Spectral analysis of Dirac operators on bounded domains.

Abstract

This thesis is devoted to the spectral study of two types of perturbation of theDirac operator, which are singular from the point of view of scaling.In the first part of this thesis, we consider the coupling of the Dirac operator with a combinationof delta-shell interactions of electrostatic, Lorentz scalar, and magnetic type supportedeither on regular compact surfaces or locally deformed hyperplanes. We develop an approachbased on regularization techniques that will allow us to describe the self-adjoint realizationof the perturbed Dirac operator for any combination of the coupling constants. We then investigatethe qualitative spectral properties of the various models using a Birman-Schwingerprinciple and a Krein-type formula relating the resolvent of the perturbed operator to thatof the free Dirac operator, and we pay special attention to the case of critical combinationsof coupling constants and those that give rise to the phenomenon of confinement.In the second part, we study the coupling of the Dirac operator with non-critical combinationsof delta interactions supported on non-regular compact surfaces. We first generalizethe results obtained in the context of regular surfaces to the case of surfaces locally coincidentwith the graph of a Lipschitz function whose gradient is bounded and has vanishing meanoscillations. For this we use some techniques from harmonic analysis, potential theory andFredholm¿s theory. Moreover, in the case of Hölder surfaces, we show how the smoothnessof the surface supporting the delta interactions affects the Sobolev regularity of the domainof the operator under consideration. In a second step, we investigate delta-interactions supportedon surfaces satisfying certain weak topological conditions. We first study the Diracoperator coupled with the electrostatic and Lorentz scalar delta-shell interactions supportedon uniformly rectifiable surfaces. Under certain conditions on the coupling constants, weprove the self-adjointness fo the perturbed operator and we establish several spectral propertiesin the Lipschitz case. In particular, we determine the essential spectrum of the perturbedoperator and we show that at most a finite number of eigenvalues can appear in the gap.Moreover, we fit these results to other delta-shell interactions and derive several models ofDirac operators that give rise to the confinement phenomenon.In the third part of this thesis, we are concerned the study of the pseudodifferentialproperties of Poincaré-Steklov (PS) operators associated with the Dirac operator with theMIT bag boundary condition. First, we show that the PS operators fit well into the frameworkof classical pseudodifferential operators. Then, we study the PS operators from the pointof view of semiclassical pseudodifferential operators, where the semiclassical parameter isgiven by the inverse of the mass. In particular, using some regularity properties of the MITbag operator, we show that the PS operators are zero-order semiclassical pseudodifferentialoperators, and we determine their semiclassical principal symbols. In a second step, we studythe Dirac operator coupled with a potential depending on an additional mass and supportedoutside a regular domain. When the additional mass is large enough, using the symboliccalculus and the properties of the PS operators, we establish a Krein-type formula relatingthe resolvent of the perturbed operator to that of the MIT bag operator. With its help, weshow that the perturbed operator converges in the norm resolvent sense towards the MITbag operator and give a sharp estimate of the convergence rate.