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

TL;DR

This paper studies Schr"odinger operators with delta-potentials on unbounded Lipschitz hypersurfaces, establishing spectral properties, uniqueness of ground states, and an optimization result for the spectrum's bottom.

Contribution

It introduces new spectral analysis results for Schr"odinger operators with delta-potentials supported on unbounded Lipschitz hypersurfaces, including spectrum determination and ground state uniqueness.

Findings

Proved the uniqueness of the ground state.

Determined the essential spectrum under certain conditions.

Established an optimization result for the spectrum's lower bound.

Abstract

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 $\mathsf{A}_\alpha$.