Completely Integrable Systems Associated with Classical Root Systems

We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete int...

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Datum:	2007
1. Verfasser:	Oshima, T.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2007
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1473692019-02-15T01:24:36Z Completely Integrable Systems Associated with Classical Root Systems Oshima, T. We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants. We review conditions supporting the conjecture and give a new condition assuring it. 2007 Article Completely Integrable Systems Associated with Classical Root Systems / T. Oshima // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 41 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81R12; 70H06 http://dspace.nbuv.gov.ua/handle/123456789/147369 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants. We review conditions supporting the conjecture and give a new condition assuring it.
format	Article
author	Oshima, T.
spellingShingle	Oshima, T. Completely Integrable Systems Associated with Classical Root Systems Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Oshima, T.
author_sort	Oshima, T.
title	Completely Integrable Systems Associated with Classical Root Systems
title_short	Completely Integrable Systems Associated with Classical Root Systems
title_full	Completely Integrable Systems Associated with Classical Root Systems
title_fullStr	Completely Integrable Systems Associated with Classical Root Systems
title_full_unstemmed	Completely Integrable Systems Associated with Classical Root Systems
title_sort	completely integrable systems associated with classical root systems
publisher	Інститут математики НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/147369
citation_txt	Completely Integrable Systems Associated with Classical Root Systems / T. Oshima // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 41 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT oshimat completelyintegrablesystemsassociatedwithclassicalrootsystems
first_indexed	2025-07-11T01:56:04Z
last_indexed	2025-07-11T01:56:04Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 061, 50 pages Completely Integrable Systems Associated with Classical Root Systems? Toshio OSHIMA Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8914, Japan E-mail: oshima@ms.u-tokyo.ac.jp URL: http://akagi.ms.u-tokyo.ac.jp/ Received December 14, 2006, in final form March 19, 2007; Published online April 25, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/061/ Abstract. We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants. We review conditions supporting the conjecture and give a new condition assuring it. Key words: completely integrable systems; Calogero–Moser systems; Toda lattices with boundary conditions 2000 Mathematics Subject Classification: 81R12; 70H06 To the memory of Professor Vadim B. Kuznetsov 1 Introduction A Schrödinger operator P = −1 2 n∑ j=1 ∂2 ∂x2 j +R(x) (1.1) with the potential function R(x) of n variables x = (x1, . . . , xn) is called completely integrable if there exist n differential operators P1, . . . , Pn such that [Pi, Pj ] = 0 (1 ≤ i < j ≤ n), P ∈ C[P1, . . . , Pn], (1.2) P1, . . . , Pn are algebraically independent. In this paper, we explicitly construct the integrals P1, . . . , Pn for completely integrable potential functions R(x) of the form R(x) = ∑ 1≤i<j≤n (u−ij(xi − xj) + u+ ij(xi + xj)) + n∑ k=1 vk(xk) (1.3) ?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:oshima@ms.u-tokyo.ac.jp http://akagi.ms.u-tokyo.ac.jp/ http://www.emis.de/journals/SIGMA/2007/061/ http://www.emis.de/journals/SIGMA/kuznetsov.html 2 T. Oshima appearing in other papers. The Schrödinger operators with these commuting differential oper- ators treated in this paper include Calogero–Moser–Sutherland systems (cf. [5, 22, 27, 28, 36]), Heckman–Opdam’s hypergeometric systems (cf. [34] for type An−1, [11] in general), their exten- sions (cf. [1, 8, 14, 23, 24, 25, 30]) and finite Toda lattices corresponding to (extended) Dynkin diagrams for classical root systems (cf. [2, 9, 10, 17, 26, 33, 38]) and those with boundary conditions (cf. [8, 13, 18, 19, 20, 21, 23, 30]). Put ∂j = ∂/∂xj for simplicity. We denote by σ(Q) the principal symbol of a differential operator of Q. For example, σ(P ) = −(1/2)(ξ21 + · · ·+ ξ2n). We note that [40] proves that the potential function is of the form (1.3) if σ(Pk) = ∑ 1≤j1<···<jk≤n ξ2j1 · · · ξ 2 jk for k = 1, . . . , n. (1.4) In this case we say that R(x) is an integrable potential function of type Bn or of the classical type. Moreover whenR(x) is symmetric with respect to the coordinate (x1, . . . , xn) and invariant under the coordinate transformation (x1, x2, . . . , xn) 7→ (−x1, x2, . . . , xn), then R(x) is determined by [32] for n ≥ 3 and by [24] for n = 2 and Pk are calculated by [29]. Classifications of the integrable potential functions under certain conditions are given in [23, 24, 25, 30, 37, 40] etc. In Section 9 we review them and we present Conjecture which claims that the potential functions given in this note exhaust those of the completely integrable systems satisfying (1.4). We also give a new condition which assures Conjecture. If vk = 0 for k = 1, . . . , n, we can expect σ(Pk) = ∑ 1≤j1<···<jk≤n ξ2j1 · · · ξ 2 jk for k = 1, . . . , n− 1 and σ(Pn) = ξ1ξ2 · · · ξn and we say the integrable potential function is of type Dn. If vk = 0 and u+ ij = 0 for k = 1, . . . , n and 1 ≤ i < j ≤ n, we can expect P1 = ∂1 + · · · + ∂n, σ(Pk) =∑ 1≤j1<···<jk≤n ξj1 · · · ξjk for k = 2, . . . , n and we say that the integrable potential function is of type An−1. Note that the integrable potential function of type An−1 or Dn is of type Bn. The elliptic potential function of type An−1 with u−ij(t) = C℘(t; 2ω1, 2ω2) + C ′, u+ ij(t) = vk(t) = 0 (C, C ′ ∈ C) (cf. [28]) and that of type Bn with u−ij(t) = v+ ij(t) = A℘(t; 2ω1, 2ω2), vk(t) = 3∑ j=0 Cj℘(t+ ωj ; 2ω1, 2ω2)− C 2 , (A, Ci, C ∈ C) introduced by [12] are most fundamental and their integrability and the integrals of higher order are established by [25, 29, 32]. Here ℘(t; 2ω1, 2ω2) is the Weierstrass elliptic function whose fundamental periods are 2ω1 and 2ω2 and ω0 = 0, ω1 + ω2 + ω3 = 0. Other potential functions are suitable limits of these elliptic potential functions, which is shown in [8, 13, 33] etc. We will study integrable systems by taking analytic continuations of the integrals given in [25] with respect to a suitable parameter, which is done for the invariant systems (of type An−1) by [32] and (of types Bn andDn) by [29] and for the systems of type An−1 by [33]. The main purpose of this note is to give the explicit expression of the operators P1, . . . , Pn in (1.2) in this unified way. Namely we construct enough commuting integrals of the non-invariant systems from those of the invariant systems given by [24, 29, 32]. Such study of the systems of types An−1, B2, Bn (n ≥ 3) and Dn are explained in Sections 3, 4, 5, 6, respectively. Completely Integrable Systems Associated with Classical Root Systems 3 Since the integrals of the system of type An−1 are much simpler than those of type Bn, we review the above analytic continuation for the systems of type An−1 in Section 3 preceding to the study for the systems of type Bn. There are many series of completely integrable systems of type B2, which we review and classify in Section 4 with taking account of the above unified way. We present 8 series of potential functions of type Bn in Section 5. There are 3 (elliptic, trigonometric or hyperbolic and rational) series of the invariant potentials of type Bn whose enough integrals are constructed by [25] and [29]. The complete integrability of the remaining 5 series of the potential functions is shown in Section 5, which is conjectured by [8] (4 series), partially proved by [18, 19, 21] (3 series) and announced by [30] (5 series). The complete integrability of two series among them seems to be first established in this note. Note that when n ≥ 3, our systems which do not belong to these 8 series of type Bn are the Calogero– Moser systems with elliptic potentials and the finite Toda lattices of type A(1) n−1, whose complete integrability is known. The main purpose of our previous study in [23, 24, 25, 30] is classification of the completely integrable systems associated with classical root systems. In this note we explicitly give integrals of all the systems classified in our previous study with reviewing known integrals. Since our expression of Pk is natural, we can easily define their classical limits without any ambiguity and get completely integrable Hamiltonians of dynamical systems together with their enough integrals. This is clarified in Section 7. In Section 8 we examine ordinary differential operators which are analogues of the Schrödinger operators studied in this note. 2 Notation and preliminary results Let {e1, . . . , en} be the natural orthonormal base of the Euclidean space Rn with the inner product 〈x, y〉 = n∑ j=1 xjyj for x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Rn. Here ej = (δ1j , . . . , δnj) ∈ Rn with Kronecker’s delta δij . Let α ∈ Rn\{0}. The reflection wα with respect to α is a linear transformation of Rn defined by wα(x) = x− 2〈α,x〉〈α,α〉 α for x ∈ Rn. Furthermore we define a differential operator ∂α by (∂αϕ)(x) = d dt ϕ(x+ tα) ∣∣∣∣ t=0 and then ∂j = ∂ej . The root system Σ(Bn) of type Bn is realized in Rn by Σ(An−1)+ = {ei − ej ; 1 ≤ i < j ≤ n}, Σ(Dn)+ = {ei ± ej ; 1 ≤ i < j ≤ n}, Σ(Bn)+S = {ek; 1 ≤ k ≤ n}, Σ(Bn)+ = Σ(Dn)+ ∪ Σ(Bn)+S , Σ(F ) = {α,−α; α ∈ Σ(F )+} for F = An−1, Dn or Bn. The Weyl groups W (Bn) of type Bn, W (Dn) of type Dn and W (An−1) of type An−1 are the groups generated by wα for α ∈ Σ(Bn), Σ(Dn) and Σ(An−1), respectively. The Weyl group 4 T. Oshima W (An−1) is naturally identified with the permutation group Sn of the set {1, . . . , n} with n elements. Let ε be the group homomorphism of W (Bn) defined by ε(w) = { 1 if w ∈W (Dn), −1 if w ∈W (Bn) \W (Dn). The potential function (1.3) is of the form R(x) = ∑ α∈Σ(Dn)+ uα(〈α, x〉) + ∑ α∈Σ(Bn)+S vβ(〈β, x〉) with functions uα and vβ of one variable. For simplicity we will denote uα(x) = u−α(x) = uα(〈α, x〉) for α ∈ Σ(Dn)+, vβ(x) = v−β(x) = vβ(〈β, x〉) for β ∈ Σ(Bn)+S , u±ij(x) = uei±ej (x), vk(x) = vek (x). Lemma 1. For a bounded open subset U of C, there exists an open neighborhood V of 0 in C such that the following statements hold. i) The function λ sinh−1 λz is holomorphically extended to (z, λ) ∈ (U \ {0}) × V and the function is 1/z when λ = 0. ii) Suppose Reλ > 0. Then the functions e2λt sinh−2 λ(z ± t) and e4λt ( sinh−2 λ(z ± t)− cosh−2 λ(z ± t) ) are holomorphically extended to (z, q) ∈ U × V with q = e−2λt and the functions are 4e∓2λz and 16e∓4λz, respectively, when q = 0. Proof. The claims are clear from λ−1 sinhλz = z + ∞∑ j=1 λ2jz2j+1 (2j + 1)! , 4e−2λt sinh2 λ(z ± t) = e±2λz(1− e−2λte∓2λz)2, sinh−2 λz − cosh−2 λz = 4 sinh−2 2λz. � The elliptic functions ℘ and ζ of Weierstrass type are defined by ℘(z) = ℘(z; 2ω1, 2ω2) = 1 z2 + ∑ ω 6=0 ( 1 (z − ω)2 − 1 ω2 ) , ζ(z) = ζ(z; 2ω1, 2ω2) = 1 z + ∑ ω 6=0 ( 1 z − ω + 1 ω + z ω2 ) , where the sum ranges over all non-zero periods 2m1ω1+2m2ω2 (m1, m2 ∈ Z) of ℘. The following are some elementary properties of these functions (cf. [41]). ℘(z) = ℘(z + 2ω1) = ℘(z + 2ω2), (2.1) ζ ′(z) = −℘(z), (2.2) (℘′)2 = 4℘3 − g2℘− g3 = 4(℘− e1)(℘− e2)(℘− e3), eν = ℘(ων) for ν = 1, 2, 3, ω3 = −ω1 − ω2 and ω0 = 0, (2.3) Completely Integrable Systems Associated with Classical Root Systems 5 ℘(2z) = 1 4 4∑ ν=0 ℘(z + ων) = (12℘(z)2 − g2)2 16℘′(z)2 − 2℘(z), ℘(z; 2ω2, 2ω1) = ℘(z; 2ω1, 2ω2), (2.4) ℘(z + ω1; 2ω1, 2ω2) = e1 + (e1 − e2)(e1 − e3) ℘(z; 2ω1, 2ω2)− e1 , (2.5) ℘(z; √ −1λ−1π,∞) = λ2 sinh−2 λz + 1 3 λ2, (2.6) ℘(z;∞,∞) = z−2, (2.7) ℘(z;ω1, 2ω2) = ℘(z; 2ω1, 2ω2) + ℘(z + ω1; 2ω1, 2ω2)− e1, (2.8)∣∣∣∣∣∣ ℘(z1) ℘′(z1) 1 ℘(z2) ℘′(z2) 1 ℘(z3) ℘′(z3) 1 ∣∣∣∣∣∣ = 0 if z1 + z2 + z3 = 0, (2.9) ℘(z; 