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

We consider the Laplacian on a class of smooth domains $Ω\subset \mathbb{R}^ν$, $ν\ge 2$, with attractive Robin boundary conditions: \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \ α>0, \] where $n$ is the outer unit normal, and study the asymptotics of its eigenvalues $E_{j}(Q^Ω_α)$ as well as some other spectral properties for $α\to+\infty$ 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 \[ E_{j}(Q^Ω_α)=-α^2+μ_j(α)+{\mathcal O}(\log α), \] where $μ_j(α)$ is the $j^{\mbox{th}}$ eigenvalue, as soon as it exists, of $-Δ_{S}-(ν-1)αH$ with $(-Δ_{S})$ and $H$ being respectively the positive Laplace-Beltrami operator and the mean curvature on $\partialΩ$. Analogous results are obtained for a class of domains with non-compact boundaries. In particular, we discuss the existence of eigenvalues in 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 $-Δ_{S}-(ν-1)αH$ enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues in the limit $α\to+\infty$ under various geometrical assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of $Q^Ω_α$ for large $α$.