Non-self-adjoint relativistic point interactions and their approximations by non-local potentials

Abstract

The research project deals with the Dirac operator with non–local potential given by the projec tion on a fixed scaled function from L 2 (R)∩ L 1 (R) multiplied by complex matrix A. The norm–resolvent limit of this not necessarily self–adjoint operator is discussed in this thesis. Furthermore, the rigorous ex pression for the norm resolvent limit is compared to the formal limit of the Dirac operator with non–local potential. This formal limit corresponds to the norm–resolvent limit. In other words, renormalization of the coupling constant does not occur. This property will lead to generalization of the definition of the Dirac operator with relativistic point interaction. Moreover, the spectrum of this newly defined operator is discussed. Remarkable spectral transitions in special cases will be presented. Finally, this spectral transition will be explained by examining ε–pseudospectrum of the operator.