Approximation of Schr\"odinger operators with $\delta$-interactions supported on hypersurfaces

TL;DR

This paper demonstrates that Schr"odinger operators with delta interactions on hypersurfaces can be approximated in the norm resolvent sense by regularized Hamiltonians, providing insights into their spectral properties.

Contribution

It introduces a method to approximate singular delta-interaction Schr"odinger operators with scaled regular potentials on hypersurfaces, extending understanding of their spectral behavior.

Findings

Approximation of delta-interaction operators by regular potentials in norm resolvent sense

Spectral consequences of the approximation are analyzed

Applicable to both bounded and unbounded hypersurfaces

Abstract

We show that a Schr\"odinger operator $A_{\delta, \alpha}$ with a $\delta$-interaction of strength $\alpha$ supported on a bounded or unbounded $C^2$-hypersurface $\Sigma \subset \mathbb{R}^d$, $d\ge 2$, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator $A_{\delta, \alpha}$ with a singular interaction is regarded as a self-adjoint realization of the formal differential expression $-\Delta - \alpha \langle \delta_{\Sigma}, \cdot \rangle \delta_{\Sigma}$, where $\alpha\colon\Sigma\rightarrow \mathbb{R}$ is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.