On the Spectrum of Odd Order Self-Adjoint Ordinary Differential Operators on the Real Line with Quasi-periodic Coefficients

Abstract

In this paper we prove that an odd order self-adjoint differential operator on the real line with quasi-periodic coefficients has only absolute continuous simple spectrum for large enough energies. Ordinary differential operators with quasi-periodic coefficients have been studied a lot during the last 20 years. These operators arise naturally from physical considerations. People have mainly been interested in the one dimensional Schro dinger operator with a quasi-periodic potential (see [Be-Te], [Din-Sin], [Eli], [Ji-Si], [Mo], [Mo-Po ], [Su], and [So-Sp]); such hamiltonians come up naturally in solid state physics (see [Pa-Fi]). In this paper, we study ordinary differential operators of higher order with quasi-periodic coefficients. Such operators also come up in physics (for example, in the study of Boussinesq's equation). article no. DE963225