Quasi-Periodic Solutions of Hamiltonian Perturbations of 2D Linear Schrodinger Equations

The general problem discussed here is the persistency of quasi-periodic solutions of linear or integrable equations after Hamiltonian perturbation. This subject is closely related to the well-known KAM-theory of invariant tori in smooth dynamical systems. The main source of new difficulties is the fact that one considers here (finite dimensional) tori in an infinite dimensional phase space. For instance, already in the study of time periodic solutions, the fact that the phase space is infinite dimensional leads to small divisor problems. Now the standard KAM-technology may be reworked in a form applicable to treat persistency problems in space dimension (1D) in the case of Dirichlet boundary conditions say, where in particular the normal frequencies of the problem are well-separated. Results along this line were obtained by S. Kuksin [Ki] and C. E. Wayne [W] among other works. We mentioned [Ki] as the main expository reference for the KAM approach. Considering PDE's in iD with periodic boundary conditions and especially in space dimension > 1, a significant new problem arises due to the presence of clusters of normal frequencies.