Abstract
This note revolves on the free Dirac operator in R3 and its δ-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich et al. [J. Math. Phys. 30(12), 2875–2882 (1989)] and by Arrizabalaga et al. [J. Math. Pures Appl. 102(4), 617–639 (2014)] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case.
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