Logarithmic lower bound on the number of nodal domains

TL;DR

This paper establishes that on negatively curved Riemann surfaces, the number of nodal domains of eigenfunctions grows at least logarithmically with eigenvalue for a density one subsequence, using advanced quantum ergodicity techniques.

Contribution

It provides the first logarithmic lower bound on the growth rate of nodal domains for eigenfunctions on negatively curved surfaces, building on recent quantum ergodicity results.

Findings

Number of nodal domains grows at least logarithmically with eigenvalue.

The result applies to a density one subsequence of eigenfunctions.

Uses new logarithmic scale quantum ergodicity theorems.

Abstract

We prove that the number of nodal domains of a density one subsequence of eigenfunctions grows at least logarithmically with the eigenvalue on negatively curved `real Riemann surfaces'. The geometric model is the same as in prior joint work with Junehyuk Jung (arXiv:1310.2919, to appear in J. Diff. Geom), where the number of nodal domains was shown to tend to infinity, but without a specified rate. The proof of the logarithmic rate uses the new logarithmic scale quantum ergodicity results of Hezari-Riviere (arXiv:1411.4078) and X. Han (arXiv:1410.3911).