2ω1, 2ω2) = − η1 ω1 + λ2 sinh−2 λz + ∞∑ n=1 8nλ2e−4nλω2 1− e−4nλω2 cosh 2nλz, η1 = ζ(ω1; 2ω1, 2ω2) = π2 ω1 ( 1 12 − 2 ∞∑ n=1 ne−4nλω2 1− e−4nλω2 ) , τ = ω2 ω1 , q = eπiτ = e−2λω2 and λ = π 2 √ −1ω1 . (2.10) Here the sums in (2.10) converge if 2Im ω2 ω1 > \|z\| \|ω1\| . Let 0 ≤ k < 2m. Then (2.10) means ℘ ( z + k m ω2; 2ω1, 2ω2 ) = − η1 ω1 + 4λ2 ( qk/me−2λz( 1− e−2λzqk/m )2 + ∞∑ n=1 qn(2−k/m)e2nλz 1− q2n ) , − η1 ω1 = 4λ2 ( 1 12 − 2 ∞∑ n=1 nq2n 1− q2n ) . Lemma 2. Let k and m be integers satisfying 0 < k < 2m. Put ℘0(z; 2ω1, 2ω2) = ℘(z; 2ω1, 2ω2) + η1 ω1 , λ = π 2 √ −1ω1 and t = q1/m = eπiω2/(mω1). Then for any bounded open set U in C × C, there exists a neighborhood V of the origin of C such that the following statements hold. i) ℘0(z; 2ω1, 2ω2)−λ2 sinh−2 λz and ℘0(z+ω1; 2ω1, 2ω2)+λ2 cosh−2 λz are holomorphic func- tions of (z, λ, q) ∈ U × V and vanish when q = 0. ii) ℘0(z + (k/m)ω2; 2ω1, 2ω2) is holomorphic for (z, λ, t) ∈ U × V and has zeros of order min{k, 2m− k} along the hyperplane defined by t = 0 and satisfies t−k℘0 ( z + k m ω2; 2ω1, 2ω2 )∣∣∣∣ t=0 = 4λ2e−2λz (0 < k < m), t−k℘0 ( z + k m ω2; 2ω1, 2ω2 )∣∣∣∣ t=0 = 8λ2 cosh 2λz (k = m), 6 T. Oshima tk−2m℘0 ( z + k m ω2; 2ω1, 2ω2 )∣∣∣∣ t=0 = 4λ2e2λz (m < k < 2m). For our later convenience we list up some limiting formula discussed above. Fix ω1 with√ −1ω1 > 0 and let ω2 ∈ R with ω2 > 0. Then λ = π/(2 √ −1ω1) > 0 and sinh2 λ(z + ω1) = − cosh2 λz, cosh 2λ(z + ω1) = − cosh 2λz, (2.11) lim λ→0 λ2 sinh−2 λz = 1 z2 , (2.12) lim N→±∞ e2λ\|N \| sinh−2 λ(z +N) = 4e∓2λz, (2.13) lim ω2→+∞ ℘0(z; 2ω1, 2ω2) = λ2 sinh−2 λz, (2.14) lim ω2→+∞ ℘0(z + ω1; 2ω1, 2ω2) = −λ2 cosh−2 λz, (2.15) lim ω2→∞ e2rλω2℘0(z + rω2; 2ω1, 2ω2) = 4λ2e−2λz if 0 < r < 1, (2.16) lim ω2→∞ e2λω2℘0(z + ω2; 2ω1, 2ω2) = 8λ2 cosh 2λz, (2.17) lim ω2→∞ e2(2−r)λω2℘0(z + rω2; 2ω1, 2ω2) = 4λ2e2λz if 1 < r < 2. (2.18) 3 Type An−1 (n ≥ 3) The completely integrable Schrödinger operator of type An−1 is of the form P = −1 2 n∑ j=1 ∂2 ∂x2 j + ∑ 1≤i<j≤n u−ij(xi − xj). Denoting uei−ej (x) = uej−ei(x) = u−ij(xi − xj), we put Pk = ∑ 0≤ν≤k/2 1 2νν!(k−2ν)!(n−k)! ∑ w∈Sn uw(e1−e2)uw(e3−e4) · · ·uw(e2ν−1−e2ν)∂w(e2ν+1) · · · ∂w(ek) = ∑ 0≤ν≤k/2 ∑ u−i1i2 · · ·u−i2ν−1i2ν ∂i2ν+1 · · · ∂ik (3.1) according to the integrals given in [25, 32]. We will examine the functions u−ij(t) which satisfy [Pi, Pj ] = 0 for 1 ≤ i < j ≤ n. (3.2) Here we note that P = P2 − 1 2 P 2 1 , P1 = ∂1 + · · ·+ ∂n, P2 = ∑ 1≤i<j≤n ∂i∂j + ∑ 1≤i<j≤n u−ij(xi − xj), P3 = ∑ 1≤i<j<k≤n ∂i∂j∂k + n∑ k=1 ∑ 1≤i<j≤n i6=k, j 6=k u−ij(xi − xj)∂k. Completely Integrable Systems Associated with Classical Root Systems 7 P4 = ∑ 1≤i<j<k<`≤n ∂i∂j∂k∂` + ∑ 1≤k<`≤n ∑ 1≤i<j≤n i6=k,`, j 6=k,` u−ij∂k∂` + ∑ 1≤i<j<k<`≤n (u−iju − k` + u−iku − j` + u−i`u − jk) = ∑ ∂i∂j∂k∂` + ∑ u−ij∂k∂` + ∑ u−iju − k`, P5 = ∑ ∂i∂j∂k∂`∂m + ∑ u−ij∂k∂`∂m + ∑ u−iju − k`∂m, P6 = ∑ ∂i∂j∂k∂`∂m∂ν + ∑ u−ij∂k∂`∂m∂ν + ∑ u−iju − k`∂m∂ν + ∑ u−iju − k`u − mν , · · · · · · · · · · · · · · · · · · Since W (An−1) is naturally isomorphic to the permutation group Sn of the set {1, . . . , n}, we will identify them. In [25, 32], the integrable potentials of type An−1 which are invariant under the action of Sn are determined and moreover (3.2) with (3.1) is proved. They are uei−ej (x) = u(〈ei − ej , x〉) (3.3) with an even function u and (Ellip-An−1) Elliptic potential of type An−1: u(t) = C℘0(t; 2ω1, 2ω2), RE(An−1;x1, . . . , xn;C, 2ω1, 2ω2) = C ∑ 1≤i<j≤n ℘0(xi − xj ; 2ω1, 2ω2), (Trig-An−1) Trigonometric potential of type An−1: u(t) = C sinh−2 λt, RT (An−1;x1, . . . , xn;C, λ) = C ∑ 1≤i<j≤n sinh−2 λ(xi − xj), (Rat-An−1) Rational potential of type An−1: u(t) = C t2 , RR(An−1;x1, . . . , xn;C) = ∑ 1≤i<j≤n C (xi − xj)2 . We review how the integrability of (Ellip-An−1) implies the integrability of other systems. Since it follows from (2.14) that lim ω2→∞ RE ( An−1;x; C λ2 , 2ω1, 2ω2 ) = RT (An−1;x;C, λ), u(t) = lim ω2→∞ C λ2 ℘0(t; 2ω1, 2ω2) = C sinh−2 λt, the integrability (3.2) for (Trig-An−1) follows from that for (Ellip-An−1) by the analytic conti- nuation of uei−ej (x) and Pk with respect to q (cf. (2.10), (3.1), (3.3) and Lemma 2 i)). The integrability for (Rat-An−1) is similarly follows in view of Lemma 1 with lim λ→0 RT (An−1;x;λ2C, λ) = RR(An−1;x;C), u(t) = lim λ→0 λ2C sinh−2 λt. This argument using the analytic continuation for the proof of (3.2) is given in [32]. As is shown [33], there are two other integrable potentials of type An−1 to which this argument can be applied. 8 T. Oshima (Toda-A(1) n−1) Toda potential of type A(1) n−1: RL(A(1) n−1;x;C, λ) = n−1∑ i=1 Ceλ(xi−xi+1) + Ceλ(xn−x1), (Toda-An−1) Toda potential of type An−1: RL(An−1;x;C, λ) = n−1∑ i=1 Ceλ(xi−xi+1). The integrability (3.2) for these potentials similarly follows in view of Lemmas 1, 2 and lim ω2→∞ RE ( An−1;x1 − 2ω2 n , . . . , xk − 2kω2 n , . . . , xn − 2ω2; e(4/n)λω2 4λ2 C, 2ω1, 2ω2 ) = RL(A(1) n−1;x;C,−2λ), uei−ej (x) = lim ω2→∞ e(4/n)λω2 4λ2 C℘0 ( xi − xj + 2(j − i)ω2 n ; 2ω1, 2ω2 ) =  Ce−2λ(xi−xi+1) if 1 < j = i+ 1 ≤ n, Ce−2λ(xn−x1) if i = 1 and j = n, 0 if 1 ≤ i < j ≤ n and j − i 6= 1, n− 1 and lim N→∞ RT ( An−1;x1 −N, . . . , xn − nN ; e2λN 4 C, λ ) = RL(An−1;x;C,−2λ), uei−ej (x) = lim N→∞ e2λN 4 C sinh−2 λ(xi − xj + (j − i)N) = { Ce−2λ(xi−xi+1) if 1 < j = i+ 1 ≤ n, 0 if 1 ≤ i < j ≤ n and j 6= i+ 1, respectively, if Reλ > 0. The restriction Reλ > 0 is removed also by the analytic continuation. Thus the following theorem is obtained by the analytic continuation of the integrals (3.1) of (Ellip-An−1) whose commutativity (3.2) is assured by [32]. Theorem 1 (An−1, [10, 11, 28, 32, 33, 34], etc.). The Schrödinger operators with the potential functions (Ellip-An−1), (Trig-An−1), (Rat-An−1), (Toda-A(1) n−1) and (Toda-An−1) are completely integrable and their integrals are given by (3.1) with uei−ej (x) in the above. Remark 1. i) There are quite many papers studying these Schrödinger operators of type An−1. The proof of this theorem using analytic continuation is explained in [33]. ii) The complete integrability (3.2) for (Ellip-An−1) is first established by [32, Theorem 5.2], whose proof is as follows. The equations [P1, Pk] = [P2, Pk] = 0 for k = 1, . . . , n are easily obtained by direct calculations with the formula (2.9) (cf. [32, Lemma 5.1]). Then the relation [P2, [Pi, Pj ]] = 0 and periodicity and symmetry of Pk imply [Pi, Pj ] = 0 (cf. [32, Lemma 3.5]). Note that the proof of the integrability given in [28, § 5 Proposition 2 and Appendix E] is not correct as is clarified in [32, Remark 3.7] (cf. [33, § 4.2]). Note that the complete integrability for (Trig-An−1) is shown in [34]. iii) If Reλ > 0, we also have lim N→∞ RL ( A (1) n−1;xk + kN ;Ce−2λN ;−2λ ) = RL(An−1;x;C,−2λ). Completely Integrable Systems Associated with Classical Root Systems 9 iv) Note that RL(An−1;xk +Nk;C, λ) = n−1∑ i=1 Ceλ(Ni−Ni+1) · eλ(xi−xi+1). Hence eλ(x1−x2) − eλ(x2−x3) gives the potential function of a completely integrable system of type An−1 with n = 3 but the potential function lim λ→0 λ−1 ( eλ(x1−x2) − eλ(x2−x3) ) = x1 − 2x2 + x3 does not give such a system because it does not satisfy (9.3) in Remark 13. Considering the limit of the parameters of the integrable potential function, we should take care of the limit of integrals. v) Let Pn(t) denote the differential operator Pn in (3.1) defined by replacing u−ij by ũ−ij = u−ij+t with a constant t ∈ C. Then Pn(t) = ∑ 0≤k≤n/2 (2k)! 2kk! Pn−2kt k with P0 = 1, [Pn(s), Pn(t)] = 0 for s, t ∈ C. (3.4) In fact, the term u−12u − 34 · · ·u − 2j−1,2j∂2j+1∂2j+2 · · · ∂n−2k appears only in the coefficient of tk in the right hand side of (3.4) and it is contained in the terms ũ−in−2k+1in−2k+2 · · · ũ−in−1in ũ−12 · · · ũ − 2j−1,2j∂2j+1 · · · ∂n−2k of Pn(t), where the number of the possibilities of these ũin−2k+1in−2k+2 · · · ũ−in−1in is (2k)!/(2kk!) because {in−2k+1, in−2k+2, . . . , in} = {n− 2k + 1, . . . , n}. vi) Since Pk−1 = (n− k + 1)[Pk, x1 + · · ·+ xn] for k = 2, . . . .n, [Pk, P2] = 0 implies [Pk−1, P2] = 0 by the Jacobi identity. Here we note that [u−12u − 34 · · ·u − 2j−1,2j∂2j+1 · · · ∂k−1∂ν , xν ] = u−12u − 34 · · ·u − 2j−1,2j∂2j+1 · · · ∂k−1 for ν = k, k + 1, . . . , n. vi) The potential functions of (Trig-An−1), (Rat-An−1) and (Toda-An−1) are specializations of more general integrable potential functions of type Bn (cf. Definition 5). In the following diagram we show the relations among integrable potentials of type An−1 by taking limits. Hierarchy of Integrable Potentials of Type An−1 (n ≥ 3) Toda-A(1) n−1 → Toda-An−1 ↗ ↗ Ellip-An−1 → Trig-An−1 → Rat-An−1 10 T. Oshima 4 Type B2 In this section we study the following commuting differential operators P and P2. P = −1 2 ( ∂2 ∂x2 + ∂2 ∂y2 ) +R(x, y), P2 = ∂4 ∂x2∂y2 + S with ordS < 4, [P, P2] = 0. (4.1) The Schrödinger operators P of type B2 in this section are known to be completely integrable. They are listed in [23, 24, 30]. We review them and give the explicit expression of P2. First we review the arguments given in [23, 24]. Since P is self-adjoint, we may assume P2 is also self-adjoint by replacing P2 by its self-adjoint part if necessary. Here for A = ∑ aij(x, y) ∂i+j ∂xi∂yj we define tA = ∑ (−1)i+j ∂i+j ∂xi∂yj aij(x, y) and A is called self-adjoint if tA = A. Lemma 3 ([23]). Suppose P and P2 are self-adjoint operators satisfying (4.1). Then R(x, y) = u+(x+ y) + u−(x− y) + v(x) + w(y), P2 = ( ∂2 ∂x∂y +u−(x−y)−u+(x+y) )2 − 2w(y) ∂2 ∂x2 −2v(x) ∂2 ∂y2 +4v(x)w(y)+T (x, y) (4.2) and the function T (x, y) satisfies ∂T (x, y) ∂x = 2(u+(x+ y)− u−(x− y))∂w(y) ∂y + 4w(y) ∂ ∂y (u+(x+ y)− u−(x− y)), ∂T (x, y) ∂y = 2(u+(x+ y)− u−(x− y))∂v(x) ∂x + 4v(x) ∂ ∂x (u+(x+ y)− u−(x− y)). (4.3) Conversely, if a function T (x, y) satisfies (4.3) for suitable functions u±(t), v(t) and w(t), then (4.1) is valid for R(x, y) and P2 defined by (4.2). Remark 2. i) If w(y) = 0, then T (x, y) does not depend on x. ii) The self-adjointness of P2 and the vanishing of the third order term of [P, P2] imply that P2 should be of the form (4.2) with a suitable function T (x, y). Then the vanishing of the first order term implies (4.3). The last claim in Lemma 3 is obtained by direct calculation. Since T (x, y) satisfying (4.3) is determined by (u−, u+; v, w) up to the difference of constants, we will write T (u−, u+; v, w) for the corresponding T (x, y) which is an element of the space of meromorphic functions of (x, y) modulo constant functions and define Q(u−, u+; v, w) by T (u−, u+; v, w) = 2(u−(x− y) + u+(x+ y))(v(x) + w(y))− 4Q(u−, u+; v, w). (4.4) Note that T (u−, u+; v, w) and Q(u−, u+; v, w) are defined only if the function T (x, y) satis- fying (4.3) exists. The following lemma is a direct consequence of (4.3) and this definition of Q. Completely Integrable Systems Associated with Classical Root Systems 11 Lemma 4 ([23]). Suppose T (u−, u+; v, w) and T (u−i , u + i ; vj , wj) are defined. Then for any C, Ci, C ′ j ∈ C the left hand sides of the following identities are also defined and T (u−(t) + C, u+(t) + C; v(t), w(t))) = T (u−(t), u+(t); v(t), w(t)), Q(u−(t), u+(t); v(t) + C,w(t) + C)) = Q(u−(t), u+(t); v(t), w(t)), Q(u−(Ct), u+(Ct); v(Ct), w(Ct)) = Q(u−(t), u+(t); v(t), w(t))\|x→Cx, y→Cy, Q  2∑ i=1 Aiu − i , 2∑ i=1 Aiu + i ; 2∑ j=1 Cjvj , 2∑ j=1 Cjwj  = 2∑ i=1 2∑ j=1 AiCjQ(u−i , u + i ; vj , wj). Hence the left hand sides give pairs P and P2 with [P, P2] = 0. For simplicity we will use the notation Q(u−, u+; v) = Q(u−, u+; v, v), Q(u; v, w) = Q(u, u; v, w), Q(u; v) = Q(u, u; v, v). (4.5) The same convention will be also used for T (u−, u+; v, w). The integrable potentials of type B2 in this note are classified into three kinds. The potentials of the first kind are the unified integrable potentials which are in the same form as those of type Bn with n ≥ 3, which we call normal integrable potentials of type B2. The integrable potentials of type B2 admit a special transformation called dual which does not exist in Bn with n ≥ 3. Hence there are normal potentials and their dual in the invariant integrable potentials of type B2. Because of this duality, there exist another kind of invariant integrable potentials of type B2, which we call special integrable potentials of type B2. In this section we present ( R(x, y), T (x, y) ) as suitable limits of elliptic functions as in the previous section since it helps to study the potentials of type Bn in Section 5. We reduce the complete integrability of the limits to that of a systems with elliptic potentials. But we can also check (4.3) by direct calculations (cf. Remark 5). 