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

Abstract

We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a 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 Calderón projectors associated with the problem and illustrate the method in two special cases: the well-known MIT bag model and an electrostatic δ-shell interaction.

The relativistic spherical δ-shell interaction in R3: Spectrum and approximation

“…Comparing this work with the general abstract theory given in [15], one could suppose that this kind of interaction is forcing g to be in H 1/2 (Σ). Indeed, recently, in [14] they proved that this conjecture is true. Moreover they also define the domain of δ-shell Dirac operator when the coupling constant λ = ±2.…”

Dirac operators with Lorentz scalar shell interactions

“…i.e. A m,±2 is the orthogonal sum of Dirac operators in Ω ± with MIT bag boundary conditions as studied, e.g., in [2,21]. Using the language of [2], for τ = 2 and m > 0 one recovers the MIT bag operator with the positive mass m in Ω + and the one with a negative mass (−m) in Ω − (and vice versa for τ = −2).…”

“…The analysis was based mostly on the usage of potential operators involving the fundamental solution of the unperturbed Dirac equation. In [7,8,21], the study was pushed further in order to understand the Sobolev regularity of functions in the domain, the δ-shell potential being then encoded by a transmission condition at the shell. Furthermore, as for Schrödinger operators with δ-potentials [6], the shell interactions in the Dirac setting can be understood as suitable limits of regular potentials localized near the surface, as it was shown recently in [18,19].…”

“…Section 2 is devoted to the definition of the operator and to the proof of the assertions (A) and (B), see Theorem 2.3. The proofs are mostly based on the use of singular integral operators previously studied in [21] and some resolvent machineries already used in a similar (but different) context in [7,8]. In Section 3 we deal with a more detailed study of the discrete spectrum.…”

Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors

“…Because the Dirac operator is an elliptic operator of order one, one expect this domain to be contained in the usual Sobolev space H 1 . Of course, it depends on the boundary conditions and it is true for the MIT bag model for sufficiently smooth domains, as proved in [4] for C 2,1 -smooth domains and in [26] for C 2 -smooth domains. Moreover, when one deals with C ∞ -smooth domains more general results can be found in [6,Thm.…”