Bounding the number of nodal domains of eigenfunctions without singular points on the square

TL;DR

This paper proves Polterovich's conjecture on the growth of nodal domains for eigenfunctions on a square, assuming the eigenfunctions lack singular points, advancing understanding of eigenfunction behavior.

Contribution

It establishes the conjecture for eigenfunctions without singular points on the square, providing a significant theoretical result in spectral geometry.

Findings

Confirmed the conjecture for eigenfunctions without singular points

Provided bounds on the number of nodal domains

Enhanced understanding of eigenfunction structure

Abstract

We prove Polterovich's conjecture concerning the growth of the number of nodal domains for eigenfunctions on a unit square domain, under the assumption that the eigenfunctions do not have any singular points.