4.1 Normal case In this subsection we study the integrable systems (4.1) with (4.2) which have natural extension to type Bn for n ≥ 3 and have the form u−(t) = Au−0 (t), u+(t) = Au+ 0 (t), v(t) = 3∑ j=0 Cjvj(t), w(t) = 3∑ j=0 Cjwj(t) with any A, C0, C1, C2, C3 ∈ C. These systems are expressed by the symbol (〈u−0 〉, 〈u + 0 〉; 〈v0, v1, v2, v3〉, 〈w0, w1, w2, w3〉). The most general system is the following (Ellip-B2) defined by elliptic functions, which is called Inozemtsev model [12]. Theorem 2 (B2: Normal case, [12, 24, 23, 30] etc.). The operators P and P2 defined by the following pairs of R(x, y) and T (x, y) satisfy (4.1) and (4.2). (Ellip-B2): (〈℘(t)〉; 〈℘(t), ℘(t+ ω1), ℘(t+ ω2), ℘(t+ ω3)〉) v(x) = 3∑ j=0 Cj℘(x+ ωj), w(y) = 3∑ j=0 Cj℘(y + ωj), 12 T. Oshima u−(x− y) = A℘(x− y), u+(x+ y) = A℘(x+ y), R(x, y) = A(℘(x− y) + ℘(x+ y)) + 3∑ j=0 Cj(℘(x+ ωj) + ℘(y + ωj)), T (x, y) = 2A(℘(x− y) + ℘(x+ y)) ( 3∑ j=0 Cj ( ℘(x+ ωj) + ℘(y + ωj) ) − 4A 3∑ j=0 Cj℘(x+ ωj)℘(y + ωj). (Trig-B2): (〈sinh−2 λt〉; 〈sinh−2 λt, sinh−2 2λt, sinh2 λt, sinh2 2λt〉) v(x) = C0 sinh−2 λx+ C1 cosh−2 λx+ C2 sinh2 λx+ 1 4 C3 sinh2 2λx, w(y) = C0 sinh−2 λy + C1 cosh−2 λy + C2 sinh2 λy + 1 4 C3 sinh2 2λy, u−(x− y) = A sinh−2 λ(x− y), u+(x+ y) = A sinh−2 λ(x+ y), R(x, y) = A ( sinh−2 λ(x− y) + sinh−2 λ(x+ y) ) + C0 ( sinh−2 λx+ sinh−2 λy ) + C1 ( cosh−2 λx+ cosh−2 λy ) + C2 ( sinh2 λx+ sinh2 λy ) + 1 4 C3 ( sinh2 2λx+ sinh2 2λy ) , T (x, y) = 2A(sinh−2 λ(x− y) + sinh−2 λ(x+ y)) × ( C0 ( sinh−2 λx+ sinh−2 λy ) + C1(cosh−2 λx+ cosh−2 λy) + C2(sinh2 λx+ sinh2 λy) + 1 4 C3(sinh2 2λx+ sinh2 2λy) ) − 4A ( C0 sinh−2 λx · sinh−2 λy − C1 cosh−2 λx · cosh−2 λy + C3(sinh2 λx+ sinh2 λy + 2 sinh2 λx · sinh2 λy) ) . (Rat-B2): (〈t−2〉; 〈t−2, t2, t4, t6〉) v(x) = C0x −2 + C1x 2 + C2x 4 + C3x 6, w(y) = C0y −2 + C1y 2 + C2y 4 + C3y 6, u−(x− y) = A (x− y)2 , u+(x+ y) = A (x+ y)2 , R(x, y) = A (x−y)2 + A (x+y)2 +C0(x−2 + y−2)+C1(x2+y2)+C2(x4+y4) + C3(x6+y6), T (x, y) = 2(u−(x− y) + u+(x+ y))(v(x) + w(y))− 4A ( C0 x2y2 + C2(x2 + y2) + C3(x4 + y4 + 3x2y2) ) = 8A 2C0 + 2C1x 2y2 + C2x 2y2(x2 + y2) + 2C3x 4y4 (x2 − y2)2 . (Toda-D(1) 2 -bry): (〈cosh 2λt〉; 〈sinh−2 λt, sinh−2 2λt〉, 〈sinh−2 λt, sinh−2 2λt〉) v(x) = C0 sinh−2 λx+ C1 sinh−2 2λx, w(y) = C2 sinh−2 λy + C3 sinh−2 2λy, u−(x− y) = A cosh 2λ(x− y), u+(x+ y) = A cosh 2λ(x+ y), R(x, y) = A cosh 2λ(x− y) +A cosh 2λ(x+ y) + C0 sinh−2 λx+ C1 sinh−2 2λx+ C2 sinh−2 λy + C3 sinh−2 2λy, Completely Integrable Systems Associated with Classical Root Systems 13 T (x, y) = 8A(C0 cosh 2λy + C2 cosh 2λx). (Toda-B(1) 2 -bry): (〈e−2λt〉; 〈e2λt, e4λt〉, 〈sinh−2 λt, sinh−2 2λt〉) v(x) = C0e 2λx + C1e 4λx, w(y) = C2 sinh−2 λy + C3 sinh−2 2λy, u−(x− y) = Ae−2λ(x−y), u+(x+ y) = Ae−2λ(x+y), R(x, y) = Ae−2λ(x−y) +Ae−2λ(x+y) + C0e 2λx + C1e 4λx + C2 sinh−2 λy + C3 sinh−2 2λy, T (x, y) = 4A ( C0 cosh 2λy + 2C2e −2λx ) . (Trig-A(1) 1 -bry): (〈sinh−2 λt〉, 0; 〈e−2λt, e−4λt, e2λt, e4λt〉) v(x) = C0e −2λx + C1e −4λx + C2e 2λx + C3e 4λx, w(y) = C0e −2λy + C1e −4λy + C2e 2λy + C3e 4λy, u−(x− y) = A sinh−2 λ(x− y), u+(x+ y) = 0, R(x, y) = A sinh−2 λ(x− y) + C0(e−2λx + e−2λy) + C1(e−4λx + e−4λy) + C2(e2λx + e2λy) + C3(e4λx + e4λy), T (x, y) = 2A sinh−2 λ(x− y)(C0 ( e−2λx + e−2λy ) + 2C1e −2λ(x+y) + C2 ( e2λx + e2λy ) + 2C3e 2λ(x+y)). (Toda-C(1) 2 ): (〈e−2λt〉, 0; 〈e2λt, e4λt〉, 〈e−2λt, e−4λt〉) v(x) = C0e 2λx + C4λx 1 , w(y) = C2e −2λy + C3e −4λy, u−(x− y) = Ae−2λ(x−y), u+(x+ y) = 0, R(x, y) = Ae−2λ(x−y) + C0e 2λx + C4λx 1 + C2e −2λy + C3e −4λy, T (x, y) = 2A ( C0e 2λy + C2e −2λx ) . (Rat-A1-bry): (〈t−2〉, 0; 〈t, t2, t3, t4〉) v(x) = C0x+ C1x 2 + C2x 3 + C3x 4, w(y) = C0y + C1y 2 + C2y 3 + C3y 4, u−(x− y) = A (x− y)2 , u+(x+ y) = 0, R(x, y) = A (x− y)2 + C0(x+ y) + C1(x2 + y2) + C2(x3 + y3) + C3(x4 + y4), T (x, y) = 2A (x− y)2 (C0(x+ y) + C1(x2 + y2) + C2xy(x+ y) + 2C3x 2y2). Remark 3. For example, (〈t−2〉, 0; 〈t, t2, t3, t4〉) in the above (Rat-A1-bry) means u−(t) = At−2, u+(t) = 0, v(t) = w(t) = C0t+ C1t 2 + C2t 3 + C3t 4 with a convention similar to that in (4.5). We will review the proof of the above theorem after certain remarks. Remark 4. All the invariant integrable potentials of type B2 together with P2 are determined by [24, 25]. They are classified into three cases. In the normal case they are (Ellip-B2), (Trig-B2) and (Rat-B2) which have the following unified expression of the invariant potentials given by [32, Lemma 7.3], where the periods 2ω1 and 2ω2 may be infinite (cf. (2.6) and (2.7)) R(x, y) = A℘(x− y) +A℘(x+ y) + C4℘(x)4 + C3℘(x)3 + C2℘(x)2 + C1℘(x) + C0 ℘′(x)2 14 T. Oshima + C4℘(y)4 + C3℘(y)3 + C2℘(y)2 + C1℘(y) + C0 ℘′(y)2 , T (x, y) = 4A(℘(x)− ℘(y))−2 ( C4℘(x)2℘(y)2 + C3 2 ℘(x)2℘(y) + C3 2 ℘(x)℘(y)2 + C2℘(x)℘(y) + C1 2 ℘(x) + C1 2 ℘(y) + C0 ) . This is the original form we found in the classification of the invariant integrable systems of type Bn (cf. [25]). Later we knew Inozemtzev model and in fact, when the periods are finite, (2.3) and (2.5) show that the above potential function corresponds to (Ellip-B2). When ω1 = ω2 =∞, ℘(t) = t−2 and (℘(x)− ℘(y))−2 = x4y4(x2 − y2)−2 and R(x, y) = A (x− y)2 + A (x+ y)2 + 1 4 (C4x −2 + C3 + C2x 2 + C1x 4 + C0x 6) + 1 4 (C4y −2 + C3 + C2y 2 + C1y 4 + C0y 6), T (x, y) = 2A(x2 − y2)−2(2C4 + C3(x2 + y2) + 2C2x 2y2 + C1x 2y2(x2 + y2) + 2C0x 4y4). We review these invariant cases discussed in [24, 32]. Owing to the identity 2(u− − u+)v′ + 4v((u−)′ − (u+)′) + ∂y(2(u− + u+)(v + w)− 4vw) = 2 ∣∣∣∣∣∣ v v′ 1 w −w′ 1 u− −(u−)′ 1 ∣∣∣∣∣∣+ 2 ∣∣∣∣∣∣ v −v′ 1 w −w′ 1 u+ (u+)′ 1 ∣∣∣∣∣∣ and (2.9), the right hand side of the above is zero and we have (4.3) when u− = C℘(x−y) + C ′, u+ = C℘(x+y) + C ′, v = C℘(x) + C ′ and w = C℘(y) + C ′ with T (x, y) = 2(u− + u+)(v + w) − 4vw. Hence with Q ( ℘(t);℘(t) ) = ℘(x)℘(y), the function T (x, y) given by (4.4) satisfies (4.3) with the above u±, v and w. Using the transformations (x, y) 7→ (x+ ωj , y + ωj), we have ℘(x+ ωj)℘(y + ωj) = Q(℘(t+ ωj);℘(t+ ωj)) = Q(℘(t);℘(t+ ωj)) (4.6) for j = 0, 1, 2, 3 because the function ℘(x ± y) does not change under these transformations (cf. (2.1)). Thus we have Theorem 2 for (Ellip-B2) in virtue of Lemma 3 and Lemma 4. Here we note that ℘ may be replaced by ℘0. By the limit under ω2 →∞, we have the following (Trig-B2) from (Ellip-B2). See the proof of [29, Proposition 6.1] for the precise argument. (Trig-B2): Q(sinh−2 λt; sinh−2 λt) = sinh−2 λx · sinh−2 λy, (4.7) Q(sinh−2 λt; cosh−2 λt) = − cosh−2 λx · cosh−2 λy, (4.8) Q(sinh−2 λt; sinh2 λt) = 0, (4.9) Q ( sinh−2 λt; 1 4 sinh2 2λt ) = sinh2 λx+ sinh2 λy + 2 sinh2 λx · sinh2 λy. (4.10) The equations (4.7), (4.8) and (4.9) correspond to (2.14), (2.15) and (2.17), respectively. More- over (2.8) should be noted and (4.10) corresponds to (2.17) with replacing (ω1, λ) by (ω1/2, 2λ). Completely Integrable Systems Associated with Classical Root Systems 15 By the limit under λ → 0, we have the following (Rat-B2) from (Trig-B2) as was shown in the proof of [29, Proposition 6.3]. Here we note (2.13) and cosh−2 λt · sinh2 λt = 1− cosh−2 λt, cosh−2 λt · sinh4 λt = −1 + cosh−2 λt+ sinh2 λt, cosh−2 λt · sinh6 λt = 1− cosh−2 λt− 2 sinh2 λt+ 1 4 sinh2 2λt, lim λ→0 λ−2j cosh−2 λt · sinh2j λt = t2j for j = 1, 2, 3, 1 (x− y)2 + 1 (x+ y)2 = 2(x2 + y2) (x2 − y2)2 . The result is as follows. (Rat-B2): Q(t−2; t−2) = x−2y−2, T (t−2; t−2) = 4(x−2 + y−2)((x− y)−2 + (x+ y)−2)− 4x−2y−2 = 4(x2 + y2)2 − 4(x2 − y2)2 x2y2(x2 − y2)2 = 16 (x2 − y2)2 , Q(t−2; t2) = 0, T (t−2; t2) = 4(x2 + y2)2 (x2 − y2)2 = 16x2y2 (x2 − y2)2 + 4, Q(t−2; t4) = x2 + y2, T (t−2, t4) = 4(x2 + y2)(x4 + y4) (x2 − y2)2 − 4(x2 + y2) = 8x2y2(x2 + y2) (x2 − y2)2 , Q(t−2; t6) = x4 + y4 + 3x2y2, T (t−2, t6) = 4(x2 + y2)(x6 + y6) (x2 − y2)2 − 4(x4 + 3x2y2 + y4) = 16x4y4 (x2 − y2)2 . This expression of T (x, y) for (Rat-B2) is also given in Remark 4. Note that we ignore the difference of constants for Q and T . Proof of Theorem 2. The three cases (Ellip-B2), (Trig-B2) and (Rat-B2) have been explai- ned. Note that if u±, v, w and T (u−, u+; v, w) (or Q(u−, u+; v, w)) are defined and they have an analytic parameter, Lemma 3 assures that their analytic continuations also define P and P2 satisfying [P, P2] = 0. (Toda-D(1) 2 -bry) ← (Ellip-B2): Replacing (x, y) by (x+ ω2, y), we have Q(cosh 2λt; 0, sinh−2 λt) = lim ω2→∞ Q ( e2λω2 8λ2 ℘0(t+ ω2); 1 λ2 ℘0(t+ ω2), 1 λ2 ℘0(t) ) = lim ω2→∞ e2λω2 8λ4 ℘0(x+ ω2)℘0(y) = cosh 2λx · sinh−2 λy, Q(cosh 2λt; 0, cosh−2 λt) = lim ω2→∞ Q ( e2λω2 8λ2 ℘0(t+ω2);− 1 λ2 ℘0(t+ω1 + ω2);− 1 λ2 ℘0(t+ω1) ) = − lim ω2→∞ e2λω2 8λ4 ℘0(x+ ω1 + ω2) · ℘0(y + ω1) = − cosh 2λx · cosh−2 λy, Q(cosh 2λt; sinh−2 λt, 0) = sinh−2 λx · cosh2 λy, 16 T. Oshima Q(cosh 2λt; cosh−2 λt, 0) = − cosh−2 λx · sinh2 λy. Hence T (cosh 2λt; 0, sinh−2 λt) = 2 ( cosh 2λ(x+ y) + cosh 2λ(x− y) ) · sinh−2 λy − 4 cosh 2λx · sinh−2 λy = 8 cosh 2λx, T (cosh 2λt; 0, cosh−2 λt) = 2 ( cosh 2λ(x+ y) + cosh 2λ(x− y) ) · cosh−2 λy + 4 cosh 2λx · cosh−2 λy = 8 cosh 2λx, T (cosh 2λt; 0, sinh−2 2λt) = T (cosh 2λt; sinh−2 2λt, 0) = 0, T (cosh 2λt; sinh−2 λt, 0) = 8 cosh 2λy. (Toda-B(1) 2 -bry)← (Toda-D(1) 2 -bry): Replacing (x, y) by (x−N, y), we have T (e−2λt; 0, sinh−2 λt) = lim N→∞ T (2e−2λN cosh 2λ(t−N); 0, sinh−2 λt) = lim N→∞ 16e−2λN cosh 2λ(x−N) = 8e−2λx, T (e−2λt; 0, sinh−2 2λt) = lim N→∞ T (2e−2λN cosh 2λ(t−N); 0, sinh−2 2λt) = 0, T (e−2λt; e2λt, 0) = lim N→∞ T ( 2e−2λN cosh 2λ(t−N); 1 4 e2λN sinh−2 λ(t−N), 0 ) = 4 cosh 2λy, T ( e−2λt; e4λt, 0 ) = lim N→∞ T ( 2e−2λN cosh 2λ(t−N); 1 4 e4λN sinh−2 2λ(t−N), 0 ) = 0. (Trig-C(1) 2 ) ← (Trig-B(1) 2 -bry): Replacing (x, y) by (x+N, y +N), T (e−2λt, 0; e2λt, 0) = lim N→∞ T (e−2λt, e−2λ(t+2N); e−2λNe2λ(t+N), 0) = lim N→∞ e−2λN cosh 2λ(y +N) = 2e2λy, T (e−2λt, 0; e4λt, 0) = lim N→∞ T (e−2λt, e−2λ(t+2N); e−4λNe4λ(t+N), 0) = 0. By the transformation (x, y, λ) 7→ (y, x,−λ), we have T (e−2λt, 0; 0, e−2λt) = 2e−2λx, T (e−2λt, 0; 0, e−4λt) = 0. (Trig-A(1) 1 -bry) ← (Trig-B2): Replacing (x, y) by (x+N, y +N), Q ( sinh−2 λt, 0; e2λt ) = lim N→∞ Q ( sinh−2 λt, sinh−2 λ(t+ 2N); 1 4 e2λN sinh−2 λ(t+N) ) = lim N→∞ 1 4 e2λN sinh−2 λ(x+N) sinh−2 λ(y +N) = 0, T ( sinh−2 λt, 0; e2λt ) = 2(e2λx + e2λy) sinh−2 λ(x− y), Q(sinh−2 λt, 0; e4λt) lim N→∞ Q(sinh−2 λt, sinh−2 λ(t+ 2N); 4e−4λN sinh2 2λ(t+N)) = 16 lim N→∞ e−4λN (sinh2 λ(x+N) + sinh2 λ(y +N) + 2 sinh2 λ(x+N) sinh2 λ(y +N)) = 2e2λ(x+y), T (sinh−2 λt, 0; e4λt) = 2 sinh−2 λ(x− y)(e4λx + e4λy)− 8e2λ(x+y) = 2sinh−2 λ(x− y)(e4λx + e4λy − e2λ(x+y)(eλ(x−y) − e−λ(x−y))2) = 4e2λ(x+y)sinh−2 λ(x− y). Completely Integrable Systems Associated with Classical Root Systems 17 (Rat-A1-bry) ← (Trig-A1-bry): Taking the limit λ→ 0, Q(t−2, 0; t) = lim λ→0 Q ( λ2 sinh−2 λt, 0; 1 2λ (e2λt − 1) ) = 0, T (t−2, 0; t) = 2(x+ y) (x− y)2 , Q(t−2, 0; t2) = lim λ→0 Q ( λ2 sinh−2 λt, 0; 1 4λ2 (e2λt + e−2λt − 2) ) = 0, T (t−2, 0; t2) = 2 x2 + y2 (x− y)2 , Q(t−2, 0; t3) = lim λ→0 Q ( λ2 sinh−2 λt, 0; 1 8λ3 (e4λt − 3e2λt − e−2λt + 3) ) = lim λ→0 2 8λ (e2λ(x+y) − 1) = 1 2 (x+ y), T (t−2, 0; t3) = 2 x3 + y3 (x− y)2 − 2(x+ y) = 2 xy(x+ y) (x− y)2 , Q(t−2, 0; t4) = lim λ→0 Q ( λ2 sinh−2 λt, 0; 1 16λ4 (e4λt + e−4λt − 4e2λt − 4e−2λt + 6) ) = lim λ→0 2 16λ2 (e2λ(x+y) + e−2λ(x+y) − 2) = 1 2 (x+ y)2, T (t−2, 0; t4) = 2 x4 + y4 (x− y)2 − 2(x+ y)2 = 4x2y2 (x− y)2 . Thus we have completed the proof of Theorem 2 by using Lemma 3 and Lemma 4. � Remark 5. Theorem 2 can be checked by direct calculations. For example, Remark 2 and the equations 2(εe−2λ(x+y) − e−2λ(x−y))(e2λx)′ + 4e2λx ∂ ∂x (εe−2λ(x+y) − e−2λ(x−y)) = 4λ(εe−2λy − e2λy)− 8λ(εe−2λy − e2λy) = ∂ ∂y (2(εe−2λy + e2λy)), 2(εe−2λ(x+y) − e−2λ(x−y))(e4λx)′ + 4e4λx ∂ ∂x (εe−2λ(x+y) − e−2λ(x−y)) = 0 with ε = 1 give T (e−2λt; e2λt, 0) and T (e−2λt; e4λt, 0) for (Trig-B2-bry). Moreover the functions T (e−2λt, 0; e2λt, 0) and T (e−2λt, 0; e4λt, 0) for (Toda-C(1) 2 ) also follow from these equations with ε = 0. 