state, described by (117) with (123). These two transformations may be applied in two different orders, but in our definition of h^{(g)} (125) as the symmetrical functions of the pairs (ϕ_{k}, f_{k}) the result should be the same. Of course, the intermediate states are different. These two different ordering are shown in Fig. (1).

In the terms of the transfer matrices and modified Q-operators the diagram means t(λ; {u_{m}, w_{m}}_{m=1}^{M}) Q_{ϕ_{1}} = Q_{ϕ_{1}} t(λ; {u′_{m}, w′_{m}}_{m=1}^{M}) ,

t(λ; {u′_{m}, w′_{m}}_{m=1}^{M}) Q_{ϕ_{2}} = Q_{ϕ_{2}} t(λ; {u′′_{m}, w′′_{m}}_{m=1}^{M}) ,

(135)

versus

t(λ; {u_{m}, w_{m}}_{m=1}^{M}) Q̃_{ϕ_{2}} = Q̃_{ϕ_{2}} t(λ; {ũ′_{m}, w̃′_{m}}_{m=1}^{M}) ,
t(λ; {ũ′_{m}, w̃′_{m}}_{m=1}^{M}) Q̃_{ϕ_{1}} = Q̃_{ϕ_{1}} t(λ; {u′′_{m}, w′′_{m}}_{m=1}^{M}) .

(136)

Since the result of both mappings is the same, two products of different Q-operators should be the same up to a multiplier. This multiplier corresponds to the normalization factors (102)

ρ_{Q_{1}}^{−1}ρ_{Q_{2}}^{−1} Q_{ϕ_{1}} Q_{ϕ_{2}} = ρ−1 ; Q̃_{2}ρ−1 ; Q̃_{1} Q̃_{ϕ_{2}} Q̃_{ϕ_{1}} . (137)

This equation is provided actually by an intertwining relation

ρ_{R_{1}}^{−1}ρ_{R_{2}}^{−1} R_{m,ϕ_{1}} R_{m,ϕ_{2}} S_{ϕ_{1},ϕ_{2}} = ρ_{R′_{1}}^{−1}ρ_{R′_{2}}^{−1} S′_{ϕ_{1},ϕ_{2}} R′_{m,ϕ_{2}} R′_{m,ϕ_{1}} , (138)

where S is a sort of the Chiral Potts Model R-matrix. These details are not essential in this paper and besides need a lot of extra notations and definitions.

Eq. (137) provides the following

Proposition 6. Operator K^{(M−1)}, given by (75), and parameterized with the help of τ_{m}^{(n)} and θ_{m}^{(n)}, defined for the minimal complete sequence (131), does not depend (up to a multiplier!) on the ordering of the data {(f_{n}, ϕ_{n})}_{n=1}^{M−1} Thus, without loss of generality, one may fix the ordering in (75) and in 𝔉_{M} as follows

ϕ_{n} = \frac{πn}{M} , n = 1, ..., M − 1 . (139)

4.5. Inhomogeneous chain. In this section we investigated and solved the problem of the isospectrality for the homogeneous initial state (23) of the quantum chain. For the chain of the size M with the periodic boundary conditions the solution is described in the terms of Hirota-type solitonic τ-functions (123) (the number of independent f_{n} is equal to M − 1).

Usual Q-operators are the particular cases of our modified Q-operators. Namely, let us consider the first step of the recursion in n (105) ones more. If ϕ in (108) and (114) is such that ϕ ∉ 𝔉_{M} mod π (this corresponds to a generic value of λ_{ϕ}) then ℤ_{M}-periodicity demands f ≡ 0, and τ′_{m} = θ′_{m} = 1. It means, Q_{ϕ}-transformation is the trivial one, and the corresponding Q_{ϕ}-operator changes nothing. Evidently, such Q-operators Q(λ_{ϕ}) commute with the transfer matrix t(λ) and form the commutative family. They are the usual Baxter Q-operators.

In general, one may start from the arbitrary initial states, when τ_{m}^{(0)} and θ_{m}^{(0)} depend on m. (13) still provides the integrability of the model. But even on the first step one gets the inhomogeneous linear recursion for (τ′_{m})^{N}, see (105). Being considered with m ∈ ℤ_{M}, this linear recursion is the null eigenvector problem for M × M matrix, constructed with the help of (τ_{m})^{N}, (θ_{m})^{N} and λ_{ϕ}^{N}. Such equations may be solved if the determinant of this M × M matrix is zero. This condition defines a hyperelliptic curve Γ_{M−1} with the genus

30 S. PAKULIAK AND S. SERGEEV

g = M − 1 in general position (see the next section for the details). (τ_{m}^{(n)})^{N} and (θ_{m}^{(n)})^{N} may be parameterized in terms of θ-functions on Jac(Γ_{M−1}). Our expressions (125) may be considered as the rational limit of hyperelliptic θ-functions. The most interesting feature of such inhomogeneous models is that one may not construct at all a commutative family of Q-operators.

5. The Baxter equation

In this section we derive the Baxter T − Q equation in the most general operator form. We will use the well known method of the triangulization of the auxiliary L-operators.

5.1. Generic inhomogeneous chain. Let us consider the inhomogeneous quantum chain, such that all u_{m}, w_{m} are in general position. Let the normalization (19) be implied nevertheless. Turn to the quantum intertwining relation (52) and formulas (53, 60, 61).

The auxiliary L-operator (49) has the degeneration point at λ = λ_{ϕ}:

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}) . (140)

Analogously, ℓ̃_{ϕ}^{−1}(λ ↦ λ_{ϕ}) also may be decomposed with the help of two vectors, orthogonal to ones in (140). In the degeneration point quantum intertwining relation (52) may be rewritten in the vector form. To do this, we will use the row vector in (140) and the column vector, orthogonal to it:

ψ_{ϕ,m}^{′∗} =^{def} (1 , −ω^{1/2} \frac{u_{ϕ,m}}{λ_{ϕ}} x_{ϕ}) and ψ′′_{ϕ,m} =^{def} ( \begin{matrix}ω^{1/2} \frac{u_{ϕ,m}}{λ_{ϕ}} x_{ϕ} \\ 1 \end{matrix} ) . (141)

Here we use the notation ψ for the column vectors with operator-valued entries, and ψ^{∗} for the similar row vectors. The following two equations are the triangular form of intertwining relation (52) (the dot stands for the matrix multiplication):

ψ_{ϕ,m}^{′∗} · ℓ_{m}(λ_{ϕ}; u_{m}, w_{m}) R_{m,ϕ} = R′_{m,ϕ} ψ_{ϕ,m+1}^{′∗} , (142)

and

ℓ_{m}(λ_{ϕ}; u′_{m}, w′_{m}) · R_{m,ϕ} ψ′′_{ϕ,m+1} = ψ′′_{ϕ,m} R′′_{m,ϕ} . (143)

Here

R′_{m,ϕ} = \frac{u_{m}}{u_{ϕ,m+1}} x_{m} R_{m,ϕ} x_{ϕ}^{−1} ,

R′′_{m,ϕ} = ω^{1/2} \frac{u_{ϕ,m+1} w_{m}}{λ_{ϕ}} z_{m} R_{m,ϕ} x_{ϕ} ,

(144)

and the recursion relations (60, 61) are implied. Equations (142, 143, 144) can be verified by the direct calculations in the component form.

