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 evolution equations and finite dimensional Hamiltonian systems on the moduli spaces of Riemann surfaces. We also investigate the phase-space geometry of these Hamiltonian systems and introduce deformations of the level sets associated to conserved quantities, which results in a new class of solutions with monodromy for N-component systems of PDEs. After constructing a variety of mechanical systems related to the spatial flows of nonlinear evolution equations, we investigate their semiclassical limits. In particular, we obtain semiclassical asymptotics for the Bloch eigenfunctions of the energy dependent Schrodinger operators, which is of importance in investigating zero-dispersion limits of N-component systems of PDEs.