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: Bibliography : Commuting differential operators of : Commuting differential operators of


1.1. Several integral systems are accidentally related to root systems. Olshanetsky-Perelomov ([OP1], [OP2]) considered integrable $ n$-particle models in dimension one arising from root systems. The systems of differential operators satisfied by zonal spherical functions give such integrable systems and these were generalized by Sekiguchi and Heckman-Opdam ([Sj], [HO]).

In [OOS] we announce a classification of integrable systems invariant under simple classical Weyl groups. The precise discussion has already been given by [OS] and [O] except for the case of type $ B_2$. As is shown in [OS], the classification problem for type $ B_2$ is reduced to a functional differential equation (1.4).

In §2 we give a complete list of solutions of this functional equation. Some solutions have already been obtained, after [OP2], by Inozemtsev [IM], [I] (See also [P]). The main result of §2 is Theorem 2.9, which is stated in §1.3 in a different form.

In §3 we examine the reducibility of the system obtained in §2. We note that if the system coincides with the system satisfied by zonal spherical functions of a semisimple Lie group, the reducibility is related to degenerate series representations.

The final draft of this paper was completed when the authors were visiting University of Leiden in the fall of 1994. The authors express their sincere gratitude to Profdrvan Dijk for his hospitality during their stay there.

1.2. Now we give a quick review of the results in [OS, §6] concerning with type $ B_2$. Let $ W(B_2)$ be the Weyl group of type $ B_2$, which is identified with the group of coordinate transformations of $ (x_1,x_2)$ generated by $ (x_1,x_2)\mapsto(x_2,x_1)$ and $ (x_1,x_2)\mapsto(x_1,-x_2)$. Consider $ W(B_2)$-invariant differential operators

(1.1) \begin{displaymath}\begin{cases}P_1 = \partial_1^2 + \partial_2^2 + R(x),\\ P_2 = \partial_1^2\partial_2^2 +\text{lower order terms} \end{cases}\end{displaymath}

which satisfies $ [P_1,P_2] = 0$ and $ {}^t P_2 = P_2$. Here we denote $ \partial_1=\frac{\partial}{\partial x_1}$ and $ \partial_2=\frac{\partial}{\partial x_2}$ for simplicity and the map $ {}^t{}$ is the anti-automorphism of the algebra of differential operators such that $ {}^t a(x) = a(x)$ for functions $ a(x)$ and $ {}^t \partial_i = -\partial_i$ for $ i=1$ and 2. We assume that the coefficients of differential operators are extended to holomorphic functions on a Zariski open subset of an open connected neighborhood of the origin of the complexification $ \mathbb{C}^2$ of $ \mathbb{R}^2$.

The operators are proved to be expressed by even functions $ u$ and $ v$ of one variable as follows ([OS, Proposition 6.3]):

(1.2) \begin{displaymath}\begin{cases}P_1 &= \partial_1^2 + \partial_2^2 + u(x_1+x_2) ...
...)\partial_2^2\\ &\quad + v(x_1)v(x_2) + T(x_1,x_2), \end{cases}\end{displaymath}

where $ T$ is determined by the following equations up to a constant.

(1.3) \begin{displaymath}\begin{cases}\partial_2T = \frac12v'(x_1)\big(u(x_1+x_2)-u(x_...
..._2)\big) + v(x_2)\big(u'(x_1+x_2)+u'(x_1-x_2)\big). \end{cases}\end{displaymath}

As the compatibility condition for the existence of the solution $ T$ of the equation (1.3), we have an equation


which have been posed in [OS, Proposition 6.3] (cf. [P, §2.2.C]).

Conversely for any solution $ (u,v)$ of (1.4) and the pair $ (P_1, P_2)$ of the operators which are given by (1.2) with

(1.4) $\displaystyle T = \frac12\Big(\partial_1^2-\partial_2^2\Big)\Big( V(x_1)\big(U(x_1+x_2)+U(x_1-x_2)\big)-G(x_1) \Big)$

under the notation in Remark 2.1 and Lemma 2.2, we have $ [P_1,P_2] = 0$.

1.3. We give a complete list of solutions of the functional equation (1.4). Remind that the Schrödinger operator $ P_1$ is explicitly expressed as in (1.2) using $ u$ and $ v$.

$ 1)$ (Trivial case) $ u=$ constant, $ v=$ an arbitrary even function,

$ 1^d)$ $ u=$ an arbitrary even function, $ v=$ constant.

Let $ \omega_1$ and $ \omega_2$ denote the primitive half periods of the Weierstrass elliptic function $ \wp(t)$ and put $ \omega_3 = -\omega_1 - \omega_2$ and $ \omega_4 = 0$.