Let further Q′_{ϕ} and Q′′_{ϕ} be the traces of the monodromies of R′_{m,ϕ} and R′′_{m,ϕ} similar to (69) and we imply the ℤ_{M} boundary conditions for the chain and recursion (60). For two vectors ψ_{ϕ,1}^{′∗} = ψ_{ϕ,M+1}^{′∗} and ψ′′_{ϕ,1} = ψ′′_{ϕ,M+1} let ψ′_{ϕ,1} and ψ_{ϕ,1}^{′′∗} be their dual vectors ψ^{′∗} · ψ′ = ψ^{′′∗} · ψ′′ = 1 , ψ^{′∗} · ψ′′ = ψ^{′′∗} · ψ′ = 0 , (145) such that the 2 × 2 unity matrix may be decomposed as follows ψ′ ψ^{′∗} + ψ′′ ψ^{′′∗} = 1 . (146) This decomposition allows one to calculate

t(λ_{ϕ}) Q_{ϕ} = tr_{ℂ^{2},ϕ} ((ψ′_{ϕ,1} ψ_{ϕ,1}^{′∗} + ψ′′_{ϕ,1} ψ_{ϕ,1}^{′′∗}) · t̂(λ_{ϕ}) Q̂_{ϕ})

= tr_{ℂ^{2},ϕ} (ψ′_{ϕ,1}ψ_{ϕ,1}^{′∗} · t̂(λ_{ϕ}) Q̂_{ϕ} + t̂(λ_{ϕ}) Q̂_{ϕ} · ψ′′_{ϕ,M+1}ψ_{ϕ,M+1}^{′′∗})
= tr_{ϕ} (ψ_{ϕ,1}^{′∗} · Q̂_{ϕ}ψ′_{ϕ,1} + ψ_{ϕ,1}^{′′∗} · Q̂_{ϕ}ψ′′_{ϕ,1}) = Q′_{ϕ} + Q′′_{ϕ} .

(147)

The last equality here is obtained using recursions (142) and (143). Eq. (147) gives the Baxter equation in its operator form t(λ_{ϕ}) Q_{ϕ} = Q_{ϕ} t′(λ_{ϕ}) = Q′_{ϕ} + Q′′_{ϕ} , (148) where t(λ) is the initial transfer matrix with the parameters u_{m}, w_{m}, and t′(λ) is the transfer matrix with the parameters u′_{m}, w′_{m}. Using (144) and the first relation of (88), one may obtain

(149) Q′_{ϕ} = ∏_{m} \frac{u_{m}}{u_{ϕ,m}} XQ_{ϕ}X^{−1} , Q′′ = ∏_{m} ω^{1/2} \frac{u_{ϕ,m}w_{m}}{λ_{ϕ}} ZQ_{ϕ}X ,

where X and Z are given by

X = ∏_{m=1}^{M} x_{m} , Z = ∏_{m=1}^{M} z_{m} . (150)

Note that in the spectral decomposition of t(λ) and t′(λ) there exists operator Y (15). Q′_{ϕ} may be rewritten as

Q′′_{ϕ} = (∏_{m=1}^{M} − \frac{u_{ϕ,m}w_{m}}{λ_{ϕ}} ) Y X^{−1} Q_{ϕ} X . (151)

Using (134), one may check YQ_{ϕ} = Q_{ϕ}Y (the matrix elements should be taken). Let now (see (19))

∏_{m=1}^{M} \frac{u_{ϕ,m}}{u_{m}} = μ_{ϕ} , ∏_{m=1}^{M} u_{ϕ,m}w_{m} = ω^{−M/2} μ_{ϕ} . (152)

The Baxter equation in the operator form becomes

(153) t(λ_{ϕ}) Q_{ϕ} = Q_{ϕ} t′(λ_{ϕ}) = μ_{ϕ}^{−1} X Q_{ϕ} X^{−1} + \frac{μ_{ϕ}}{(−ω^{1/2}λ_{ϕ})^{M}} Y X^{−1} Q_{ϕ} X .

The natural question arises: what is μ_{ϕ}? To answer it, consider (53) in the degeneration point λ = λ_{ϕ}, applied to the classical monodromy matrix T̂, eqs. (20) and (66). Using the formulas like (142) and (143) for the classical counterpart, one obtains

\begin{matrix}(1 , \frac{u_{ϕ,1}^{N}}{λ_{ϕ}^{N}} ) · T̂(λ_{ϕ}^{N}) = ∏_{m=1}^{M} \frac{u_{m}^{N}}{u_{ϕ,m+1}^{N}} · (1 , \frac{u_{ϕ,M+1}^{N}}{λ_{ϕ}^{N}} ) , \\ T̂(λ_{ϕ}^{N}) · ( \begin{matrix}−\frac{u_{ϕ,M+1}^{N}}{λ_{ϕ}^{N}} \\ 1 \end{matrix} ) = ( \begin{matrix}−\frac{u_{ϕ,1}^{N}}{λ_{ϕ}^{N}} \\ 1 \end{matrix} ) ∏_{m=1}^{M} (−\frac{u_{ϕ,m+1}^{N}w_{m}^{N}}{λ_{ϕ}^{N}} ) . \end{matrix} (154)

When the ℤ_{M} boundary condition u_{ϕ,M+1} = u_{ϕ,1} is imposed, one may see that both μ_{ϕ}^{−N} and μ_{ϕ}^{N}/(λ_{ϕ}^{N})^{M} are the eigenvalues of the classical monodromy matrix, i.e.

J(λ_{ϕ}^{N}, μ_{ϕ}^{N}) =^{def} det (μ_{ϕ}^{−N} − T̂(λ_{ϕ}^{N})) = 0 (155)

or

A(λ_{ϕ}^{N}) + D(λ_{ϕ}^{N}) = \frac{1}{μ_{ϕ}^{N}} + \frac{μ_{ϕ}^{N}}{λ_{ϕ}^{NM}} . (156)

Hence (λ_{ϕ}^{N}, μ_{ϕ}^{N}) is the point of the genus g = M − 1 hyperellitic curve Γ_{M−1}, defined by (155). As it was mentioned in the section (4.5) on page 29, all u_{m}^{N}, w_{m}^{N}, u_{m}^{′N} and w_{m}^{′N} may be parameterized in the terms of θ-functions on Jac(Γ_{M−1}).

Operator Q_{ϕ} makes the isospectrality transformation (148) of the initial inhomogeneous quantum chain. So it may be decomposed similar to (41). If |Ψ_{t}⟩ is the complete basis of the eigenvectors of t(λ) then

Q_{ϕ} = ∑_{t} |Ψ_{t}⟩ q_{t,ϕ} ⟨Ψ′_{t}| . (157)

Using further the explicit formula (134) for the matrix elements of Q_{ϕ} and the properties (82) of w_{p}-functions, one may calculate (taking the matrix elements)

X Q_{ϕ} X^{−1} = Q_{ϕ}[y_{1,m} ↦ ω^{−1}y_{1,m}] . (158)

and besides

\begin{matrix}Y Q_{ϕ} = Q_{ϕ} Y = \\ ∏_{m∈ℤ_{M}} (−ω^{−1/2} \frac{y_{1,m}}{y_{3,m}} \frac{1 − ωx_{3,m}}{1 − ωx_{1,m}} ) · Q_{ϕ}[x_{1,m}, y_{1,m}, x_{3,m} ↦ ωx_{1,m}ω^{−1}y_{1,m}, ωx_{3,m}] . \end{matrix} (159)

The square braces in the right hand sides of these two equations show the changes of the Fermat curve points. The rest x_{j,m} and y_{j,m} are the same in the left and right hand sides.

A glance to the parametrization (132) shows that the simultaneous change of the phases of all, say, y_{1,m} is equivalent to the change of λ_{ϕ} while δ_{ϕ}, μ_{ϕ} and w_{ϕ} remain unchanged. Using (158), we may conclude

XQ_{ϕ}X^{−1} = ∑_{t} |Ψ_{t}⟩ q_{t,ϕ}(λ_{ϕ} ↦ ω^{−1}λ_{ϕ}) ⟨Ψ′_{t}| . (160)

Besides, both |Ψ_{t}⟩ and ⟨Ψ′_{t}| are the eigenvectors of Y, Y |Ψ_{t}⟩ = |Ψ_{t}⟩ ω^{γ}. The operator equation (153) may be rewritten as the functional equation

t(λ) q_{t}(λ, μ) = μ^{−1} q_{t}(ω^{−1}λ, μ) + \frac{μ}{(−ω^{1/2}λ)^{N}} ω^{γ} q_{t}(ωλ, μ) , (161)

where (λ^{N}, μ^{N}) ∈ Γ_{M−1}, eq. (155), and in the decomposition t(λ) = ∑_{m=0}^{M} λ^{−m}t_{m}, t_{M} = (−κ)^{M}ω^{γ}.

