On the number of nodal domains of toral eigenfunctions (Q345051)

The aim of the paper is to give asymptotic formulas for the number of nodal domains of real valued toral Laplace eigenfunctions, i. e. functions \(f:\mathbb{T}^{2} \rightarrow \mathbb{R}\) satisfying for some \(E>0\) the equation \( \Delta f + 4 \pi^{2}Ef=0\). The main result states that for generic eigenfunctions (which are subject to some constraints on the energy level \(E\) and on the coefficients of their representation as a trigonometric polynomial) the ratio between the number of nodal domains and the energy \(E\) is asymptotically equal to the Nazarov-Sodin constant [\textit{F. Nazarov} and \textit{M. Sodin}, Zh. Mat. Fiz. Anal. Geom. 12, No. 3, 205--278 (2016; Zbl 1358.60057)]. As a corollary, one obtains an optimal lower bound for the number of nodal domains of generic eigenfunctions.