Generalized interactions supported on hypersurfaces

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.