4.2 Special case In this subsection we study the integrable systems (4.1) with (4.2) which are of the form R(x, y) = u−(x− y) + u+(x+ y) + v(x) + w(y), u−(t) = 1∑ j=0 Aju − j (t), u+(t) = 1∑ j=0 Aju + j (t), v(t) = 1∑ j=0 Cjvj(t), w(t) = 1∑ j=0 Cjwj(t) with A0, A1, C0, C1 ∈ C. The most general system (Ellip-B2-S) in the following theorem is presented by [24] as an elliptic generalization of (Trig-B2-S) found by [32]. 18 T. Oshima Theorem 3 (B2 : Special case, [23, 24, 30] etc.). The operators P and P2 defined by the following pairs of R(x, y) and T (x, y) satisfy (4.1) and (4.2). (Ellip-B2-S): (〈℘(t; 2ω1, 2ω2), ℘(t;ω1, 2ω2)〉; 〈℘(t;ω1, 2ω2), ℘(t;ω1, ω2)〉) v(x) = C0℘(x;ω1, 2ω2) + C1℘(x;ω1, ω2), w(y) = C0℘(y;ω1, 2ω2) + C1℘(y;ω1, ω2), u−(x− y) = A0℘(x− y; 2ω1, 2ω2) +A1℘(x− y;ω1, 2ω2), u+(x+ y) = A0℘(x+ y; 2ω1, 2ω2) +A1℘(x+ y;ω1, 2ω2), R(x, y) = A0℘(x− y; 2ω1, 2ω2) +A0℘(x+ y; 2ω1, 2ω2) +A1℘(x− y;ω1, 2ω2) +A1℘(x+ y;ω1, 2ω2) + C0℘(x;ω1, 2ω2) + C0℘(y;ω1, 2ω2) + C1℘(x;ω1, ω2) + C1℘(y;ω1, ω2), T (x, y) = 2(A0℘(x− y; 2ω1, 2ω2) +A0℘(x+ y; 2ω1, 2ω2) +A1℘(x− y;ω1, 2ω2) +A1℘(x+ y;ω1, 2ω2)) × (C0℘(x;ω1, 2ω2) + C0℘(y;ω1, 2ω2) + C1℘(x;ω1, ω2) + C1℘(y;ω1, ω2)) − 4A0C0 1∑ j=0 ℘(x+ ωj ; 2ω1, 2ω2) · ℘(y + ωj ; 2ω1, 2ω2) − 4A0C1 3∑ j=0 ℘(x+ ωj ; 2ω1, 2ω2)℘(y + ωj ; 2ω1, 2ω2) − 4A1C0℘(x;ω1, 2ω2)℘(y;ω1, 2ω2) − 4A1C1 1∑ j=0 ℘(x+ ω2j ;ω1, 2ω2)℘(y + ω2j ;ω1, 2ω2). (Trig-B2-S): (〈sinh−2 λt, sinh−2 2λt〉; 〈sinh−2 2λt, sinh2 2λt〉) v(x) = C0 sinh−2 2λx+ C1 sinh2 2λx, w(y) = C0 sinh−2 2λy + C1 sinh2 2λy, u−(x− y) = A0 sinh−2 λ(x− y) +A1 sinh−2 2λ(x− y), u+(x+ y) = A0 sinh−2 λ(x+ y) +A1 sinh−2 2λ(x+ y), R(x, y) = A0 sinh−2 λ(x+ y) +A0 sinh−2 λ(x− y) +A1 sinh−2 2λ(x+ y) +A1 sinh−2 2λ(x− y) + C0 sinh−2 2λx+ C0 sinh−2 2λy + C1 sinh2 2λx+ C1 sinh2 2λy, T (x, y) = 2(A0 sinh−2 λ(x+ y) +A0 sinh−2 λ(x− y) +A1 sinh−2 2λ(x+ y) +A1 sinh−2 2λ(x− y))(C0 sinh−2 2λx+ C0 sinh−2 2λy + C1 sinh2 2λx + C1 sinh2 2λy)−A0C0 ( sinh−2 λx · sinh−2 λy + cosh−2 λx · cosh−2 λy ) − 4A0C1(sinh2 λx+ sinh2 λy + 2 sinh2 λx · sinh2 λy) − 4A1C0 sinh−2 2λx · sinh−2 2λy. (Rat-B2-S): (〈t−2, t2〉; 〈t−2, t2〉) v(x) = C0x −2 + C1x 2, w(y) = C0y −2 + C1y 2, u−(x− y) = A0 (x− y)2 +A1(x− y)2, u+(x+ y) = A0 (x+ y)2 +A1(x+ y)2, R(x, y) = A0 (x− y)2 + A0 (x+ y)2 +A1(x− y)2 +A1(x+ y)2 + C0 x2 + C0 y2 + C1x 2 + C1y 2, T (x, y) = 16A0C0 + 16A0C1x 2y2 (x2 − y2)2 + 16A1C1x 2y2. Completely Integrable Systems Associated with Classical Root Systems 19 (Toda-D(1) 2 -S-bry): (〈cosh 2λ, cosh 4λt〉; 〈sinh−2 2λt〉, 〈sinh−2 2λt〉) v(x) = C0 sinh−2 2λx, w(y) = C1 sinh−2 2λy, u−(x− y) = A0 cosh 2λ(x− y) +A1 cosh 4λ(x− y), u+(x+ y) = A0 cosh 2λ(x+ y) +A1 cosh 4λ(x+ y), R(x, y) = A0 cosh 2λ(x− y) +A0 cosh 2λ(x+ y) +A1 cosh 4λ(x− y) +A1 cosh 4λ(x+ y) + C0 sinh−2 2λx+ C1 sinh−2 2λy, T (x, y) = 8A1(C0 cosh 4λy + C1 cosh 4λx). (Toda-B(1) 2 -S-bry): (〈e−2λt, e−4λt〉; 〈e4λt〉, 〈sinh−2 2λt〉) v(x) = C0e 4λx, w(y) = C1 sinh−2 2λy, u−(x− y) = A0e −2λ(x−y) +A1e −4λ(x−y), u+(x+ y) = A0e −2λ(x+y) +A1e −4λ(x+y), R(x, y) = A0e −2λ(x−y)+A0e −2λ(x+y)+A1e −4λ(x−y)+A1e −4λ(x+y)+C0e 4λx+C1 sinh−2 2λy, T (x, y) = 4A1 ( C0 cosh 4λy + 2C1e −4λx ) . (Toda-C(1) 2 -S): (〈e−2λt, e−4λt〉, 0; 〈e4λt〉, 〈e−4λt〉) v(x) = C0e 4λx, w(y) = C1e −4λy, u−(x− y) = A0e −2λ(x−y) +A1e −4λ(x−y), u+(x+ y) = 0, R(x, y) = A0e −2λ(x−y) +A1e −4λ(x−y) + C0e 4λx + C1e −4λy, T (x, y) = 2A1 ( C0e 4λy + C1e −4λx ) . (Trig-A1-S-bry): (〈sinh−2 λt, sinh−2 2λt〉, 0; 〈e−4λt, e4λt〉) v(x) = C0e −4λx + C1e 4λx, w(y) = C0e −4λy + C1e 4λy, u−(x− y) = A0 sinh−2 λ(x− y) +A1 sinh−2 2λ(x− y), u+(x+ y) = 0, R(x, y) = A0 sinh−2 λ(x− y)+A1 sinh−2 2λ(x− y)+C0e −4λx+C0e −4λy+C1e 4λx+C1e 4λy, T (x, y) = 2A1 sinh−2 2λ(x− y)(C0e −4λx + C0e −4λy + C1e 4λx + C1e 4λy) + 4A0 sinh−2 λ(x− y)(C0e −2λ(x+y) + C1e 2λ(x+y)). (Ratd-D2-S-bry): (〈t2, t4〉; 〈t−2〉, 〈t−2〉) v(x) = C0x −2, w(y) = C1y −2, u−(x− y) = A0(x− y)2 +A1(x− y)4, u+(x+ y) = A0(x+ y)2 +A1(x+ y)4, R(x, y) = 2A0(x2 + y2) + 2A1(x4 + 6x2y2 + y4) + C0 x2 + C1 y2 , T (x, y) = 32A1(C0y 2 + C1x 2). Proof. (Ellip-B2-S): We have the following from (4.6), Lemma 4 and (2.8). Q(℘(t;ω1, 2ω2);℘(t;ω1, 2ω2)) = ℘(x;ω1, 2ω2)℘(y;ω1, 2ω2), Q(℘(t; 2ω1, 2ω2);℘(t;ω1, 2ω2)) = Q(℘(t; 2ω1, 2ω2);℘(t; 2ω1, 2ω2) + ℘(t2A1 sinh−2 2λ(x− y)(C0 + ω1; 2ω1, 2ω2)) = ℘(x; 2ω1, 2ω2)℘(y; 2ω1, 2ω2) + ℘(x+ ω1; 2ω1, 2ω2)℘(y + ω1; 2ω1, 2ω2), Q(℘(t;ω1, 2ω2);℘(t;ω1, ω2)) = ℘(x;ω1, 2ω2)℘(y;ω1, 2ω2) 20 T. Oshima + ℘(x+ ω2;ω1, 2ω2)℘(y + ω2;ω1, 2ω2), Q(℘(t; 2ω1, 2ω2);℘(t;ω1, ω2)) = 3∑ j=0 ℘(x+ ωj ; 2ω1, 2ω2)℘(y + ωj ; 2ω1, 2ω2). (Rat-B2-S) is given in [32, (7.13)] but it is easy to check (4.3) or prove the result as a limit of (Trig-B2-S). (Ratd-D2-S-bry) follows from T (t2; t−2, 0) = 0 and T (t4; t−2, 0) = 32y2 (cf. Re- mark 2 i)). Moreover (Trig-B2-S), (Toda-D(1) 2 -S-bry), (Toda-C(1) 2 -S) and (Trig-A1-S-bry) are obtained from the corresponding normal cases together with Lemma 4. For example, Q for (Trig-B2-S) is given by (4.7), (4.9) and Q(sinh−2 λt; sinh−2 2λt) = Q ( sinh−2 λt; 1 4 sinh−2 λt− 1 4 cosh−2 λt ) = 1 4 (sinh−2 λt sinh−2 λy + cosh−2 λx cosh−2 λy), Q(sinh−2 2λt; sinh−2 2λt) = sinh−2 2λx · sinh−2 2λy, Q(sinh−2 2λt; sinh2 2λt) = 0. Thus we get Theorem 3 from Theorem 2. � 4.3 Duality Definition 1 (Duality in B2, [23]). Under the coordinate transformation (x, y) 7→ (X,Y ) = ( x+ y√ 2 , x− y√ 2 ) the pair (P, P 2−P2) also satisfies (4.1), which we call the duality of the commuting differential operators of type B2. Denoting ∂x = ∂/∂x, ∂y = ∂/∂y and put L = P 2 − P2 − ( 1 2 ∂2 x − 1 2 ∂2 y + w − v )2 + u−(∂x + ∂y)2 + u+(∂x − ∂y)2. Then the order of L is at most 2 and the second order term of L is −(u+ + u− + v + w)(∂2 x + ∂2 y)− 2(u− − u+)∂x∂y + 2w∂2 x + 2v∂2 y − (w − v)(∂2 x − ∂2 y) + u−(∂x + ∂y)2 + u+(∂x − ∂y)2 = 0. Since L is self-adjoint, L is of order at most 0 and the 0-th order term of L is −1 2 (∂2 x + ∂2 y)(u+ + u− + v + w) + (u+ + u− + v + w)2 − 4vw − T − ∂x∂y(u− − u+) − 1 2 (∂2 x − ∂2 y)(w − v) = (u+ + u− + v + w)2 − 4vw − T and therefore we have the following proposition. Proposition 1 ([23, 24]). i) By the duality in Definition 1 the pair ( R(x, y), T (x, y) ) changes into ( R̃(x, y), T̃ (x, y) ) with R̃(x, y) = v ( x+ y√ 2 ) + w ( x− y√ 2 ) + u+( √ 2x) + u−( √ 2y), Completely Integrable Systems Associated with Classical Root Systems 21 T̃ (x, y) = R̃(x, y)2 − 4v ( x+ y√ 2 ) w ( x− y√ 2 ) − T ( x+ y√ 2 , x− y√ 2 ) . ii) Combining the duality with the scaling map R(x, y) 7→ c−2R(cx, cy), the following pair( Rd(x, y), T d(x, y) ) defines commuting differential operators if so does ( R(x, y), T (x, y) ) . This Rd(x, y) is also called the dual of R(x, y), Rd(x, y) = v(x+ y) + w(x− y) + u+(2x) + u−(2y), T d(x, y) = Rd(x, y)2 − 4v(x+ y)w(x− y)− T (x+ y, x− y). Remark 6. i) We list up the systems of type B2 given in Sections 4.1 and 4.2: (u−(t), u+(t); v(t), w(t)) Symbol (〈℘(t)〉; 〈℘(t), ℘(t+ ω1), ℘(t+ ω2), ℘(t+ ω3)〉), (Ellip-B2) (〈℘(t; 2ω1, 2ω2), ℘(t;ω1, 2ω2)〉; 〈℘(t;ω1, 2ω2), ℘(t;ω1, ω2)〉), (Ellip-B2-S) (〈sinh−2 λt〉; 〈sinh−2 λt, sinh−2 2λt, sinh2 λt, sinh2 2λt〉), (Trig-B2) (〈sinh−2 λt, sinh−2 2λt〉; 〈sinh−2 2λt, sinh2 2λt〉), (Trig-B2-S) (〈t−2〉; 〈t−2, t2, t4, t6〉), (Rat-B2) (〈t−2, t2〉; 〈t−2, t2〉), (Rat-B2-S) (〈cosh 2λt〉; 〈sinh−2 λt, sinh−2 2λt〉, 〈sinh−2 λt, sinh−2 2λt〉), (Toda-D(1) 2 -bry) (〈coshλt, cosh 2λt〉; 〈sinh−2 λt〉, 〈sinh−2 λt〉), (Toda-D(1) 2 -S-bry) (〈e−2λt〉; 〈e2λt, e4λt〉, 〈sinh−2 λt, sinh−2 2λt〉), (Toda-B(1) 2 -bry) (〈e−λt, e−2λt〉; 〈e2λt〉, 〈sinh−2 λt〉), (Toda-B(1) 2 -S-bry) (〈e−2λt〉, 0; 〈e2λt, e4λt〉, 〈e−2λt, e−4λt〉), (Toda-C(1) 2 ) (〈e−2λt, e−4λt〉, 0; 〈e4λt〉, 〈e−4λt〉), (Toda-C(1) 2 -S) (〈sinh−2 λt〉, 0; 〈e−2λt, e−4λt, e2λt, e4λt〉), (Trig-A1-bry) (〈sinh−2 λt, sinh−2 2λt〉, 0; 〈e−4λt, e4λt〉), (Trig-A1-S-bry) (〈t−2〉, 0; 〈t, t2, t3, t4〉), (Rat-A1-bry) (〈t−2〉, 〈t−2〉; 〈t2, t4〉). (Rat-D2-S-bry) The first 6 cases above are classified by [24] as invariant systems. The systems such that at least two of u±, v and w are not entire are classified by [23]. (Trig-∗) and (Toda-∗) in the above are classified by [30] as certain systems with periodic potentials. We do not put (〈t−2, t2〉, 0; 〈t, t2〉) in the list which corresponds to (Rat-A1-S-bry) because its dual defines a direct sum of trivial operators (cf. Section 9). ii) Since 1− sinh−2 t+4 sinh−2 2t = coth2 t = t2− (2/3)t4 + o(t4) and sinh2 t = t2 +(2/3)t4 + o(t4), we have sinh2 2λx+sinh2 2λy−2 coth2 λ(x−y)−2 coth2 λ(x+y) = 8λ4(x2 +y2)2 +o(λ4). Hence the potential function A0 (x− y)2 + A0 (x+ y)2 + C0 x2 + C0 y2 + C1(x2 + y2) +A1(x2 + y2)2 is an analytic continuation of that of (Trig-B2-S) but this is not a completely integrable potential function of type B2 iii) The dual is indicated by superfix d. For example, the dual of (Ellip-B2) is denoted by (Ellipd-B2) whose potential function is R(x, y) = A℘(2x) +A℘(2y) + 3∑ j=0 Cj(℘(x− y + ωj) + ℘(x+ y + ωj)) 22 T. Oshima and the dual of (Toda-C(1) 2 ) is (〈e−2λt, e−4λt〉, 〈e2λt, e4λt〉; 0, 〈e−4λt〉) (Todad-C(1) 2 ) since the dual of ( u−(t), u+(t); v(t), w(t) ) is ( w(t), v(t);u+(2t), u−(2t) ) . Similarly (〈℘(t;ω1, 2ω2), ℘(t;ω1, ω2)〉; 〈℘(2t; 2ω1, 2ω2), ℘(2t;ω1, 2ω2)〉). (Ellipd-B2-S) Since 4℘(2t; 2ω1, 2ω2) = ℘(t;ω1, ω2) etc., (Ellipd-B2-S) coincides with (Ellip-B2-S) by replacing (ω1, ω2) by (2ω2, ω1). Then we have the following diagrams and their duals, where the arrows with double lines represent specializations of parameters. For example, “Trig-B2 5:3⇒ Trig-BC2-reg” means that 2 parameters (coupling constants C2 and C3) out of 5 in the potential function (Trig-B2) are specialized to get the potential function (Trig-BC2-reg) with 3 parameters. For the normal cases see Definition 5 and the diagrams in the last part of the next section (type Bn). Hierarchy of Normal Integrable Potentials of type B2 Trig-BC2-reg → Toda-D2-bry C2=C3=0 ⇑ 5:3 C0=C1=0 ⇑ 5:3 Trig-B2 → Toda-B(1) 2 -bry 5:3⇒ C0=C1=0 Toda-B(1) 2 ↗ ↗ ↓ Ellip-B2 → Toda-D(1) 2 -bry Toda-C(1) 2 5:3⇒ C0=C1=0 Toda-BC2 ↘ Trig-B2 → Trig-A1-bry 5:3⇒ C2=C3=0 Trig-A1-bry-reg ↘ ↘ Rat-B2 → Rat-A1-bry Hierarchy of Special Integrable Potentials of type B2 Trig(d)-B2-S-reg → Toda(d)-D2-S-bry Ratd-D2-S-bry C1=0 ⇑ 4:3 C0=0 ⇑ 4:3 ↗ Trig(d)-B2-S → Toda(d)-B(1) 2 -S-bry 4:3⇒ C1=0 Toda(d)-B(1) 2 -S ↗ ↗ ↓ Ellip-B2-S→ Toda(d)-D(1) 2 -S-bry Toda(d)-C(1) 2 -S 4:3⇒ C0=0 Toda(d)-B2-S ↘ Trig(d)-B2-S → Trig(d)-A1-S-bry 4:3⇒ C1=0 Trig(d)-A1-S-bry-reg ↘ Rat-B2-S Definition 2. We define some potential functions as specializations. (Trig-B2-S-reg) Trigonometric special potential of type B2 with regular boundary conditions is (Trig-B2-S) with C1 = 0. (Toda-D2-S-bry) Toda special potential of type D2 with boundary conditions is (Toda-B(1) 2 -S-bry) with C0 = 0. (Toda-B(1) 2 -S) Toda special potential of type B(1) 2 is (Toda-B(1) 2 -S-bry) with C1 = 0. Completely Integrable Systems Associated with Classical Root Systems 23 (Toda-B2-S) Toda special potential of type B2 is (Toda-C(1) 2 -S-bry) with C0 = 0. (Trig-A1-S-bry-reg) Trigonometric special potential of type A1 with regular boundary conditions is (Trig-A1-S-bry) with C1 = 0. Remark 7. We have some equivalences as follows: (Ellip-B2-S) = (Ellipd-B2-S), (Rat-B2-S) = (Ratd-B2-S), (Trig-BC2-reg) = (Trigd-B2-S-reg), (Trig-A1-bry-reg) = (Todad-D2-S-bry), (Toda-D2-bry) = (Trigd-A1-S-bry-reg), (Toda-B(1) 2 ) = (Todad-B(1) 2 -S), (Toda-BC2) = (Todad-B2-S). 