Nodal count for Dirichlet-to-Neumann operators with potential
Asma Hassannezhad, David A. Sher · 2022
We consider Dirichlet-to-Neumann operators associated to ∆+ q on a Lipschitz domain in a smooth manifold, where q is an L ∞ potential.We prove a Courant-type bound for the nodal count of the extensions u k of the kth Dirichlet-to-Neumann eigenfunctions φ k to the interior satisfying (∆ + 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 ∆.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.