Ergodic properties and Weyl M-functions for random linear Hamiltonian systems

This paper provides a topological and ergodic analysis of random linear Ham iltonian systems. We consider a class of Hamiltonian equations presenting a bsolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L-1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at +/-infinity define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrodinger equation a t the first point of the spectrum.