Shell Interactions for Dirac Operators

In this notes we gather the latest results on spectral theory for the coupling H + V, where H = i alpha . del + m beta is the free Dirac operator in 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 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-lambda, which depend on a parameter lambda is an element of R. Finally, we prove 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). The ball is the unique optimizer of this inequality.