An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators

In this article we investigate spectral properties of the coupling $H+V_\lambda$, where $H=-i\alpha\cdot\nabla +m\beta$ is the free Dirac operator in $\mathbb R^3$, $m>0$ and $V_\lambda$ is an electrostatic shell potential (which depends on a parameter $\lambda\in\mathbb R$) located on the boundary of a smooth domain in $\mathbb R^3$. Our main result is an isoperimetric-type inequality for the admissible range of $\lambda$'s for which the coupling $H+V_\lambda$ generates pure point spectrum in $(-m,m)$. That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman-Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible $\lambda$'s, and we use this to relate the endpoints of the admissible range of $\lambda$'s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.