A strategy for self-adjointness of Dirac operators: Applications to the MIT bag model and delta-shell interactions

TL;DR

This paper presents a new method to establish the self-adjointness of Dirac operators with boundary conditions, with applications to the MIT bag model and delta-shell interactions, using layer potentials and Calderón projectors.

Contribution

It introduces a novel approach involving layer potentials and Calderón projectors to prove self-adjointness of Dirac operators with boundary and transmission conditions.

Findings

Successfully applied to MIT bag model

Extended to electrostatic delta-shell interactions

Provides a unified framework for boundary conditions

Abstract

We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a $\mathcal{C}^2$-compact surface without boundary. To do so we are lead to study the layer potential induced by the Dirac system as well as to define traces in a weak sense for functions in the appropriate Sobolev space. Finally, we introduce Calder\'on projectors associated with the problem and illustrate the method in two special cases: the well-known MIT bag model and an electrostatic $\delta$-shell interaction.