5 Type Bn (n ≥ 3) In this section we construct integrals of the completely integrable systems of type Bn appearing in the following diagram. The diagram is also given in [8, Figure III.1], where (Toda-B(1) n -bry) is missing. The most general system (Ellip-Bn) is called Inozemtzev model (cf. [12]). Hierarchy of Integrable Potentials with 5 parameters (n ≥ 2) Toda-D(1) n -bry → Toda-B(1) n -bry → Toda-C(1) n ↗ ↗ Ellip-Bn → Trig-Bn → Rat-Bn ↘ ↘ Trig-An−1-bry → Rat-An−1-bry 5.1 Integrable potentials Definition 3. The potential functions R(x) of (1.1) are as follows: Here A, C0, C1, C2 and C3 are any complex numbers. (Ellip-Bn) Elliptic potential of type Bn: ∑ 1≤i<j≤n A(℘(xi − xj ; 2ω1, 2ω2) + ℘(xi + xj ; 2ω1, 2ω2)) + n∑ k=1 3∑ j=0 Cj℘(xk + ωj ; 2ω1, 2ω2), (Trig-Bn) Trigonometric potential of type Bn:∑ 1≤i<j≤n A(sinh−2 λ(xi − xj) + sinh−2 λ(xi + xj)) + n∑ k=1 ( C0 sinh−2 λxk + C1 cosh−2 λxk + C2 sinh2 λxk + C3 4 sinh2 2λxk ) , (Rat-Bn) Rational potential of type Bn: ∑ 1≤i<j≤n ( A (xi − xj)2 + A (xi + xj)2 ) + n∑ k=1 (C0x −2 k + C1x 2 k + C2x 4 k + C3x 6 k), 24 T. Oshima (Trig-An−1-bry) Trigonometric potential of type An−1 with boundary conditions: ∑ 1≤i<j≤n A sinh−2 λ(xi − xj) + n∑ k=1 (C0e −2λxk + C1e −4λxk + C2e 2λxk + C3e 4λxk), (Toda-B(1) n -bry) Toda potential of type B(1) n with boundary conditions: n−1∑ i=1 Ae−2λ(xi−xi+1)+Ae−2λ(xn−1+xn)+C0e 2λx1 +C1e 4λx1 +C2 sinh−2 λxn+C3 sinh−2 2λxn, (Toda-C(1) n ) Toda potential of type C(1) n : n−1∑ i=1 Ae−2λ(xi−xi+1) + C0e 2λx1 + C1e 4λx1 + C2e −2λxn + C3e −4λxn , (Toda-D(1) n -bry) Toda potential of type D(1) n with boundary conditions: n−1∑ i=1 Ae−2λ(xi−xi+1) +Ae−2λ(xn−1+xn) +Ae2λ(x1+x2) + C0 sinh−2 λx1 + C1 sinh−2 2λx1 + C2 sinh−2 λxn + C3 sinh−2 2λxn, (Rat-An−1-bry) Rational potential of type An−1 with boundary conditions: ∑ 1≤i<j≤n A (xi − xj)2 + n∑ k=1 (C0xk + C1x 2 k + C2x 3 k + C3x 4 k). Remark 8. In these cases the Schrödinger operator P is of the form P = −1 2 n∑ j=1 ∂2 ∂x2 j +R(x), R(x) = ∑ 1≤i<j≤n (uei−ej (x) + uei+ej (x)) + 3∑ j=0 Cjv j(x), vj(x) = n∑ k=1 vj ek (x). Here ∂auα(x) = ∂bv j β(x) = 0 if a, b ∈ Rn satisfy 〈a, α〉 = 〈b, β〉 = 0. The complete integrability of the invariant systems (Ellip-Bn), (Trig-Bn) and (Rat-Bn) is established by [29]. We review their integrals and then we prove that the other five systems are also completely integrable by constructing enough integrals, which is announced by [30]. The complete integrability of (Trig-An−1-bry), (Toda-C(1) n ), (Toda-D(1) n -bry) and (Rat-An−1-bry) is presented as an unknown problem by [8] and then that of (Toda-B(1) n -bry), (Toda-C(1) n ) and (Toda-D(1) n -bry) are established by [18, 19, 21] using R-matrix method. The compete integra- bility of (Trig-An−1-bry) and (Rat-An−1-bry) seems to have not been proved. Definition 4 ([25, 29]). Let uα(x) and T o I (vj) are functions given for α ∈ Σ(Dn) and subsets I = {i1, . . . , ik} ⊂ {1, . . . , n} such that uα(x) = u−α(x) and ∂yuα = 0 for y ∈ Rn with 〈α, y〉 = 0. Completely Integrable Systems Associated with Classical Root Systems 25 Define a differential operator Pn(C) = n∑ k=0 1 k!(n− k)! ∑ w∈Sn ( q{w(1),...,w(k)}(C) ·∆2 {w(k+1),...,w(n)} ) by ∆{i1,...,ik} = ∑ 0≤j≤[ k 2 ] 1 2kj!(k − 2j)! (5.1) × ∑ w∈W (Bk) ε(w)w(uei1 −ei2 (x)uei3 −ei4 (x) · · ·uei2j−1 −ei2j (x) · ∂i2j+1∂i2j+2 · · · ∂ik), q{i1,...,ik}(C) = k∑ ν=1 ∑ I1q···qIν={i1,...,ik} TI1 · · ·TIν , (5.2) TI = (−1)#I−1 ( CSo I − 3∑ j=0 CjT o I (vj) ) , (5.3) where So {i1,i2,...,ik} = 1 2 ∑ w∈W (Bk) w(uei1 −ei2 (x)uei2 −ei3 (x) · · ·ueik−1 −eik (x)), So ∅ = 0, So {k} = 1, So {i,j} = 2uei−ej (x) + 2uei+ej (x), T o ∅(vj) = 0, T o {k}(v j) = 2vj ek (x) for 1 ≤ k ≤ n, q∅ = 1, q{i} = T{i}, q{i1i2} = T{i1}T{i2} + T{i1,i2}, . . . In the above, we identify W (Bk) with the reflection group generated by weik and weiν−eiν+1 (ν = 1, . . . , k− 1). The sum in (5.2) runs over all the partitions of the set I and the order of the subsets I1, . . . , Iν is ignored. Replacing ∂j by ξj for j = 1, . . . , n in the definition of ∆{i1,...,ik} and Pn(C), we define functions ∆̄{i1,...,ik} and P̄n(C) of (x, ξ), respectively. We will define uα(x) and T o I (vj) so that [Pn(C), Pn(C ′)] = 0 for C, C ′ ∈ C. (5.4) Then putting qo I = qI ∣∣ C=0 , Pn = Pn(0) = n∑ k=0 1 k!(n− k)! ∑ w∈Sn (qo {w(1),...,w(k)}∆ 2 {w(k+1),...,w(n)}), (5.5) Pn−j = n∑ i=j n∑ k=i (−1)i−j i!(k−i)!(n−k)! ∑ w∈Sn ∑ I1q···qIj=w({1,...,i}) So I1 · · ·S o Ij qo w({i+1,...,k})∆ 2 w({k+1,...,n}), (5.6) we have Pn(C) = n∑ j=0 CjPn−j and (1.4) and then [Pi, Pj ] = 0 for 1 ≤ i < j ≤ n, −P1 2 = −1 2 n∑ j=1 ∂2 ∂x2 j + ∑ 1≤i<j≤n (uei−ej (x) + uei+ej (x)) + 3∑ j=0 Cjv j(x). (5.7) 26 T. Oshima Remark 9. i) When n = 2, T (x, y) in the last section corresponds to T12, namely T (x1, x2) = T12 ∣∣ C=0 . ii) Put U(x) = ∑ 1≤i<j≤n (u−ij(xi − xj) + u+ ij(xi + xj)) and V (x) = n∑ k=1 vk(xk) in (1.3) and let TI(U ;V ) be the corresponding TI given by (5.3). Then [29, Remark 4.3] says TI(c0U ; c1V + c2W ) = c#I−1 0 c1TI(U ;V ) + c#I−1 0 c2TI(U ;W ) for ci ∈ C. (5.8) iii) The definition (5.3) may be replaced by TI = (−1)#I−1 ( C #I∑ ν=1 ∑ I1q···qIν=I (−2λ)ν−1(ν − 1)! · So I1 · · ·S o Iν − 3∑ j=0 CjT o I (vj) ) (5.9) because λ can be any complex number in [29, Lemma 5.2 ii)] when v = C. Note that we fixed λ = 1 in [29]. Combining (5.9) and (5.8), we may put TI = (−1)#I−1 ( CSo I + c′ ∑ ν≥2 cν−1 ∑ I1q···qIν=I So I1 · · ·S o Iν − 3∑ j=0 CjT o I (vj) ) for any c, c′ ∈ C and hence TI = (−1)#I−1 ( CSo I + ∑ ν≥2 cν ∑ I1q···qIν=I So I1 · · ·S o Iν − 3∑ j=0 CjT o I (vj) ) for any c2, c3, . . . ∈ C. Theorem 4 (Ellip-Bn, [25], [29, Theorem 7.2]). Put uei±ej (x) = A℘0(xi ± xj ; 2ω1, 2ω2) for 1 ≤ i < j ≤ n, vj ek (x) = ℘0(xk + ωj ; 2ω1, 2ω2) for 1 ≤ k ≤ n and 0 ≤ j ≤ 3 and T o I (vj) = #I∑ ν=1 ∑ I1q···qIν=I (−A)ν−1(ν − 1)! · SI1(v j) · · ·SIν (vj), (5.10) S{i1,...,ik}(v j) = ∑ w∈W (Bk) vj w(ei1 )(x)uw(ei1 −ei2 )(x)uw(ei2 −ei3 )(x) · · ·uw(eik−1 −eik )(x). (5.11) Then (5.4) holds. Example 1. Put vj k = vj ek , ṽk = 3∑ j=0 Cjv j k, and w±ij = u−ij ± u + ij . Then ∆{1} = ∂1, ∆{1,2} = ∂1∂2 + u−12 − u + 12 = ∂1∂2 + w−12, ∆{1,2,3} = ∂1∂2∂3 + w−12∂3 + w−23∂1 + w−13∂2, Completely Integrable Systems Associated with Classical Root Systems 27 ∆{1,2,3,4} = ∂1∂2∂3∂4 + w−34∂1∂2 + w−24∂1∂3 + w−23∂1∂4 + w−14∂2∂3 + w−13∂2∂4 + w−12∂3∂4 + w−12w − 34 + w−13w − 24 + w−14w − 23, S1(vj) = 2vj 1, S{1,2}(v j) = 2vj 1u − 12 + 2vj 1u + 12 + 2vj 2u − 12 + 2vj 2u + 12 = 2(vj 1 + vj 2)(u − 12 + u+ 12), S{1,2,3}(v j) = 2vj 1u − 12u − 23 + 2vj 1u − 12u + 23 + 2vj 1u + 12u − 23 + 2vj 1u + 12u + 23 + · · · = 2(vj 1 + vj 3)w + 12w + 23 + 2(vj 1 + vj 2)w + 23w + 13 + 2(vj 2 + vj 3)w + 12w + 13, T{1} = CS{1} − 3∑ j=0 CjT o {1}(v j) = C − 2ṽ1, T{1,2} = −CSo {1,2} + 3∑ j=0 CjT o {1,2}(v j), q{1} = T{1}, q{1,2} = T{1}T{2} + T{1,2}, q{1,2,3} = T{1}T{2}T{3} + T{1,2}T{3} + T{1,3}T{2} + T{1}T{2,3} + T{1,2,3}. If T o(vj) and Si1,...,ik(vj) are given by (5.10) and (5.11), then T o {1}(v j) = 2vj 1, T o {1,2}(v j) = S{1,2}(v j)−AT o {1}T o {2} = 2(vj 1 + vj 2)w + 12 − 4Avj 1v j 2, T o {1,2,3}(v j) = S{1,2,3}(v j)− 2A(vj 1S{2,3}(v j) + vj 2S{1,3}(v j) + vj 3S{1,2}(v j)) + 16A2vj 1v j 2v j 3. In particular, if n = 2, then P2(C) = ∆2 {1,2} + q{1}∆ 2 {2} + q{2}∆ 2 {1} + q{1,2} = ( ∂1∂2 + u−12 − u + 12 )2 + T{1}∂ 2 2 + T{2}∂ 2 1 + T{1}T{2} − CSo {1,2} + 3∑ j=0 CjT o {1,2}(v j) = ( ∂1∂2 + u−12 − u + 12 )2 + (C − 2ṽ1)∂2 2 + (C − 2ṽ2)∂2 1 + (C − 2ṽ1)(C − 2ṽ2) − 2C(u−12 + u+ 12) + 2(ṽ1 + ṽ2)(u−12 + u+ 12)− 4A 3∑ j=0 Cjv j 1v j 2 = C2 − 2P · C + P2 with P = −1 2 (∂2 1 + ∂2 2) + ṽ1 + ṽ2 + u−12 + u+ 12, P2 = ( ∂1∂2+u−12−u + 12 )2−2ṽ1∂2 2−2ṽ2∂2 1 +4ṽ1ṽ2+2(ṽ1+ṽ2)(u−12+u+ 12)−4A 3∑ j=0 Cjv j 1v j 2, which should be compared with (4.2), (4.4) and (4.6). In general P1 = n∑ k=1 (∆2 {k} + qo {k})− ∑ 1≤i<j≤n So {i,j} = n∑ k=1 (∂2 k − 2ṽj k)− 2 ∑ 1≤i<j≤n w+ ij , P2 = ∑ 1≤i<j≤n ∆2 {i,j} + ∑ 1≤i≤n 1≤j≤n, i6=j n∑ j=1 qo {i}∆ 2 {j} + ∑ 1≤i<j≤n qo {i,j} 28 T. Oshima − ∑ 1≤i<j≤n 1≤k≤n, k 6=i, j So {i,j}(∆ 2 {k} + qo {k}) + ∑ 1≤i<j≤n i<k<`≤n j 6=k, ` So {i,j}S o {k,`} + ∑ 1≤i<j<k≤n So {i,j,k} = ∑ (∂i∂j + w−ij) 2 − ∑ 2ṽi∂ 2 j + ∑ 4ṽiṽj + ∑ CjT o {k,`}(v j) − 2 ∑ w+ ij(∂ 2 k − 2ṽk) + 4 ∑ w+ ijw + k` + 2 ∑ w+ ijw + jk. (5.12) Here if (5.10) and (5.11) are valid, then T o {k,`}(v j) = 2(vj k + vj ` )w + k` − 4Avj kv j ` . (5.13) The commuting operator P3 of the 6-th order is P3 = ∑ ∆2 {i,j,k} + ∑ qo {k}∆ 2 {i,j} + ∑ qo {i,j,k} − ∑ So {k,`}∆ 2 {i,j} − ∑ So {k,`}q o {i}∆ 2 {j} − ∑ So {k,`}q o {i,j} + ∑ So {i,j,k,`,m} + ∑ So {i,j,k,`}S o {µ,ν} + ∑ So {i,j,k}S o {`,µ,ν} + ∑ So {i,j,k}S o {`,m}S o {µ,ν} + ∑ So {i1,i2}S o {j1,j2}S o {k1,k2}S o {`1,`2}. In Theorem 4 the Schrödinger operator is P = −1 2 n∑ k=0 ∂2 ∂x2 k +A ∑ 1≤i<j≤n (℘(xi − xj) + ℘(xi + xj)) + 3∑ j=0 Cj n∑ k=1 ℘(xk + ωj) and the operator P2 satisfying [P, P2] = 0 is given by (5.12) and (5.13) with ṽk = 3∑ ν=0 Cν℘(xk + ων), vj k = ℘(xk + ωj), w±ij = A(℘(xi − xj)± ℘(xi + xj)). 5.2 Analytic continuation of integrals Theorem 5 (Toda-D(1) n -bry). For uei−ej (x) = { Ae−2λ(xi−xi+1) (j = i+ 1), 0 (\|j − i\| > 1), uei+ej (x) =  Ae2λ(x1+x2) (i+ j = 3), Ae−2λ(xn−1+xn) (i+ j = 2n− 1), 0 (i+ j /∈ {3, 2n− 1}), (5.14) v0 k(x) = δ1k sinh−2 λx1, v1 k(x) = δ1k sinh−2 2λx1, v2 k(x) = δnk sinh−2 λxn, v3 k(x) = δnk sinh−2 2λxn, we have commuting integrals Pj by (5.5), (5.1), (5.2), (5.3) and So {k} = 1 for 1 ≤ k ≤ n, So I = 0 if I 6= {k, k + 1, . . . , `} for 1 ≤ k < ` ≤ n, So {k,k+1,...,`} = 2A`−k+1(e−2λ(xk−x`) + δ1ke 2λ(x1+x`) + δ`ne −2λ(xk+xn)), T 0 {k}(v j) = 2vj k(x) for 0 ≤ j ≤ 3, k = 1, . . . , n, T 0 I (v0) = 0 if I 6= {1, . . . , k} for k = 1, . . . , n, T 0 I (v2) = 0 if I 6= {k, . . . , n} for k = 1, . . . , n, Completely Integrable Systems Associated with Classical Root Systems 29 T 0 I (v1) = T 0 I (v3) = 0 if #I > 1, T 0 {1,...,k}(v 0) = 8Ak−1(e2λxk + δkne −2λxn) for k ≥ 2, T 0 {n−k+1,...,n}(v 2) = 8Ak−1(e−2λxn−k+1 + δkne 2λx1) for k ≥ 2. Proof. Put x̃ = ( x1 − 1− 1 n− 1 ω2, . . . , xk − k − 1 n− 1 ω2, . . . , xn − n− 1 n− 1 ω2 ) , ũei∓ej (x̃) = A e2λω2/(n−1) 4λ2 ℘0 ( xi − i− 1 n− 1 ω2 ∓ ( xj − j − 1 n− 1 ω2 ) ; 2ω1, 2ω2 ) , ṽj k(x̃) = (−1)j λ2 ℘0 ( xk − k − 1 n− 1 ω2 + ωj ; 2ω1, 2ω2 ) for 0 ≤ j ≤ 3, 1 ≤ k ≤ n. When ω2 →∞, ũei∓ej (x̃) and ṽ` k (` = 0, 1, 2, 3) converge to uei∓ej (x) in (5.14) and v0 k(x) = δ1k sinh−2 λx1, v1 k(x) = δ1k cosh−2 λx1, v2 k(x) = δnk sinh−2 λxn, v3 k(x) = δnk cosh−2 λxn, respectively. Under the notation in Theorem 4, let S̃I(ṽ`) and T̃ o I (ṽ`) be the functions defined in the same way as SI(v`) and T o I (v`), respectively, where (uei∓ej (x), v ` k(x)) are replaced by (ũei∓ej (x), ṽ ` k(x̃)). Then by taking the limits for ω2 →∞, T̃ o I (ṽ`) converge to the following T̄ o I (v`) T̄ 0 I (v0) = T̄ 0 I (v1) = 0 if I 6= {1, . . . , k} for k = 1, . . . , n, T̄ 0 I (v2) = T̄ 0 I (v3) = 0 if I 6= {k, . . . , n} for k = 1, . . . , n. If k ≥ 2, then T̄ 0 {1,...,k}(v 0) = lim ω2→∞ #I∑ ν=1 ∑ I1q···qIν=I ( −Ae 2λω2/(n−1) 4λ2 )ν−1 (ν − 1)!S̃I1(ṽ 0) · · · S̃Iν (ṽ0) = lim ω2→∞ S̃{1,...,k}(ṽ 0)− lim ω2→∞ S̃{1}(ṽ 0)A e2λω2/(n−1) 4λ2 S̃{2,...,k}(ṽ 0) = 2Ak−1 sinh−2λx1(e−2λ(x1−xk)+ e2λ(x1+xk) + δkne 2λ(x1−xn) + δkne −2λ(x1+xn)) − 2 sinh−2 λx1 · 2Ak−1(e2λxk + δkne −2λxn) = 8Ak−1(e2λxk + δkne −2λxn), T̄ 0 {1,...,k}(v 1) = 2Ak−1 cosh−2λx1(e−2λ(x1−xk)+ e2λ(x1+xk)+ δkne 2λ(x1−xn)+ δkne −2λ(x1+xn)) + 4Ak−1 cosh−2 λx1(e2λxk + δkne −2λxn) = 8Ak−1(e2λxk + δkne −2λxn), T̄ 0 {n−k+1,...,n}(v 2) = 8Ak−1(e−2λxn−k+1 + δkne 2λx1), T̄ 0 {n−k+1,...,n}(v 3) = 8Ak−1(e−2λxn−k+1 + δkne 2λx1). Replacing v1 and v3 by (1/4)(v0 − v1) and (1/4)(v2 − v3), respectively, we have the theorem by the analytic continuation given in Lemma 2. � As is proved by [29], suitable limits of the functions in Theorem 4 give the following theorem. Theorem 6 (Trig-Bn, [29, Proposition 6.1]). For complex numbers λ, C, C0, . . ., C3 and A with λ 6= 0, we have (5.7) by putting uei±ej (x) = A sinh−2 λ(xi ± xk), 30 T. Oshima v0 ek (x) = sinh−2 λxk, v1 ek (x) = cosh−2 λxk, v2 ek (x) = sinh2 λxk, v3 ek (x) = 1 4 sinh2 2λxk and TI = (−1)#I−1(CSo I − C0T o I (v0)− C1T o I (v1)− C2SI(v2)− C3SI(v3) + 2C3 ∑ I1qI2=I (SI1(v 2)SI2(v 2) + SI1(v 2)So I2 + So I1SI2(v 2))), T o I (v0) = #I∑ ν=1 ∑ I1q···qIν=I (−A)ν−1(ν − 1)!SI1(v 0) · · ·SIν (v0), T o I (v1) = #I∑ ν=1 ∑ I1q···qIν=I Aν−1(ν − 1)!SI1(v 1) · · ·SIν (v1). Theorem 7 (Trig-An−1-bry). For uei−ej (x) = A sinh−2 λ(xi − xj), uei+ej (x) = 0, v0 ek (x) = e−2λxk , v1 ek (x) = e−4λxk , v2 ek (x) = e2λxk , v3 ek (x) = e4λxk we have (5.7) by putting TI =(−1)#I−1 ( CSo I − 3∑ j=0 CjSI(vj)+ 2 ∑ I1qI2=I (C1SI1(v 0)SI2(v 0)+C3SI1(v 2)SI2(v 2)) ) . Proof. Putting ũei±ej = A sinh−2 λ ( (xi +N)± (xj +N) ) , ṽ0 k = 1 4 e2λN sinh−2 λ(xk +N), ṽ1 k = 1 4 e4λN sinh−2 2λ(xk +N) = 1 16 e4λN ( sinh−2 2λ(xk +N)− cosh−2 