Local smoothing for the Schr\"odinger equation with a prescribed loss
Analysis of PDEs
2012-01-30 v2 Spectral Theory
Abstract
We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schr\"odinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss.
Keywords
Cite
@article{arxiv.1103.3908,
title = {Local smoothing for the Schr\"odinger equation with a prescribed loss},
author = {Hans Christianson and Jared Wunsch},
journal= {arXiv preprint arXiv:1103.3908},
year = {2012}
}
Comments
26 pages, incorporated referee's comments