Quantum unique ergodicity and the number of nodal domains of eigenfunctions

TL;DR

This paper proves that eigenfunctions on certain negatively curved surfaces, including Hecke--Maass forms, have an increasing number of nodal domains as their eigenvalues grow large, under quantum ergodicity conditions.

Contribution

It establishes the growth of nodal domains for eigenfunctions on negatively curved surfaces satisfying quantum unique ergodicity, extending previous results to a broader class.

Findings

Number of nodal domains increases with eigenvalue

Results apply to Hecke--Maass forms and symmetric eigenfunctions

Supports quantum unique ergodicity implications

Abstract

We prove that the Hecke--Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $+\infty$. More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which Quantum Unique Ergodicity holds.