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.

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Date:	2007
Main Author:	Takemura, K.
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Language:	English
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spelling	nasplib_isofts_kiev_ua-123456789-1478242025-02-10T01:24:29Z Towards Finite-Gap Integration of the Inozemtsev Model Takemura, K. 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. This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The author would like to thank the referees for valuable comments. 2007 Article Towards Finite-Gap Integration of the Inozemtsev Model / K. Takemura // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 49 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81R12; 33E10; 34M35 https://nasplib.isofts.kiev.ua/handle/123456789/147824 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	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.
format	Article
author	Takemura, K.
spellingShingle	Takemura, K. Towards Finite-Gap Integration of the Inozemtsev Model Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Takemura, K.
author_sort	Takemura, K.
title	Towards Finite-Gap Integration of the Inozemtsev Model
title_short	Towards Finite-Gap Integration of the Inozemtsev Model
title_full	Towards Finite-Gap Integration of the Inozemtsev Model
title_fullStr	Towards Finite-Gap Integration of the Inozemtsev Model
title_full_unstemmed	Towards Finite-Gap Integration of the Inozemtsev Model
title_sort	towards finite-gap integration of the inozemtsev model
publisher	Інститут математики НАН України
publishDate	2007
url	https://nasplib.isofts.kiev.ua/handle/123456789/147824
citation_txt	Towards Finite-Gap Integration of the Inozemtsev Model / K. Takemura // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 49 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT takemurak towardsfinitegapintegrationoftheinozemtsevmodel
first_indexed	2025-12-02T11:37:13Z
last_indexed	2025-12-02T11:37:13Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 038, 17 pages Towards Finite-Gap Integration of the Inozemtsev Model? Kouichi TAKEMURA Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan E-mail: takemura@yokohama-cu.ac.jp Received October 31, 2006, in final form February 07, 2007; Published online March 02, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/038/ Abstract. 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–Suther- land model and the Inozemtsev model, and discuss some approaches to the finite-gap inte- gration for multivariable models. Key words: finite-gap integration; Inozemtsev model; Heun’s equation; Darboux transfor- mation 2000 Mathematics Subject Classification: 81R12; 33E10; 34M35 1 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 differen- tial 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(( d dw )2 + ( γ w + δ w − 1 + ε w − t ) d dw + αβw − q w(w − 1)(w − t) ) f̃(w) = 0, (1.1) with the condition γ + δ + ε = 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. If there exists an odd-order differential operator A = (d/dx)2g+1 + 2g−1∑ j=0 bj(x) (d/dx)2g−1−j such that [A,−d2/dx2 +q(x)] = 0, then q(x) is called the algebro-geometric finite-gap potential. Note that the equation [A,−d2/dx2+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 −d2/dx2 + q(x) where q(x) is an algebro-geometric finite- gap potential. Originally, the finite-gap property is a notion related to spectra. Let H be the ?This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html mailto:takemura@yokohama-cu.ac.jp http://www.emis.de/journals/SIGMA/2007/038/ http://www.emis.de/journals/SIGMA/kuznetsov.html 2 K. Takemura operator −d2/dx2 + q(x), and the set σb(H) be defined as follows: E ∈ σb(H) ⇔ Every solution to (H − E)f(x) = 0 is bounded on x ∈ R. If the closure of the set σb(H) can be written as σb(H) = [E0, E1] ∪ [E2, E3] ∪ · · · ∪ [E2g,∞), where E0 < E1 < · · · < E2g, 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 algebro- geometric 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 (H − E)f(x) = ( − d2 dx2 + 3∑ i=0 li(li + 1)℘(x + ωi)− E ) f(x) = 0, (1.2) where ℘(x) is the Weierstrass ℘-function with periods (2ω1, 2ω3), ω0(= 0), ω1, ω2(= −ω1 − ω3), ω3 are half-periods, and li (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)− ℘(ω1))/(℘(ω2)− ℘(ω1)). 