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

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