Title: Optimization of the lowest eigenvalue for the Schrödinger operator with a $δ$-potential supported on a hyperplane

Abstract: We consider the self-adjoint Schr\"odinger operator in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $\delta$-potential supported on a hyperplane $\Sigma\subseteq\mathbb{R}^d$ of strength $\alpha=\alpha_0+\alpha_1$, where $\alpha_0\in\mathbb{R}$ is a constant and $\alpha_1\in L^p(\Sigma)$ is a nonnegative function. As the main result, we prove that the lowest spectral point of this operator is not smaller than that of the same operator with potential strength $\alpha_0+\alpha_1^*$, where $\alpha_1^*$ is the symmetric decreasing rearrangement of $\alpha_1$. The proof relies on the Birman-Schwinger principle and the reduction to an analogue of the P\'{o}lya-Szeg\H{o} inequality for the relativistic kinetic energy in $\mathbb{R}^{d-1}$.