Essential Spectrum of Schrödinger Operators with $\delta$-Interactions on Unbounded Hypersurfaces

Let $\Gamma$ be a simply connected unbounded $C^{2}$-hypersurface in $\mathbb{R}^{n}$ such that $\Gamma$ divides $\mathbb{R}^{n}$ into two unbounded domains $D^{\pm}$. We consider the essential spectrum of Schrödinger operators on $\mathbb{R}^{n}$ with surface $\delta_{\Gamma}$-interactions which can be written formally as
$$ H_{\Gamma}=-\Delta+W-\alpha_{\Gamma}\delta_{\Gamma}, $$
where $-\Delta$ is the nonnegative Laplacian in $\mathbb{R}^{n}$, $W\in L^{\infty}(\mathbb{R}^{n})$ is a real-valued electric potential, $\delta_{\Gamma}$ is the Dirac $\delta$-function with the support on the hypersurface $\Gamma$ and $\alpha_{\Gamma}\in L^{\infty}(\Gamma)$ is a real-valued coupling coefficient depending of the points of $\Gamma$. We realize $H_{\Gamma}$ as an unbounded operator $\mathcal{A}_{\Gamma}$ in $L^{2}(\mathbb{R}^{n})$ generated by the Schrödinger operator
$$ H_{\Gamma}=-\Delta+W\qquad \text{on}\quad \mathbb{R}^{n}\setminus\Gamma $$
and Robin-type transmission conditions on the hypersurface $\Gamma$. We give a complete description of the essential spectrum of $\mathcal{A}_{\Gamma}$ in terms of the limit operators generated by $A_{\Gamma}$ and the Robin transmission conditions.