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 l0, l1, l2, l3 are equal to zero, equation (1.2) is called Lamé’s equation. Ince [19] established in 1940 that if n ∈ Z≥1, ω1 ∈ R \ {0} and ω3 ∈ √ −1R \ {0}, then the potential of Lamé’s operator − d2 dx2 + n(n + 1)℘(x + ω3), is finite-gap. In the late 1980s, Treibich and Verdier [48] found that the method of finite-gap integration is applicable for the case l0, l1, l2, l3 ∈ Z. Namely, they showed that the potential in equation (1.2) is an algebro-geometric finite-gap potential if li ∈ Z for all i ∈ {0, 1, 2, 3}. There- fore the potential 3∑ i=0 li(li + 1)℘(x + ωi) 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 ty- pe BCN [20] is a quantum mechanical system with N -particles whose Hamiltonian is given by H = − N∑ j=1 ∂2 ∂x2 j + 2l(l + 1) ∑ 1≤j<k≤N (℘(xj − xk) + ℘(xj + xk)) + N∑ j=1 3∑ i=0 li(li + 1)℘(xj + ωi), where l and li (i = 0, 1, 2, 3) are coupling constants. It is known that the Inozemtsev model of type BCN is completely integrable. More precisely, there exist operators of the form Hk = Towards Finite-Gap Integration of the Inozemtsev Model 3 N∑ j=1 ( ∂ ∂xj )2k +(lower terms) (k = 2, . . . , N) such that [H,Hk] = 0 and [Hk1 ,Hk2 ] = 0 (k, k1, k2 = 2, . . . , N). Note that the Inozemtsev model of type BCN is a universal completely integrable model of quantum mechanics with BN 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 BC1 is equivalent to solving Heun’s equation. By the trigonometric limit (τ(= ω3/ω1) → √ −1∞), we obtain the trigonometric Calogero– Moser–Sutherland model. The trigonometric model is well-studied by multivariable orthogo- nal 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 BCN 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. 2 Finite-gap integration of Heun’s equation 2.1 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(l0,l1,l2,l3) where H(l0,l1,l2,l3) = − d2 dx2 + 3∑ i=0 li(li + 1)℘(x + ωi). (2.1) 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 φ0(x) be an eigenfunction of the operator H = −d2/dx2 + q(x) corresponding to an eigenvalue E0, i.e.( − d2 dx2 + q(x) ) φ0(x) = E0φ0(x). For this case, the potential q(x) is written as q(x) = (φ′0(x)/φ0(x))′ + (φ′0(x)/φ0(x))2 + E0. By setting L = d/dx − φ′0(x)/φ0(x) and L† = −d/dx − φ′0(x)/φ0(x), we have the factorization H − E0 = L†L. We set H̃ = LL† + E0. Then we have H̃ = −d2/dx2 + q(x) − 2(φ′0(x)/φ0(x))′ and the relation H̃L = 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 operator H̃ 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. 4 K. Takemura Proposition 1 (cf. [3]). Suppose that the operator H = −d2/dx2 + q(x) preserves an n-dimen- sional space U of functions. Let L be the monic differential operator of order n which annihilates all functions in U , and write L = ( d dx )n + n∑ i=1 ci(x) ( d dx )n−i . Set H̃ = −d2/dx2 + q(x) + 2c′1(x). Then we have H̃L = LH. We call the operator L in Proposition 1 the generalized Darboux transformation or the Darboux–Crum transformation. For the case n = 1, let φ0(x) be a non-zero function in U , then U = Cφ0(x), the operator which annihilates φ0(x) is given by L = d/dx − φ′0(x)/φ0(x) and the operator H̃ is given by H̃ = H − 2(φ′0(x)/φ0(x))′. Hence the proposition reproduces the Darboux transformation for the case n = 1. We apply Proposition 1 to Heun’s equation. For this purpose, we recall