An Isoperimetric-Type Inequality for Electrostatic Shell Interactions for Dirac Operators
Abstract
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.