Note that equation (161) does not follow from (153) directly. In general, coefficients q_{t,ϕ} = ⟨Ψ_{t}|Q_{ϕ}|Ψ′_{t}⟩ in (153) may have the following structure

q_{t ϕ} = q_{t}(λ_{ϕ}, μ_{ϕ}; {u_{m}, w_{m}}_{m=1}^{M}, {u′_{m}, w′_{m}}_{m=1}^{M}) . (162)

Since the matrix elements of Q_{ϕ} are the rational functions of λ_{ϕ} and of μ_{ϕ}, q_{t} is a rational function of λ_{ϕ} and μ_{ϕ}. Therefore due to (160) the matrix elements of (153) between ⟨Ψ_{t}| and |Ψ′_{t}⟩ are the system of N linear equations for q_{t}(ω^{k}λ_{ϕ}, μ_{ϕ}; ...), k ∈ ℤ_{N}. Since (λ_{ϕ}^{N}, μ_{ϕ}^{N}) ∈ Γ_{M−1}, the rank of this linear system is equal to N −1 for generic λ_{ϕ} (see [23] for the details). Hence the linear system has a unique solution in the class of the rational functions of λ_{ϕ}, μ_{ϕ} up to a multiplier, depending on μ_{ϕ} in general. We will show further that such multiplier may be fixed by an extra relation provided by (159)).

Actually, there are two spectral curves in our considerations. The first one is the classical spectral curve Γ_{M−1} with the point (λ_{ϕ}^{N}, μ_{ϕ}^{N}) (155). But the point (λ_{ϕ}, μ_{ϕ}) belongs to the quantum curve, which is N^{2}-sheeted covering of the classical one.

The considerations above prove the following

Proposition 7. For any isospectral transformation {u_{m}, w_{m}}_{m=1}^{M} ↦ {u′_{m}, w′_{m}}_{m=1}^{M} the coefficients q_{t} of the corresponding modified Q-operator, decomposed as in (157), depend only on the moduli of the spectral curve and on the point on the spectral curve:

q_{t,ϕ} = q_{t}(λ_{ϕ}, μ_{ϕ}) , (163)

and do not depend on “classical times” (i.e. a point on the jacobian), entering to a parameterization of u_{m}, w_{m}, u′_{m}, w′_{m}. In particular, for the homogeneous initial state, q_{t} does not depend on the amplitudes of the solitonic states involved.

This proposition clarifies the derivation of (161) from (153).

5.2. Baxter equation for the homogeneous chain. All the formulas from the previous subsection are valid for the homogeneous initial state (23). Arbitrary value of λ_{ϕ} provides the trivial action of the functional counterpart Q_{ϕ} of Q_{ϕ}, see discussion after proposition 5 on page 25. Such Q-operators commute with t(λ) and form the commutative family. Explicit form of this Q-operator is given by (134) with the homogeneous parameterization p_{1,m} = p_{1}, p_{2,m} = p_{2}, p_{3,m} = p_{3},

\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}} = w_{ϕ} , \frac{y_{1}}{y_{3}} = ω^{1/2} \frac{λ_{ϕ}}{κδ_{ϕ}} , \end{matrix} (164)

where δ_{ϕ}, λ_{ϕ} and w_{ϕ} are given by (30, 107, 121) with generic value of ϕ. Due to (164) Q = Q(λ_{ϕ}, δ_{ϕ}) . (165) Several properties of this Q-operator may be derived with the help of the matrix elements of (134) and (164), and with the help of (82). In particular, (158) and (159) become

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

(166)

The second equation is the consequence of (159) since x_{1}, y_{1}, x_{3} ↦ ωx_{1}, ω^{−1}y_{1}, ωx_{3} ⇔ δ_{ϕ} ↦ ωδ_{ϕ} . (167) Using (166) and (153), 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}) , (168)

where δ_{ϕ}-argument of all Q-s remains unchanged. Recall that t and Q may be diagonalized simultaneously. Comparing (168) and (153), we conclude μ_{ϕ} ≡ δ_{ϕ}^{M}. Further we will omit the subscribe ϕ. Let t(λ) and q_{t}(λ, δ) be the eigenvalues of t(λ) and Q(λ) for the same eigenvector, then (168) provides the functional equation

t(λ)q_{t}(λ, δ) = δ^{−M}q_{t}(ω^{−1}λ, δ) + (−ω^{−1/2} \frac{δ}{λ})^{M} ω^{γ}q_{t}(ωλ, δ) . (169)

Here q_{t} is a meromorphic function on the Baxter curve (see the set of definitions (31, 107) etc.):

(δ^{−1}, δ^{∗−1}) ∈ Γ_{B} ⇔ \frac{1}{δ^{N}} + \frac{1}{δ^{∗N}} = 1 − κ^{N} \frac{1}{δ^{N}δ^{∗N}} , λ = δδ^{∗} . (170)

More detailed investigation of (169) in the spirit of Proposition 7 on page 34 shows that for generic λ any solution of (169) such that t(λ) is a polynomial and q_{t}(λ, δ) is a meromorphic function on the curve (170), gives the eigenvalue of t(λ) and Q(λ). The second equation of (166) provides the equation, fixing the δ ambiguity of the solution of the Baxter equation (169):

ω^{γ} q_{t}(λ, δ) = (−\frac{λ}{δ} \frac{1 − δ}{κ − ω^{1/2}δ} )^{M} q_{t}(λ, ωδ) . (171)

5.3. Evolution operator. Q_{ϕ}-transformation and its quantum counterpart Q_{ϕ} may be interpreted as a kind of evolution, depending strongly on their spectral parameter λ_{ϕ}. Evolution in more usual sense may be obtained in the limit when λ_{ϕ}^{−1} ↦ 0. Such a phenomenon, when a “physical” evolution operator is Q-operator in the singular point, appeared for example in the quantum Liouville model [9].

Evolution transformation 𝔈 for the quantum relativistic Toda chain may be produced by non-local similarity transformation of the quantum L-operators (1):

𝔈(ℓ_{m})(λ) = A_{m}(λ)^{−1} ℓ_{m}(λ) A_{m+1}(λ) , (172)

where 2 × 2 matrix A_{m} has the operator-valued entries:

A_{m}(λ) = ( \begin{matrix}1 & −λ^{−1}w_{m−1}^{−1} \\ w_{m} & −ω^{−1/2}κλ^{−1} \end{matrix} ) . (173)

The similarity transformations do not change the transfer matrix of the model with the periodical boundary conditions. The explicit form of 𝔈-transformation may be obtained from (172):

𝔈(u_{m}) = ω^{−1/2}w_{m}^{−1} ,
𝔈(w_{m}) = (1 − ω^{−1/2}κw_{m}^{−1}w_{m−1})^{−1}ω^{1/2}w_{m−1}u_{m}w_{m+1}(1 − ω^{−1/2}κw_{m+1}^{−1}w_{m}) .

