Convergence of Schrödinger operators on domains with scaled resonant potentials
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918304018333696 |
|---|---|
| author | Lotoreichik, Vladimir Post, Olaf |
| author_facet | Lotoreichik, Vladimir Post, Olaf |
| contents | We consider Schrödinger operators on a bounded, smooth domain of dimension $d \ge 2$ with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to analyse the convergence of these operators as the scaling parameter tends to zero. If the scaled potential is resonant, the limit in strong resolvent sense is a Robin Laplacian with boundary coefficient expressed in terms of the mean curvature of the boundary. A counterexample shows that norm resolvent convergence cannot hold in general in this setting. If the scaled potential is non-resonant and satisfies an explicit assumption on the smallness of the negative part, the limit in strong resolvent sense is the Dirichlet Laplacian. We conjecture that we can drop this additional assumption in the non-resonant case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of Schrödinger operators on domains with scaled resonant potentials Lotoreichik, Vladimir Post, Olaf Spectral Theory Mathematical Physics Analysis of PDEs We consider Schrödinger operators on a bounded, smooth domain of dimension $d \ge 2$ with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to analyse the convergence of these operators as the scaling parameter tends to zero. If the scaled potential is resonant, the limit in strong resolvent sense is a Robin Laplacian with boundary coefficient expressed in terms of the mean curvature of the boundary. A counterexample shows that norm resolvent convergence cannot hold in general in this setting. If the scaled potential is non-resonant and satisfies an explicit assumption on the smallness of the negative part, the limit in strong resolvent sense is the Dirichlet Laplacian. We conjecture that we can drop this additional assumption in the non-resonant case. |
| title | Convergence of Schrödinger operators on domains with scaled resonant potentials |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2505.02480 |