Abstract
In this article, Dirac operators A_{eta , tau } coupled with combinations of electrostatic and Lorentz scalar delta -shell interactions of constant strength eta and tau , respectively, supported on compact surfaces Sigma subset mathbb {R}^3 are studied. In the rigorous definition of these operators, the delta -potentials are modeled by coupling conditions at Sigma . In the proof of the self-adjointness of A_{eta , tau }, a Krein-type resolvent formula and a Birman–Schwinger principle are obtained. With their help, a detailed study of the qualitative spectral properties of A_{eta , tau } is possible. In particular, the essential spectrum of A_{eta , tau } is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of A_{eta , tau } is computed and it is discussed that for some special interaction strengths, A_{eta , tau } is decoupled to two operators acting in the domains with the common boundary Sigma .
Highlights
Working with the equations of motion, there is a particular interest to find solutions which are exact and which correspond to specific physical systems
Our goal is to study Dirac operators acting in L2(R3)4 which are formally given by
After presenting some preliminary material on integral operators which are associated to the Green function of the resolvent of the free Dirac operator, we introduce in Sect. 3 the operator Aη,τ in a mathematically rigorous way via the coupling condition (1.3)
Summary
Working with the equations of motion, there is a particular interest to find solutions which are exact and which correspond to specific physical systems. While some of the interesting properties of Aη,τ like the decoupling of the operator to two Dirac operators acting in ± for interaction strengths satisfying η2 − τ 2 = −4c2 were observed in [20], compare Lemma 3.1, others like, e.g., unexpected spectral effects for η2 − τ 2 = 4c2 could not be seen with this approach due to the decomposition to the spherical harmonics. It took 25 years until Dirac operators with singular interactions supported on more general surfaces in R3 were studied. + ηδ and which gives another justification that the jump condition (1.3) models the δ-potential correctly
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
