An eigenvalue inequality for Schr\"odinger operators with $\delta$ and $\delta'$-interactions supported on hypersurfaces

TL;DR

This paper establishes strict eigenvalue inequalities for Schr"odinger operators with delta and delta-prime interactions on hypersurfaces, highlighting geometric conditions where inequalities are strict even at boundary cases.

Contribution

It proves strict eigenvalue inequalities under certain conditions and identifies geometries where inequalities hold even at the borderline case.

Findings

Eigenvalue inequalities are strict when eta < 4/ alpha on a nonempty, open subset.

Special geometries like broken lines or cones can have strict inequalities at the boundary case.

The results extend understanding of spectral properties of Schr"odinger operators with surface interactions.

Abstract

We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with a $\delta$-interaction of strength $\alpha$ and a $\delta'$-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 $0 < \beta \leq 4/\alpha$ 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 $\beta < 4 / \alpha$ 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 $\beta = 4 / \alpha$.