the quasi-solvability of Heun’s equation. If a finite-dimensional space is invariant under an action of the Hamiltonian H of a model, the model is partially solvable by diagonalizing the matrix representation of H. This situation is called quasi-solvable. On the quasi-solvability of Heun’s equation, we have Proposition 2 ([43], Proposition 5.1). Let αi be a number such that αi = −li or αi = li + 1 for all i ∈ {0, 1, 2, 3}, and set d = − 3∑ i=0 αi/2. Suppose that d ∈ Z≥0, and let Vα0,α1,α2,α3 be the d + 1-dimensional space spanned by{ Φ̂(℘(x))℘(x)n } n=0,...,d , where Φ̂(z) = (z − e1)α1/2(z − e2)α2/2(z − e3)α3/2. Then the operator H(l0,l1,l2,l3) (see equa- tion (2.1)) preserves the space Vα0,α1,α2,α3. By applying Propositions 1 and 2, we obtain the following proposition after some calculations: Proposition 3 ([46]). Let αi be a number such that αi = −li or αi = li+1 for all i ∈ {0, 1, 2, 3}, and set d = − 3∑ i=0 αi/2. Suppose that d ∈ Z≥0, and let Lα0,α1,α2,α3 be the monic differential operator of order d + 1 which annihilates the space Vα0,α1,α2,α3. Then we have H(α0+d,α1+d,α2+d,α3+d)Lα0,α1,α2,α3 = Lα0,α1,α2,α3H (l0,l1,l2,l3). We construct an odd-degree differential operator A which commutes with H by composing the operators Lα0,α1,α2,α3 . We define the operator L̃α0,α1,α2,α3 as follows: L̃α0,α1,α2,α3 =  Lα0,α1,α2,α3 , 3∑ i=0 αi/2 ∈ Z≤0; L1−α0,1−α1,1−α2,1−α3 , 3∑ i=0 αi/2 ∈ Z≥2; 1, otherwise. Towards Finite-Gap Integration of the Inozemtsev Model 5 Set le0 = (−l0 + l1 + l2 + l3)/2, le1 = (l0 − l1 + l2 + l3)/2, le2 = (l0 + l1 − l2 + l3)/2, le3 = (l0 + l1 + l2 − l3)/2, lo0 = (l0 + l1 + l2 + l3 + 1)/2, lo1 = (l0 + l1 − l2 − l3 − 1)/2, lo2 = (l0 − l1 + l2 − l3 − 1)/2, lo3 = (l0 − l1 − l2 + l3 − 1)/2. The following proposition is proved by applying Proposition 3 four times: Proposition 4 ([46]). Assume li ∈ Z≥0 for i = 0, 1, 2, 3. If l0 + l1 + l2 + l3 is even, we set A = L̃−le3,le2+1,le1+1,−le0 L̃−l1,l0+1,−l3,l2+1L̃−le0,−le1,le2+1,le3+1L̃−l0,−l1,−l2,−l3 , (2.2) while if l0 + l1 + l2 + l3 is odd, we set A = L̃lo2+1,−lo3,−lo0,−lo1 L̃−l1,−l0,l3+1,−l2L̃−lo0,lo1+1,−lo2,−lo3 L̃l0+1,−l1,−l2,−l3 . (2.3) We then have that the operator A commutes with H(l0,l1,l2,l3), i.e., AH(l0,l1,l2,l3) = H(l0,l1,l2,l3)A. It is known that, if l0, l1, l2, l3 ∈ Z≥0, then there exist four invariant spaces of the operator H(l0,l1,l2,l3), which we consider in Section 2.3, and the four operators in Proposition 4 are related to the four spaces. Let ki be the rearrangement of li such that k0 ≥ k1 ≥ k2 ≥ k3(≥ 0). Set g =  k0, l0 + l1 + l2 + l3 : even, k0 + k3 ≥ k1 + k2; (k0 + k1 + k2 − k3)/2, l0 + l1 + l2 + l3 : even, k0 + k3 < k1 + k2; k0, l0 + l1 + l2 + l3 : odd, k0 ≥ k1 + k2 + k3 + 1; (k0 + k1 + k2 + k3 + 1)/2, l0 + l1 + l2 + l3 : odd, k0 < k1 + k2 + k3 + 1. Then g ∈ Z≥0 and the degree of the operator A is 2g + 1. For the case l0 = 2, l1 = l2 = l3 = 0, we have g = 2 and the operator A is expressed as L2,−1,−1,0L1,−2,1,0L0,2,−1,−1L−2,0,0,0 = ( d dx + 1 2 ℘′(x) ℘(x)− e1 + 1 2 ℘′(x) ℘(x)− e2 )( d dx + ℘′(x) ℘(x)− e1 − 1 2 ℘′(x) ℘(x)− e2 ) × ( d dx − ℘′(x) ℘(x)− e1 + 1 2 ℘′(x) ℘(x)− e2 + 1 2 ℘′(x) ℘(x)− e3 ) × (( d dx )2 − 1 2 ( ℘′(x) ℘(x)− e1 + ℘′(x) ℘(x)− e2 + ℘′(x) ℘(x)− e3 ) d dx ) = ( d dx )5 − 15℘(x) ( d dx )3 − 45 2 ℘′(x) ( d dx )2 − 9 ( 5℘(x)2 − 3 4 g2 ) d dx . 