The Meromorphic Solutions of the Bruschi–Calogero Equation

Abstract

We give all the meromorphic functions defined near the origin 0 e C satisfying a functional equation investigated by Bruschi and Calogero [1], [2]. § 0. Introduction It is an important problem to find a Lax pair L and M whose equations of motion are equivalent to the Lax equation [10], [11], [12]. In order to prove their complete integrability it is convenient to use a Lax representation. The systems of Calogero-Sutherland type, which describe one-dimensional n-particle dynamics, are defined by the following Hamiltonian where the potential U has the form Communicated by T. Kawai, July 8, 1999. Revised November 1, 1999. 1991 Mathematics Subject Classification: Primary, 58F07; Secondary, 33E05, 35Q58, 39B02, 30D05. * Department of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan. e-mail: kawazumi@ms.u-tokyo.ac.jp ** Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 0600810, Japan. e-mail: shibu@math.sci.hokudai.ac.jp 86 NARIYA KAWAZUMI AND YOUICHI SHIBUKAWA n 2 U(qlt . . . , qn) = g j QJ = 0;(0, ;' = 1, 2, . . . , n. k=i k*j Bruschi and Calogero [1] discovered a representation of the equations of motion of the system (0.1) in the Lax form L= [L,M], where L and M are the n X n matrices, Ljk = dM+d-d n Mjk = 6jk £ m = m^j Here the function a (x) is a solution of the following functional equation of addition type (0.2) oGOa'GO-a'GOaCy) = (a(x+y)-a(x)a(y)) 0?GO-7?(30), which we call the Bruschi-Calogero equation. The function v(x) is given by SOLUTIONS OF BRUSCHI-CALOGERO EQUATION 87