Mahdi Zreik will defend his thesis on Monday, 13th May

• The defence will take place at the Conference Room at the Institute of Mathematics of Bordeaux (IMB), with collaboration from the University of the Basque Country (UPV/EHU) and BCAM

Mahdi Zreik embarked on his academic journey by graduating with a bachelor’s degree in mathematics from the University of Lebanon in 2018. He subsequently earned his master’s degree in Pure Mathematics from the same university a year later. Continuing his education, he pursued a master’s degree in Fundamental and Applied Mathematics, with a specialisation in Analysis and Probability, at the University of Nantes, France.

Currently, Zreik works as a PhD candidate at the Basque Center for Applied Mathematics (BCAM) in the field of Applied Mathematics and scientific computing, where he joined in October 2023.

His thesis, titled "Spectral Properties of Dirac Operators on Some Domains," is under the expert guidance of Vincent Bruneau (UB) & Luis Vega (BCAM&UPV/EHU).

The defence is scheduled for Monday, 13th May at the University of Bordeaux at 14:00h.

On behalf of all members of BCAM, we would like to wish Mahdi the best of luck in defending his thesis.

Abstract

This thesis mainly focused on the spectral analysis of perturbation models of the free Dirac operator, in 2-D and 3-D space. More precisely, this thesis is divided into two parts: Dirac operator with MIT bag conditions  and Dirac operator coupled with delta shell interactions. Most of these studies are conducted through the analysis of the resolvents of these operators. In the first part, we introduce the Poincaré-Steklov (PS) operators, which appear naturally in the study of Dirac operators with MIT bag boundary conditions, and analyze them from a microlocal point of view. Secondly, our study focuses on the three-dimensional Dirac operator coupled with a singular delta interactions, in which we establish an approximation of the confining version of Dirac operator coupled with purely Lorentz scalar delta shell interactions. This first part deals with the large mass limit (supported on a fixed domain and a domain whose thickness tends to zero). In the second part of this thesis, we also generalize an approximation of the non-confining version of Dirac operator coupled with a singular combination of electrostatic and Lorentz scalar 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 supported on non-smooth curves, the self-adjointness of the realization of the Dirac operator under consideration, in Sobolev space of order one-half.