2.2 Application of finite-gap property We investigate Heun’s equation in the elliptic form (H − E) f(x) = 0, H = − d2 dx2 + u(x), u(x) = 3∑ i=0 li(li + 1)℘(x + ωi), (2.4) 6 K. Takemura by applying the finite-gap integration, which is based on the commutativity of H (= −d2/dx2 + 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 A = (−1)g g∑ j=0 ( ãj(x) d dx + b̃j(x) ) Hg−j , where ã0(x) = 1. We have 0 = [(−1)gA,H] = g∑ j=0 [ ãj(x) d dx + b̃j(x),− d2 dx2 + u(x) ] Hg−j = g∑ j=0 ( ãj(x)u′(x) + 2ã′j(x) d2 dx2 + (ã′′j (x) + 2b̃′j(x)) d dx + b̃′′j (x) ) Hg−j = g∑ j=0 ( 2ã′j(x)(−H + u(x)) + (ã′′j (x) + 2b̃′j(x)) d dx + ãj(x)u′(x) + b̃′′j (x) ) Hg−j = g∑ j=0 ( (ã′′j (x) + 2b̃′j(x)) d dx − 2ã′j+1(x) + 2ã′j(x)u(x) + ãj(x)u′(x) + b̃′′j (x) ) Hg−j . Hence we obtain b̃j(x) = −ã′j(x)/2 + cj , ã′′′j (x)− 4u(x)ã′j(x) + 4ã′j+1(x)− 2u′(x)ãj(x) = 0 for some constants cj (j = 0, . . . , g). Therefore we have Proposition 5. Set ã0(x) = 1 and ãg+1(x) = 0. The operator A may be expressed in the form A = (−1)g  g∑ j=0 { ãj(x) d dx − 1 2 ( d dx ãj(x) )} Hg−j + g∑ j=0 cjH g−j  , (2.5) for some functions ãj(x) (j = 1, . . . , g) and constants cj (j = 0, . . . , g), where the functions ãj(x) (j = 0, . . . , g) satisfy ã′′′j (x)− 4u(x)ã′j(x) + 4ã′j+1(x)− 2u′(x)ãj(x) = 0. (2.6) We define a function Ξ(x,E) which plays the important role for the solutions and the mon- odromy of Heun’s equation. Set Ξ(x,E) = g∑ i=0 ãg−i(x)Ei. (2.7) 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.,( d3 dx3 − 4 (u(x)− E) d dx − 2u′(x) ) Ξ(x,E) = 0. (2.8) On the basis of Proposition 5 and the function Ξ(x,E) in equation (2.7), we have Towards Finite-Gap Integration of the Inozemtsev Model 7 Proposition 6 ([46]). (i) The constants cj (j = 1, . . . , g) in equation (2.5) are all zero. (ii) The function Ξ(x, E) is even doubly-periodic and expressed as Ξ(x,E) = c0(E) + 3∑ i=0 li−1∑ j=0 b (i) j (E)℘(x + ωi)li−j , (2.9) where the coefficients c0(E) and b (i) j (E) are polynomials in E. The coefficients do not have com- mon divisors and the polynomial c0(E) is monic. We have g = degE c0(E) and the coefficients satisfy degE b (i) j (E) < g for all i and j. For the case l0 = 2, l1 = l2 = l3 = 0, we have Ξ(x,E) = E2 + 3E℘(x) + 9℘(x)2 − 9 4g2. where ei = ℘(ωi) and g2 = −4(e1e2 + e2e3 + e3e1). 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 Q(E) = Ξ(x,E)2 ( E − 3∑ i=0 li(li + 1)℘(x + ωi) ) + 1 2 Ξ(x,E) d2Ξ(x, E) dx2 − 1 4 ( dΞ(x,E) dx )2 . (2.10) 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 l0 = 2, l1 = l2 = l3 = 0, we have Q(E) = (E2 − 3g2) 3∏ i=1 (E − 3ei). The following proposition on the integral representation of a solution to equation (2.4) was obtained in [42]: Proposition 7 ([42], Proposition 3.7). Let Ξ(x,E) be the doubly periodic function defined in equation (2.7) and Q(E) be the monic polynomial defined in equation (2.10). Then the function Λ(x,E) = √ Ξ(x,E) exp ∫ √ −Q(E)dx Ξ(x,E) is a solution to the differential equation (2.4). If the value E satisfies Q(E) 6= 0, then the functions Λ(x,E) and Λ(−x,E) form a basis of solutions to equation (2.4). Since equation (2.4) is doubly-periodic, the functions Λ(x + 2ωk, E) and Λ(−(x + 2ωk), E) are also solutions to equation (2.4). We consider the monodromy on the functions Λ(x,E) and Λ(−x,E). Note that, if Λ(x + 2ωk, E) is expressed as B(E)Λ(x,E), then Λ(−(x + 2ωk), E) is expressed as B(E)−1Λ(−x,E). We will express B(E) as a hyperelliptic integral of second kind. We rewrite the function Ξ(x,E) and define a(E) as follows: Ξ(x,E) = c(E) + 3∑ i=0 li−1∑ j=0 a (i) j (E) ( d dx )2j ℘(x + ωi), a(E) = 3∑ i=0 a (i) 0 (E). 8 K. Takemura Proposition 8 ([44], Theorem 3.7). Assume li ∈ Z≥0 (i = 0, 1, 2, 3). Let E0 be a value such that Q(E0) = 0. Then Λ(x + 2ωk, E0) = (−1)qkΛ(x,E0) for qk ∈ {0, 1} (k = 1, 3) and we have Λ(x + 2ωk, E) = (−1)qkΛ(x, E) exp −1 2 ∫ E E0 −2ηka(Ẽ) + 2ωkc(Ẽ)√ −Q(Ẽ) dẼ  , where ηk = ζ(ωk) (k = 1, 3) and ζ(x) is the Weierstrass zeta function. For the case l0 = 2, l1 = l2 = l3 = 0, we set E0 = √ 3g2. Then q1 = q3 = 0 and the function a(E) and c(E) are determined as c(E) = E2 − 3 2g2, a0(E) = 3E. Hence we have Λ(x + 2ωk, E) = Λ(x,E) exp −1 2 ∫ E √ 3g2 −6Ẽηk + (2Ẽ2 − 3g2)ωk√ −(Ẽ2 − 3g2) 3∏ i=1 (Ẽ − 3ei) dẼ  . 