Symmetry, Integrability and Geometry: Methods and Applications Towards Finite-Gap Integration of the Inozemtsev Model ⋆

The Inozemtsev model is considered to be a multivaluable generalization of Heun's equation. We review results on Heun's equation, the elliptic Calogero-Moser-Sutherland model and the Inozemtsev model, and discuss some approaches to the finite-gap integration for multivariable models.


Introduction
Differential equations defined on a complex domain frequently appear in mathematics and physics. One of the most important differential equations is the Gauss hypergeometric differential equation. Mathematically, it is a standard form of the second-order differential equation with three regular singularities on the Riemann sphere. Global properties of the solutions, i.e., the monodromy, often play decisive roles in applications to physics and mathematics.
A canonical form of a Fuchsian equation with four singularities is given by Heun's equation which is written as with the condition γ + δ + ǫ = α + β + 1. This equation appears in several topics in physics, i.e., astrophysics, crystalline materials and so on (see [37] and references therein). Although the problem of describing the monodromy of Heun's equation is much more difficult than that of the hypergeometric equation, Heun's equation has been studied from several viewpoints. A method of finite-gap integration is available on a study of Heun's equation, and consequently we have some formulae on the monodromy.
Note that the equation [A, −d 2 /dx 2 +q(x)] = 0 is equivalent to the function q(x) being a solution to a stationary higher-order KdV equation (see [10]). In our setting, finite-gap integration is a method for analysis of the operator −d 2 /dx 2 + q(x) where q(x) is an algebro-geometric finitegap potential. Originally, the finite-gap property is a notion related to spectra. Let H be the operator −d 2 /dx 2 + q(x), and the set σ b (H) be defined as follows: If the closure of the set σ b (H) can be written as where E 0 < E 1 < · · · < E 2g , i.e., the number of bounded bands is finite, then q(x) is called the finite-gap potential. It was established in the 1970s that, under the assumption that q(x) is a periodic, smooth, real function, the potential q(x) is finite-gap if and only if q(x) is algebrogeometric finite-gap. On the approach by the finite-gap integration for Heun's equation, it is essential to transform Heun's equation into a form with the elliptic function. The transformed equation is written as where ℘(x) is the Weierstrass ℘-function with periods (2ω 1 , 2ω 3 ), ω 0 (= 0), ω 1 , ω 2 (= −ω 1 − ω 3 ), ω 3 are half-periods, and l i (i = 0, 1, 2, 3) are coupling constants. Here the variables w and x in equations (1.1), (1.2) are related by w = (℘(x) − ℘(ω 1 ))/(℘(ω 2 ) − ℘(ω 1 )). For details of the transformation, see [34,38,43]. The expression in terms of the elliptic function was already discovered in the 19th century, and some results which may relate to finite-gap integration were found in that era. For the case when three of l 0 , l 1 , l 2 , l 3 are equal to zero, equation (1.2) is called Lamé's equation. Ince [19] established in 1940 that if n ∈ Z ≥1 , ω 1 ∈ R \ {0} and ω 3 ∈ √ −1R \ {0}, then the potential of Lamé's operator is finite-gap. In the late 1980s, Treibich and Verdier [48] found that the method of finite-gap integration is applicable for the case l 0 , l 1 , l 2 , l 3 ∈ Z. Namely, they showed that the potential in equation (1.2) is an algebro-geometric finite-gap potential if l i ∈ Z for all i ∈ {0, 1, 2, 3}. Therefore the potential 3 i=0 l i (l i + 1)℘(x + ω i ) is called the Treibich-Verdier potential. Subsequently several others [17,38,42,44,45,46] have produced more precise statements and concerned results on this subject. Namely, integral representations of solutions [38,42], the Bethe Ansatz [17,42], the global monodromy in terms of the hyperelliptic integrals [44], the Hermite-Krichever Ansatz [45] and a relationship with the Darboux transformation [46] were studied. In this paper, we discuss some approaches to finite-gap integration for multivariable cases. A multivariable generalization of Heun's equation is given by the Inozemtsev model, which is a generalization of the Calogero-Moser-Sutherland model. The Inozemtsev model of type BC N [20] is a quantum mechanical system with N -particles whose Hamiltonian is given by where l and l i (i = 0, 1, 2, 3) are coupling constants. It is known that the Inozemtsev model of type BC N is completely integrable. More precisely, there exist operators of the form H k = N j=1 ∂ ∂x j 2k + (lower terms) (k = 2, . . . , N ) such that [H, H k ] = 0 and [H k 1 , H k 2 ] = 0 (k, k 1 , k 2 = 2, . . . , N ). Note that the Inozemtsev model of type BC N is a universal completely integrable model of quantum mechanics with B N symmetry, which follows from the classification due to Ochiai, Oshima and Sekiguchi [30]. For the case N = 1, the potential coincides with the Treibich-Verdier potential and the spectral problem for the Inozemtsev model of type BC 1 is equivalent to solving Heun's equation.
