Schr\"odinger operators with $\delta$-potentials supported on unbounded Lipschitz hypersurfaces

In this note we consider the self-adjoint Schr"odinger operator ${\mathsf{A}}_{\alpha }$ in $L^2({\mathbb{R}}^d)$, $d\ge 2$, with a $\delta$-potential supported on a Lipschitz hypersurface $\mathrm{\Sigma }\subseteq {\mathbb{R}}^d$ of strength $\alpha\in L^p(\Sigma)+L^\infty(\Sigma)$. We show the uniqueness of the ground state and, under some additional conditions on the coefficient $\alpha$ and the hypersurface $\mathrm{\Sigma }$, we determine the essential spectrum of ${\mathsf{A}}_{\alpha }$. In the special case that $\mathrm{\Sigma }$ is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schr"{o}dinger operator as Birman-Schwinger operator. As an application we prove an optimization result for the bottom of the spectrum of ${\mathsf{A}}_{\alpha }$.