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. Proposition 9 ([42], Theorem 3.12). (i) If the value E satisfies Q(E) 6= 0, then there exists t1, . . . , tn and C such that tj 6= tj′ (j 6= j′), tj 6∈ ω1Z + ω3Z and Λ(x,E) is expressed as Λ(x,E) = C l∏ j=1 σ(x + tj) σ(x)l0σ1(x)l1σ2(x)l2σ3(x)l3 exp ( −x l∑ i=1 ζ(tj) ) , (2.11) where σ(x) is the Weierstrass sigma function and σi(x) (i = 1, 2, 3) are the co-sigma functions defined by σi(z) = exp(−ηiz)σ(z + ωi)/σ(ωi). (ii) The function Λ̃(x) = l∏ j=1 σ(x + tj) σ(x)l0σ1(x)l1σ2(x)l2σ3(x)l3 exp(cx), (2.12) with the condition tj 6= tj′ (j 6= j′) and tj 6∈ ω1Z + ω3Z is an eigenfunction of the operator H (see equation (2.4)), if and only if tj (j = 1, . . . , l) and c satisfy the relations, ∑ k 6=j ζ(−tj + tk)− l0ζ(−tj)− 3∑ i=1 li(ζ(−tj + ωi)− ζ(ωi)) = −c, (j = 1, . . . , l), (2.13) (1− δl0,0) c + l∑ j=1 ζ(tj)  = 0, (1− δli,0) c + lζ(ωi) + l∑ j=1 ζ(−ωi + tj)  = 0, (i = 1, 2, 3). Towards Finite-Gap Integration of the Inozemtsev Model 9 The eigenvalue E is given by E = −c2 + (l0l1 + l2l3)e1 + (l0l2 + l1l3)e2 + (l0l3 + l1l2)e3 − 3∑ i=1 liηi(2c + lηi) − l∑ j=1 3∑ i=0 li(℘(tj − ωi)− ζ(tj − ωi)2) + ∑ j<k (℘(tj − tk)− ζ(tj − tk)2). Equation (2.13) is called the Bethe Ansatz equation for the Inozemtsev model of type BC1 (see [42]). Note that Gesztesy and Weikard [17] obtained similar results. The monodromy of the function Λ̃(x) in equation (2.12) is written as Λ̃(x + 2ωk) = exp(2ηk(t1 + · · ·+ tl) + 2ωk(c− ζ(t1)− · · · − ζ(tl)))Λ̃(x) for k = 1, 2, 3. In order to describe the proposition on the Hermite–Krichever Ansatz, we define Φi(x, α) = σ(x + ωi − α) σ(x + ωi) exp(ζ(α)x), (i = 0, 1, 2, 3). Proposition 10 ([45]). There exist polynomials P1(E), . . . , P6(E) such that, if P2(E) 6= 0, then Λ(x,E) is written as Λ(x,E) = exp (κx)  3∑ i=0 li−1∑ j=0 b̃ (i) j ( d dx )j Φi(x, α)  (2.14) for some values b̃ (i) j (i = 0, . . . , 3, j = 0, . . . , li − 1), α and κ. The values α and κ are expressed as ℘(α) = P1(E) P2(E) , ℘′(α) = P3(E) P4(E) √ −Q(E), κ = P5(E) P6(E) √ −Q(E). For the periodicity of the function Λ(x,E), we have Λ(x + 2ωk, E) = exp(−2ηkα + 2ωkζ(α) + 2κωk)Λ(x,E) for k = 1, 3. Note that α in equation (2.14) and tj in equation (2.11) satisfy the relation α = − l∑ j=1 tj . To calculate the polynomials P1(E), . . . , P6(E), it is effective to apply the notions “twisted Heun polynomial” and “theta-twisted Heun polynomial” (see [45]). If l0 = 2, l1 = l2 = l3 = 0, then the values α and κ are expressed as ℘(α) = e1 − (E − 3e1)(E + 6e1)2 9(E2 − 3g2) , κ = 2 3(E2 − 3g2) √ −Q(E). 2.3 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: 10 K. Takemura Proposition 11 ([44], Proposition 3.2). LetH andAbe the operators defined by equation (2.4) and equations (2.2), (2.3), and Q(E) be the polynomial defined in equation (2.10). Then A2 + Q(H) = 0. It is known that, if l0, l1, l2, l3 ∈ Z≥0, then there exist four invariant subspaces with respect to the action of the operator H. We describe the spaces more precisely. Let Vα0,α1,α2,α3 be the space defined in Proposition 2 and Uα0,α1,α2,α3 =  Vα0,α1,α2,α3 , 3∑ i=0 αi/2 ∈ Z≤0; V1−α0,1−α1,1−α2,1−α3 , 3∑ i=0 αi/2 ∈ Z≥2; {0}, otherwise. If l0, l1, l2, l3 ∈ Z≥0 and l0 + l1 + l2 + l3 is even, then the operator H preserves the space V = U−l0,−l1,−l2,−l3 ⊕ U−l0,−l1,l2+1,l3+1 ⊕ U−l0,l1+1,−l2,l3+1 ⊕ U−l0,l1+1,l2+1,−l3 , (2.15) and also preserves the components in equation (2.15). If l0, l1, l2, l3 ∈ Z≥0 and l0 + l1 + l2 + l3 is odd, then the operator H preserves the space V = U−l0,−l1,−l2,l3+1 ⊕ U−l0,−l1,l2+1,−l3 ⊕ U−l0,l1+1,−l2,−l3 ⊕ Ul0+1,−l1,−l2,−l3 , (2.16) and also preserves the components in equation (2.16). Then we have Proposition 12 ([46]). (i) The operator A annihilates any elements in the space V . (ii) The monic characteristic polynomial of the space V with respect to the action of H coincides with Q(E). Note that the operator A was constructed by composing the generalized Darboux transfor- mations which are related to the spaces in the components of equation (2.15) or equation (2.16). 