By the trigonometric limit (τ (= ω 3 /ω 1 ) → √ −1∞), we obtain the trigonometric Calogero-Moser-Sutherland model. The trigonometric model is well-studied by multivariable orthogonal polynomials (i.e., the Jack polynomial and the multivariable Jacobi polynomial). Vadim Kuznetsov and his collaborators studied multivariable orthogonal polynomials from the aspects of separation of variables [25], the Pfaff lattice [1] and the Q operator [24]. Note that in the paper [26] relationships among separation of variables, integral transformations and Lamé's (Heun's) differential equation were discussed. Applications of these technique for models with elliptic potentials are anticipated.
Although the Inozemtsev model of type BC N is much more difficult than the trigonometric one, some approaches (perturbation from the trigonometric limit, quasi-solvability etc.) were introduced. Now we hope to develop analysis of this model by finite-gap integration. Although we can regard several works to be on this direction, in my opinion, they are still far from complete understanding of the model. We may consider the multivariable Darboux transformation as a possible approach, but we should develop it in the future. This paper is organized as follows. In Section 2, we review the finite-gap integration of Heun's equation. An approach by the Darboux transformation is introduced. In Section 3, we collect results on the Calogero-Moser-Sutherland model and the Inozemtsev model. In Section 4, we discuss some approaches to finite-gap integration for those models.

Darboux transformations and Heun's equation
We consider the finite-gap property of Heun's equation in the elliptic form. It is known that the Treibich-Verdier potential is algebro-geometric finite-gap, i.e., there exists a differential operator A of odd order which commutes with the operator H (l 0 ,l 1 ,l 2 ,l 3 ) where In this subsection, we construct an odd-order differential operator A by composing the Darboux-Crum transformation which we will explain below. We review the Darboux transformation. Let φ 0 (x) be an eigenfunction of the operator H = −d 2 /dx 2 + q(x) corresponding to an eigenvalue E 0 , i.e.
For this case, the potential q(x) is written as , we have the factorization H − E 0 = L † L. We setH = LL † + E 0 . Then we haveH = −d 2 /dx 2 + q(x) − 2(φ ′ 0 (x)/φ 0 (x)) ′ and the relationHL = LH. Hence, if φ(x) is an eigenfunction of the operator H corresponding to the eigenvalue E, then Lφ(x) is an eigenfunction of the operatorH corresponding to the eigenvalue E. This transformation is called the Darboux transformation. We generalize the operator L to be the differential operator of higher order in the following proposition.
Proposition 1 (cf. [3]). Suppose that the operator H = −d 2 /dx 2 + q(x) preserves an n-dimensional space U of functions. Let L be the monic differential operator of order n which annihilates all functions in U , and write Then we havẽ We call the operator L in Proposition 1 the generalized Darboux transformation or the Darboux-Crum transformation. For the case n = 1, let φ 0 (x) be a non-zero function in U , then U = Cφ 0 (x), the operator which annihilates φ 0 (x) is given by Hence the proposition reproduces the Darboux transformation for the case n = 1.
By applying Propositions 1 and 2, we obtain the following proposition after some calculations: , and let L α 0 ,α 1 ,α 2 ,α 3 be the monic differential operator of order d + 1 which annihilates the space V α 0 ,α 1 ,α 2 ,α 3 . Then we have We construct an odd-degree differential operator A which commutes with H by composing the operators L α 0 ,α 1 ,α 2 ,α 3 . We define the operatorL α 0 ,α 1 ,α 2 ,α 3 as follows: otherwise. Set The following proposition is proved by applying Proposition 3 four times: We then have that the operator A commutes with H (l 0 ,l 1 ,l 2 ,l 3 ) , i.e., It is known that, if l 0 , l 1 , l 2 , l 3 ∈ Z ≥0 , then there exist four invariant spaces of the operator H (l 0 ,l 1 ,l 2 ,l 3 ) , which we consider in Section 2.3, and the four operators in Proposition 4 are related to the four spaces. Let Then g ∈ Z ≥0 and the degree of the operator A is 2g + 1.
