Inner radius of nodal domains of quantum ergodic eigenfunctions

TL;DR

This paper improves lower bounds on the inner radius of nodal domains for quantum ergodic eigenfunctions, showing logarithmic and polynomial enhancements depending on the manifold's properties, using recent small-scale equidistribution results.

Contribution

It provides new lower bounds on the inner radius of nodal domains for quantum ergodic eigenfunctions, leveraging recent small-scale equidistribution techniques.

Findings

Logarithmic improvements for negatively curved manifolds.

Polynomial improvements for toral eigenfunctions in dimensions ≥ 3.

Results apply to a full density subsequence of eigenfunctions.

Abstract

In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results (arXiv:1606.02057), which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by the results of Han and Hezari-Rivi\`ere small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get logarithmic improvements on the inner radius of such manifolds. We also get improvements for manifolds with ergodic geodesic flows. In addition using the small scale equidistribution results of Lester-Rudnick, one gets polynomial betterments of for toral eigenfunctions in dimensions $n \geq 3$. The results work only for a full density subsequence of eigenfunctions.