3 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 equa- tion in the elliptic form. For this purpose, we introduce the quantum mechanical systems. 3.1 The elliptic Calogero–Moser–Sutherland model The elliptic Calogero–Moser–Sutherland model (or the elliptic Olshanetsky–Perelomov mo- del [31]) of type AN−1 is a quantum many-body system whose Hamiltonian is given as follows: H = −1 2 N∑ i=1 ∂2 ∂x2 i + l(l + 1) ∑ 1≤i<j≤N ℘(xi − xj), where ℘(x) is the Weierstrass elliptic function. For the case N = 2, the model reproduces Lamé’s equation by setting x1 − x2 = x and restricting to the line x1 + x2 = 0. This model is known to be completely integrable, i.e., there exist N -algebraically independent commuting operators Pk (k = 1, . . . , N) which commute with the Hamiltonian H. Namely, by setting Pk = ∑ 0≤j≤[k/2] (l(l + 1))j 2jj!(k − 2j)! ∑ σ∈SN σ(℘(x1 − x2)℘(x3 − x4) · · · Towards Finite-Gap Integration of the Inozemtsev Model 11 × ℘(x2j−1 − x2j)∂2j+1∂2j+2 · · · ∂k), (3.1) where SN is the symmetric group, [x] is the integral part of x and ∂i = ∂/∂xi, we have [Pk,H] = 0 (1 ≤ k ≤ N) and [Pk, Pk′ ] = 0 (1 ≤ k, k′ ≤ N) (see [30]). The Hamiltonian H is expressed as H = P2 − 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, Htrig = −1 2 N∑ i=1 ∂2 ∂x2 i + π2l(l + 1) ∑ 1≤i<j≤N 1 sin2(π(xi − xj)) . The eigenstates of the Calogero–Sutherland model are described by the Jack polynomial J ( 1 l+1 ) λ (X) (λ ∈MN ) (see [39]), i.e., Htrig(J ( 1 l+1 ) λ (X)∆(X)l+1) = (e0 + 2π2E [ 1 l+1 ] λ )J ( 1 l+1 ) λ (X)∆(X)l+1, where Xi = exp ( 2π √ −1xi ) , MN = {λ = (λ1, λ2, . . . , λN )\|i > j ⇒ λi − λj ∈ Z≥0}, e0 = 1 6π2(l+1)2N(N2−1), ∆(X) = (X1X2 · · ·XN ) 1−N 2 ∏ i<j (Xi−Xj) and E [α] λ = N∑ i=1 λ2 i + N∑ i=1 N+1−2i α λi. 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 N - particle 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. Fix the parameters N and l. We set m = lN(N−1)/2. Let c : {1, . . . ,m} → {1, . . . , N} be the unique non-decreasing function such that c−1(j) has (N−j)l elements. Let εi (1 ≤ i ≤ N) be an orthonormal basis of RN with an inner product (·, ·). Set αi = εi−εi+1, h∗ = { N∑ i=1 xiεi\| N∑ i=1 xi = 0}, pi = i(2N − i − 1)l/2 and Vi = {pi−1 + 1, pi−1 + 2, . . . , pi} (1 ≤ i ≤ N − 1). Let W be the set of maps w = (w1, . . . , wN ) (wi : Vi → {i, i + 1, . . . , N − 1}) such that #{w−1 i (j)} = l for 1 ≤ i ≤ j ≤ N − 1. For w = (w1, . . . , wN−1) ∈ W , let Fw be the set of maps f = (f1, . . . , fN−2) (fi : Vi+1 → Vi) such that (i) fi is injective (ii) If fi(x) = y then wi+1(x) = wi(y). Set θ1(x) = 2 ∞∑ n=1 (−1)n−1 exp ( τπ √ −1(n− 1/2)2 ) sin(2n− 1)πx, θ(x) = θ1(x) θ′1(0) , σλ(x) = θ′(0)θ(x− λ) θ(x)θ(λ) . For ξ ∈ h∗, we introduce the functions Φτ (t1, . . . , tm) and ω(t;x) as follows Φτ (t1, . . . , tm) = e2π √ −1(ξ, ∑ j tjαc(j)) × ∏ 1≤j≤(N−1)l θ(tj)−lN ∏ c(i)=c(j) i<j θ(ti − tj)2 ∏ \|c(i)−c(j)\|=1 i<j θ(ti − tj)−1, 12 K. Takemura ω(t;x) = e2π √ −1(ξ, ∑ i xiεi) ∑ w∈W ∑ f∈Fw N−1∏ i=1 pi∏ k=pi−1+1 σxi−xwi(k)+1 (tk − tfi(k)), where t0 = 0, f0(k) = 0. Then we have Proposition 13 ([12, 13, 11]). If (t01, . . . , t 0 m) satisfy the following Bethe Ansatz equations, ∂Φτ ∂ti \|(t01,...,t0m) = 0 (1 ≤ i ≤ m), the function ω(t0;x) is an eigenfunction of the Hamiltonian H with the eigenvalue 2π2(ξ, ξ)− 2π √ −1 ∂ ∂τ S(t01, . . . , t 0 m; τ)− l(l + 1)(N − 1)Nη, where S(t1, . . . , tm; τ) = ∑ i<j (αc(i), αc(j)) log θ(ti − tj)− ∑ c(i)=1 lN log θ(ti), η = π2 ( 1 6 − 4 ∞∑ n=1 pn 1− pn ) and p = exp ( 2π √ −1τ ) . 