(174)

One may check directly that 𝔈-transformation is the canonical one. As usual, at the root of unity we have to separate

𝔈(f) = E E(f) E^{−1} , (175) where E is the finite dimensional operator, and E acts on Nth powers of the Weyl elements u_{m}^{N}, w_{m}^{N} as follows

\{ \begin{matrix}E(u_{m}^{N}) = − \frac{1}{w_{m}^{N}} , \\ E(w_{m}^{N}) = − u_{m}^{N}w_{m}^{N}w_{m−1}^{N} \frac{w_{m+1}^{N} + κ^{N}w_{m}^{N}}{w_{m}^{N} + κ^{N}w_{m−1}^{N}} . \end{matrix} (176)

For the given set of {u_{m}, w_{m}}_{m=1}^{M} parameterized in the terms of τ_{m} and θ_{m} (117), denote τ_{m} = τ_{0,m} , θ_{m} = τ_{1,m} , (177) where we have introduced the extra auxiliary subscript of τ-function. The first equation in (176) yields E(τ_{l,m}) = τ_{l+1,m} , (178) and the second equation in (176) gives the following “equation of motion” τ_{l,m+1}^{N}τ_{l,m−1}^{N} + κ^{N}(τ_{l,m}^{N})^{2} = (1 + κ^{N})τ_{l+1,m}^{N}τ_{l−1,m}^{N} . (179)

Let τ_{0,m}^{N} = τ_{m}^{N} is given by the n-solitonic expression (123). Then the complete solution of (179) is given by (τ_{l,m}^{(n)})^{N} = h^{(n)}({f_{k}e^{2imϕ_{k}+2ilψ_{k}}}_{k=1}^{n}) , (180) where

e^{2iψ_{k}} = s_{k}(1) = \frac{Δ_{k}^{∗}}{Δ_{k}} \frac{Δ_{k} − 1}{Δ_{k}^{∗} − 1} , (181)

see (125,126) for the notations. The substitution (180) makes (179) the identity (see the Appendix).

Comparing (180) with the definition of h^{(g)} and τ_{m}^{(g)} (123,125), we see that in fact τ_{l,m}^{(n)} = τ_{m}^{(n+l)} , (182) where the initial set of {ϕ_{k}, f_{k}}_{k=1}^{n} for h^{(n)} and τ_{m}^{(n)} is enlarged formally by {ϕ′_{k}, f′_{k}}_{k=n+1}^{n+l} with f′_{k} ≡ 0 and all ϕ′_{k} equal and obeying formally δ_{ϕ′_{k}}^{∗N} = 1 ⇒ δ_{ϕ′_{k}}^{−1} = λ_{ϕ′_{k}}^{−1} = 0 . (183)

38 S. PAKULIAK AND S. SERGEEV

This establishes the relation between the evolution (174,176) and Q_{ϕ}, Q_{ϕ}. In particular, the finite dimensional counterpart E of the evolution operator 𝔈 may be obtained in the limit

E = lim_{δ=∞, δ^{∗}=1} Q(δ, δ^{∗}) . (184)

Eq. (134) gives the explicit form of the matrix elements of E on the l-th step of the evolution

⟨α|E_{l}|β⟩ = ∏_{m=1}^{M} ω^{(α_{m}−β_{m+1})β_{m}} w_{p_{2,l,m}}(β_{m+1} − β_{m}) , (185)

where

x_{2,l,m} = ω^{−1/2}κ \frac{τ_{l,m}^{2}}{τ_{l,m−1}τ_{l,m+1}} , y_{2,l,m} = κ′ \frac{τ_{l−1,m}τ_{l+1,m}}{τ_{l,m−1}τ_{l,m+1}} (186)

with

κ^{′N} = 1 + κ^{N} . (187)

In particular, N-bilinear relation (179) allows one to parameterize the Fermat relation for the coordinates of p_{j,m}. Let

\overline{θ}_{m} = τ_{−1,m} . (188)

Then

\begin{matrix}y_{1,m}^{(n)} = ω^{1/2} \frac{λ_{n}w_{ϕ_{n}}κ′}{κδ_{n}} \frac{\overline{θ}_{m}^{(n)}θ_{m−1}^{(n−1)}}{τ_{m}^{(n)}τ_{m−1}^{(n−1)}} , \\ y_{2,m}^{(n)} = κ′ \frac{\overline{θ}_{m}^{(n)}θ_{m}^{(n)}}{τ_{m−1}^{(n)}τ_{m+1}^{(n)}} , y_{3,m}^{(n)} = w_{ϕ_{n}}κ′ \frac{\overline{θ}_{m}^{(n)}θ_{m}^{(n−1)}}{τ_{m+1}^{(n)}τ_{m−1}^{(n−1)}} . \end{matrix} (189)

6. Quantum separation of variables

Turn to the last subject of this paper – the quantum separation of variables, or functional Bethe ansatz.

In the previous sections we have established the following. Let t(λ) ≡ t^{(0)}(λ) be the transfer matrix of the homogeneous quantum chain of the length M. There exists the family of isospectral to t(λ) transfer matrices t^{(M−1)}(λ), parameterized by (M − 1) complex variables f_{1}, ..., f_{M−1} – the amplitudes of Hirota-type solitons in the minimal set (131). The isospectrality means that for any generic values of f_{1}, ..., f_{M−1} t^{(M−1)}(λ) has the same eigenvalues as the initial t(λ). Therefore only the eigenvectors |Ψ_{t}^{(M−1)}⟩ depend on f_{1}, ..., f_{M−1}. We may choose the values of f_{1}, ..., f_{M−1} in an appropriate way, such that |Ψ_{t}^{(M−1)}⟩ would become more simple then the initial unknown |Ψ_{t}⟩. Since the similarity matrix K^{(M−1)} is known explicitly, any exact information about |Ψ^{(M−1)}⟩ solves the eigenstate problem for the initial homogeneous quantum relativistic Toda chain.

6.1. Null subspaces of b(λ). Now we have to understand what special values of f_{1}, ..., f_{M−1} are the most useful ones. To do this, we should turn back to the degenerated ℓ̃_{ϕ}-operator and write out equations (142) and (143) for the monodromies of ℓ and R

\begin{matrix}ψ_{ϕ,1}^{′∗} · t̂(λ_{ϕ}) Q̂_{ϕ} = Q̂′_{ϕ} ψ_{ϕ,M+1}^{′∗} , \\ t̂(λ_{ϕ}) Q̂_{ϕ} · ψ′′_{ϕ,M+1} = ψ′′_{ϕ,1} Q̂′′_{ϕ} . \end{matrix} (190)

In components,

(a(λ_{ϕ}) − ω^{1/2} \frac{u_{ϕ,1}}{λ_{ϕ}} x_{ϕ}c(λ_{ϕ})) Q̂_{ϕ} = Q̂′_{ϕ} ,

(b(λ_{ϕ}) − ω^{1/2} \frac{u_{ϕ,1}}{λ_{ϕ}} x_{ϕ}d(λ_{ϕ})) Q̂_{ϕ} = −ω^{1/2} \frac{u_{ϕ,M+1}}{λ_{ϕ}} Q̂′_{ϕ} x_{ϕ} ,

(191)

and

a(λ_{ϕ})Q̂_{ϕ} ω^{1/2} \frac{u_{ϕ,M+1}}{λ_{ϕ}} x_{ϕ} + b(λ_{ϕ}) Q̂_{ϕ} = ω^{1/2} \frac{u_{ϕ,1}}{λ_{ϕ}} x_{ϕ}Q̂′′_{ϕ} ,

c(λ_{ϕ})Q̂_{ϕ} ω^{1/2} \frac{u_{ϕ,M+1}}{λ_{ϕ}} x_{ϕ} + d(λ_{ϕ}) Q̂_{ϕ} = Q̂′′_{ϕ} .

