Semiclassical second microlocal propagation of regularity and integrable systems

TL;DR

This paper develops a second-microlocal calculus in the semiclassical setting to analyze Lagrangian regularity propagation, with applications to integrable systems and eigenfunction behavior on invariant tori.

Contribution

It introduces a new semiclassical second-microlocal calculus and proves propagation theorems for Lagrangian regularity, extending classical results to integrable systems.

Findings

Propagation of Lagrangian regularity on invariant tori is established.

Lagrangian regularity either fills the entire torus or is absent, under nondegeneracy.

The calculus generalizes Bony's homogeneous Lagrangian calculus to the semiclassical setting.

Abstract

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold $X$ with respect to a Lagrangian submanifold of $T^*X.$ The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of $T^*\mathbb{R}^n,$ and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to H\"ormander's theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g. eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.