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Published in:
2017 | OriginalPaper | Chapter
Shell Interactions for Dirac Operators
Author : Naiara Arrizabalaga
Published in: Advances in Quantum Mechanics
Publisher: Springer International Publishing
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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.