Bethe Ansatz for the Ruijsenaars Model of BC_1-Type

We consider one-dimensional elliptic Ruijsenaars model of type $BC_1$. It is given by a three-term difference Schr\"odinger operator $L$ containing 8 coupling constants. We show that when all coupling constants are integers, $L$ has meromorphic eigenfunctions expressed by a variant of Bethe ansatz. This result generalizes the Bethe ansatz formulas known in the $A_1$-case.


Introduction
The quantum Ruijsenaars model [14] in the simplest two-body case reduces to the following difference operator acting on functions of one variable: where σ(z) is the Weierstrass σ-function, m is the coupling parameter, and T γ stands for the shift operator acting by (T γ f )(z) = f (z + γ). This operator, which first appeared in E. Sklyanin's work [15,16], can be viewed as a difference version of the Lamé operator −d 2 /dz 2 +m(m+1)℘(z). It was observed by Krichever-Zabrodin [13] and by Felder-Varchenko [8,7], that in the special case of integer coupling parameter the operator (1) shares many features with the Lamé operator.
In particular, when m ∈ Z + they both have meromorphic Bloch eigenfunctions which can be given explicitly by a suitable Bethe ansatz. This reflects the well-known fact that the Lamé operator is finite-gap for integer m (see e.g. [6] for a survey of the finite-gap theory). The Lamé operator has the following generalization closely related to the Heun's equation: where ω p are the half-periods of ℘(z). It can be viewed as a BC 1 -generalization of the Lamé operator. Again, for integer coupling parameters g p this operator is finite-gap, as was discovered by Treibich-Verdier [19], see also [17,18] for the detailed study of (2). The operator (2) has a multivariable generalization known as the Inozemtsev model [9]. A relativistic version of the Inozemtsev model (≡ BC n version of the Ruijsenaars model) was suggested by J.F. van Diejen [4,5], see also [10,11]. In the simplest one-variable case it takes the form of a three-term difference operator where c(z) is given explicitly in (8) below, and the notations are explained at the beginning of the next section. Therefore, the operator (3) should be viewed both as a difference analogue of (2) and a BC 1version of the Ruijsenaars model (1). In the trigonometric limit it coincides with the Askew-Wilson difference operator [1]. (Notice that (3) contains eight parameters µ p , µ ′ p , compared to the four in the Askew-Wilson operator and in (2).) Therefore, it is natural to expect that (3) and (1) should have similar properties. This is indeed the case, as we will demonstrate below. Our main result says that in the case of integer coupling parameters

Ruijsenaars operator of type BC 1
Let L be the operator (3)-(4) with the coefficient c(z) given by where c p looks as follows: Here the permutations π p are the same as in (7).

Symmetries of L
It is obvious from the formula (8) for c(z) that c(−z) = c(z). As a result, L is invariant under (z ↔ −z). Next, a direct computation shows that the function c(z) = c µ (z) is covariant under the shifts by half-periods, namely: Let us write L = L µ to indicate dependence on µ. Combining (6) and (10), we obtain that where µ = π r (µ) is the same as in (6), (10). This implies that for any ω ∈ Γ where λ(ω) := n 1 λ 1 + n 2 λ 2 if ω = n 1 ω 1 + n 2 ω 2 , and µ in the right-hand side is defined as

Bethe ansatz
Providing the coupling constants satisfy (5), put m = 3 p=0 (m p + m ′ p ) and consider the following function ψ(z) depending on the parameters t 1 , . . . , t m , k ∈ C: Let us impose m relations onto these parameters as follows: (Here s = 0, . . . , 3.) We will refer to (13)- (14) as the Bethe ansatz equations, or simply the Bethe equations. Explicitly, they look as follows: Now we can formulate the main result of this paper.
The proof will be given in the next section.

Remark 2.
To compute the corresponding eigenvalue, one evaluates the expression Lψ/ψ at any suitable point z. For instance, a convenient choice is z = 2γm 0 (provided m 0 > 0), because then the first term in Lψ vanishes.
Remark 3. If some of the coupling parameters m p , m ′ p vanish, then the corresponding sets of the Bethe equations are not present in (13)- (14). For example, in the case when the only nonzero parameter is m 0 = m, the Bethe equations take the form: In that form (seemingly different from [13,8]) they appeared in [21].

