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

TL;DR

This paper analyzes the eigenvalue asymptotics of a Robin Laplacian with a large parameter on smooth domains, introducing an effective Hamiltonian that captures the spectral behavior as the parameter grows large, with applications to spectral gaps.

Contribution

It introduces an effective Hamiltonian for Robin Laplacians with large boundary parameter, providing precise eigenvalue asymptotics and spectral gap analysis on various domains.

Findings

Eigenvalues asymptotically behave as -α^2 plus eigenvalues of an effective Hamiltonian.

Effective Hamiltonian involves Laplace-Beltrami operator and mean curvature on the boundary.

Spectral gaps can be identified for large boundary parameters under certain geometrical conditions.

Abstract

We consider the Laplacian on a class of smooth domains $\Omega\subset \mathbb{R}^{\nu}$, $\nu\ge 2$, with attractive Robin boundary conditions: \[ Q^\Omega_\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text{ on } \partial\Omega, \ \alpha>0, \] where $n$ is the outer unit normal, and study the asymptotics of its eigenvalues $E_{j}(Q^\Omega_\alpha)$ as well as some other spectral properties for $\alpha\to+\infty$ 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 \[ E_{j}(Q^\Omega_\alpha)=-\alpha^2+\mu_j(\alpha)+{\mathcal O}(\log \alpha), \] where $\mu_j(\alpha)$ is the $j^{\mbox{th}}$ eigenvalue, as soon as it exists, of $-\Delta_{S}-(\nu-1)\alpha H$ with $(-\Delta_{S})$ and $H$ being respectively the positive Laplace-Beltrami operator and the mean curvature on $\partial\Omega$. 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 $-\Delta_{S}-(\nu-1)\alpha H$ enters the framework of semi-classical Schr\"odinger operators on manifolds, and we provide the asymptotics of its eigenvalues in the limit $\alpha\to+\infty$ under various geometrical assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of $Q^\Omega_\alpha$ for large $\alpha$.