Anatomy of quantum chaotic eigenstates

TL;DR

This survey reviews analytical methods for understanding quantum eigenstates in chaotic systems, focusing on semiclassical limits, statistical properties, and deviations in special models, highlighting both macroscopic and microscopic descriptions.

Contribution

It compiles and discusses various analytical approaches and models for describing quantum chaotic eigenstates, emphasizing statistical and structural properties in the semiclassical regime.

Findings

Random wave models align well with numerical data

Specific systems like arithmetic ones deviate from random models

Descriptions span macroscopic averages and microscopic structures

Abstract

The eigenfunctions of quantized chaotic systems cannot be described by explicit formulas, even approximate ones. This survey summarizes (selected) analytical approaches used to describe these eigenstates, in the semiclassical limit. The levels of description are macroscopic (one wants to understand the quantum averages of smooth observables), and microscopic (one wants informations on maxima of eigenfunctions, "scars" of periodic orbits, structure of the nodal sets and domains, local correlations), and often focusses on statistical results. Various models of "random wavefunctions" have been introduced to understand these statistical properties, with usually good agreement with the numerical data. We also discuss some specific systems (like arithmetic ones) which depart from these random models.