From nodal points to non-equidistribution at the Planck scale

TL;DR

This paper observes that Laplacian eigenfunctions do not evenly distribute at the Planck scale, especially near points of large eigenfunction values, challenging assumptions of uniform distribution at quantum scales.

Contribution

It highlights the failure of equidistribution of Laplacian eigenfunctions at the Planck scale and near high-value points, providing new insights into quantum chaos and eigenfunction behavior.

Findings

Eigenfunctions fail equidistribution at the Planck scale

Equidistribution also fails near points of large eigenfunction values

Challenges assumptions of uniform eigenfunction distribution at quantum scales

Abstract

In this note, we make an observation that Laplacian eigenfunctions fail equidistribution at the Planck scale. Furthermore, equidistribution at the same scale also fails around the points where the eigenfunctions have large values.