Shell Interactions for Dirac Operators

Abstract

In this notes we gather the latest results on spectral theory for the coupling H + V, where H = −iα ⋅ ∇ + mβ is the free Dirac operator in $\mathbb{R}^{3}$, m > 0 and V is a measure-valued potential. The potentials under consideration are given in terms of surface measures on the boundary of bounded regular domains in $\mathbb{R}^{3}$. We give three main results. We study the self-adjointness. We give a criterion for the existence of point spectrum, with applications to electrostatic shell potentials, V _λ , which depend on a parameter $\lambda \in \mathbb{R}$. Finally, we prove 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). The ball is the unique optimizer of this inequality.