Floquet theory for linear differential equations with meromorphic solutions

Abstract

If A is a !-periodic matrix Floquet's theorem states that the dif- ferential equation y 0 = Ay has a fundamental matrix P (x) exp(Jx) where J is constant and P is !-periodic, i.e., P (x) = P (e 2 ix=! ). We prove here that P is rational if A is bounded at the ends of the period strip and if all solutions of y 0 = Ay are meromorphic. This version of Floquet's theorem is important in the study of certain in- tegrable systems. In the early 1880s Floquet established his celebrated theorem on the structure of solutions of periodic dieren tial equations (see (4) and (5)). It is interesting to note that modern day versions of the theorem consider the case where the independent variable is real and the coecien ts are, say, piecewise continuous (cf. Magnus and Winkler (13) or Eastham (3)) while in Floquet's original work, due to the inuence of Fuchs (6), the independent variable is complex, the coecien ts are analytic save for isolated points 1 , and the general solution is explicitly required to be single-valued (it will then also be analytic except at isolated singularities.) It is also interesting to realize that Floquet's theorem comes shortly after Hermite had established an analogue theorem for Lam e's equation (see (12)): for every value of z the solutions of y 00 n(n + 1)}(x)y = zy are doubly periodic of the second kind 2 . The proof of this theorem relied on the fact that, because of the particular coecien t n(n + 1) in Lam e's equation, the general solution is single- valued. Shortly after this Picard extended this nding to other equations with doubly periodic coecien ts and single-valued general solutions ((16), (17), (18)). It appears, however, that Floquet's work is independent of Hermite and Picard. Another relative of Floquet's original theorem was discovered at about the same time by Halphen (11): If the coecien ts of a linear homogeneous dieren tial expres- sion are rational functions which are bounded at innit y, if the leading coecien t is one, and if the general solution is meromorphic, then there is a fundamental sys- tem of solutions whose elements are of the form R(x) exp(x ) where R is a rational function and a certain complex number.