Nodal Domains on Surfaces under Perturbation: Upper Semicontinuity, Courant-Sharpness, and Boundary Intersections
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915990136160256 |
|---|---|
| author | Maji, Saikat Mukherjee, Mayukh Saha, Soumyajit |
| author_facet | Maji, Saikat Mukherjee, Mayukh Saha, Soumyajit |
| contents | We study how the number of nodal domains of eigenfunctions of Schrödinger operators $-Δ_{g_t}+V_t$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nodal Domains on Surfaces under Perturbation: Upper Semicontinuity, Courant-Sharpness, and Boundary Intersections Maji, Saikat Mukherjee, Mayukh Saha, Soumyajit Spectral Theory Analysis of PDEs Differential Geometry We study how the number of nodal domains of eigenfunctions of Schrödinger operators $-Δ_{g_t}+V_t$ 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. |
| title | Nodal Domains on Surfaces under Perturbation: Upper Semicontinuity, Courant-Sharpness, and Boundary Intersections |
| topic | Spectral Theory Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2507.04928 |