Tesis
THESIS DEFENSE -- High-energy Eigenfunctions of the Laplacian: Localization Properties and Relation with Berry’s Random Wave Model
Ponente: Alba Dolores García Ruiz (ICMAT-UAM)Director/es: Alberto Enciso Carrasco (ICMAT-CSIC) y Daniel Peralta Salas (ICMAT-CSIC)Fecha: martes 15 de julio de 2025 - 11:30Lugar: Aula Naranja, ICMAT
Resumen:
Berry's random wave conjecture posits that high-energy eigenfunctions of chaotic systems behave like random monochromatic waves at the Planck scale. One notable consequence is that, at an appropriate scale, solutions to the Helmholtz equation can be locally approximated by high-energy eigenfunctions around a large set of points in the manifold. This phenomenon, sometimes referred to as \emph{inverse localization}, has been investigated in various settings, with both positive and negative results depending on the context. In this thesis, we further explore the inverse localization property, its connection to Berry's random wave conjecture, and its implications for the study of nodal sets of eigenfunctions.
The first part of the thesis focuses on compact manifolds without boundary. Specifically, Chapter 4 examines the relationship between two formulations of Berry's conjecture and the inverse localization property; Chapter 5 establishes an analogous version of this property for parametric families of operators in hyperbolic space; and Chapter 6 classifies all flat tori according to whether they satisfy or not this property. The second part of the thesis considers Euclidean domains whose associated classical billiard dynamics are integrable. We analyze the validity of the inverse localization property in this setting, obtaining both positive and negative results, some of which have implications for the understanding of Berry’s conjecture for integrable dynamical systems. Additionally, we explore applications of these findings to the structure of nodal sets and the distribution of critical points of eigenfunctions with Dirichlet or Neumann boundary conditions.
