Spreading of Lagrangian Regularity on Rational Invariant Tori

Let Ph 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 Ph vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes uh of Ph. We show that on a nondegenerate rational invariant torus, Lagrangian regularity of uh (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional “nongeometric” manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics.