Nodal Domains on Surfaces under Perturbation: Upper Semicontinuity, Courant-Sharpness, and Boundary Intersections

TL;DR

This paper investigates how the number of nodal domains of eigenfunctions on surfaces behaves under smooth perturbations, establishing upper semicontinuity, stability, and applications to Courant-sharp metrics and boundary intersections.

Contribution

It provides new results on the upper semicontinuity and stability of nodal domain counts under perturbations, including branch-free spectral cluster analysis and boundary intersection prescriptions.

Findings

Nodal domain count is upper semicontinuous under perturbations.

No new local nodal domains are created near critical points.

Constructs metrics that are Courant-sharp up to any finite level.

Abstract

We study how the number of nodal domains of eigenfunctions of Schr\"odinger operators $-\Delta_{g_t}+V_t$ on closed surfaces changes under smooth perturbations of $(g_t,V_t)$ along convergent eigenbranches. Locally, near each nodal critical point of the limit eigenfunction, we give a sector/graph count showing that no new local domains can be created and that vanishing orders cannot increase. Globally, we prove upper semicontinuity of the nodal domain count; in the noncritical case the count is stable. The result is branch-free on spectral clusters. At the wavelength scale, new closed nodal loops cannot be created. We also treat localised (topology-changing) perturbations: the count inside the unperturbed core cannot increase. As applications, we construct metrics on any closed surface that are Courant-sharp up to an arbitrary finite level and prescribe $2n_i$ boundary intersections on each boundary component. An appendix records a uniform (wavelength-scale) lower bound on the inner radius of nodal domains along the branch.