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Non-concentration of quasimodes for integrable systems
- Publication Year :
- 2010
-
Abstract
- We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set $WF_h$ of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of $WF_h$ along positive codimension sub-tori of a Liouville torus $\mathcal{L}$ if there exist rational relations among the frequencies of the flow on $\mathcal{L}.$ We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. The main tools are the spreading of Lagrangian regularity on $\mathcal{L}$ previously shown by Vasy and the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode.<br />Comment: One addition to acknowledgments; to appear in Comm. PDE
- Subjects :
- Mathematics - Analysis of PDEs
Mathematical Physics
Mathematics - Spectral Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1008.4396
- Document Type :
- Working Paper