Spectral properties of Dirac operators on certain domains

Abstract

We are interested in spectral study of perturbations of Dirac operators on certaindomains. The majority of the studies carried out in this thesis are established through the study ofthe resolvents of these operators. On one hand, we introduce Poincaré-Steklov (PS) operators, whichappear naturally in the study of Dirac operators with MIT bag boundary conditions, and analyzethem from a microlocal point of view (Chapter 2). On the other hand, our study focuses on thethree-dimensional Dirac operator coupled with a singular delta interactions: Chapter 3 is devoted toan approximation of the confining version of Dirac operator coupled with purely Lorentz scalar deltashell interactions. Chapters 2 and 3 deal with the large mass limit (supported on a fixed domain anda domain whose thickness tends to zero). Chapter 4 also generalizes an approximation of the nonconfiningversion of Dirac operator coupled with a singular combination of electrostatic and Lorentzscalar delta interactions by a Dirac operator with regular local interaction. Finally, in two-dimension,we develop a new technique that allows us to prove, for combinations of delta interactions supportedon non-smooth curves, the self-adjointness of the realization of the Dirac operator underconsideration, in Sobolev space of order one-half.