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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2411.01594 |
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Sommario:
- 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.