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

In this article we investigate spectral properties of the coupling $H+V_λ$, where $H=-iα\cdot\nabla +mβ$ is the free Dirac operator in $\mathbb R^3$, $m>0$ and $V_λ$ is an electrostatic shell potential (which depends on a parameter $λ\in\mathbb R$) 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 $λ$'s for which the coupling $H+V_λ$ 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 $λ$'s, and we use this to relate the endpoints of the admissible range of $λ$'s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.