MICROLOCAL SINGULARITIES AND SCATTERING THEORY FOR SCHRODINGER EQUATIONS ON MANIFOLDS

Here we review several recent results on the propagation of microlocal singularities for (1) the solutions to Schrodinger equations; and (2) scattering matrices for Schrodinger operators on manifolds. These results are both closely related to a construction of classical mechanical scattering theory on manifolds, and scattering type time evolutions. We first recall the basic ideas of scattering theories, both classical mechanical and quantum mechanical ones. Then we construct a classical mechanical scattering theory on asymptotically conic manifolds. By using different quantizations, we obtain two different sets of microlocal results described above.