An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter
| hal.structure.identifier | Laboratoire de Mathématiques d'Orsay [LM-Orsay] | |
| dc.contributor.author | PANKRASHKIN, Konstantin | |
| hal.structure.identifier | Équipe EDP et Physique Mathématique | |
| dc.contributor.author | POPOFF, Nicolas | |
| dc.date.accessioned | 2024-04-04T03:17:35Z | |
| dc.date.available | 2024-04-04T03:17:35Z | |
| dc.date.created | 2015 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/194318 | |
| dc.description.abstractEn | We consider the Laplacian on a class of smooth domains Ω ⊂ R ν , ν ≥ 2, with attractive Robin boundary conditions: Q Ω α u = −∆u, ∂u ∂n = αu on ∂Ω, α > 0, where n is the outer unit normal, and study the asymptotics of its eigenvalues Ej(Q Ω α) as well as some other spectral properties for α tending to +∞. We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact C 2 boundaries and fixed j, we show that Ej(Q Ω α) = −α 2 + µj (α) + O(log α), where µj(α) is the j th eigenvalue, as soon as it exists, of −∆S−(ν−1)αH with (−∆S) and H being respectively the positive Laplace-Beltrami operator and the mean curvature on ∂Ω. Analogous results are obtained for a class of domains with non-compact boundaries. In particular, we discuss the existence of eigenvalues for non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions. The effective Hamiltonian −∆S − (ν − 1)αH enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues for large α under various geometric assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of Q Ω α for large α. | |
| dc.language.iso | en | |
| dc.title.en | An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Théorie spectrale [math.SP] | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-01202601 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01202601v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=PANKRASHKIN,%20Konstantin&POPOFF,%20Nicolas&rft.genre=preprint |
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