Invariant subspaces
The idea of the proof of the theorem is that applying L to ψ will not destroy the conditions (13)- (14), cf. [2,3]. We begin with two elementary results about a three-term difference operator with meromorphic coefficients: Suppose that a, b, c are regular at z ∈ 2γZ, apart from z = 0 where a, b have simple poles. Furthermore, suppose that res z=0 (a + b) = 0 (17) and that for some m ∈ Z + the following is true: Proof . For D ′ = aT 2γ + bT −2γ this is precisely Lemma 2.2 from [2], thus D ′ (Q m ) ⊆ Q m . On the other hand, the conditions on c in (18) imply trivially that cQ m ⊆ Q m .
For the next lemma, we assume that: (1) a is regular at z ∈ γ + 2γZ apart from a simple pole at z = −γ; (2) b is regular at z ∈ γ + 2γZ apart from a simple pole at z = γ; (3) c is regular at z ∈ γ + 2γZ apart from simple poles at z = ±γ. Also, suppose that (The last condition makes sense because (19) implies that a + b + c is regular at z = ±γ.) In addition to that , assume that for some m ∈ Z + the following is true: Lemma 2 (cf. Let us apply these facts to the Ruijsenaars operator (3) with integer coupling parameters (6). Below we always assume that the step γ is irrational, i.e. γ / ∈ Q ⊗ Z Γ = Qω 1 + Qω 2 . We proceed by defining Q as the space of entire functions ψ(z) satisfying the following conditions for every ω ∈ ω s + 2Γ (s = 0, . . . , 3): Here m stands as before for m = 3 p=0 (m p + m ′ p ), and the constant η(ω) is defined for ω = n 1 ω 1 + n 2 ω 2 as η(ω) = n 1 η 1 + n 2 η 2 . Proof . First, by applying Corollaries 1, 2 to the Ruijsenaars operator, we obtain that L preserves the spaces Q m 0 and Q ′ m ′ 0 (in the notations of Lemmas (1), (2)). Note that in doing so, we only have to check the vanishing of a(z) as required in Corollaries 1, 2, and the conditions on the residues (19). This is where the formula (9) becomes crucial. Finally, in order to show that L preserves similar conditions at other points ω ∈ Γ, one applies the formula (11).
Entire functions in F α 1 ,α 2 m (m > 0) are known as theta-functions of order m (with characteristics), each of them being a constant multiple of (12), for appropriate t 1 , . . . , t m , k.
Now, a simple check shows that in the case (6)  (m p + m ′ p ): Combining this with Proposition 1, we conclude that L preserves the space of theta-functions of order m satisfying the conditions (23)-(24): Proof of the Theorem 1. Take a solution (t 1 , . . . , t m , k) to the Bethe equations and the corresponding function ψ (12). Clearly, ψ belongs to F α 1 ,α 2 m for some α 1 , α 2 . The Bethe equations give the conditions (23)-(24) only for ω = ω s , but the rest follows from the translation properties of ψ. Thus, ψ belongs to the space F α 1 ,α 2 m ∩ Q. By (25), ψ := Lψ also belongs to this space. Now we use the following fact (whose proof will be given below): Lemma 3. For any two functions ψ, ψ ∈ F α 1 ,α 2 m ∩ Q, their quotient ψ/ψ is an even elliptic function, i.e. it belongs to C(℘(z)).
By this lemma, if ψ/ψ is not a constant, then its poles must be invariant under z → −z, thus there exist at least two of t 1 , . . . , t m such that their sum belongs to 2Γ.
Proof of the lemma. Take any two functions ψ, ψ in F α 1 ,α 2 m ∩ Q and put f := ψ/ψ. Note that f is an elliptic function of degree ≤ m (because its denominator and numerator have m zeros in the fundamental region). Let us label m pairs of points ω s ± 2jγ, ω s ± (2j − 1)γ as P ± l with l = 1, . . . , m, then the properties of ψ, ψ imply that f satisfies the conditions We may assume that f is regular in at least one of the half-periods ω s , otherwise switch to 1/f = ψ/ ψ. Let us anti-symmetrize f to get g(z) := f (z) − f (−z), which will be odd elliptic, of degree ≤ 2m. It is clear that g also satisfies the conditions (26). At the same time, it is antisymmetric under any of the transformations z → 2ω s − z (s = 0, . . . , 3). Altogether this implies that g must vanish at each of the 2m points P ± l . Finally, it must vanish at one of the half-periods (where f was regular). So g has ≥ 2m + 1 > deg(g) zeros, hence g = 0, f (z) ≡ f (−z), and we are done.
The above argument, however, would not work if one or both of the functions ψ, ψ vanish at some of the points P ± l . Indeed, then we cannot claim that f is regular at those points, so some of the conditions (26) would not hold for f . In that case, we can argue as follows. Let ψ λ denote the linear combination ψ λ = ψ + λ ψ. Then ψ λ , ψ µ for generic λ, µ will have zero of the same multiplicity at any given point P ± l . Thus, choosing λ, µ appropriately, we can always achieve that ψ λ /ψ µ = 0, ∞ at every of these 2m points. Let r be the number of those pairs (P + l , P − l ) where ψ λ , ψ µ vanish. Then we have that their ratio f := ψ λ /ψ µ still satisfies the conditions (26) at the remaining m − r pairs of points and has degree ≤ m − r due to the cancelation of the zeros in the denominator and numerator of f . Thus, the previous argument applies and gives that f is even. Therefore, ψ/ψ is even.

