  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 $\alpha\in
L^p(\Sigma)+L^\infty(\Sigma)$. We show the uniqueness of the ground state and,
under some additional conditions on the coefficient $\alpha$ and the
hypersurface $\Sigma$, we determine the essential spectrum of
$\mathsf{A}_\alpha$. In the special case that $\Sigma$ 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 $\mathsf{A}_\alpha$.
