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Local smoothing for the Schrödinger equation with a prescribed loss
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 6, December 2013
- pp. 1601-1632
- 10.1353/ajm.2013.0047
- Article
- Additional Information
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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 Schrödinger 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.