For the case l 0 = 2, l 1 = l 2 = l 3 = 0, we have g = 2 and the operator A is expressed as

Application of finite-gap property
We investigate Heun's equation in the elliptic form by applying the finite-gap integration, which is based on the commutativity of H (= −d 2 /dx 2 + u(x)) and an odd-order differential operator A.
Since A is a monic differential operator of order 2g + 1, it can be expressed in the form whereã 0 (x) = 1. We have Hence we obtaiñ for some constants c j (j = 0, . . . , g). Therefore we have Proposition 5. Setã 0 (x) = 1 andã g+1 (x) = 0. The operator A may be expressed in the form for some functionsã j (x) (j = 1, . . . , g) and constants c j (j = 0, . . . , g), where the functionsã j (x) (j = 0, . . . , g) satisfỹ We define a function Ξ(x, E) which plays the important role for the solutions and the monodromy of Heun's equation. Set It follows from equation (2.6) that Ξ(x, E) satisfies a differential equation satisfied by products of any pair of the solutions to equation (2.4), i.e., On the basis of Proposition 5 and the function Ξ(x, E) in equation (2.7), we have where the coefficients c 0 (E) and b The coefficients do not have common divisors and the polynomial c 0 (E) is monic. We have g = deg E c 0 (E) and the coefficients satisfy where e i = ℘(ω i ) and g 2 = −4(e 1 e 2 + e 2 e 3 + e 3 e 1 ).
Note that the function Ξ(x, E) can be also obtained as the function satisfying equation (2.8) and Proposition 6 (ii) (see [42]).
We can derive an integral formula for a solution to equation (2.4) in terms of the doubly periodic function Ξ(x, E). Set It is shown by differentiating the right-hand side of equation (2.10) and applying equation (2.8) that Q(E) is independent of x. Thus Q(E) is a monic polynomial in E of degree 2g + 1, which follows from the expression for Ξ(x, E) given by equation (2.9). For the case l 0 = 2, l 1 = l 2 = l 3 = 0, we have The following proposition on the integral representation of a solution to equation (2.4) was obtained in [42]: is a solution to the differential equation (2.4).
For the case l 0 = 2, l 1 = l 2 = l 3 = 0, we set E 0 = √ 3g 2 . Then q 1 = q 3 = 0 and the function a(E) and c(E) are determined as Hence we have We review the propositions related with the Bethe Ansatz (Proposition 9) and the Hermite-Krichever Ansatz (Proposition 10), which are also reductions of the finite-gap property.

11)
where σ(x) is the Weierstrass sigma function and σ i (x) (i = 1, 2, 3) are the co-sigma functions defined by (ii) The functioñ with the condition t j = t j ′ (j = j ′ ) and t j ∈ ω 1 Z + ω 3 Z is an eigenfunction of the operator H (see equation (2.4)), if and only if t j (j = 1, . . . , l) and c satisfy the relations, The eigenvalue E is given by Equation (2.13) is called the Bethe Ansatz equation for the Inozemtsev model of type BC 1 (see [42]). Note that Gesztesy and Weikard [17] obtained similar results. The monodromy of the functionΛ(x) in equation (2.12) is written as In order to describe the proposition on the Hermite-Krichever Ansatz, we define

Relationship among commuting operators
We review a relationship among the operators H, A, the polynomial Q(E) and the invariant subspaces. On the operators H and A, we have the following relation: It is known that, if l 0 , l 1 , l 2 , l 3 ∈ Z ≥0 , then there exist four invariant subspaces with respect to the action of the operator H. We describe the spaces more precisely. Let V α 0 ,α 1 ,α 2 ,α 3 be the space defined in Proposition 2 and otherwise.
