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We study semiclassical sequences of distributions u h associated with a Lagrangian submanifold of phase space L ⊂ T ∗ X . If u h is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on L , then the asymptotics of u h are well understood by work of Arnol'd, provided L projects to X with a stable simple Lagrangian singularity. We establish sup-norm estimates on u h under much more general hypotheses on the rate at which it is concentrating on L (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians. [ABSTRACT FROM AUTHOR]

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 a single closed orbit if all frequencies of the flow are rationally related. We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. Indeed, in the two-dimensional case, we show that WF h must fill out an entire Lagrangian torus. The main tools are the spreading of Lagrangian regularity previously shown by the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode. These yield a unique continuation theorem for the principal symbol of Lagrangian quasimode, which is the principal new result of the paper. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]