Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow
The paper treats a dispersive evolution equation on a manifold especially a Schrödinger evolution equation on a complete Riemannian manifold, and discusses the relationship between (microlocal) smoothing effects and the globl behavior of the Hamilton flow of the principal symbol. The main results are: (i) propagation of smoothing effects along the Hamilton flow in the forward direction; (ii) lack of smoothing effects at every point \(z\) such that the total times the orbits spend in each neighborhood of \(z\) are unbounded; (iii) existence of smoothing effects at every backwards-nontrapped point under the assumption of the existence of a suitable strictly convex function near infinity. The part (iii) requires global conditions; its applications include the Schrödinger equations associated with an asymptotically Euclidean metric (of long-range metric perturbation), an asymptotically flat, conformally compact metric, a generalized scattering metric, and a metric of separation of variables near infinity.
dispersive evolution equation on a manifold
scientific article; zbMATH DE number 1546886