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

Abstract

We consider the Laplacian on a class of smooth domains Ω⊂Rν, ν≥2, with attractive Robin boundary conditions:QαΩu=−Δu,∂u∂n=αu on ∂Ω,α>0, where n is the outer unit normal, and study the asymptotics of its eigenvalues Ej(QαΩ) 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 C2 boundaries we show that, for each fixed j,Ej(QαΩ)=−α2+μj(α)+O(log⁡α), where μj(α) is the jth eigenvalue of the operator −ΔS−(ν−1)αH with (−ΔS) 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 −ΔS−(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 QαΩ for large α.