Shell interactions for Dirac operators: on the point spectrum and the confinement

Abstract

Spectral properties and the confinement phenomenon for the couplingH+Varestudied, whereH=-ia·¿+mßis the free Dirac operator inR3andVis a measure-valuedpotential. The potentialsVunder consideration are given in terms of surface measures onthe boundary of bounded regular domains inR3.A criterion for the existence of point spectrum is given, with applications to electrostaticshell potentials. In the case of the sphere, an uncertainty principle is developed and itsrelation with some eigenvectors of the coupling is shown.Furthermore, a criterion for generating confinement is given. As an application, someknown results about confinement on the sphere for electrostatic plus Lorentz scalar shellpotentials are generalized to regular surfaces.