Spectral theory of second-order almost periodic differential operators and its relation to classes of nonlinear evolution equations

Abstract

We study the most general second-order self-adjoint linear operatorL with Bohr almost periodic coefficients and the corresponding system of differential equations. We consider the Lyapounov and rotation numbers and show that they determine a holomorphic functionw strictly related to the spectral properties ofL, which are then exhaustively investigated. We then define a Hamiltonian structure on the space of the coefficients ofL and prove thatw provides an infinite number of conserved quantities. The connection with classes of integrable nonlinear evolution equations is finally discussed.