Publications

Local smoothing for the Schrodinger equation on a multi-warped product manifold with inflection-transmission trapping with H. Christianson, accepted Proceedings of the American Mathematical Society

Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schrödinger equation. Surprisingly little is known about solutions to the Schrödinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of H. Christianson and J. Metcalfe  on inflection-transmission type trapping on warped product manifolds to the case of multi-warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has only one infinite end and only inflection-transmission type trapping.

Thesis 

Local smoothing for the Schrödinger equation on a multi-warped product manifold with degenerate trapping.

The thesis extends the work from above to the case where there are two infinite ends. This situation is interesting because there is degenerate trapping on an interval rather than discrete points. However, the trapping at this points is in a single direction. I show that due to this fact, we still have the same gain in local smoothing as in the case of the surface of revolution with a single degenerate trapping point case covered by H. Christianson and J. Wunsch.