Local smoothing for the Schrödinger equation with a prescribed loss

Details Local smoothing for the Schrödinger equation with a prescribed loss Local smoothing for the Schrödinger equation with a prescribed loss

2011-03-21

arxiv.org/abs/1103.3908v2

Mathematics

Analysis of PDEs

26 pages, incorporated referee's comments

Scientific paper

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.

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