Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow

Analytic smoothing effect for the Schrödinger equation with long‐range perturbation

“…Instead, a local smoothing effect has been used to study the local smoothness of solutions to Schrödinger equations. The smoothing effects for the Schrödinger equation has been a very rich source of recent investigation; see e.g., [1,2,3,4,5,6,7,9,10,11,12,13,14,17,19,20,21,22,23,24,26,27,28,29,30]. In particular, Craig, Kappeler, and Strauss [1] showed that this effect may be considered as a microlocal phenomenon, and this observation inspired a series of investigations, both in the C ∞ case (in particular, [5,26]) and in the analytic case [21,22,23].…”

On the decay properties of solutions to a class of Schrödinger equations

“…as a consequence of its dispersive character and the decay assumptions on the data has also been studied in several works; see [2], [3], and the references therein.…”

Global parametrices and dispersive estimates for variable coefficient wave equations

“…Heuristically this is a reflection of the fact that waves move at speed O(1) and thus spend a time O(R) within a bounded spatial ball of radius R. These are the counterpart of the so called local smoothing estimates for the Schrödinger equation. See, e.g., [23], [39], [4], [7], and [5]. A significant difference is that, in the case of the Schrödinger equation the speed is proportional to the frequency; therefore one also gains half a derivative in the estimates.…”