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Local smoothing for the Schr\'odinger equation with a prescribed loss
- Publication Year :
- 2011
-
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.<br />Comment: 26 pages, incorporated referee's comments
- Subjects :
- Mathematics - Analysis of PDEs
Mathematics - Spectral Theory
35J10, 35B34
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1103.3908
- Document Type :
- Working Paper