(192)

Using relation R_{m,ϕ}x_{m} = x_{ϕ}R_{m,ϕ} (see the (88)) one may obtain from (144)

Q̂′_{ϕ} = (∏_{m=1}^{M} \frac{u_{m}}{u_{ϕ,m+1}} ) X Q̂_{ϕ} X^{−1} x_{1}x_{ϕ}^{−1} , (193)

and

Q̂′′_{ϕ} = (∏_{m=1}^{M} ω^{1/2} \frac{u_{ϕ,m+1}w_{m}}{λ_{ϕ}} ) Z Q̂_{ϕ} X x_{1}^{−1}x_{ϕ} . (194)

Let the initial chain is the homogeneous one. If ϕ ∈ 𝔉_{M}, then

u_{ϕ,m}^{N} = u_{ϕ,M+m}^{N} = (−)^{N−1}δ_{ϕ}^{N} \frac{1 − f_{ϕ}e^{2iϕ(m−1)}}{1 − f_{ϕ}e^{2iϕm}} . (195)

40 S. PAKULIAK AND S. SERGEEV

Note,

∏_{m∈Z_{M}} u_{ϕ,m} = (−ω^{−1/2}δ_{ϕ})^{M} (196)

does not depend on f_{ϕ}. Consider the limit when f_{ϕ} ↦ 1. Due to (196) Q̂_{ϕ}, Q̂′_{ϕ} and Q̂′′_{ϕ} are regular, but u_{ϕ,1} = u_{ϕ,M+1} = 0 , (197) and (191,192) give

(198) a(λ_{ϕ}) Q̃_{ϕ} = Q̃′_{ϕ} , d(λ_{ϕ}) Q̃_{ϕ} = Q̃′′_{ϕ} , b(λ_{ϕ}) Q̃_{ϕ} = 0 . The last expression is important. Taking the trace over ϕ-th space, one obtains b(λ_{ϕ}) Q_{ϕ} = 0 if f_{ϕ} = 1 . (199) Recall, the phase of λ_{ϕ} is the parameters of Q_{ϕ}.

Turn now to the operator K^{(M−1)} and consider the limit when all f_{n} ↦ 1, n = 1, ..., M −1. Each Q_{ϕ_{n}} in (75) depends essentially on the phase of λ_{ϕ_{n}}, n = 1, ..., M − 1. Since K^{(M−1)} is the symmetrical operator with respect to any permutation of {(f_{n}, ϕ_{n})}_{n=1}^{M−1}, see Proposition 6 on page 29, the following proposition holds Proposition 8. Let λ_{ϕ_{n}} be the spectral parameter of Q_{ϕ_{n}} in the decomposition (75) of the minimal K^{(M−1)} (131). Then b(λ_{ϕ_{n}}) K^{(M−1)} = 0 n = 1, ..., M − 1 if all f_{n} ↦ 1 . (200)

The structure of eigenvectors of b(λ) was described by eq. (46). Comparing (46) and (200), and taking into account YQ_{ϕ} = Q_{ϕ}Y for any Q_{ϕ}, we may conclude that in the limit f_{k} ↦ 1 ∀k

K^{(M−1)}({λ_{ϕ_{k}}}_{k=1}^{M−1}) = ∑_{γ∈ℤ_{N}} |{λ_{k}}_{k=1}^{M−1}, γ⟩ ⟨χ_{γ}| . (201)

Here |{λ_{k}}_{k=1}^{M−1}, γ⟩ are the eigenvectors of b̃ (45,46), and χ_{γ} are some vectors, such that ⟨χ_{γ}| Y = ω^{γ} ⟨χ_{γ}|, where Y is given by (15).

¿From the other side, using the natural decomposition (71) and (41), we should have for the generic values of f_{k}

K^{(M−1)}({λ_{ϕ_{k}}}_{k=1}^{M−1}) = ∑_{t} |Ψ_{t}⟩ ( ∏_{k=1}^{M−1} q_{t}(λ_{ϕ_{n}}, δ_{ϕ_{n}})) ⟨Ψ_{t}^{(M−1)}| , (202)

where the summation is taken over all the eigenvalues t of t. Functions q_{t}(λ, δ) obey the Baxter equation (169) and due to proposition (7) on page (34) do not depend on f_{1}, ..., f_{M−1}. Comparing two decompositions (201) and (202) at f_{k} ↦ 1, we may conclude that the eigenvectors ⟨Ψ_{t}^{(M−1)}| stick together in this limit

⟨Ψ_{t}^{(M−1)}| ↦ ⟨χ_{γ}| ∀ t = {t_{1}, ..., t_{M−1}, t_{M} = (−κ)^{M}ω^{γ}} . (203)

So the functional Bethe ansatz formula (48) follows from (201) and (202) up to a normalization. In addition, we know explicitly the corresponding projector K^{(M−1)}.

Detailed investigation shows that the explicit form of ⟨χ_{γ}| depends on a limiting procedure f_{n} ↦ 1. The reason is that when f_{n} ↦ 1, n = 1, ..., g, then τ_{m}^{(g)} ↦ 0, m = 0, ..., g − 1. This follows from the recursive definition of h^{(g)} (125). The proof is based on Proposition 10 on page 47 of the Appendix. Because of the parameterization of u_{m}^{(n)} and p_{j,m}^{(n)} contains the ratios of τ_{m}^{(n)}, the ambiguity arises. Some of u_{m}^{(n)} and p_{j,m}^{(n)} of the form 0/0 depends on the details of the limiting procedure.

6.2. Limiting procedure. In this subsection we describe one of the possible limiting procedures. Most simple K^{(M−1)}({λ_{k}}) appears when we choose u_{m}^{(M−1)} = 0 for m = 1...M − 1. Let us consider the set of infinitely small numbers,

ε_{k} ↦ 0 , k = 1...M − 1 , (204)

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 , (205)

u_{M}^{(M−1)} = −ω^{−1/2} \frac{(δ_{1}...δ_{M−1})^{M−1}}{ε_{1}...ε_{M−1}} . (206)

Recursion (60,61) with

κ_{ϕ_{n}} = \frac{κ}{δ_{n}} (207)

42 S. PAKULIAK AND S. SERGEEV

yields

u_{m}^{(n)} = −ω^{−1/2} \frac{ε_{n−m+1}}{δ_{1}...δ_{n}} + o(ε_{1}) , m = 1...n (208)

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

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 . (209)

In the limit ε_{k} = 0 we may choose

\frac{τ_{n}^{(n)}}{θ_{n}^{(n)}} = ∏_{k=1}^{n} ω^{−1/2} \frac{δ_{k}}{λ_{k}w_{k}} , (210)

where w_{k} are given by (121), and

θ_{m}^{(n)} = (ω^{1/2}κ)^{(n−m)(n−m−1)/2} \frac{θ_{n}^{(n)}}{(δ_{1}...δ_{n})^{n−m}} . (211)

Nth powers of these formulas follow from Proposition 12 on page (48) of the Appendix. Eq.

(211) gives

\frac{w_{m+1}^{(n)}}{w_{m}^{(n)}} = ω^{1/2}κ , m = 0...n − 1 . (212)

In the limit when ϵ_{k} ↦ 0 the regular p_{j,m}^{(n)}, j = 1, 2, 3, in Q_{ϕ_{n}} (134), entering the decomposition of K^{(M−1)}, are given by (132). Below we will list the singular p_{j,m}^{(n)}, using notations (83,84).

• \underline{m = 1...n − 1}:

p_{3,m}^{(n)} = q_{1} , p_{2,m}^{(n)} = O(p_{1,m}^{(n)}) , x_{1,m}^{(n)} = \frac{1}{ω^{1/2}κ} \frac{ε_{n−m+1}}{ε_{n−m}} . (213)

• \underline{m = n}: p_{1,n}^{(n)} = q_{0} , p_{2,n}^{(n)} = q_{∞} ∀ n (214)

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

p_{3,M−1}^{(M−1)} = q_{∞} . (215)

• \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} , (216)

and p_{2,M−1}^{(n)} = p_{3,M−1}^{(n)} = q_{∞} ∀ n . (217)

• \underline{m = M}: p_{1,M}^{(n)} = q_{∞} , p_{2,M}^{(n)} = p_{3,M}^{(n)} = q_{0} ∀ n. (218) Substituting these expressions into (134), one obtains the following form of Nth 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} (219)

and for the last (M − 1)-th Q-matrix

⟨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}) . (220)

Here Φ is given by (81). Explicit form of the modified Q-operators (219) and (220) allows one to prove the following Proposition 9. K^{(M−1)}({λ_{k}}_{k=1}^{M−1}) (z_{m} − z_{m+1}) = 0 , m ∈ ℤ_{M} . (221) Proof. The matrix elements of Q^{(n)} depend on β_{M} − α_{M}, therefore Q^{(n)} z_{M} = z_{M} Q^{(n)} ∀n. The set of delta-symbols in each Q^{(n)} gives Q^{(n)} z_{m} = z_{m−1} Q^{(n)}, m = 2, ..., n + 1. Also the

44 S. PAKULIAK AND S. SERGEEV

matrix elements of each Q^{(n)} depend on β_{1} + β_{M}, hence Q^{(n)} (z_{1} − z_{M}) = 0 ∀n. Therefore ∀m

(222) K^{(M−1)} (z_{m} − z_{M}) = Q^{(1)}...Q^{(M−m)} (z_{1} − z_{M}) Q^{(M−m+1)}...Q^{(M−1)} = 0.

