An eigenvalue inequality for Schr\"odinger operators with $\delta$ and ${\delta }^{\mathrm{\prime }}$-interactions supported on hypersurfaces

We consider self-adjoint Schr"odinger operators in $L^2({\mathbb{R}}^d)$ with a $\delta$-interaction of strength $\alpha$ and a ${\delta }^{\mathrm{\prime }}$-interaction of strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and ${\beta }^{-1}$ are bounded, real-valued functions. It is known that the inequality $0<\beta \le 4\mathrm{/}\alpha$ implies inequality of the eigenvalues of these two operators below the bottoms of the essential spectra. We show that this eigenvalue inequality is strict whenever $\beta <4\mathrm{/}\alpha$ on a nonempty, open subset of the hypersurface. Moreover, we point out special geometries of the interaction support, such as broken lines or infinite cones, for which strict inequality of the eigenvalues even holds in the borderline case $\beta =4\mathrm{/}\alpha$.