We consider Schrödinger equations with variable coefficients, which are long-range type perturbations of the flat Laplacian on . We characterize the wave front set of solutions to Schrödinger equations in terms of the initial state. Then it is shown that the singularities propagates along the classical flow, and results are formulated in a semiclassical setting. Methods analogous to the long-range scattering theory, in particular a modified free propagator, are employed.
Shu NAKAMURA. "Semiclassical singularities propagation property for Schrödinger equations." J. Math. Soc. Japan 61 (1) 177 - 211, January, 2009. https://doi.org/10.2969/jmsj/06110177