On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon

Abstract

Let $\Omega \subset {\mathbb{R}}^2$ be the exterior of a convex polygon whose side lengths are ${\ell }_1,...,{\ell }_M$. For $\alpha >0$, let $H_{\alpha }^{\Omega }$ denote the Laplacian in $\Omega$, $u\mapsto -\Delta u$, with the Robin boundary conditions $\partial u/\partial \nu =\alpha u$, where $\nu$ is the exterior unit normal at the boundary of $\Omega$. We show that, for any fixed $m\in \mathbb{N}$, the $m$th eigenvalue $E_m^{\Omega }(\alpha )$ of $H_{\alpha }^{\Omega }$ behaves as \[ E^Ω_m(α)=-α^2+μ^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrtα}\Big) \quad {as $\alpha$ tends to $+\infty$}, \] where ${\mu }_m^D$ stands for the $m$th eigenvalue of the operator $D_1\oplus ...\oplus D_M$ and $D_n$ denotes the one-dimensional Laplacian $f\mapsto -f"$ on $(0,{\ell }_n)$ with the Dirichlet boundary conditions.

10 pages. To appear in Nanosystems: Physics, Chemistry, Mathematics. Minor revision: misprints corrected, references updated