If l 0 , l 1 , l 2 , l 3 ∈ Z ≥0 and l 0 + l 1 + l 2 + l 3 is even, then the operator H preserves the space and also preserves the components in equation (2.15). If l 0 , l 1 , l 2 , l 3 ∈ Z ≥0 and l 0 + l 1 + l 2 + l 3 is odd, then the operator H preserves the space

Results on the Calogero-Moser-Sutherland model and the Inozemtsev model
We are going to consider multidimensional generalizations of Lamé's equation and Heun's equation in the elliptic form. For this purpose, we introduce the quantum mechanical systems.

The elliptic Calogero-Moser-Sutherland model
The elliptic Calogero-Moser-Sutherland model (or the elliptic Olshanetsky-Perelomov model [31]) of type A N −1 is a quantum many-body system whose Hamiltonian is given as follows: where ℘(x) is the Weierstrass elliptic function. For the case N = 2, the model reproduces Lamé's equation by setting x 1 − x 2 = x and restricting to the line x 1 + x 2 = 0. This model is known to be completely integrable, i.e., there exist N -algebraically independent commuting operators P k (k = 1, . . . , N ) which commute with the Hamiltonian H. Namely, by setting (3.1) where S N is the symmetric group, [x] is the integral part of x and ∂ i = ∂/∂x i , we have [P k , H] = 0 (1 ≤ k ≤ N ) and [P k , P k ′ ] = 0 (1 ≤ k, k ′ ≤ N ) (see [30]). The Hamiltonian H is expressed as H = P 2 − P 2 1 /2. By the trigonometric limit (τ → √ −1∞) of the elliptic Calogero-Moser-Sutherland model where (1, τ ) is the basic periods of the elliptic function, we obtain (up to an additive scalar) the Hamiltonian of the trigonometric Calogero-Moser-Sutherland model, .
The eigenstates of the Calogero-Sutherland model are described by the Jack polynomial [39]), i.e., In particular, the ground-state is given by ∆(X) l+1 . Several properties of the Jack polynomial were studied. Vadim Kuznetsov and his collaborators studied the Jack polynomial and related polynomials from the aspects of separation of variable [25], the Pfaff lattice [1] and the Q operator [24].
In contrast with the trigonometric models, the elliptic models are less investigated and the spectra or the eigenfunctions are not sufficiently analyzed. There is, however, some important progress due to Felder and Varchenko. They introduced the Bethe Ansatz method for the Nparticle elliptic Calogero-Moser model with the coupling constant l a positive integer. Note that Hermite essentially introduced the Bethe Ansatz method for the case N = 2 and l ∈ Z (see [49]), and Dittrich and Inozemtsev [9] did it for the case N = 3 and l = 1 in a different representation.
For ξ ∈ h * , we introduce the functions Φ τ (t 1 , . . . , t m ) and ω(t; x) as follows where t 0 = 0, f 0 (k) = 0. Then we have Proposition 13 ([12, 13, 11]). If (t 0 1 , . . . , t 0 m ) satisfy the following Bethe Ansatz equations, the function ω(t 0 ; x) is an eigenfunction of the Hamiltonian H with the eigenvalue lN log θ(t i ), Therefore, if we find solutions to the Bethe Ansatz equations, we can investigate the Calogero-Moser-Sutherland model in more detail. There are two things to be considered for applying Proposition 13 to the spectral problem of the elliptic Calogero-Moser-Sutherland model. The first one is to find the condition when the eigenfunctions obtained by the Bethe Ansatz method are connected to square-integrable eigenstates and the second one is how the solutions of the Bethe Ansatz equation behave.
On the first question, the condition is described as the parameter ξ belonging to some lattice (the weight lattice of type A N −1 ). By symmetrizing or anti-symmetrizing the function ω(t 0 ; x), we obtain square-integrable eigenstates, although we must check that they are identically zero or not.
On the second question, we consider the solution at p = exp(2π √ −1τ ) = 0 (the case of the trigonometric limit τ → √ −1∞) and look into the behavior where p is near 0, because it is hopeful to solve the Bethe Ansatz equations for the trigonometric case in contrast to being hopeless directly for the elliptic case. A key tool to connect the trigonomertic solutions to the elliptic solutions is the implicit function theorem. Thus we construct the square-integrable eigenstates and obtain the main result in [40], which gives a sufficient condition for regular convergence of the perturbation expansion. In particular, for the case N = 2, l ∈ Z ≥1 and the case N = 3, l = 1, we have convergence of the perturbation series for all eigenstates related to the Jack polynomial.
