ENERGY-DEPENDENT SCHRODINGER-OPERATORS AND COMPLEX HAMILTONIAN-SYSTEMS ON RIEMANN SURFACES

We use so-called energy-dependent Schrodinger operators to establish a link between special classes of solutions of N-component systems of e volution equations and finite dimensional Hamiltonian systems on the m oduli spaces of Riemann surfaces. We also investigate the phase-space geometry of these Hamiltonian systems and introduce deformations of th e level sets associated to conserved quantities, which results in a ne w class of solutions with monodromy for N-component systems of PDEs. A fter constructing a variety of mechanical systems related to the spati al flows of nonlinear evolution equations, we investigate their semicl assical limits. In particular, we obtain semiclassical asymptotics for the Bloch eigenfunctions of the energy dependent Schrodinger operator s, which is of importance in investigating zero-dispersion limits of N -component systems of PDEs.