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 AN−1). By symmetrizing or anti-symmetrizing the function ω(t0;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(t0, x) written in the form of the Bethe Ansatz is also an eigenfunction of the higher commuting operators P3, . . . , PN (see also [35]). Towards Finite-Gap Integration of the Inozemtsev Model 13 3.2 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 BCN [20] is a quantum mechanical system with N -particles whose Hamiltonian is given by H = − N∑ j=1 ∂2 ∂x2 j + 2l(l + 1) ∑ 1≤j<k≤N (℘(xj − xk) + ℘(xj + xk)) (3.2) + N∑ j=1 3∑ i=0 li(li + 1)℘(xj + ωi), where l and li (i = 0, 1, 2, 3) are coupling constants. This is also a generalization of the elliptic Calogero–Moser–Sutherland model of type BCN . The Inozemtsev model of type BCN is completely integrable, i.e., there exist N algebraically independent mutually commuting differential operators Pk (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 BCN (resp. the elliptic Calogero–Moser–Sutherland model of type AN ) is a universal completely integrable model of quantum mechanics with the symmetry of the Weyl group of type BN (resp. type AN ), 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 BCN 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 BCN , and we can investigate the Inozemtsev model of type BCN by perturbation from the trigonometric model [23, 47]. A method of quasi-solvability is available on the Inozemtsev model of type BCN . 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 BCN . We now describe the finite- dimensional 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, bi (i = 0, 1, 2, 3) be the numbers which satisfy a ∈ {−l, l+1} and bi ∈ {−li/2, (li + 1)/2} (i = 0, 1, 2, 3). Set Φ(z) = ∏ 1≤j<k≤N (zj − zk)a N∏ j=1 3∏ i=1 (zj − ei)bi . Assume that d = −((N − 1)a + b0 + b1 + b2 + b3) is a non-negative integer. Let W sym d be the space spanned by Φ(℘(x1), ℘(x2), . . . , ℘(xN )) ∑ σ∈SN ℘(x1)mσ(1)℘(x2)mσ(2) · · ·℘(xN )mσ(N) such that mi ∈ {0, 1, . . . , d} for all i. Then we have H ·W sym d ⊂ W sym d . The quasi-solvability was extended to the commuting differential operators. 14 K. Takemura Proposition 15 ([41], Theorem 3.3). Assume that d = −((N − 1)a + b0 + b1 + b2 + b3) is a non-negative integer. Then Pk ·W sym d ⊂ W sym d for k = 1, 2, . . . , N , where W sym d is the finite- dimensional space defined in Proposition 14 and Pk are the commuting differential operators which ensure the complete integrability. By the quasi-solvability, finitely-many eigenvalues and eigenfunctions are calculated by diag- onalizing the commuting matrices simultaneously for the case that the assumption of Proposi- tion 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, b0, b1 in Proposition 14 are chosen as a = l + 1, b0 = (l0 + 1)/2 and b1 = (l1 + 1)/2 are square-integrable. It seems that an explicit expression of the Bethe Ansatz as Proposition 13 for the Inozemtsev model of type BCN 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 BCN , though these Bethe Ansatz equations would be rather complicated. For a special case of the BC2 case, this scheme is worked out explicitly in [6, § 6]. We hope to investigate the Bethe Ansatz for the Inozemtsev model of type BCN to study the model in more detail. 4 Towards finite-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 in- troduced the notion “algebraic integrability”. The Schrödinger operator L = − N∑ i=1 ∂2/∂x2 i + u(x1, . . . , xN ) is called completely integrable [8], if there exist N commuting operators L1 = L,L2, . . . , LN with algebraically independent constant highest symbols s1(ξ) (= ξ2 1 + · · ·+ ξ2 N ), s2(ξ), . . . , sN (ξ) (ξj = √ −1∂/∂xj). For example, the Calogero–Moser–Sutherland models are completely integrable. On the model of type AN , the highest symbols of Pk (see equation (3.1)) are written as ((− √ −1)k/k!) ∑ i1<i2<···<ik ξi1ξi2 · · · ξik . The Inozemtsev model of type BCN 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 L0 commuting with Li (i = 1, . . . , N) and the highest symbol s0(ξ) of L0 takes the distinct values at the roots of the equations si(ξ) = Ei (i = 1, . . . , N) for almost all Ei. On Heun’s equation in the elliptic form, if l0, l1, l2, l3 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 