Planck-scale mass equidistribution of toral Laplace eigenfunctions

Abstract

We study the small scale distribution of the $L^2$-mass of eigenfunctions of the Laplacian on the the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick showed the existence of a density one subsequence whose $L^2$-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the $L^2$-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.

35 pages

References in corpus (6)

• The quantum unique ergodicity conjecture for thin sets
• Small scale quantum ergodicity in negatively curved manifolds
• On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus
• Small scale equidistribution of eigenfunctions on the torus
• $L^p$ norms, nodal sets, and quantum ergodicity
• Small scale equidistribution of random eigenbases

Cited by in corpus (10)

• Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms
• Asymptotic distribution of nodal intersections for arithmetic random waves
• On the Random Wave Conjecture for Dihedral Maaß Forms
• CLT for Planck scale mass distribution of toral Laplace eigenfunctions
• The defect of toral Laplace eigenfunctions and Arithmetic Random Waves
• Small-scale equidistribution for random spherical harmonics
• Mass distribution for toral eigenfunctions via Bourgain's de-randomisation
• Small scale equidistribution for a point scatterer on the torus
• Quantum ergodicity for shrinking balls in arithmetic hyperbolic manifolds
• Concentration for nodal component count of Gaussian Laplace eigenfunctions