Continuous limit
As remarked in [4], the operator (3) with the coupling parameters (6) in the continuous limit γ → 0 turns into the BC 1 -version (2) of the Lamé operator: and the coupling parameters g p are given by g p := m p + m ′ p .
Theorem 2. Suppose the parameters t 1 , . . . , t m , k satisfy the Bethe equations (28) and the conditions t i + t j / ∈ 2ω 1 Z + 2ω 2 Z for 1 ≤ i = j ≤ m. Then the function w −1 (z)ψ(z) given by (12) with w as in (27), is an eigenfunction of the operator (2). This theorem is proved analogously to Theorem 1. σ(z + t j ). The Bethe ansatz equations (28) for k, t 1 , . . . , t m in this case reduce to: We should note that this particular form of Bethe equations differs from the classical result by Hermite [20]. For instance, in Hermite's equations one discards the points with t i = t j mod 2Γ, while in Theorem 2 we discard the points with t i = −t j mod 2Γ. Thus, comparing Theorem 2 with the Hermite's result, we conclude that (29) must be equivalent to Hermite's equations [20] provided t i ± t j / ∈ 2Γ for i = j. The same remark applies to the equations (28) when compared to the Bethe ansatz in, e.g., [18].

Spectral curve
Let us say few words about the structure of the solution set X ⊂ C m × C to the Bethe equations (13)- (14). We will skip the details, since the considerations here are parallel to those in [8,7,13,21].
As a result, X is invariant under these transformations. Also, multiplying ψ(z) by e πiz/γ does not affect the Bethe equations, because such an exponential factor is (anti)periodic under the shifting of z by multiples of γ. Therefore, X is invariant under the shifts of k by πi/γ: (t 1 , . . . , t m , k) → (t 1 , . . . , t m , k + πi/γ).
Finally, ψ does not change under permutation of t 1 , . . . , t m , so X is invariant under such permutations. Let X denote the quotient of X by the group generated by all of the above transformations. Explicitly, let b s,j (t 1 , . . . , t m ) and b ′ s,j (t 1 , . . . , t m ) denote the left-hand side of equations (15) and (16). (Here s = 0, . . . , 3 and j = 1, . . . , m s or j = 1, . . . , m ′ s , respectively.) Introduce the variable q := e 2γk . Then X is described by the equations Excluding the q-variable from the equations (30), we may think of X as an algebraic subvariety in the symmetric product S m E of m copies of the elliptic curve E = C/2Γ where Γ = Zω 1 + Zω 2 .
(See, however, Remark 4 below.) Counting the number of equations, we conclude that every irreducible component of X has dimension ≥ 1. Since we are interested (cf. Theorem 1) in those points (t 1 , . . . , t m ) of X where t i + t j / ∈ 2Γ, we should restrict ourselves to the open part Y ⊂ X, lying in We need to show that Y is nonempty and one-dimensional. To this end, one easily observes from the equations (15)-(16) that the closure Y of Y in S m E contains the points P + = (P + 1 , . . . , P + m ) and P − = (P − 1 , . . . , P − m ), in the notations of the proof of the Lemma 3. These 'infinite' points correspond to q → 0, ∞ in (30). Similarly to [7], lemma 3.2, one shows that near P ± the variety Y looks like a smooth curve, with q ±1 being a local parameter. One can show that Y is an irreducible, projective curve, and it should be regarded as the 'spectral curve' for the operator L. For every (t 1 , . . . , t m ) ∈ Y \ {P + , P − }, the corresponding value of q = e 2γk is determined from (30), and the corresponding ψ(z) is unique, up to a factor of the form e πiN z/γ . We have Lψ = ǫψ, with the eigenvalue ǫ being a single-valued function on Y which has two simple poles at P ± . There is an involution ν on Y , which sends (t 1 , . . . , t m ) to (−t 1 , . . . , −t m ) and the corresponding ψ(z) to ψ(−z); note that ν(P + ) = P − . The function ǫ is ν-invariant, and takes each its value exactly twice on Y . It is straightforward to compute the asymptotics of ǫ and ψ near P ± . Finally, for generic value of ǫ, the eigenspace of meromorphic functions {f : Lf = ǫf } is spanned by the corresponding ψ(z), ψ(−z) over the field K of 2γ-periodic meromorphic functions of z.
Remark 4. Note that in the case when all m ′ s = 0, the second set (14) of the Bethe equations is absent, thus a shift k → k + πi 2γ is also allowed. In that case the coefficient c(z) (8) vanishes, so L has two terms only, and it is easy to see that the transformation ψ → e πiz 2γ ψ changes the sign of the eigenvalue ǫ. As a result, the subvariety of S m E which was obtained by excluding q from (30), will be a quotient of X by Z 2 , rather than X itself (cf. [8,7,13,21]).