Non-concentration of quasimodes for integrable systems

TL;DR

This paper proves that, under certain conditions, eigenfunctions of integrable systems do not concentrate along specific sub-tori in phase space, highlighting limitations on their localization properties.

Contribution

It demonstrates the non-concentration of quasimodes on invariant tori in integrable systems under non-degeneracy conditions, extending understanding of eigenfunction behavior.

Findings

Eigenfunctions do not concentrate along rational sub-tori of Liouville tori.

Spreading of Lagrangian regularity prevents localization of quasimodes.

Analysis of higher order transport equations supports non-concentration results.

Abstract

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 positive codimension sub-tori of a Liouville torus $\mathcal{L}$ if there exist rational relations among the frequencies of the flow on $\mathcal{L}.$ We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. The main tools are the spreading of Lagrangian regularity on $\mathcal{L}$ previously shown by Vasy and the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode.