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

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.