AN , Braverman, Etingof and Gaitsgory [4] obtained algebraic integrability. More precisely, they established that, if l is a positive integer, then the operator H = −1 2 N∑ i=1 ∂2 ∂x2 i + l(l + 1) ∑ 1≤i<j≤N ℘(xi − xj) is algebraically integrable by applying the Bethe Ansatz due to Felder–Varchenko (see Sec- tion 3.1) and the differential Galois theory. In [6], Chalykh, Etingof and Oblomkov proved Towards Finite-Gap Integration of the Inozemtsev Model 15 that the Chalykh–Veselov conjecture is true and the Inozemtsev model of type BCN (see equa- tion (3.2)) is also algebraically integrable, if l, l0, l1, l2, l3 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. The explicit expressions of the extra commuting operators were obtained and investigated by Oblomkov, Khodarinova and Prikhodsky [29, 21, 22] for the case l = 1 on the Calogero–Moser– Sutherland model of type A3 and the case l = l0 = 1, l1 = l2 = l3 = 0 on the Inozemtsev model of type BC2. On the Calogero–Moser–Sutherland model of type A3 with l = 1, the Hamiltonian and commuting operators which guarantee the complete integrability are given as H = −(∂2 1 + ∂2 2 + ∂2 3)/2 + 2(℘12 + ℘23 + ℘31), P1 = ∂1 + ∂2 + ∂3, P3 = ∂1∂2∂3 + 2℘12∂3 + 2℘23∂1 + 2℘31∂2, (see equation (3.1)) where we have used the notations ∂i = ∂/∂xi and ℘ij = ℘(xi − xj). The additional commuting operators are written as I12 = (∂1 − ∂3)2(∂2 − ∂3)2 − 8℘23(∂1 − ∂3)2 − 8℘13(∂2 − ∂3)2 + 4(℘12 − ℘13 − ℘23)(∂1 − ∂3)(∂2 − ∂3)− 2(℘′12 + ℘′13 + 6℘′23)(∂1 − ∂3) − 2(−℘′12 + 6℘′13 + ℘′23)(∂2 − ∂3)− 2℘′′12 − 6℘′′13 − 6℘′′23 + 4(℘2 12 + ℘2 13 + ℘2 23) + 8(℘12℘13 + ℘12℘23 + 7℘13℘23), I23 and I31, which are written by permuting the indices. Then any non-symmetric linear com- bination of them, e.g., L4 = I12 + 2I23 would fit into the definition of algebraic integrability (see [21]). Explicit expressions of the extra commuting operators for the models which have symmetry of the deformed root system A3(m) or B2(l,m) were also obtained. 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 con- structed 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 Hamiltonian H̃(= 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 BCN , if the value d(= −((N−1)a+b0+b1+b2+b3)) (a ∈ {−l, l+1}, bi ∈ {−li/2, (li+1)/2} (i = 0, 1, 2, 3)) in the assumption of Proposition 14 is equal to zero. Then the alternate Hamil- tonian H̃ with respect to H in equation (3.2) would be written as H̃ = − N∑ j=1 ∂2 ∂x2 j + 2a(a + 1) ∑ j<k (℘(xj − xk) + ℘(xj + xk)) + N∑ j=1 3∑ i=0 2bi(2bi + 1)℘(xj + ωi). In the moment of writing, we do not know an operator L which directly intertwines the oper- ators H and H̃ as H̃L = LH. 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[49] Whittaker E.T., Watson G.N., A course of modern analysis, 4th ed., Cambridge University Press, New York, 1962. http://arxiv.org/abs/math.CA/0306242 http://arxiv.org/abs/solv-int/9701004 http://arxiv.org/abs/q-alg/9705006 http://arxiv.org/abs/math-ph/0007036 http://arxiv.org/abs/math-ph/0511015 http://arxiv.org/abs/math.CA/0109149 http://arxiv.org/abs/math.QA/0002104 http://arxiv.org/abs/math.QA/0205274 http://arxiv.org/abs/math.CA/0103077 http://arxiv.org/abs/math.CA/0112179 http://arxiv.org/abs/math.CA/0201208 http://arxiv.org/abs/math.CA/0406141 http://arxiv.org/abs/math.CA/0508093 http://arxiv.org/abs/nlin.SI/0303005 1 Introduction 2 Finite-gap integration of Heun's equation 2.1 Darboux transformations and Heun's equation 2.2 Application of finite-gap property 2.3 Relationship among commuting operators 3 Results on the Calogero-Moser-Sutherland model and the Inozemtsev model 3.1 The elliptic Calogero-Moser-Sutherland model 3.2 The Inozemtsev model 4 Towards finite-gap integration of the Inozemtsev model References

Towards Finite-Gap Integration of the Inozemtsev Model

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