2λ(xk +N) ) , ṽ2 k = 4e−2λN sinh2 λ(xk +N), ṽ3 k = 4e−4λN sinh2 2λ(xk +N), x̃ = (x1 +N,x2 +N, . . . , xn +N) under the notation in Theorem 6, we have (ūei−ej , ūei+ej , v̄ 0 k, v̄ 1 k, v̄ 2 k, v̄ 3 k) := lim N→∞ (ũei−ej , ũei+ej , ṽ 0 k, ṽ 1 k, ṽ 2 k, ṽ 3 k) = (A sinh−2 λ(xi − xk), 0, e−2λxk , e−4λxke2λxk , e4λxk), lim N→∞ 1 4 e2λNT o I (v0)(x̃) = S̄I(v̄0), lim N→∞ 1 16 e4λN ( T o I (v0)(x̃)− T o I (v1)(x̃)) = S̄I(v̄1)− 2 ∑ I1qI2=I S̄I1(v̄ 0)S̄I2(v̄ 0), lim N→∞ 4e−2λNT o I (v2)(x̃) = S̄I(v̄2), lim N→∞ 4e−2λNT o I (v3)(x̃) = S̄I(v̄3)− 2 ∑ I1qI2=I S̄I1(v̄ 2)S̄I2(v̄ 2). Here S̄I(v̄`) are defined by (5.11) with uei±ej and v` ek replaced by ūei±ej and v̄` ek , respectively. Then the theorem is clear. � Completely Integrable Systems Associated with Classical Root Systems 31 Theorem 8 (Toda-B(1) n -bry). For the potential function defined by uei−ej (x) = { Ae−2λ(xi−xi+1) if j = i+ 1, 0 if 1 ≤ i < i+ 1 < j ≤ n, uei+ej (x) = { Ae−2λ(xn−1+xn) if i = n− 1, j = n, 0 if 1 ≤ i < j ≤ n and i 6= n− 1, (5.15) v0 k(x) = δk1e 2λx1 , v1 k(x) = δk1e 4λx1 , v2 k(x) = δkn sinh−2 λxn, v3 k(x) = δkn sinh−2 2λxn, we have (5.7) with So {k} = 1 for 1 ≤ k ≤ n, So I = 0 if I 6= {k, k + 1, . . . , `} for 1 ≤ k < ` ≤ n, So {k,k+1,...,`} = 2A`−k+1(e−2λ(xk−x`) + δ`ne −2λ(xk+xn)), T 0 {k}(v j) = 2vj k(x) for 0 ≤ j ≤ 3, k = 1, . . . , n, T 0 I (v0) = 0 if I 6= {1, . . . , k} for k = 1, . . . , n, T 0 I (v2) = 0 if I 6= {k, . . . , n} for k = 1, . . . , n, T 0 I (v1) = T 0 I (v3) = 0 if #I > 1, T 0 {1,...,k}(v 0) = 2Ak−1(e2λxk + δkne −2λxn) for k ≥ 2, T 0 {n−k+1,...,n}(v 2) = 8Ak−1e−2λxn−k+1 for k ≥ 2. Proof. Suppose Reλ > 0. In (Toda-D(1) n -bry) put x̃ = (x1 − (n− 1)N, . . . , xk − (n− k)N, . . . , xn − (n− n)N), ũei−ej = { Ae−2λNe−2λ(xi−(n−i)N−xi+1+(n−i−1)N) (j = i+ 1), 0 (\|j − i\| > 1), ũei+ej =  Ae−2λNe2λ(x1−(n−1)N+x2−(n−2)N) (i+ j = 3), Ae−2λNe−2λ(xn−1−N+xn) (i+ j = 2n− 1), 0 (i+ j /∈ {3, 2n− 1}), ṽ0 k = δ1k e2λ(n−1)N 4 sinh−2 λ(x1 − (n− 1)N), ṽ1 k = δ1k e4λ(n−1)N 4 sinh−2 2λ(x1 − (n− 1)N), ṽ2 k = δnk sinh−2 λxn, ṽ3 k = δnk sinh−2 2λxn and we have (5.15) by the limit N → ∞. Moreover for k ≥ 2, it follows from Theorem 5 and (5.8) that T̃{1,...,k}(ṽ 1) = T̃{1,...,k}(ṽ 3) = 0, T̃{1,...,k}(ṽ 0) = 1 4 e2λ(n−1)N (Ae−2λN )k−1(8e2λ(xk−(n−k)N) + 8δkne −2λxn) = 2Ak−1(e2λxk + δkne −2λxn), T̃{n−k+1,...,n}(ṽ 2) = (Ae−2λN )k−1(8e−2λ(xn−k+1−(k−1)N) + 8δ1ke 2λ(x1−(n−1)N)) = 8Ak−1e−2λxn−k+1 , which implies the theorem. � 32 T. Oshima Theorem 9 (Toda-C(1) n ). For the potential function defined by uei−ej (x) = { Ae−2λ(xi−xi+1) if j = i+ 1, 0 if 1 ≤ i < i+ 1 < j ≤ n, uei+ej (x) = 0 for 1 ≤ i < j ≤ n, v0 k(x) = δk1e 2λx1 , v1 k(x) = δk1e 4λx1 , v2 k(x) = δkne −2λxn , v3 k(x) = δkne −4λxn , we have (5.7) with So {k} = 1 for 1 ≤ k ≤ n, So I = 0 if I 6= {k, k + 1, . . . , `} for 1 ≤ k < ` ≤ n, So {k,k+1,...,`} = 2A`−k+1e−2λ(xk−x`), T 0 {k}(v j) = 2vj k(x) for 0 ≤ j ≤ 3, k = 1, . . . , n, T 0 I (v0) = 0 if I 6= {1, . . . , k} for k = 1, . . . , n, T 0 I (v2) = 0 if I 6= {k, . . . , n} for k = 1, . . . , n, T 0 I (v1) = T 0 I (v3) = 0 if #I > 1, T 0 {1,...,k}(v 0) = 2Ak−1e2λxk for k ≥ 2, T 0 {n−k+1,...,n}(v 2) = 2Ak−1e−2λxn−k+1 for k ≥ 2. Proof. Substituting xk by xk +R for k = 1, . . . , n and multiplying v0 k, v 1 k, v 2 k and v3 k by e−2λR, e−4λR, (1/4)e2λR and (1/4)e4λR, respectively, we have the claim from Theorem 8. � Theorem 10 (Rat-An−1-bry). We have (5.7) if uei−ej (x) = A (xi − xj)2 , uei+ej (x) = 0, vj k(x) = xj+1 k , TI = (−1)#I−1 ( CSo I − 3∑ j=0 CjSI(vj) + ∑ I1qI2=I C1(SI1(v 0)So I2 + So I1SI1(v 0)) + ∑ I1qI2=I C3(SI1(v 1)So I2 + SI1(v 0)SI2(v 0) + So I1SI2(v 1)) ) . Proof. Put ũei−ej = λ2 sinh−2 λ(xi − xj), ũei+ej = 0, ṽ0 k = 1 2λ (e2λxk − 1), ṽ1 k = 1 4λ2 (e2λxk + e−2λxk − 2), ṽ2 k = 1 8λ3 (e4λxk−3e2λxk−e−2λxk + 3), ṽ3 k = 1 16λ4 (e4λxk +e−4λxk−4e2λxk−4e−2λxk +6). Then taking λ→ 0 we have the required potential function. Owing to (Trig-An−1-bry) and Remark 9, we have lim λ→0 S̃I (∑ ṽj k ) = S̄I (∑ xj+1 k ) , lim λ→0 λ2 1 8λ3 ( S̃I1 (∑ e2λxk ) S̃I2 (∑ e2λxk ) − 4S̃o I1S̃ o I2 ) Completely Integrable Systems Associated with Classical Root Systems 33 = 1 2 ( S̄I1 (∑ xk ) S̄o I2 + S̄o I1S̄I2 (∑ xk )) , lim λ→0 λ2 1 16λ4 ( S̃I1 (∑ e2λxk ) S̃I2 (∑ e2λxk ) + S̃I1 (∑ e−2λxk ) S̃I2 (∑ e−2λxk ) − 8S̃o I1S̃ o I2 ) = 1 2 ( S̄I1 (∑ x2 k ) S̄o I2 + S̄I1 (∑ xk ) S̄I2 (∑ xk ) + S̄o I1S̄I2 (∑ xk )) . and thus the theorem. � As is proved by [29], suitable limits of the functions in Theorem 6 give the following theorem. Theorem 11 (Rat-Bn, [29, Proposition 6.3]). Put uei−ej (x) = A (xi − xj)2 , uei+ej (x) = A (xi + xj)2 v0 k(x) = x−2 k , v1 k(x) = x2 k, v2 k(x) = x4 k, v3 k(x) = x6 k. Then (5.7) holds with TI = (−1)#I−1 ( CSo I − C0T o I (v0)− C1SI(v1)− C2SI(v2) + 2C2 ∑ I1qI2=I (SI1(v 1)So I2 + So I1SI2(v 1))− 2C3S o I (v3) + C3 ∑ I1qI2=I (SI1(v 1)SI2(v 1) + 2SI1(v 2)So I2 + 2So I1SI2(v 2)) − 24C3 ∑ I1qI2qI3=I (SI1(v 1)So I2S o I3 + So I1SI2(v 1)So I3 + So I1S o I2SI3(v 1)) ) , T o I (v0) = #I∑ ν=1 ∑ I1q···qIν=I (−A)ν−1(ν − 1)! · SI1(v 0) · · ·SIν (v0). Definition 5. We define some potential functions as specializations of potential functions in Definition 3. (Trig-An−1-bry-reg) Trigonometric potential of type An−1 with regular boundary conditions is (Trig-An−1-bry) with C2 = C3 = 0. (Trig-An−1) Trigonometric potential of type An−1 is (Trig-An−1-bry) with C0 = C1 = C2 = C3 = 0. (Trig-BCn-reg) Trigonometric potential of type BCn with regular boundary conditions is (Trig- Bn) with C2 = C3 = 0. (Toda-Dn-bry) Toda potential of type Dn with boundary conditions is (Toda-B(1) n -bry) with C0 = C1 = 0. (Toda-B(1) n ) Toda potential of type B(1) n is (Toda-B(1) n -bry) with C2 = C3 = 0. (Toda-D(1) n ) Toda potential of type D(1) n is (Toda-D(1) n -bry) with C0 = C1 = C2 = C3 = 0. (Toda-An−1) Toda potential of type An−1 is (Toda-C(1) n ) with C0 = C1 = C2 = C3 = 0. (Toda-BCn) Toda potential of type Bn is (Toda-C(1) n ) with C0 = C1 = 0. (Ellip-Dn) Elliptic potential of type Dn is (Ellip-Bn) with C0 = C1 = C2 = C3 = 0. (Trig-Dn) Trigonometric potential of type Dn is (Trig-Bn) with C0 = C1 = C2 = C3 = 0. 34 T. Oshima (Rat-Dn) Rational potential of type Dn is (Rat-Bn) with C0 = C1 = C2 = C3 = 0. (Toda-Dn) Toda potential of type Dn is (Toda-B(1) n -bry) with C0 = C1 = C2 = C3 = 0. (Rat-Bn-2) Rational potential of type Bn-2 is (Rat-Bn) with C2 = C3 = 0. (Rat-An−1-bry2) Rational potential of type An−1 with 2-boundary conditions is (Rat-An−1-bry) with C2 = C3 = 0. In this case, we may assume C0 = 0 or C1 = 0 by the transformation xk 7→ xk + c (k = 1, . . . , n) with a suitable c ∈ C. Then we have the following diagrams for n ≥ 3. Note that we don’t write all the arrows in the diagrams (ex. (Toda-Dn-bry) → (Toda-BCn)) and the meaning of the symbol 5:3⇒ is same as in the diagram for type B2. Namely, 5 parameters (coupling constant) in the potential are reduced to 3 parameters by a certain restriction. Hierarchy of Elliptic-Trigonometric-Rational Integrable Potentials Rat-Bn-2 Ellip-Dn ⇑5:3 ↓ Rat-Bn ↖ Trig-Dn → Rat-Dn ↑ ⇑3:1 Ellip-Bn → Trig-Bn 5:3⇒ Trig-BCn-reg Ellip-An−1 ↓ ↓ ↓ Trig-An−1-bry 5:3⇒ Trig-An−1-bry-reg 3:1⇒ Trig-An−1 ↓ ↓ ↓ Rat-An−1-bry 5:3⇒ Rat-An−1-bry2 3:1⇒ Rat-An−1 Hierarchy of Toda Integrable Potentials Trig-BCn-reg → Toda-Dn-bry 3:1⇒ Toda-Dn ⇑5:3 ⇑5:3 ⇑3:1 Trig-Bn → Toda-B(1) n -bry 5:3⇒ Toda-B(1) n ↗ ↗ ↘ Ellip-Bn → Toda-D(1) n -bry 5:1⇒ Toda-D(1) n Toda-C(1) n ⇒5:1 ↗ ⇓5:3 Ellip-Dn Trig-An−1 Toda-BCn ↗ ↘ ⇓3:1 Ellip-An−1 → Toda-A(1) n−1 → Toda-An−1 6 Type Dn (n ≥ 3) Theorem 12 (Type Dn). The Schrödinger operators (Ellip-Dn), (Trig-Dn), (Rat-Dn), (Toda- D (1) n ) and (Toda-Dn) are in the commutative algebra of differential operators generated by P1, P2, . . . , Pn−1 and ∆{1,...,n} which are the corresponding operators for (Ellip-Bn), (Trig-Bn), (Rat-Bn), (Toda-D(1) n -bry), (Toda-Dn-bry) with C0 = C1 = C2 = C3 = 0, respectively. Proof. This theorem is proved by [29] in the cases (Ellip-Dn), (Trig-Dn), (Rat-Dn). Other two cases have been defined by suitable analytic continuation and therefore the claim is clear. � Completely Integrable Systems Associated with Classical Root Systems 35 Remark 10. In the above theorem we have Pn = ∆2 {1,...,n} because qo I = 0 if I 6= ∅. Then [Pj , Pn] = 0 implies [Pj ,∆{1,...,n}] = 0. Hierarchy of Integrable Potentials of Type Dn (n ≥ 3) Toda-D(1) n → Toda-Dn ↗ ↗ Ellip-Dn → Trig-Dn → Rat-Dn 7 Classical limits For functions f(ξ, x) and g(ξ, x) of (ξ, x) = (ξ1, . . . , ξn, x1, . . . , xn), we define their Poisson bracket by { f, g } = n∑ k=1 ( ∂f ∂ξk ∂g ∂xk − ∂g ∂ξk ∂f ∂xk ) . Theorem 13. Put P̄ (ξ, x) = −1 2 n∑ k=1 ξ2k +R(x). Then for the integrable potential function R(x) given in this note, the functions P̄k(ξ, x) and ∆̄{1,...,n}(ξ, x) of (ξ, x) defined by replacing ∂ν by ξν (ν = 1, . . . , n) in the definitions of Pk and ∆{1,...,n} in Sections 3, 4 and 5 satisfy{ P̄i(ξ, x), P̄j(ξ, x) } = { P̄ (ξ, x), P̄k(ξ, x) } = 0 for 1 ≤ i < j ≤ n and 1 ≤ k ≤ n. Hence P̄ (ξ, x) are Hamiltonians of completely integrable dynamical systems. Moreover if the potential function R(x) is of type Dn, then{ ∆̄{1,...,n}(ξ, x), P̄k(ξ, x) } = { ∆̄{1,...,n}(ξ, x), P̄ (ξ, x) } = 0 for 1 ≤ k ≤ n. Proof. If R(x) is a potential function of (Ellip-An−1), (Ellip-Bn) or (Ellip-Dn), the claim is proved in [29, 32]. Since the claim keeps valid under suitable holomorphic continuations with respect to the parameters which are given in the former sections, we have the theorem. � Remark 11. Since our operators Pk are expressed by operators P ν k = ∑ i pν k,i(x)q ν k,i(∂) such that the polynomials qν k,i(∂) satisfy [pν k,i(x), q ν k,i(∂)] = 0, there is no ambiguity in the definition of the classical limits by replacing ∂ν by ξν . In another word, if we have given the above integrals P̄j(x, ξ) of the classical limit, we have a natural unique quantization of them. 8 Analogue for one variable Putting n = 1 for the Schrödinger operator P of type An in Section 3 or of type Bn in Section 5, we examine the ordinary differential equation Pu = Cu with C ∈ C (cf. [41, § 10.6]). We will write the operators Q = P − C. (Ellip-B1) The Heun equation (cf. [32, § 8], [41, pp. 576]) −1 2 d2 dt2 + 3∑ j=0 Cj℘(t+ ωj)− C. 36 T. Oshima (Ellip-A1) The Lamé equation −1 2 d2 dt2 +A℘(t)− C. (Trig-BC1-reg) The Gauss hypergeometric equation −1 2 d2 dt2 + C0 sinh2 λt + C1 sinh2 2λt − C. (Trig-A1) The Legendre equation −1 2 d2 dt2 + C0 sinh2 λt − C. (Trig-B1) with C0 = C1 = C3 = 0. The (Modified) Mathieu equation −1 2 d2 dt2 + C2 cosh 2λt− C. (Rat-B1-2) Equation of the paraboloid of revolution −1 2 d2 dt2 + C0 t2 + C1t 2 − C. This is the Weber equation if C0 = 0. With s = t2 and using the unknown function t 1 2u, the above equation is reduced to the Whittaker equation: −1 2 d2 ds2 + C ′ 0 s2 + C ′ 1 s − C ′. (Rat-A0-bry2) with C2 = C3 = 0: −1 2 d2 dt2 + C0t+ C1t 2 − C. If C1 6= 0, this is transformed into the Weber equation under the coordinate s = t+ C2/(2C1). If C1 = 0, this is the Stokes equation which is reduced to the Bessel equation. In particular, the Airy equation corresponds to C = C1 = 0. (Toda-BC1) −1 2 d2 dt2 + C0e −2t + C1e −4t − C, which is transformed into (Rat-B1-2) by putting s = e−t. In particular (Toda-A1) −1 2 d2 dt2 + C0e −2t − C is reduced to the Bessel equation. (Rat-A1) the Bessel equation −1 2 d2 dt2 + C0 t2 − C. Completely Integrable Systems Associated with Classical Root Systems 37 In fact, the equation −u′′/2 + C0u/t 2 = Cu is equivalent to( d2 dt2 + 1 t d dt − 2C0 + 1/4 t2 + 2C ) t−1/2u = 0 since t− 1 2 ◦ d 2 dt2 ◦ t 1 2 = d2 dt2 + 1 t d dt − 1 4t2 . Hence if C 6= 0, the function v = t−1/2u satisfies the following Bessel equation with s = √ −2Ct: d2v ds2 + 1 s dv ds − ( 1− C0 + 1/8 Cs2 ) v = 0. Hierarchy of ordinary differential equations Heun 5:2⇒ Lamé → Legendre ↘ ⇑3:2 ↘ Mathieu 2:5⇐ Trig-B1 5:3⇒ Gauss Bessel (Stokes) 2:1→ Airy ↓5:4 ↘ ↘ ⇑3:2 ↑2:1 Rat-A0-bry Rat-B1 5:3⇒ Whittaker 3:2→ Weber 9 A classification We present a conjecture which characterizes the systems listed in this note. Let P be the Schrödinger operator with the expression (1.1) and consider the condition there exist P1, . . . , Pn such that  P ∈ C[P1, . . . , Pn], [Pi, Pj ] = 0 (1 ≤ i < j ≤ n), σ(Pk) = ∑ 1≤j1<···<jk≤n ξ2j1 · · · ξ 2 jk (1 ≤ k ≤ n). (9.1) Note that all the completely integrable systems given in Sections 3, 4 or 5 satisfy this condition. Conjecture. Suppose P satisfies (9.1). Under a suitable affine transformation of the coordinate x ∈ Cn which keeps the algebra C [ n∑ k=1 ∂2 k , n∑ k=1 ∂4 k , . . . , n∑ k=1 ∂2n k ] invariant, P is transformed into an integrable Schrödinger operator studied in Sections 3, 4 or 5, (namely u±ij, vk and wk in (1.3) are suitable analytic continuations of the corresponding functions of the invariant elliptic systems) or in general a direct sum of such operators and/or trivial operators (A1) d2 dx2 + v(x) with arbitrary functions v(x) of one variable. Here the direct sum of the two operators Pj(x, ∂x) = ∑ α∈{0,1,...