Length of the Instability Intervals for Hill’s Equation MALIK MAMODE Laboratoire de Ge ´nie Industriel, Universite´ de La Re ´union, 15 Avenue Rene´ Cassin, BP 7151, 97715 St Denis Cedex 09, La Re´union, France. e-mail: malik.mamode@univ-reunion.fr (Received: 19 September 2003) Abstract. The length of instability intervals is investigated for the Hill equation y 00 þ xðx  2pðxÞÞy ¼ 0, where pðxÞ has an infinite Fourier series of coefficients c n . For any small , it is shown that these lengths are completely characterized by the c n ’s. Mathematics Subject Classifications (2000). 34-XX, 34Dxx, 34Exx, 34D20, 34E05. Key words. Hill equation, discriminant, instability interval. 1. Introduction In this Letter, we consider the following Hill equation: y 00 ðxÞ¼xðx  2pðxÞÞyðxÞ; ð1Þ where x,  are real parameters and pðxÞ a real-valued locally integrable and periodic function. Without loss of generality, pðxÞ may be assumed to be of period 1 and of average value 1. It is known that for any  and function pðxÞ, the x-axis consists of intervals of stability in which all solutions of (1) are bounded and intervals of insta- bility in which (1) has an unbounded solution. Our aim in this letter is to determine the length Dx n of the instability intervals when  is small and pðxÞ has an infinite Fourier series pðxÞ¼ X þ1 n¼1 c n e 2ipnx ; c 0 ¼ 1: Such a problem has been extensively examined in the literature by, among many others, Erde´lyi [1] and the result obtained in [2] by Levy and Keller when pðxÞ has a finite Fourier series. It is known that the behavior of the sequence Dx n for a fixed , is sensitive to smoothness properties of pðxÞ but to the author’s knowledge, a simple connection between Dx n and the Fourier coefficients c n has never been established. In this Letter, we show that the length of the nth instability interval for (1) is given by Dx n ¼ 2jjjc n jþ Oð 2 Þ; if c n 6¼ 0, ja n j 2 þ Oð 3 Þ; if c n ¼ 0,  ð2Þ Letters in Mathematical Physics 67: 95–102, 2004. Ó 2004 Kluwer Academic Publishers. Printed in the Netherlands. 95