# Semiclassical behaviour of quantum eigenstates

## TL;DR

This paper explores how classical dynamical systems influence quantum eigenstates in the semi-classical limit, contrasting integrable and chaotic systems using semiclassical measures and microlocal analysis.

## Contribution

It provides a comparative analysis of classical-quantum correspondence in integrable versus chaotic systems using semiclassical measures.

## Key findings

- Classical dynamics significantly affect quantum eigenstate behaviour.
- Microlocal analysis tools help illustrate classical-quantum correspondence.
- Differences between integrable and chaotic systems are highlighted.

## Abstract

Given a quantum Hamiltonian, we explain how the dynamical properties of the underlying classical system affect the behaviour of quantum eigenstates in the semi-classical limit. We study this problem via the notion of semiclassical measures. We mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. We recall standard tools from microlocal analysis and from dynamical systems. We show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.