On the geometry of nodal sets of eigenfunctions of fractional powers of the laplace operator

Abstract

This work studies the geometric properties of eigenfunctions of fractional powers of the self-adjoint Laplace
operator. The sturucture of nodal domains bounded by nodal surfaces, on which the eigenfunction vanishes,
is investigated. The behavior of these nodal domains is analyzed depending on the eigenvalue. The paper
derives estimates showing that the width of the nodal domain decreases as the eigenvalue increases. From
these estimates, it follows that the width of the nodal domain tends to zero as the eigenvalue increases.