}nj aα(x)∂α x of x ∈ Cnj for j = 1, 2 means the operator P1(x, ∂x) + P2(y, ∂y) of (x, y) ∈ Cn1+n2 . We review known conditions assuring this conjecture and give another condition (cf. Theo- rem 19 and Remark 17). We also review related results on the classification of completely integrable quantum systems associated with classical root systems. 38 T. Oshima Remark 12. The condition there exists P2 such that [P, P2] = 0 and σ(P2) = ∑ 1≤i<j≤n ξ2i ξ 2 j (9.2) may be sufficient to assure the claim of the conjecture. Remark 13 (Type A2). If n = 2 and if there exists P3 satisfying σ(P3) = ξ1ξ2 + ξ2ξ3 + ξ3ξ1 and [P, P3] = [∂1 + ∂2 + ∂3, P ] = [∂1 + ∂2 + ∂3, P3] = 0, then Conjecture is true. In fact this case is reduced to solving the equation∣∣∣∣∣∣∣ u(x) u′(x) 1 v(y) v′(y) 1 w(z) w′(z) 1 ∣∣∣∣∣∣∣ = 0 for x+ y + z = 0 (9.3) for three unknown functions u(t), v(t) and w(t), which is solved by [3, 4]. Here u(t) = ue1−e2(t), v(t) = ue2−e3(t) and w(t) = ue1−e3(−t). 9.1 Pairwise interactions and meromorphy Theorem 14 ([40]). The potential function R(x) of P satisfying (9.1) is of the form R(x) = ∑ α∈Σ(Bn)+ uα(〈α, x〉) (9.4) with meromorphic functions uα(t) of one variable. Remark 14. i) The condition (9.2) assures R(x) = ∑ α∈Σ(Bn)+ uα(〈α, x〉) + ∑ 1≤i<j<k≤n Cijkxixjxk with Cijk ∈ C and thus the above theorem is proved in the invariant case (cf. Section 9.2) by [32] or in the case of Type B2 by [23] or in the case of Type An−1. This theorem is proved in [40] by using [P, P2] = [P, P3] = 0. ii) Suppose n = 2 and the operators P = −1 2 ( ∂2 ∂x2 1 + ∂2 ∂x2 2 ) +R(x1, x2), T = m∑ i=0 cj ∂m ∂xi 1∂x m−i 2 + ∑ i+j≤m−2 Ti,j(x1, x2) ∂i+j ∂xi 1∂x j 2 satisfy [P, T ] = 0 and σm(T ) /∈ C[σ(P )]. Here cj ∈ C. Then [31, Theorem 8.1] shows that there exist functions uν,i(t) of one variable such that R(x1, x2) = L∑ ν=1 mν−1∑ i=0 (bνx1 + aνx2)iuν,i(aνx1 − bνx2) by putting( ξ ∂ ∂τ − τ ∂ ∂ξ ) m∑ i=0 ciξ m−iτ i = L∏ ν=1 (aνξ − bντ)mν . Here (aν , bν) ∈ C2 \ {(0, 0)} and aνbµ 6= aµbν if µ 6= ν. Completely Integrable Systems Associated with Classical Root Systems 39 Definition 6. By the expression (9.4), put S = {α ∈ Σ(Bn)+ ; u′α 6= 0} and let W (S) be the Weyl group generated by {wα ; α ∈ S} and moreover put S̄ = W (S)S. Theorem 15 ([23] for Type B2, [40] in general). If the root system S̄ has no irreducible component of rank one, then (9.2) assures that any function uα(t) extends to a meromorphic function on C. Remark 15 ([32, (6.4)–(6.5)], [40, § 3]). The condition (9.2) is equivalent to Sij = Sji (1 ≤ i < j ≤ n) (9.5) with Sij = ( ∂2 i vi(xi) + ∑ ν∈I(i,j) ∂2 i ( u+ iν(xi + xν) + u−iν(xi − xν) )) (u+ ij(xi + xj)− u−ij(xi − xj)) + 3 ( ∂ivi(xi) + ∑ ν∈I(i,j) ∂i ( u+ iν(xi + xν) + u−iν(xi − xν) )) (∂iu + ij(xi + xj)− ∂iu − ij(xi − xj)) + 2 ( vi(xi) + ∑ ν∈I(i,j) (u+ iν(xi + xν) + u−iν(xi − xν)) ) (∂2 i u + ij(xi + xj)− ∂2 i u − ij(xi − xj)) + ∑ ν∈I(i,j) (∂2 i u + iν(xi + xν)− ∂2 i u − iν(xi − xν))(u+ jν(xj + xν)− u−jν(xj − xν)). Here I(i, j) = {1, 2, . . . , n} \ {i, j}. Lemma 5. Suppose P satisfies (9.2) and (9.4). Let S0 be a subset of S̄ such that S0 ⊂ m∑ i=1 Rei and S̄ \ S0 ⊂ n∑ i=m+1 Rei with a suitable m. Then the Schrödinger operator P ′ = −1 2 m∑ i=1 ∂2 i + ∑ α∈S0∩S u(〈α, x〉) on Rm admits a differential operator P ′ 2 on Rm satisfying [P ′, P ′ 2] = 0 and σ(P ′ 2) = m∑ 1≤i<j≤n ξ2i ξ 2 j , that is, the condition (9.2) with replacing P by P ′. Proof. This lemma clearly follows from the equivalent condition (9.5) given in Remark 15. � 9.2 Invariant case Theorem 16 ([24, 25, 29, 32]). Assume that P in (1.1) is invariant under the Weyl group W = W (An−1), W (Bn) or W (Dn) with n ≥ 3, or W = W (B2). If we have (1.2) with P1 = ∂1 + ∂2 + · · ·+ ∂n if W = W (An−1), 40 T. Oshima σ(Pk) =  ∑ 1≤j1<j2<···<jk≤n ξj1ξj2 · · · ξjk if W = W (An−1) and 1 ≤ k ≤ n,∑ 1≤j1<j2<···<jk≤n ξ2j1ξ 2 j2 · · · ξ 2 jk if W = W (Bn) and 1 ≤ k ≤ n,∑ 1≤j1<j2<···<jk≤n ξ2j1ξ 2 j2 · · · ξ 2 jk if W = W (Dn) and 1 ≤ k < n, σ(Pn) = ξ1ξ2 · · · ξn if W = W (Dn), Conjecture is true. Remark 16. The condition [P, P1] = [P, P3] = 0 if W = W (An−1), [P, P2] = 0 if W = W (Bn) or W = W (Dn) together with (9.4) is sufficient for the proof of this theorem. 9.3 Enough singularities Put Ξ = {α ∈ Σ(Bn)+ ; uα(t) is not entire}. Theorem 17. i) ([23]) Suppose n = 2 and let S̄ be of type B2. If #Ξ ≥ 2, then Conjecture is true. ii) ([40]) If S̄ is of type An−1 or of type Bn and moreover the reflections wα for α ∈ Ξ generate W (An−1) or W (Bn), respectively, then Conjecture is true. This theorem follows from the following key Lemma. Lemma 6 ([23, 37, 40]). Suppose (9.1) and moreover that there exist α and β in S such that α 6= β, 〈α, β〉 6= 0 and uα(t) has a singularity at t = t0. Then uα(t − t0) is an even function with a pole of order two at the origin and uwα(β) ( t− 2t0 〈α, γ〉〈α, α〉 ) = uβ(t) if wα(β) ∈ Σ(Bn)+, u−wα(β) ( −t+ 2t0 〈α, γ〉〈α, α〉 ) = uβ(t) if − wα(β) ∈ Σ(Bn)+. (9.6) Corollary 1. Suppose the assumption in Lemma 6. i) If uα(t) has another singularity at t1 6= t0, then uγ ( t+ 2(t1 − t0) 〈α, γ〉〈α, α〉 ) = uγ(t) for γ ∈ S. (9.7) ii) Assume that uα has poles at 0, t0 and t1 such that t0 and t1 are linearly independent over R. Then uβ(t) is a doubly periodic function and therefore uβ(t) has poles and hence uα(t) is also a doubly periodic function. We may moreover assume that uβ has a pole at 0 by a parallel transformation of the variable x. Case I: Suppose α = ei − ej, β = ej − ek with 1 ≤ i < j < k ≤ n. uei−ej (t) = uej−ek (t) = uei−ek (t) = C℘(t; 2ω1, 2ω2) + C ′ with suitable C, C ′ ∈ C, which corresponds to (Ellip-A2). Case II: Suppose α = ei − ej and β = ej with 1 ≤ i < j ≤ n. Then ( uei−ej (t), uei+ej (x), uei(t), uej (t) ) is (Ellip-B2), (Ellip-B2-S) or (Ellipd-B2). Completely Integrable Systems Associated with Classical Root Systems 41 iii) If S̄ is of type An−1 or Bn or Dn and one of uα(t) is a doubly periodic function with poles, then P transforms into (Ellip-An−1) or (Ellip-Bn) or (Ellip-Dn) under a suitable parallel transformation on Cn. Proof of Corollary 1. i) is a direct consequence of Lemma 6. iii) follows from ii). We have only to show ii). Case I: It follows from (9.6) that uα(t) = uβ(t) = uei−ek (t) and they are even functions. Let Γ2ω1,2ω2 = {2m1ω1 + 2m2ω2 ; m1,m2 ∈ Z} be the set of poles of uα. Then (9.7) implies uβ(t+ 2ω1) = uβ(t+ 2ω2) = uβ(t). Since 2ω1 and 2ω2 are periods of ℘(t) and there exists only one double pole in the fundamental domain defined by these periods, we have the claim. Case II: It follows from (9.6) that uei−ej (t) = uei+ej (t) and uei(t) = uek (t) and they are even functions. Let Γ2ω1,2ω2 be the poles of uei−ej (t). Then (9.7) means uei(t+2ω1) = uei(t+2ω2) = uei(t). Considering the poles of uei−ek (t) with (9.7), we have four possibilities of poles of uei : (Case II-0): Γ2ω1,2ω2 , (Case II-1): Γ2ω1,2ω2 ⋃ (ω1 + Γ2ω1,2ω2), (Case II-2): Γ2ω1,2ω2 ⋃ (ω2 + Γ2ω1,2ω2), (Case II-3): Γ2ω1,2ω2 ⋃ (ω1 + Γ2ω1,2ω2) ⋃ (ω2 + Γ2ω1,2ω2). Here we note that (Case II-1) changes into (Case II-2) if we exchange ω1 and ω2. Then we have (Case II-0): uei−ej (t+ 4ω1) = uei−ej (t+ 4ω2) = u(t), (Case II-2): uei−ej (t+ 4ω1) = uei−ej (t+ 2ω2) = u(t), (Case II-3): uei−ej (t+ 2ω1) = uei−ej (t+ 2ω2) = u(t). Thus (Case II-0), (Case II-2) and (Case II-3) are reduced to (Ellipd-B2), (Ellip-B2-S) and (Ellip-B2), respectively. � Let H be a finite set of mutually non-parallel vectors in Rn and suppose P = −1 2 n∑ j=1 ∂2 j +R(x), R(x) = ∑ α∈H Cα 〈α, α〉〈α, x〉2 + R̃(x). Here Cα are nonzero complex numbers and R̃(x) is real analytic at the origin. We assume that H is irreducible, namely, Rn = ∑ α∈H Rα, ∅ 6= ∀H′ & H ⇒ ∃α ∈ H′ and ∃β ∈ H \ H′ with 〈α, β〉 6= 0. Definition 7. The potential function R(x) of a Schrödinger operator is reducible if R(x) and Rn is decomposed as R(x) = R1(x) +R2(x) and Rn = V1 ⊕ V2 such that 0 ( V1 ( Rn, V2 = V ⊥ 1 , ∂v2R1(x) = ∂v1R2(x) = 0 for ∀ v2 ∈ V2 and ∀ v1 ∈ V1. If R(x) is not reducible, R(x) is called to be irreducible. Theorem 18 ([37]). Suppose n ≥ 2 and there exists a differential operator Q with [P,Q] = 0 whose principal symbol does not depend on x and is not a polynomial of n∑ i=0 ξ2i . Put W = {wα;α ∈ H}. If 2Cα 6= k(k + 1) for k ∈ Z and α ∈ H, (9.8) then W is a finite reflection group and σ(Q) is W -invariant. 42 T. Oshima 9.4 Periodic potentials The following theorem is a generalization of the result in [30]. Theorem 19. Assume R(x) is of the form (9.4) with meromorphic functions uα(t) on C and R ( x+ 2π √ −1α 〈α, α〉 ) = R(x) for α ∈ Σ(Bn) (9.9) and moreover assume that the root system S̄ does not contain an irreducible component of type B2 or even if S̄ contains an irreducible component S̄2 = {±ei ± ej ,±ei,±ej} of type B2, the origin s = 0 is not an isolated essential singularity of uα(log s) for α ∈ S̄2 ⋂ Σ(Dn)+. Then Conjecture is true. Remark 17. i) The integrable systems classified in this note which satisfy the assumption of Theorem 19 under a suitable coordinate system are (Ellip-∗) and (Trig-∗) and (Toda-∗), which are the systems given in this note whose potential functions are not rational. ii) The assumption (9.9) implies that uα(log s) is a meromorphic function on C \ {0} for any α ∈ Σ(Bn)+. It means that the corresponding Schrödinger operator is naturally defined on the Cartan subgroup of Sp(n,C) with a meromorphic potential function. Lemma 7. Assume n = 2, (9.2), (9.9), S̄ is of type B2 and moreover uα(log s) are holomorphic for α ∈ Σ(B2)+ and 0 < \|s\| � 1. If the origin is at most a pole of uβ(log s) for β ∈ Σ(D2)+, the origin is also at most a pole of uα(log s) for α ∈ Σ(B2)+. Proof. Use the notation as in (4.2). Put u−(log s) = U− 0 + ∞∑ ν=r νU− ν s ν , u+(log s) = U+ 0 + ∞∑ ν=m νU+ ν s ν , v(log s) = V0 + ∞∑ ν=−∞ νVνs ν , w(log s) = W0 + ∞∑ ν=−∞ νWνs ν . with U− ν , U + ν , Vν , Wν ∈ C, rm 6= 0 and (U− r , U + m) 6= 0. Then as is shown in [30] the condition for the existence of T (x, y) in (4.3) is equivalent to pq(2p− q)(p− q)(V2p−qU + q−p + VqU − p−q +Wq−2pU + p −WqU − p ) = 0 for p, q ∈ Z. (9.10) Hence if p < r and p < m, p(p− k)(p+ k)k(Vp+kU + −k + Vp−kU − k ) = 0 for k ∈ Z. Case U− r 6= 0: Put k = r. Suppose q is negative with a sufficiently large absolute value. Then Vq = (−U+ −r/U − r )Vq+2r, which implies Vq = 0 since ∞∑ ν=−∞ νVνs ν converges for 0 < \|s\| � 1. Suppose q is negative with a sufficiently large absolute value compared to p. Then by the relation Wq−2pU + p −WqU − p = 0 we similarly conclude Wq = 0. Case U+ m 6= 0: Putting k = −m, we have the same conclusion as above in the same way. � Completely Integrable Systems Associated with Classical Root Systems 43 Proof of Theorem 19. Lemma 5 assures that we may assume S̄ is an irreducible root system. We may moreover assume that the rank of S̄ is greater than one. Suppose that there exists γ ∈ S such that the origin is neither a removable singularity nor an isolated singularity of uγ(log s). Then uγ(t) is a doubly periodic function with poles. Owing to Corollary 1, ∑ α∈S0 uα(〈α, x〉) is reduced to the potential function of (Ellip-An−1) or (Ellip-Bn) or (Ellip-Dn). Thus we may assume that the origin is a removable singularity or an isolated singularity