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 $\Omega\subset $\mathbb{R}$^{\nu}$,$\nu\ge 2$, with attractive Robin boundary conditions: $$Q_{\alpha }^{\mathrm{\Omega }}u=-\mathrm{\Delta }u,{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }n}}=\alpha u\text{ on }\mathrm{\partial }\mathrm{\Omega },\alpha >0,$$ where $n$ is the outer unit normal, and study the asymptotics of its eigenvalues $E_j(Q_{\alpha }^{\mathrm{\Omega }})$ as well as some other spectral properties for $\alpha \to +\mathrm{\infty }$ 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 $$E_j(Q_{\alpha }^{\mathrm{\Omega }})=-{\alpha }^2+{\mu }_j(\alpha )+\mathcal{O}(\log \alpha ),$$ where ${\mu }_j(\alpha )$ is the $j^{th}$ eigenvalue, as soon as it exists, of $-{\mathrm{\Delta }}_S-(\nu -1)\alpha H$ with $(-{\mathrm{\Delta }}_S)$ and $H$ being respectively the positive Laplace-Beltrami operator and the mean curvature on $\mathrm{\partial }\mathrm{\Omega }$. 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 $-{\mathrm{\Delta }}_S-(\nu -1)\alpha H$ enters the framework of semi-classical Schr"odinger operators on manifolds, and we provide the asymptotics of its eigenvalues in the limit $\alpha \to +\mathrm{\infty }$ under various geometrical assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of $Q_{\alpha }^{\mathrm{\Omega }}$ for large $\alpha$.