# Semiclassical second microlocalization at linear coisotropic submanifolds in the torus

## TL;DR

This paper develops a refined semiclassical second microlocal calculus for linear coisotropic submanifolds in the torus, enabling detailed analysis of wavefront sets and propagation of coisotropic regularity for eigenfunctions and quasimodes.

## Contribution

It introduces a new second microlocal calculus for coisotropic submanifolds in the torus and establishes propagation theorems for coisotropic wavefronts, extending classical microlocal analysis.

## Key findings

- Proves propagation theorems analogous to Hörmander's for coisotropic wavefronts.
- Shows invariance of coisotropic wavefront under Hamiltonian flow expansions.
- Provides tools for analyzing regularity and singularities of eigenfunctions and quasimodes.

## Abstract

We develop a semiclassical second microlocal calculus of pseudodifferential operators associated to linear coisotropic submanifolds $\mathcal{C}\subset T^* \mathbb{T}^n$, where $\mathbb{T}^n = \mathbb{R}^n / \mathbb{Z}^n$. First microlocalization is localization in phase space $T^* \mathbb{T}^n$; second microlocalization is finer localization near a submanifold of $T^* \mathbb{T}^n$. Our second microlocal operators test distributions on $\mathbb{T}^n$ (e.g., Laplace eigenfunctions) for a coisotropic wavefront set, a second microlocal measure of absence of \emph{coisotropic regularity}. This wavefront set tells us where, in the coisotropic, and in what directions, approaching the coisotropic, a distribution lacks coisotropic regularity. We prove propagation theorems for coisotropic wavefront that are analogous to H\"{o}rmander's theorem for pseudodifferential operators of real principal type. Furthermore, we study the propagation of coisotropic regularity for quasimodes of semiclassical pseudodifferential operators. We Taylor expand the relevant Hamiltonian vector field, partially in the characteristic variables, at the spherical normal bundle of the coisotropic. Provided the principal symbol is real valued and depends only on the fiber variables in the cotangent bundle, and the subprincipal symbol vanishes, we show that coisotropic wavefront is invariant under the first two terms of this expansion.