□

So, in our limiting procedure, χ_{γ} is defined by

⟨χ_{γ}| (z_{m} − z_{m+1}) = 0 , ⟨χ_{γ}| Y = ω^{γ} ⟨χ_{γ}| . (223)

Hence the matrix element of χ_{γ} is

⟨χ_{γ}|α⟩ = χ_{γ}(\overline{α}) ≡ χ_{γ}(\overline{α} mod N), (224)

where

\overline{α} =^{def} ∑_{m∈ ℤ_{M}} α_{m} . (225)

Discrete function χ_{γ}(\overline{α}) is defined by

\frac{χ_{γ}(\overline{α} + M)}{χ_{γ}(\overline{α})} = (−ω^{−1/2})^{M} ω^{γ−\overline{α}} . (226)

Solution of (226) depends strongly on M mod N.

7. Discussion

In this paper we investigated one of the simplest integrable model, associated with the local Weyl algebra at the root of unity. All such models always contain a classical discrete dynamic of parameters. Nontrivial solution of this classical part of the model provides the solution of the isospectrality problem of the spin part. Well known nowadays results by Sklyanin, Kuznetsov at al reveal that in the classical limit of the usual Toda chain (and many other models) Baxter’s quantum Q-operator is related to the Bäcklund transformation of the classical system, see e.g. [1, 25, 26, 23, 27] etc. In our case we have the Bäcklund transformation of the classical counterpart and modified Q-operator in the quantum space simultaneously. It is unusual that solving the quantum isospectrality problem, we miss the commutativity of the modified Q-operators.

Our results are 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 (75). We hope that the solution of the isospectrality problem will help to solve the model with arbitrary N in the thermodynamic limit exactly.

As one particular application of the isospectrality 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 functional Bethe ansatz. In this paper we have established it explicitly, but for the product of modified operators Q.

Note in conclusion that this method may be applied to any model, based on the local Weyl algebra. One may mention the chiral Potts model [16, 17] and the Zamolodchikov-Bazhanov-Baxter model in the vertex formulation [18]. Moreover, 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 [15].

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.

Appendix A. Fay’s identity in the rational limit

It is well known what is the rational limit of Θ-function on a generic algebraic curve. Θ-function by definition is

(227) Θ^{(g)}(z) = ∑_{n∈ℤ^{g}} e^{iπ(n,Ωn) + 2iπ(n,z)} ,

where z = (z_{1}, ..., z_{g}) ∈ ℂ^{g} and the imaginary part of the symmetrical period matrix Ω is positively defined. Θ-function may be defined recursively as

Θ^{(g)}(z) = ∑_{n_{g}∈ℤ} Θ^{(g−1)}(z′ + Ω′_{g}n_{g}) e^{iπΩ_{g,g}n_{g}^{2} + 2iπn_{h}z_{g}} , (228)

where z′ = (z_{1}, ..., z_{g−1}) and Ω′_{g} = (Ω_{1,g}, ..., Ω_{g−1,g}).

The rational limit corresponds to

e^{iπΩ_{n,n}+2iπz_{n}} = −f_{n} , (229)

and

e^{iπΩ_{n,n}} ↦ 0 , e^{iπΩ_{k,n}} ↦ \frac{(q_{k} − q_{n}) (p_{k} − p_{n})}{(q_{k} − p_{n}) (p_{k} − q_{n})} . (230)

46 S. PAKULIAK AND S. SERGEEV

Explicit details of the limiting procedure are rather tedious. We should better invent all the necessary components in the rational limit directly and formulate the rational limit of the Fay identity. The proof of all the propositions is analogous to that in the algebraic geometry with the elementary replacement of the usual Θ-function methods to the analysis of the polynomial structure and intensive use of the Main Algebraic Theorem.

A.1. Rational limit of arbitrary algebraic curve jacobian. Let {p_{n}, q_{n}}_{n=1}^{g} be a set of 2g complex parameters. “Θ-function” in the rational limit is a function of g free complex arguments f_{1}, ..., f_{g}, the set {p_{n}, q_{n}}_{n=1}^{g} is its parameters. “Θ-function” may be defined recursively with respect to g as follows

\begin{matrix}Θ^{(0)} = 1 , Θ^{(1)}(f_{1}) = 1 − f_{1} , \\ Θ^{(2)}(f_{1}, f_{2}) = 1 − f_{1} − f_{2} + d_{1,2}f_{1}f_{2} , \end{matrix} (231)

and so on,

Θ^{(g)}({f_{n}}_{n=1}^{g}) = Θ^{(g−1)}({f_{n}}_{n=1}^{g−1}) − f_{g} Θ^{(g−1)}({f_{n}d_{n,g}}_{n=1}^{g−1}) . (232)

The recursion follows from (228), where n_{g} = 0, 1. Parameters {p_{n}, q_{n}}_{n=1}^{g} enter the definition of the phase shifts

d_{n,m} = \frac{(p_{n} − p_{m}) (q_{n} − q_{m})}{(p_{n} − q_{m}) (q_{n} − p_{m})} . (233)

It is useful to introduce function σ_{n}(z)

σ_{n}(z) = \frac{p_{n} − z}{q_{n} − z} . (234)

Let by definition

H^{(g)}({f_{n}}_{n=1}^{g}) = Θ^{(g)}({f_{n} ∏_{m≠n} σ_{n}(q_{m})}_{n=1}^{g}) . (235)

Recursion definition of H^{(g)} is

H^{(g)}({f_{n}}_{n=1}^{g}) = H^{(g−1)}({f_{n}σ_{n}(q_{g})}_{n=1}^{g−1}) − f_{g} ∏_{m=1}^{g−1} σ_{g}(q_{m}) H^{(g−1)}({f_{n}σ_{n}(p_{g})}_{n=1}^{g−1}) , (236)

where σ_{n}(q_{g})d_{n,g} = σ_{n}(p_{g}) is taken into account.

Proposition 10. (Zeros of Θ-function)

H^{(g)}({f_{n}}_{n=1}^{g}) = 0 ⟺ f_{n} = ∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1} , (237)

where z_{m} ∈ ℂ, m = 1, ..., g − 1.

Proof. We will use the mathematical induction method. This proposition is obviously valid for g = 1. Suppose, one has H^{(g−1)}({∏_{m=1}^{g−2} σ_{n}(z_{m})^{−1}}_{n=1}^{g−1}) = 0 ∀ z_{1}...z_{m}, p_{1}...p_{g−1}, q_{1}...q_{g−1} for some g. Consider now H^{(g)}. Due to recursion relation (236)

H^{(g)}(\{∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}\}_{n=1}^{g}) = \frac{P_{g}(z_{1}, ..., z_{m})}{∏_{n<n′}(q_{n} − q_{n′}) ∏_{n=1}^{g} ∏_{m=1}^{g−1}(p_{n} − z_{m})} , (238)

where P_{g} is a polynomial of the power g with respect to each z_{m}.

Consider the case when z_{g−1} = q_{g}, so ∏_{m} σ_{g}(z_{m})^{−1} = 0. From recursion relation (236) and the supposition of the induction it follows that H^{(g)} = 0, so due to the symmetry of H^{(g)} we obtain

H^{(g)}({∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}}_{n=1}^{g−1}) = 0 if z_{m} = q_{n} ∀ n, m . (239)

Next consider the case when z_{g−1} ↦ p_{g}, hence ∏ σ_{n}(z_{m}) ∼ (p_{g} − z_{g−1})^{−1} ↦ ∞. Due to recursion relation (236) and the supposition of the induction it follows that H^{(g)} is regular in z_{g−1} = p_{g}. Again, due to the symmetry of H^{(g)}

Res_{z_{m}=p_{n}}H^{(g)}({∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}}_{n=1}^{g}) = 0 ∀ n, m . (240)

Therefore,

P_{g}(z_{1}...z_{g−1}) ∼ ∏_{n=1}^{g} ∏_{m=1}^{g−1} (p_{n} − z_{m}) (q_{n} − z_{m}) . (241)

This product is the polynomial of the power 2g with respect to each z_{m}, and this contradicts with the structure of P_{g}, therefore P_{g} ≡ 0. Hence we prove

H^{(g)}({∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}}_{n=1}^{g}) = 0 ∀ z_{1}, ..., z_{g−1} . (242)

Backward, consider equation H^{(g)}({f_{n}}_{n=1}^{g}) = 0. Since H^{(g)} is the first power polynomial with respect to each f_{n}, solution of H^{(g)} = 0 is a rational ℂ^{g−1} variety, so as f_{n} = ∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}. □

Proposition 11. (Casoratti determinant representation)

H^{(g)}(\{f_{n}\}_{n=1}^{g}) = \frac{det |q_{j}^{i−1} − f_{j} p_{j}^{i−1}|_{i,j=1}^{g}}{∏_{i>j} (q_{i} − q_{j})} . (243)

Proof. Evidently, zeros of the right hand side of (243) correspond to the case when the columns of the matrix |q_{j}^{i−1} − f_{j} p_{j}^{i−1}|_{i,j=1}^{g} are linearly dependent, i.e. ∃ c_{i}, i = 1, ..., g:

∑_{i=1}^{g} c_{i} q_{j}^{i−1} = f_{j} ∑_{i=1}^{g} c_{i} p_{j}^{i−1} . (244)

Let ∑_{i=1}^{g} c_{i} z^{i−1} = ∏_{m=1}^{g−1} (z − z_{m}), then the determinant is zero if and only if

f_{j} = ∏_{m=1}^{g−1} \frac{q_{j} − z_{m}}{p_{j} − z_{m}} = ∏_{m=1}^{g−1} σ_{j}(z_{m})^{−1} . (245)

Therefore, due to proposition (10) the determinant is proportional to H^{(g)}. The denominator in the right hand side of (243) is the subject of normalization when f_{j} = 0. □

Proposition 12. (Theta-functions and prime forms)

H^{(g)}(\{f_{n} = σ_{n}(λ) ∏_{m=1}^{g} σ_{n}(z_{m})^{−1}\}_{n=1}^{g}) = (∏_{m=1}^{g} \frac{λ − z_{m}}{λ − q_{m}} ) H^{(g)}(\{f_{n} = ∏_{m=1}^{g} σ_{n}(z_{m})^{−1}\}_{n=1}^{g}) . (246)

Proof. A simple application of the recursion relation (236) to H^{(g+1)}({...}) = 0, where p_{g+1} = ∞ and q_{g+1} = λ. □

A.2. The Fay identity.

Proposition 13. (Fay’s identity)

\begin{matrix}(A − D)(C − B) H^{(g)}(\{f_{n} \frac{σ_{n}(A)}{σ_{n}(B)}\}_{n=1}^{g}) H^{(g)}(\{f_{n} \frac{σ_{n}(C)}{σ_{n}(D)}\}_{n=1}^{g}) + \\ (A − B)(D − C) H^{(g)}(\{f_{n} \frac{σ_{n}(A)}{σ_{n}(D)}\}_{n=1}^{g}) H^{(g)}(\{f_{n} \frac{σ_{n}(C)}{σ_{n}(B)}\}_{n=1}^{g}) = \\ (A − C)(D − B) H^{(g)}(\{f_{n}\}_{n=1}^{g}) H^{(g)}(\{f_{n} \frac{σ_{n}(A)σ_{n}(C)}{σ_{n}(B)σ_{n}(D)}\}_{n=1}^{g}) . \end{matrix} (247)

Proof. Let LHS({f_{n}}_{n=1}^{g}) be the left hand side of (247). It is a second order polynomial with respect to each of f_{1}, ..., f_{g}. Using proposition (12), it may be shown that

\begin{matrix}LHS(\{f_{n} = \frac{σ_{n}(B)σ_{n}(D)}{σ_{n}(A)σ_{n}(C)} ∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}\}_{n=1}^{g}) = \\ LHS(\{f_{n} = ∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}\}_{n=1}^{g}) = 0 \end{matrix} (248)

for any set of z_{1}, ..., z_{g−1}. Due to Proposition (10), it means unambiguously

LHS(\{f_{n}\}) = const H^{(g)}(\{f_{n}\}_{n=1}^{g}) H^{(g)}(\{f_{n} \frac{σ_{n}(A)σ_{n}(C)}{σ_{n}(B)σ_{n}(D)}\}_{n=1}^{g}) . (249)

Taking, for example, f_{n} = \frac{σ_{n}(D)}{σ_{n}(A)} ∏_{m=1}^{g−1} σ_{n}(z_{m})^{−1}, we fix the constant and obtain finally the given form (247) of the coefficients. □ A.3. Application to the relativistic Toda chain. Let all parameters {p_{n}, q_{n}}_{n=1}^{∞} be related by p_{n} + q_{n} = 1 − κ^{N}p_{n}q_{n} ∀ n , (250) so that

p_{n} = \frac{1}{Δ_{ϕ_{n}}} , q_{n} = \frac{1}{Δ_{ϕ_{n}}^{∗}} (251)

in the notations of (30). Note, σ_{n}(−1/κ^{N}) = σ_{n}(0) σ_{n}(1) , (252) and in the Δ, Δ^{∗}-notations (126) e^{2iϕ_{n}} = σ_{n}(0)^{−1} , s_{n}(1) = σ_{n}(1) , s_{n}(Δ_{m}^{∗}) = σ_{n}(q_{m}) , s_{n}(Δ_{m}) = σ_{n}(p_{m}) . (253) Besides, when {p_{n}, q_{n}} obey (251), one has H^{(g)} ≡ h^{(g)} (see (125)).

Taking z_{1} = ... = z_{m} = 1 and z_{m+1} = ... = z_{g−1} = ∞ in proposition 10 on page 47, one obtains h^{(g)}({e^{2imϕ_{n}}}_{n=1}^{g}) = 0 , m = 0, ..., g − 1 . (254)

Consider discrete equations (120) and (122) with m ∈ ℤ and n ∈ ℤ_{+}. Omitting the index n in τ-functions, we rewrite these relations as follows

\begin{matrix}(δ_{g}δ_{g}^{∗} τ′_{m} τ_{m} θ_{m−1})^{N} = (δ_{g} τ′_{m−1} τ_{m} θ_{m})^{N} \\ + (δ_{g}^{∗} τ′_{m+1} τ_{m−1} θ_{m−1})^{N} + (κ τ′_{m} τ_{m−1} θ_{m})^{N} \end{matrix} (255)

and

(θ′_{m})^{N} = (\frac{w_{ϕ}}{τ_{m−1}} )^{N} ((θ_{m}τ′_{m−1})^{N} − (δ_{g}^{∗}θ_{m−1}τ′_{m})^{N}) . (256)

Proposition 14. Let (τ_{m})^{N} = H^{(g−1)}({f_{n}e^{2iϕ_{n}m}}_{n=1}^{g−1}) , (θ_{m})^{N} = H^{(g−1)}({f_{n}e^{2iϕ_{n}m}σ_{n}(1)}_{n=1}^{g−1}) . (257) Then the complete solution of equations (255) and (256) are (τ′_{m})^{N} = H^{(g)}({f_{n}e^{2iϕ_{n}m}}_{n=1}^{g}) , (θ′_{m})^{N} = H^{(g−1)}({f_{n}e^{2iϕ_{n}m}σ_{n}(1)}_{n=1}^{g}) , (258) where e^{2iϕ_{g}} = (δ_{g}/δ_{g}^{∗})^{N}, all f_{1}, ..., f_{g−1} are the same in (257) and (258), and f_{g} is arbitrary extra complex number.

