|DATE||December 07 (Fri), 2018|
|TITLE||[Analysis & PDE Learning Seminar] On local smoothing estimates for Schr?dinger equations with variable coefficients|
|ABSTRACT||For the Schr?dinger operator, the principal symbol can be considered as a metric and the bicharacteristic flow gives the geodesics on the tangent bundle of the corresponding Riemannian manifold. It is well-known that these types of estimates fail even if we localize in time if there are trapped geodesics.
In this talk, I consider the nontrapping and asymptotically flat symbol and prove the local smoothing estimates. The proof proceeds by the positive commutator method using pseudodifferential operators adapted to the geometry of bicharacteristic flow.