Note that this idea can be interpreted to consider the elliptic Calogero-Moser-Sutherland model by perturbation from the trigonometric Calogero-Moser-Sutherland model. Convergence for the general cases was proved in [23] by another method. Namely, by applying Kato-Rellich theory, we have convergence of the perturbation series in p for l ≥ 0 and arbitrary N . The eigenvalues and the eigenfunctions are calculated as power series by a standard algorithm of perturbation. Remark that Fernandez, Garcia and Perelomov [14] derived a fully explicit formula for second order in p, and Langmann [27,28] obtained another algorithm for constructing the eigenfunctions and the eigenvalues as formal power series of p, which also gives a formula for all orders in p.
On the Bethe Ansatz for the elliptic Calogero-Moser-Sutherland model, there are some problems to be solved. For example, it has not been shown at the moment of writing that the eigenfunction ω(t 0 , x) written in the form of the Bethe Ansatz is also an eigenfunction of the higher commuting operators P 3 , . . . , P N (see also [35]).

The Inozemtsev model
We now introduce a quantum mechanical system that is a multidimensional generalization of Heun's equation in the elliptic form.
The Inozemtsev model of type BC N [20] is a quantum mechanical system with N -particles whose Hamiltonian is given by where l and l i (i = 0, 1, 2, 3) are coupling constants. This is also a generalization of the elliptic Calogero-Moser-Sutherland model of type BC N . The Inozemtsev model of type BC N is completely integrable, i.e., there exist N algebraically independent mutually commuting differential operators P k (k = 1, . . . , N ) (higher commuting Hamiltonians) which commute with the Hamiltonian of the model, and Oshima [32] described the commuting operators explicitly. Note that the Inozemtsev model of type BC N (resp. the elliptic Calogero-Moser-Sutherland model of type A N ) is a universal completely integrable model of quantum mechanics with the symmetry of the Weyl group of type B N (resp. type A N ), which follows from the classification due to Ochiai, Oshima and Sekiguchi [30,33]. For the case N = 1, the operator (3.2) appears in the elliptic form of Heun's equation (2.4). Therefore the Inozemtsev model of type BC N is regarded as a multidimensional generalization of Heun's equation.
On the trigonometric limit τ → √ −1∞, we obtain the trigonometric Calogero-Moser-Sutherland model of type BC N , and we can investigate the Inozemtsev model of type BC N by perturbation from the trigonometric model [23,47].
A method of quasi-solvability is available on the Inozemtsev model of type BC N . Finkel et al. studied quasi-solvable models in [15,16], and they found several quasi-exactly solvable many-body systems including the Inozemtsev model of type BC N . We now describe the finitedimensional spaces which are related to the quasi-solvability. The quasi-solvability with respect to the Hamiltonian H was established in [16] and reformulated in [41]. Proposition 14 ([16,41]). Let a, b i (i = 0, 1, 2, 3) be the numbers which satisfy a ∈ {−l, l + 1} and b i ∈ {−l i /2, (l i + 1)/2} (i = 0, 1, 2, 3). Set such that m i ∈ {0, 1, . . . , d} for all i. Then we have The quasi-solvability was extended to the commuting differential operators.
is the finitedimensional space defined in Proposition 14 and P k are the commuting differential operators which ensure the complete integrability.
By the quasi-solvability, finitely-many eigenvalues and eigenfunctions are calculated by diagonalizing the commuting matrices simultaneously for the case that the assumption of Proposition 14 is true. The eigenfunctions obtained by the quasi-solvability may not be square-integrable in general, although the eigenfunctions for the case that the parameters a, b 0 , b 1 in Proposition 14 are chosen as a = l + 1, b 0 = (l 0 + 1)/2 and b 1 = (l 1 + 1)/2 are square-integrable.
It seems that an explicit expression of the Bethe Ansatz as Proposition 13 for the Inozemtsev model of type BC N and corresponding conformal field theory are not known in the moment of writing, although Chalykh, Etingof and Oblomkov [6] gave a general recipe for calculating the Bloch eigenfunctions. They showed that these are parametrized by a certain algebraic variety (the Hermite-Bloch variety) which can be computed. This would lead to a version of the Bethe Ansatz for the models including the Inozemtsev model of type BC N , though these Bethe Ansatz equations would be rather complicated. For a special case of the BC 2 case, this scheme is worked out explicitly in [6, § 6].