Proof. Index m enters (257) as a scale of f_{n}, so later we may consider only m = 0.

Note that due to (236) and the fact σ_{g}(q_{g})^{−1} = 0, one has H^{(g)}({f_{n}σ_{n}(q_{g})^{−1}}_{n=1}^{g}) ≡ H^{(g−1)}({f_{n}}_{n=1}^{g−1}) . (259)

For given set of {f_{n}}_{n=1}^{g} consider two copies of Fay’s identity: the first one with

A = 0 , B = q_{g} , C = 1 , D = −\frac{1}{κ^{N}} , (260)

and the second one with

A = ∞ , B = q_{g} , C = −\frac{1}{κ^{N}} , D = 1 . (261)

Condition (250) is implied. Excluding H^{(g)}({f_{n}σ_{n}(1)^{−1}}_{n=1}^{g}) from both these equation, one obtains exactly (255) with the identification (257,258). Equation (256) corresponds to the Fay identity with A = 0 , B = q_{g} , C = 1 , D = ∞ . (262)

□

Note in conclusion, evolution equation (179) corresponds to the Fay identity with

A = ∞ , B = 0 , C = −\frac{1}{κ^{N}} , D = 1 . (263)

References

1. E. K. Sklyanin 1985 The quantum Toda chain Lecture Notes in Physics 226 196-233

2. S. N. Ruijsenaars 1990 Relativistic Toda Systems Commun. Math. Phys. 133 217-247

3. Yu. B. Suris 1990 Discrete time generalized Toda lattices: complete integrability and relation with relativistic Toda lattices Phys. Lett. A145 113–119

4. A. Kundu 1994 Generation of a quantum integrable class of discrete time of relativistic periodic Toda chains Phys. Lett. A 190 79-84

5. V. B. Kuznetsov, A. V. Tsiganov 1994 Separation of variables for the quantum relativistic Toda lattices J. Math. Sci. 80 (1996) no. 3 1802–1810

6. G. Pronko and S. Sergeev 2000 Relativistic Toda Chain Preprint nlin.SI/0009027, to appear in Journal of Applied Mathematics

7. L.D. Faddeev 1995 Discrete Heisenberg-Weyl group and modular group Lett. Math. Phys. 34 249-254 8. L.D. Faddeev 1999 Modular double of quantum group Conférence Moshé Flato 1999 Vol. I (Dijon)

149–156 Math. Phys. Stud. 21 Kluwer Acad. Publ. Dordrecht 2000

9. L.D. Faddeev, R.M. Kashaev and A.Yu. Volkov 2001 Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality Commun.Math.Phys. 219 199-219

10. S. Kharchev, D. Lebedev and M. Semenov-Tian-Shansky 2001 Unitary representations of U_{q}(𝔰𝔩(2, ℝ), the modular double, and the multiparticle q-deformed Toda chains Comm. Math. Phys. 225 (2002) 573– 609

11. L.D. Faddeev and R.M. Kashaev 1994 Quantum Dilogarithm Mod. Phys. Lett. A9 427-434

12. V.V. Bazhanov and N.Yu. Reshetikhin 1995 Remarks on the quantum dilogarithm J. Phys. A: Math. Gen. 28 2217-2226

13. V. V. Bazhanov and N. Yu. Reshetikhin 1995 Chiral Potts model and discrete Sine-Gordon model at roots of unity unpublished

14. V. Bazhanov, A. Bobenko and N. Reshetikhin 1996 Quantum discrete sine-Gordon model at roots of 1 Comm. Math. Phys. 175 377-400

15. S. Sergeev 1999 Quantum 2+1 evolution model J. Phys. A: Math. Gen. 32 5693-5714 S. Sergeev 2000 Solitons in a 3d integrable model Phys. Lett. A 265 364-368 S. Sergeev 2000 On exact solution of a Classical 3d integrable model Journal of Nonlinear Mathematical Physics 7 57-72 S. Sergeev 2000 Auxiliary transfer matrices for three dimensional integrable models Theoretical and Mathematical Physics 124 1187-1201

52 S. PAKULIAK AND S. SERGEEV

S. Sergeev 2000 Some properties of local linear problem’s matrices Notes of PDMI 269 292-307 (in russian)

16. R.J. Baxter, J.H.H. Perk and H. Au-Yang 1998 New solutions of the star-triangle relations for the chiral Potts models Phys. Lett. A128 138-142

17. V. V. Bazhanov and Yu. G. Stroganov 1990 Chiral Potts model as a descendant of the six-vertex model J. Stat. Phys. 59 799-817

18. S. M. Sergeev, V. V. Mangazeev and Yu. G. Stroganov 1996 Vertex reformulation of the Bazhanov – Baxter model J. Stat. Phys. 82 31-50

19. Yu. G. Stroganov, 2nd phd thesis

20. S. Pakuliak and S. Sergeev 2001 Relativistic Toda chain at root of unity Preprint ITEP-TH-86/00

nlin.SI/0101024

21. S. Pakuliak and S. Sergeev 2001 Relativistic Toda chain at root of unity II. Modified Q-operator Preprint

MPI 2001 – 65 nlin.SI/0107062

22. S. Pakuliak and S. Sergeev 2001 Relativistic Toda chain at root of unity III. Relativistic Toda chain hierarhy Preprint MPI 2001 – 66 nlin.SI/0107063

23. E. K. Sklyanin 1990 Functional Bethe Aansatz in Integrable and Superintegrable Systems, ed. by B. A. Kupershmidt, pp. 8-33, Singapore: World Scientific E. K. Sklyanin 1992 Quantum Inverse Scattering Method. Selected Topics In: “Quantum Group and Quantum Integrable Systems” (Nankai Lectures in Methematical Physics), ed. by Mo-Lin Ge, pp. 63-97, Singapore: World Scientific E. K. Sklyanin 1995 Separation of variables. New trends Progr. Theor. Suppl 118 35-60

24. S. Kharchev, and D. Lebedev 1999 Integral representation for the eigenfunctions of a quantum periodic Toda chain Letters in Mathematical Physics 50 53-77

25. E. K. Sklyanin 2000 Bäcklund transformations and Baxter’s Q-operator. Integrable systems: from classcal to quantum (Montréal, QC, 1999) 227–250 CRM Proc. Lecture Notes 26. Amer.Math. Soc. Providence RI 2000

26. V. B. Kuznetsov and E. K. Sklyanin 1998 On Bäcklund transformations for many-body systems J. Phys. A: Math. Gen. 31 2241-2251

27. V. B. Kuznetsov, M. Salerno and E. K. Sklyanin 2000 Quantum Bäcklund transformation for the integrable DST model J. Phys. A: Math. Gen. 33 171-189

28. V. Tarasov 1993 Cyclic monodromy matrices for sl(n) trigonometric R-matrices Commun. Math. Phys. 158 459-484

Stanislav Pakuliak, Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Moscow reg., Russia

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

Sergei Sergeev, Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Moscow reg., Russia and Max Plank Institute fur Mathematic, Vivatsgasse 7, D-53111, Bonn, Germany

E-mail address: sergeev@thsun1.jinr.ru, sergeev@mpim-bonn.mpg.de