Spectral Analysis of Partial Differential Equations

In this workshop talks on discrete spectra, including Lieb-Thirring estimates and properties of resonances were presented. The connection between continuous and discrete spectra was discussed. Various characteristics of continuous spectra in the context of random or magnetic operators were investigated. The Liouville Theorem, the problem of absolute continuity, and the classical problem of homogenization for perodic operators were treated

Anatomy of quantum chaotic eigenstates

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.Comment: Corrected typos, added a few references and updated some result