Spreading of Lagrangian regularity on rational invariant tori
Abstract
Let $P_h$ be a selfadjoint semiclassical pseudodifferential operator on a manifold $M$ such that the bicharacteristic flow of the principal symbol on $T^*M$ is completely integrable and the subprincipal symbol of $P_h$ vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes $u_h$ of $P_h.$ We show that on a nondegenerate rational invariant torus, Lagrangian regularity of $u_h$ (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional ``diffractive'' manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606495
 Bibcode:
 2006math......6495W
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Spectral Theory;
 35P20;
 81Q20;
 58F07
 EPrint:
 Revised version: proof of Theorem A pruned, some examples added, hypotheses clarified