Local limits of high energy eigenfunctions on integrable billiards

TL;DR

This work analyzes whether inverse localization, the idea that high-energy eigenfunctions locally approximate any Helmholtz solution, holds in integrable billiards. Using the classification of integrable polygons and spectral formulas for rectangles and the three integrable triangles, plus separable models for balls and ellipses, the authors prove a dichotomy: certain integrable polygons (notably all irrational rectangles and most ellipses) fail to exhibit local limits matching random monochromatic waves, while integrable triangles and rational rectangles possess strong inverse localization under symmetry constraints. The results extend to higher dimensions and to almost-integrable polygonal billiards, and they have consequences for nodal sets and critical points, including the construction of eigenfunctions with highly intricate nodal topologies. Overall, the study indicates that Berry’s random wave picture does not generically describe local high-energy limits in integrable systems, and it highlights a rich interplay between domain geometry, boundary conditions, and spectral localization.

Abstract

Berry's random wave conjecture posits that high energy eigenfunctions of chaotic systems resemble random monochromatic waves at the Planck scale. One important consequence is that, at the Planck scale around "many" points in the manifold, any solution to the Helmholtz equation $Δ\varphi+\varphi =0$ can be approximated by high energy eigenfunctions. This property, sometimes called inverse localization, has useful applications to the study of the nodal sets of eigenfunctions. Alas, the only manifold for which the local limits of a sequence of high energy eigenfunctions are rigorously known to be given by random waves is the flat torus $(\mathbf{R}/\mathbf{Z})^2$, which is certainly not chaotic. Our objective in this paper is to study the validity of this "inverse localization" property in the class of integrable billiards, exploiting the fact that integrable polygonal billiards are classified and that Birkhoff conjectured that ellipses are the only smooth integrable billiards. Our main results show that, while there are infinitely many integrable polygons exhibiting good inverse localization properties, for "most" integrable polygons and ellipses, this property fails dramatically. We thus conclude that, in a generic integrable billiard, the local limits of Dirichlet and Neumann eigenfunctions do not match random waves, as one might expect in view of Berry's conjecture. Extensions to higher dimensions and nearly integrable polygons are discussed too.

Figures (11)

• Figure 1: The integrable polygonal billiards.
• Figure 2: Examples of some almost integrable billiards drawn in lattices.
• Figure 1.3: Circular and elliptic domains
• Figure 2.1: Triangular Coordinate System
• Figure 2.2: Change of coordinates in the equilateral triangle given by $\overline{z}_1=z_1-\frac{1}{2}$ and $\overline{z}_2=\frac{\sqrt{3}}{2}-z_2$

Theorems & Definitions (90)

• Conjecture : Berry's random wave conjecture
• Definition 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2