Generalized interactions supported on hypersurfaces

TL;DR

This paper studies a broad class of singular Schrödinger operators with interactions supported on hypersurfaces, generalizing previous models like delta interactions, and explores their spectral properties and inequalities.

Contribution

It introduces a generalized framework for singular interactions on hypersurfaces, extending prior models and analyzing their spectral characteristics and operator inequalities.

Findings

Spectral properties depend on the four parameters characterizing the interaction.

Operator inequalities relate different coupling configurations on the same hypersurface.

Implications for spectral bounds and stability of the operators.

Abstract

We analyze a family of singular Schr\"odinger operators with local singular interactions supported by a hypersurface $\Sigma \subset \mathbb{R}^n, n \ge 2$, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of $\Sigma$ the interaction is characterized by four real parameters, the earlier studied case of $\delta$- and $\delta'$-interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.