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

We study the most general second-order self-adjoint linear operator L 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 function w strictly related to the spectral properties of L, which are then exhaustively investigated. We then define a Hamiltonian structure on the space of the coefficients of L and prove that w provides an infinite number of conserved quantities. The connection with classes of integrable nonlinear evolution equations is finally discussed.