$ 2)$ (Elliptic case) For $ \omega_1$, $ \omega_2 < \infty$

    $\displaystyle \allowdisplaybreaks$ $\displaystyle \left\{\begin{array}{ll} u(t) &= C_6\wp(t) + C_7,\\ v(t) &= \sum_{i=1}^4 C_i\wp(t+\omega_i) + C_5, \end{array}\right.$

$ 2^d)$

      $\displaystyle \left\{\begin{array}{ll} u(t) &= \sum_{i=1}^4 C_i\wp(t+\omega_i) + C_5,\\ v(t) &= C_6 \wp(2t) + C_7. \end{array}\right.$

$ 2)'$ (Trigonometric case)

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_6 \sinh^{-2}\lambda t + C_7,\\...
...da t + C_3 \sinh^2 \lambda t + C_4 \sinh^2 2\lambda t + C_5, \end{array}\right.$

$ 2^d)'$

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_1 \sinh^{-2}\lambda t + C_2 \s...
...2 2\lambda t + C_5\\ v(t) &= C_6 \sinh^{-2}2\lambda t + C_7. \end{array}\right.$

$ 2)''$ (Rational case)

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_6t^{-2} + C_7,\\ v(t) &= C_1t^{-2} + C_2 + C_3t^2 + C_4t^4 + C_5t^6, \end{array}\right.$

$ 2^d)''$

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_1t^{-2} + C_2 + C_3t^2 + C_4t^4 + C_5t^6,\\ v(t) &= C_6t^{-2} + C_7. \end{array}\right.$

$ 3)$ (Elliptic case) For $ \omega_1$, $ \omega_2 < \infty$

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_1\big(\wp(\frac t2 + \omega_1)...
...(t) + C_3,\\ v(t) &= C_4\wp(t) + C_5\wp(t + \omega_3) + C_6. \end{array}\right.$

$ 3)'$ (Trigonometric case)

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_1\sinh^{-2}\frac\lambda2 t + C...
... v(t) &= C_4\sinh^{-2}\lambda t + C_5\sinh^2\lambda t + C_6, \end{array}\right.$

$ 3^d)'$

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_4\sinh^{-2}\lambda t + C_5\sin...
...) &= C_1\sinh^{-2}\lambda t + C_2\sinh^{-2}2\lambda t + C_3. \end{array}\right.$

$ 3)''$ (Rational case)

      $\displaystyle \left\{\begin{array}{ll} u(t) &= C_1t^{-2} + C_2 + C_3t^2,\\ v(t) &= C_4t^{-2} + C_5 + C_6t^2. \end{array}\right.$

1.4. Although we deal with the commuting differential operators of type $ B_2$ with the Weyl group symmetry in the main body of this paper, we will give a brief summary of the related works.

The commuting differential operators of type $ A$ have been studied very well. The commuting differential operators of type $ A$ with the Weyl group invariant condition are classified in [OS]. This work is generalized to the commuting differential operators of type $ A_2$ without Weyl group invariant condition

\Delta_1 = \partial_1 + \partial_2 + \partial...
...l_1\partial_2\partial_3 +\text{lower order terms}.

To classify the potential function $ R(x)$, we may assume that $ {}^t \Delta_3= - \Delta_3$. Then there exist one-variable functions $ u_1=u_1(x_2-x_3)$, $ u_2=u_2(x_3-x_1)$ and $ u_3=u_3(x_1-x_2)$ such that $ R(x) = -u_1-u_2-u_3$, and

(1.5) $\displaystyle \left\vert \begin{array}{lll} 1 & 1 & 1\\ u_1(x) & u_2(y) & u_3(z) \\ u'_1(x) & u'_2(y) & u'_3(z) \\ \end{array} \right\vert = 0$   for $\displaystyle x+y+z=0.$

For the Weyl group invariant case, we have $ u_1(z)=u_2(z)=u_3(z)$ and the proof of this fact is given in Proposition 4.2 (with $ m=3$) of [OS], which is valid for the general case with no change. For the Weyl group invariant case, the functional differential equation (1.6) is solved in [WW] and the solution is a Weierstrass elliptic function $ \wp$. The corresponding potential $ R(x)$ is of Calogero-Moser type. For the general case, the equation (1.6) is solved in [BP] and [BB]. Besides the $ \wp$ solutions, we also have solutions expressed by exponential functions. The corresponding potential is known as of type periodic/non-periodic Toda, which can be regarded as a degenerating limit of a Weyl group invariant potential [vD].

For type $ B_2$, the classification of the commuting differential operators (1.1) without the Weyl group symmetry has not been done yet. It is known that the similar functional differential equation (see (2.3)) is related to such operators. The following results are obtained in [Oc]:

(i) We have the expression of the (non Weyl group invariant) operators $ P_1$ and $ P_2$ by using four functions $ u_1=u_1(x_1+x_2)$, $ u_2=u_2(x_1-x_2)$, $ v_1=v_1(x_1)$ and $ v_2=v_2(x_2)$ with one-variable. Actually, if we replace $ u(x_1+x_2)$ by $ u_1(x_1+x_2)$, $ u(x_1-x_2)$ by $ u_2(x_1-x_2)$, and so on, the formula (1.2) is also valid for non-invariant operators. These functions satisfy the functional differential equation like (1.4).

(ii) Suppose $ P_1$ be non-trivial (cf. Lemma 2.4 i)). If $ P_1$ is holomorphic at some point, then $ P_1$ and $ P_2$ can be meromorphically continued to whole plane $ {\mathbb{C}}^2$. The orders of poles of $ P_1$ are at most two.

(iii) Suppose, moreover, that $ v_2(z)$ has poles at three points $ z=z_1,z_2,z_3$ such that $ z_1-z_2$ and $ z_2-z_3$ are linearly independent over $ \mathbb{Q}$. Then the function $ v_2$ can be expressed as

$\displaystyle v_2(z) = \sum_{i=1}^4 C_i \wp(z+\omega_i) + C_5,

with an elliptic function $ \wp$ and constants $ C_1,\dots,C_5$.

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: Bibliography : Commuting differential operators of : Commuting differential operators of
Nobuki Takayama Heisei 15-9-23.