An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter

Konstantin Pankrashkin (1) , Nicolas Popoff (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Équipe EDP et Physique Mathématique

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.

Domaines: Théorie spectrale [math.SP]

Konstantin Pankrashkin, Nicolas Popoff. An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter. 2015. ⟨hal-01202601⟩