Spreading of Lagrangian regularity on rational invariant tori

Details Spreading of Lagrangian regularity on rational invariant tori Spreading of Lagrangian regularity on rational invariant tori

2006-06-20

arxiv.org/abs/math/0606495v2

Mathematics

Analysis of PDEs

Revised version: proof of Theorem A pruned, some examples added, hypotheses clarified

Scientific paper

Let $P_h$ be a self-adjoint 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.

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