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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2411.01594 |
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| _version_ | 1866912103050248192 |
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| author | Lo, Chia-Chun |
| author_facet | Lo, Chia-Chun |
| contents | We show that the spectrum of a Schrödinger eigenvalue problem posed on a closed Riemannian manifold $M$ with non-negative potential can be approached by that of Robin eigenvalue problems with constant positive boundary parameter posed on a sequence of domains in $M$. We construct these Robin problems by means of a homogenisation procedure. We show a similar result for compact manifolds with non-empty boundary and sign-indefinite potential; in this case the Robin boundary parameter can be taken to be constant on each boundary component and to have constant magnitude. As an application, we prove a flexibility result for optimal Schrödinger potentials: for certain problems where it is known that there exists some potential $V$ which extremises some Schrödinger eigenvalue, we show that this extremal eigenvalue is also approached by the corresponding eigenvalues for a sequence of smooth potentials which remain bounded away from $V$ in some dual Sobolev space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01594 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homogenisation for the Robin eigenvalue problem on manifolds and flexibility of optimal Schrödinger potentials Lo, Chia-Chun Spectral Theory Analysis of PDEs 35P15, 58C40 (Primary), 35B27, 47A75 (Secondary) We show that the spectrum of a Schrödinger eigenvalue problem posed on a closed Riemannian manifold $M$ with non-negative potential can be approached by that of Robin eigenvalue problems with constant positive boundary parameter posed on a sequence of domains in $M$. We construct these Robin problems by means of a homogenisation procedure. We show a similar result for compact manifolds with non-empty boundary and sign-indefinite potential; in this case the Robin boundary parameter can be taken to be constant on each boundary component and to have constant magnitude. As an application, we prove a flexibility result for optimal Schrödinger potentials: for certain problems where it is known that there exists some potential $V$ which extremises some Schrödinger eigenvalue, we show that this extremal eigenvalue is also approached by the corresponding eigenvalues for a sequence of smooth potentials which remain bounded away from $V$ in some dual Sobolev space. |
| title | Homogenisation for the Robin eigenvalue problem on manifolds and flexibility of optimal Schrödinger potentials |
| topic | Spectral Theory Analysis of PDEs 35P15, 58C40 (Primary), 35B27, 47A75 (Secondary) |
| url | https://arxiv.org/abs/2411.01594 |