We hope to investigate the Bethe Ansatz for the Inozemtsev model of type BC N to study the model in more detail.

Towards f inite-gap integration of the Inozemtsev model
In Section 2, we reviewed the finite-gap integration of Heun's equation, and observed that the existence of commuting operator of odd order plays important roles.
For a multidimensional generalization of finite-gap integration, Chalykh and Veselov introduced the notion "algebraic integrability". The Schrödinger operator L = − N i=1 ∂ 2 /∂x 2 i + u(x 1 , . . . , x N ) is called completely integrable [8], if there exist N commuting operators L 1 = L, L 2 , . . . , L N with algebraically independent constant highest symbols s 1 (ξ) (= ξ 2 1 + · · · + ξ 2 N ), s 2 (ξ), . . . , s N (ξ) (ξ j = √ −1∂/∂x j ). For example, the Calogero-Moser-Sutherland models are completely integrable. On the model of type A N , the highest symbols of P k (see equation (3.1)) are written as ((− √ −1) k /k!) i 1 <i 2 <···<i k ξ i 1 ξ i 2 · · · ξ i k . The Inozemtsev model of type BC N is also completely integrable. The operator L is called algebraically integrable in the sense of [8], if L is completely integrable and there exists one more operator L 0 commuting with L i (i = 1, . . . , N ) and the highest symbol s 0 (ξ) of L 0 takes the distinct values at the roots of the equations s i (ξ) = E i (i = 1, . . . , N ) for almost all E i . On Heun's equation in the elliptic form, if l 0 , l 1 , l 2 , l 3 are integers, then it is algebraically integrable, because there exists a commuting operator A of odd order. Thus we expect that algebraically integrable Schrödinger operators also have rich properties.
Chalykh and Veselov conjectured [7] that the Calogero-Moser-Sutherland model with integral coupling constants are algebraically integrable. For the case of type A N , Braverman, Etingof and Gaitsgory [4] obtained algebraic integrability. More precisely, they established that, if l is a positive integer, then the operator is algebraically integrable by applying the Bethe Ansatz due to Felder-Varchenko (see Section 3.1) and the differential Galois theory. In [6], Chalykh, Etingof and Oblomkov proved that the Chalykh-Veselov conjecture is true and the Inozemtsev model of type BC N (see equation (3.2)) is also algebraically integrable, if l, l 0 , l 1 , l 2 , l 3 are all integers. Their method relies on results on the differential Galois theory obtained in [4] and the local triviality of the monodromy.
On an application of the algebraic integrability, the eigenfunctions of the Baker-Akhiezer (Bloch) type are considered. We expect further studies for applications of the algebraic integrability on the Calogero-Moser-Sutherland models and the Inozemtsev models.
Another possible method for constructing extra commuting operators is the multidimensional Darboux transformation, because the commuting operator for the case of Heun's equation is constructed by composing the (generalized) Darboux transformations. Multidimensional Darboux transformations were studied from several viewpoints [2,18,36,5]. In [2], based on the existence of an explicit eigenfunction of the Hamiltonian H(= H (0) ) with a certain eigenvalue, an alternate HamiltonianH(= H (N ) ), matrix valued operators H (i) (i = 1, . . . , N − 1) and supersymmetry operators Q − j+1,j and Q + j,j+1 (j = 0, . . . , N − 1) which connect H (j) and H (j+1) were introduced and studied. On the other hand, we know an explicit eigenfunction of the Inozemtsev model of type BC N , if the value d(= −((N −1)a+b 0 +b 1 +b 2 +b 3 )) (a ∈ {−l, l+1}, b i ∈ {−l i /2, (l i +1)/2} (i = 0, 1, 2, 3)) in the assumption of Proposition 14 is equal to zero. Then the alternate Hamil-tonianH with respect to H in equation (3.2) would be written as In the moment of writing, we do not know an operator L which directly intertwines the operators H andH asHL = LH. We expect applications of the multidimensional Darboux transformation for the analysis of the elliptic Calogero-Moser-Sutherland model or the Inozemtsev model.