LOCALIZATION OF EIGENFUNCTIONS OF ELLIPTIC OPERATORS (Research in teams)

Vibrations and waves play a major role in many fields of physics and of engineering, whether they are of acoustic, mechanical, optical or quantum nature. In many cases, the understanding of a vibrational system can be brought back to its spectral properties, i.e. the properties of the eigenvalues and eigenfunctions of the related wave operator. In disordered systems or in complex geometry, these eigenfunctions may exhibit a very peculiar characteristic called localization: the spatial distribution of the eigenfunctions can be strongly uneven, most of the energy being stored in a very small subregion of the entire domain. This phenomenon can have important consequences on the macroscopic behavior of physical systems, as for instance the metalinsulator transition in disordered alloys or the enhanced damping of waves achieved by complex geometries. In terms of mathematics, the central question is the spatial behavior of the eigenfunctions of a divergence form elliptic operator, e.g., L = −divA∇ + V , in a bounded domain Ω, with various types of boundary conditions. In particular, one aims to predict and to quantify localization of the eigenfunctions triggered by irregularities of the coefficients of the elliptic matrix A = A(x), of the potential V , or of the domain Ω. Given an obviously extended range of applications, many ad hoc approaches have been developed to treat particular instances of this problem, in particular, Laplacian on various peculiarly shaped domains (e.g., fractals), or Hamiltonian L = −∆ +V with a disordered potential (e.g., semiclassical theory and the studies surrounding Anderson localization). However, there has been no overreaching theory which would address occurrence and frequency of localized eigenfunctions, their specific spatial location and severity of localization in the general scenario. In particular, interplay between the influence of A, V , and Ω, seemed largely out of reach.