of uα(log s) for any α ∈ S. Let α, β ∈ S ⋂ Σ(Dn) with α 6= β and 〈α, β〉 6= 0. Put γ = wαβ or γ = −wαβ so that γ ∈ Σ(Dn)+. Then [30] shows that u(t) = uα(t), v(t) = uβ(t) and w(t) = uγ(−t) satisfy (9.3). Then Remark 13 says that the origin is at most a pole of u(log s), v(log s) and w(log s). Let α ∈ S ⋂ Σ(Dn) and β ∈ S \ Σ(Dn) with 〈α, β〉 6= 0. Let W be the reflection group generated by wα and wβ and put So = W{α, β} ⋂ Σ(Bn). Then [30] also shows that R(x) = ∑ γ∈So uγ(〈γ, x〉) defines an integrable potential function of type B2. Hence Lemma 7 assures that the origin is at most a pole of uα(log s) for α ∈ So. Since S is irreducible, the origin is at most a pole of uα(log s) for α ∈ S. Then Theorem 19 follows from [30]. � 9.5 Uniqueness We give some remarks on the operator which commutes with the Schrödinger operator P . Remark 18 ([32, Lemma 3.1 ii)]). If differential operators Q and Q′ satisfy [Q,Q′] = 0, σ(Q′) = n∑ j=1 ξN j and ord(Q) ≤ N −2, then Q has a constant principal symbol, that is, σ(Q) does not depend on x. Hence if there exist differential operators Q1, . . . , Qn with constant principal symbols such that σ(Q1), . . . , σ(Qn) are algebraically independent and moreover they satisfy [Qi, Qj ] = 0 for 1 ≤ i < j ≤ n, then any operator Q satisfying [Q,Qj ] = 0 for j = 1, . . . , n has a constant principal symbol. In particular, if a differential operator Q satisfies [Q,Pk] = 0 for Pk in (1.2) and (1.4) with k = 1, . . . , n, then σ(Q) does not depend on x. Remark 19. Assume that a differential operator Q commutes with a Schrödinger operator P and moreover assume that there exist linearly independent vectors cj ∈ Cn for j = 1, . . . , n such that the operator is invariant under the parallel transformations x 7→ x + cj for j = 1, . . . , n. Then σ(Q) does not depend on x (cf. [32, Lemma 3.1 i)]). Furthermore assume that P is of type (Ellip-F ) or (Trig-F ) or (Rat-F ) with F = An−1 or Bn or Dn. If the condition (9.8) holds or Q is W (F )-invariant, it follows from Theorem 18 or [29, Proposition 3.6] that Q is in the ring C[P1, . . . , Pn] generated by the W (F )-invariant commuting differential operators. If the condition (8.11) is not valid, σ(Q) is not necessarily W (F )-invariant (cf. [7, 35, 39]). Remark 20 ([32, Theorem 3.2]). Let P be the Schrödinger operator in Theorem 16. Un- der the notation in Theorem 16 suppose Pk are W -invariant for 1 ≤ k ≤ n. Then the ring C[P1, . . . , Pn] is uniquely determined by P and Q, where Q = P3 if W = W (An−1) and Q = P2 if W = W (Bn) or W (Dn). 44 T. Oshima Remark 21. If Pc = −(1/2) n∑ j=1 ∂2 j + cR(x) is a Schrödinger operator with a coupling constant c ∈ C such that Pc admits a non-trivial commuting differential operator Qc of order four for any c ∈ C, then the operator Pc may be a system stated in Conjecture under a suitable coordinate system. The following example satisfies neither this condition nor the condition (9.8). It does not admit commuting differential operators (1.2) satisfying (1.4) if m 6= 0, −1. Example 2. It is shown in [6, 35] that the Schrödinger operator P = −1 2 n∑ j=1 ∂2 ∂x2 j + ∑ 1≤i<j<n m(m+ 1) (xi − xj)2 + n−1∑ i=1 m+ 1 (xi − √ mxn)2 is completely integrable for any m and algebraically integrable if m is an integer. The following example shows that the Schrödinger operator P does not necessarily determine the commuting system C[P1, . . . , Pn]. Example 3. Let α, β, γ and λ be complex numbers. Put (A0, A1, C0, C1) = (α, γ/2−λ/2, β, λ) for (Rat-B2-S) in Theorem 3 (cf. [32, Remark 3.7]). Then the Schrödinger operator Pα,β,γ = −1 2 ( ∂2 ∂x2 + ∂2 ∂y2 ) + (x2 + y2) ( 2α (x2 − y2)2 + β x2y2 + γ ) commutes with Qα,β,γ,λ = ( ∂2 ∂x∂y + 4αxy (x2 − y2)2 − 2(γ − λ)xy )2 − 2 ( β y2 + λy2 ) ∂2 ∂x2 − 2 ( β x2 + λx2 ) ∂2 ∂y2 + 4 ( β x2 + λx2 )( β y2 + λy2 ) + 16αλx2y2 + 16αβ (x2 − y2)2 + 8λ(γ − λ)x2y2 for any λ ∈ C. Note that [Qα,β,γ,λ, Qα,β,γ,λ′ ] 6= 0 if λ 6= λ′ and these operators are W (B2)- invariant. The half of the coefficient of the term λ of Qα,β,γ,λ considered as a polynomial function of λ is Sα,β,γ = − ( y ∂ ∂x − x ∂ ∂y )2 + 2α ( xy (x− y)2 − xy (x+ y)2 ) + 2β ( y2 x2 + x2 y2 ) + 4γx2y2. In particular, P = −(1/2)(∂2 x + ∂2 y) + γ(x2 + y2) commutes with ∂x∂y − 2γxy and x∂y − y∂x. Note that if R(x) is a polynomial function on Cn, the condition [ −(1/2) n∑ j=1 ∂2 j +R(x), Q ] = 0 for a differential operator Q implies that the coefficients of Q are polynomial functions (cf. [32, Lemma 3.4]). 9.6 Regular singularities Definition 8 ([16]). Put ϑk = tk∂/∂tk and Yk = {t = (t1, . . . , tn) ∈ Cn ; tk = 0}. Then a differential operator Q of the variable t is said to have regular singularities along the set of walls {Y1, . . . , Yn} if Q = q(ϑ1, . . . , ϑn) + n∑ k=1 tkQk(t, ϑ). Completely Integrable Systems Associated with Classical Root Systems 45 Here q is a polynomial of n variables and Qk are differential operators with the form Qk(t, ϑ) = ∑ aα(t)ϑα1 1 · · ·ϑ αn n and aα(t) are analytic at t = 0. In this case we define σ∗(Q) = q(ξ1, . . . , ξn) and σ∗(Q) is called the indicial polynomial of Q. Theorem 20. Let R(t) be a holomorphic function defined on a neighborhood of the origin of Cn. Let Q1 and Q2 be differential operators of t which have regular singularities along the set of walls {Y1, . . . , Yn}. Suppose σ∗(Q1) = σ∗(Q2) and [Q1, P ] = [Q2, P ] = 0 with the Schrödinger operator P = −1 2 ( ϑ2 n + n−1∑ j=1 ( ϑj+1 − ϑj )2)+R(t). Then Q1 = Q2. Proof. Put tj = e−(xj−xj+1) for j = 1, . . . , n − 1 and tn = e−xn . Then ∂j = ϑj+1 − ϑj for j = 1, . . . , n− 1 and ∂n = −ϑn. Under the coordinate system x = (x1, . . . , xn) Remark 19 says that Q1−Q2 has a constant principal symbol, which implies Q1 = Q2 because σ∗(Q1−Q2) = 0. � A more general result than this theorem is given in [31]. The following corollary is a direct consequence of this theorem. Corollary 2. Put tj = e−λ(xj−xj+1) for j = 1, . . . , n − 1 and tn = e−λxn. Suppose P is the Schrödinger operator of type (Trig-An−1), (Trig-An−1-bry-reg), (Trig-BCn-reg), (Trig-Dn), (Toda-An−1), (Toda-BCn) or (Toda-Dn). i) P and Pk for k = 1, . . . , n have regular singularities along the set of walls {Y1, . . . , Yn}. ii) Let Q be a differential operator which has regular singularities along the set of walls {Y1, . . . , Yn} and satisfies [Q,P ] = 0. If σ∗(Q) = σ∗(Q̃) for an operator Q̃ ∈ C[P1, . . . , Pn], then Q = Q̃. Remark 22. i) This corollary assures that certain radial parts of invariant differential operators on a symmetric space correspond to our completely integrable systems with regular singularities and the map σ∗ corresponds to the Harish-Chandra isomorphism (cf. [31]). ii) The system (Trig-BCn-reg) is Heckman–Opdam’s hypergeometric system [11] of type BCn. Since (Trig-BCn-reg) is a generalization of Gauss hypergeometric system related to the root system Σ(Bn), the systems in the following diagram are considered to be generalizations of Gauss hypergeometric system and its limits (cf. Section 8). They form a class whose eigenfunctions should be easier to be analyzed than those of other systems in this note. Hierarchy starting from (Trig-BCn-reg) Toda-An−1 ↗ Rat-Dn Trig-An−1 → Rat-An−1 ⇑3:1 ⇑3:1 ⇑3:1 Rat-Bn-2 Trig-An−1-bry-reg → Rat-An−1-bry2 ↑ ↗ Trig-BCn-reg → Toda-Dn-bry → Toda-BCn ⇓3:1 ⇓3:1 ⇓3:1 Trig-Dn → Toda-Dn → Toda-An−1 ↓ ↘ Trig-An−1 Rat-Dn 46 T. Oshima 9.7 Other forms If a Schrödinger operator P is in the commutative algebra D = C[P1, . . . , Pn], then the differential operator P̃ := ψ(x)−1P ◦ ψ(x) with a function ψ(x) is in the commutative algebra D̃ = C[ψ(x)−1P1 ◦ ψ(x), . . . , ψ(x)−1Pn ◦ ψ(x)] of differential operators. Then P̃ = −1 2 n∑ j=1 ∂2 ∂x2 j + n∑ j=1 aj(x) ∂ ∂xj + R̃(x), (9.11) ∂ψ(x) ∂xj = aj(x) for j = 1, . . . , n. (9.12) Conversely, if a function ψ(x) satisfies (9.12) for a differential operator P̃ of the form (9.11), then P = ψ(x)P̃ ◦ ψ(x)−1 is of the form (1.1), which we have studied in this note. If ψ(x) is a function satisfying 1 2ψ(x) n∑ j=1 ∂2ψ ∂x2 j (x) = R(x), then P̃ = ψ(x)−1 ( −1 2 n∑ j=1 ∂2 j +R(x) ) ◦ ψ(x) = −1 2 n∑ j=1 ∂2 j − ψ(x)−1 n∑ j=1 ∂ψ ∂xj (x)∂j . Note that e−φ(x)∂e φ(x) ∂xj = ∂φ(x) ∂xj , e−φ(x) n∑ j=1 ∂2eφ(x) ∂x2 j = n∑ j=1 ∂2φ(x) ∂x2 j + n∑ j=1 ( ∂φ(x) ∂xj )2 . Putting φ(x) = m ∑ 1≤i<j≤n log sinhλ(xi − xj), we have ∂φ(x) ∂xk = λm ∑ 1≤i≤n, i6=k cothλ(xk − xi), n∑ j=1 ∂2φ(x) ∂x2 j + n∑ k=1 (∂φ(x) ∂xk )2 = −2λ2m ∑ 1≤i<j≤n sinh−2 λ(xi − xj) + 2λ2m2 ∑ 1≤i<j≤n coth2 λ(xi − xj) + λ2m2n(n− 1)(n− 2) 3 = 2λ2m(m− 1) ∑ 1≤i<j≤n sinh−2 λ(xi − xj) + λ2m2n(n2 − 1) 3 since cothα · cothβ + cothβ · coth γ + coth γ · cothα = −1 if α+ β + γ = 0. Hence P̃ = −1 2 n∑ j=1 ∂2 j −m ∑ 1≤i<j≤n λ cothλ(xi − xj)(∂i − ∂j), Completely Integrable Systems Associated with Classical Root Systems 47 ψ(x) = ∏ 1≤i<j≤n λm sinhm λ(xi − xj), ψ(x) ◦ P̃ ◦ ψ−1(x) = −1 2 n∑ j=1 ∂2 j + ∑ 1≤i<j≤n m(m− 1)λ2 sinh2 λ(xi − xj) + m2n(n2 − 1)λ2 6 (9.13) and P̃ is transformed into the Schrödinger operator of type (Trig-An−1). Now we put φ(x) = m0 ∑ 1≤i<j≤n (log sinhλ(xi − xj) + log sinhλ(xi + xj)) +m1 ∑ 1≤k≤n log sinhλxk +m2 ∑ 1≤k≤n log sinh 2λxk and we have ∂φ(x) ∂xk = λm0 ∑ 1≤i≤n, i6=k (cothλ(xk + xi) + cothλ(xk − xi)) + λm1 cothλxk + 2λm2 coth 2λxk, cothλxk coth 2λxk = 1 + 1 2 sinh−2 λxk,∑ {i,j,k}=I (2 cothλ(xk + xi) cothλ(xk − xi) + 2 cothλ(xk + xi) cothλ(xk − xj) + cothλ(xk + xi) cothλ(xk + xj) + cothλ(xk − xi) cothλ(xk − xj)) = ∑ {i,j,k}=I (cothλ(xk + xi) cothλ(xk + xj) + cothλ(xi − xj) cothλ(xi + xk) + cothλ(xj − xi) cothλ(xj + xk)) + ∑ {i,j,k}=I (cothλ(xk − xi) cothλ(xk − xj)) = 8 for I ⊂ {1, . . . , n} with #I = 3, cothλ(xk + xi) + cothλ(xk − xi) = sinh 2λxk sinhλ(xk + xi) sinhλ(xk − xi) , cosh 2λxk − cosh 2λxi sinhλ(xk + xi) sinhλ(xk − xi) = 2 cosh2 λxk − 2 cosh2 λxi sinhλ(xk + xi) sinhλ(xk − xi) = 2, n∑ k=1 ( ∂φ(x) ∂xk )2 = 2λ2m2 0 ∑ 1≤i<j≤n (coth2 λ(xi − xj) + coth2 λ(xi + xj)) + λ2m2 1 n∑ k=1 coth2 λxk + 4λ2m2 2 n∑ k=1 coth2 2λxk + 2λ2m1m2 n∑ k=1 sinh−2 λxk + 4λ2m2 0n(n− 1)(n− 2) 3 + 2λ2m0(m1 + 2m2)n(n− 1) + 4λ2m1m2n, n∑ j=1 ∂2φ(x) ∂x2 j + n∑ k=1 ( ∂φ(x) ∂xk )2 = 2λ2m0(m0 − 1) ∑ 1≤i<j≤n (sinh−2 λ(xi−xj) + sinh−2 λ(xi+xj)) + λ2m1(m1 + 2m2 − 1) n∑ k=1 sinh−2 λxk + 4λ2m2(m2 − 1) n∑ k=1 sinh−2 2λxk + λ2 (( 2 3 m0(2n− 1) + 2m1 + 4m2 ) m0(n− 1) + (m1 + 2m2)2 ) n. 48 T. Oshima Hence P̃ = −1 2 n∑ j=1 ∂2 j − n∑ k=1 λ ( ∑ 1≤i<j≤n m0(cothλ(xi − xk) + cothλ(xi + xk)) +m1 cothλxk + 2m2 coth 2λxk ) ∂k, ψ(x) = ∏ 1≤i<j≤n (sinhm0 λ(xi − xj) sinhm0 λ(xi + xj)) n∏ k=1 sinhm1 λxk n∏ k=1 sinhm2 2λxk (9.14) and P̃ is transformed into the Schrödinger operator of type (Trig-BCn-reg): ψ(x) ◦ P̃ ◦ ψ−1(x) = −1 2 n∑ j=1 ∂2 j +m0(m0 − 1) ∑ 1≤i<j≤n ( λ2 sinh2 λ(xi−xj) + λ2 sinh2 λ(xi+xj) ) + n∑ k=1 m1(m1 + 2m2 − 1)λ2 2 sinh2 λxk + n∑ k=1 2m2(m2 − 1)λ2 sinh2 2λxk + λ2 ( m2 0 3 (2n− 1)(n− 1) +m0(m1 + 2m2)(n− 1) + (m1 + 2m2)2 2 ) n. Remark 23. As is shown in [15, Theorem 5.24 in Ch. II], the operator (9.13) or (9.14) gives the radial part of the differential equation satisfied by the zonal spherical function of a Riemannian symmetric space G/K of the non-compact type which corresponds to the Laplace–Beltrami operator on G/K. 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[40] Wakida S., Quantum integrable systems associated with classical Weyl groups, Master Thesis, University of Tokyo, 2004. [41] Whittaker E.T., Watson G.N., A course of modern analysis, 4th ed., Cambridge University Press, 1927. http://arxiv.org/abs/math.AP/0611899 http://arxiv.org/abs/math-ph/0303025 http://arxiv.org/abs/math-ph/0309011 1 Introduction 2 Notation and preliminary results 3 Type An-1 (n3) 4 Type B2 4.1 Normal case 4.2 Special case 4.3 Duality 5 Type Bn (n3) 5.1 Integrable potentials 5.2 Analytic continuation of integrals 6 Type Dn (n3) 7 Classical limits 8 Analogue for one variable 9 A classification 9.1 Pairwise interactions and meromorphy 9.2 Invariant case 9.3 Enough singularities 9.4 Periodic potentials 9.5 Uniqueness 9.6 Regular singularities 9.7 Other forms References

Completely Integrable Systems Associated with Classical Root Systems

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