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.