Non-concentration of Quasimodes for Integrable Systems

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 WFh of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of WFh 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 WFh 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.