Nodal count for Dirichlet-to-Neumann operators with potential

TL;DR

This paper extends Courant's nodal domain theorem to Dirichlet-to-Neumann operators with potential on Lipschitz domains, using duality with Robin problems, providing new bounds on eigenfunction nodal counts.

Contribution

It introduces a Courant-type bound for Dirichlet-to-Neumann operators with potential, expanding classical results from harmonic to more general elliptic operators.

Findings

Established a Courant-type bound for nodal counts

Extended classical nodal domain results to operators with potential

Utilized duality between Steklov and Robin problems

Abstract

We consider Dirichlet-to-Neumann operators associated to $\Delta+q$ on a Lipschitz domain in a smooth manifold, where $q$ is an $L^{\infty}$ potential. We prove a Courant-type bound for the nodal count of the extensions $u_k$ of the $k$th Dirichlet-to-Neumann eigenfunctions $\phi_k$ to the interior satisfying $(\Delta+q)u_k=0$. The classical Courant nodal domain theorem is known to hold for Steklov eigenfunctions, which are the harmonic extension of the Dirichlet-to-Neumann eigenfunctions associated to $\Delta$. Our result extends it to a larger family of Dirichlet-to-Neumann operators. Our proof makes use of the duality between the Steklov and Robin problems.