On the spectral properties of Dirac operators with electrostatic δ-shell interactions

Abstract

In this paper the spectral properties of Dirac operators Aη with electrostatic δ-shell interactions of constant strength η supported on compact smooth surfaces in R 3 are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of Aη are investigated. In particular, it turns out that the discrete spectrum of Aη inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of Aη and the free Dirac operator A 0 is trace class, and in the nonrelativistic limit Aη converges in the norm resolvent sense to a Schrödinger operator with an electric δ-potential of strength η.2010 Mathematics Subject Classification. Primary 81Q10; Secondary 35Q40. Key words and phrases. Dirac operator; existence and completeness of wave operators; finite discrete spectrum; nonrelativistic limit; quasi boundary triple; self-adjoint extension; shell interaction. 1 corresponding Birman-Schwinger principle; for more details and additional results see Theorem 4.4.Theorem 1.1. Let η ∈ R \ {±2c} and let A η be the Dirac operator with an electrostatic δ-shell interaction of strength η. Then the essential spectrum is given byand the discrete spectrum in the gap (−mc 2 , mc 2 ) is finite, that is,The next result on the trace class property of the difference of the third powers of the resolvents of A η and A 0 has important consequences for mathematical scattering theory. In particular, it follows that the wave operators for the scattering system {A η , A 0 } exist and are complete and that the absolutely continuous parts of A η and A 0 are unitarily equivalent. For more details see Theorem 4.6, where also a trace formula in terms of the Weyl function and its derivatives is provided.Theorem 1.2. Let η ∈ R\{±2c}, let A η be the Dirac operator with an electrostatic δ-shell interaction and let λ ∈ ρ(A η ) ∩ ρ(A 0 ). Then the operatorbelongs to the trace class ideal.

Limiting absorption principle and scattering matrix for Dirac operators with δ-shell interactions

“…While, as mentioned above, the spectral properties of A η,τ were investigated, there are hardly no results on scattering theory. Only the existence and completeness of the wave operators was shown in the case of electrostatic δ-shell interactions (τ = 0) in [9] under C ∞ -smoothness assumptions on the surface Γ; this result was extended in [10,Proposition 4.7] for combinations of electrostatic and scalar potentials. For this reason, we are concerned in this work with the direct scattering problem for the couple (A η,τ , A 0 ).…”

“…In this section we recall the definition and some of the basic properties of Dirac operators which are coupled with a combination of electrostatic and Lorentz scalar δ-shell potentials, as they were treated, e.g., in [5,9,10]. Let ν be the unit normal vector field at Γ pointing outwards of Ω + .…”

“…This problem has been recently reconsidered in [4,5,6], where in the case of a C 2 -surface the self-adjointness and several properties of Dirac operators with electrostatic δ-perturbations are derived. An alternative construction of Dirac operators with electrostatic and Lorentz scalar δ-shell interactions was proposed in [9] and further developed in [10,11]. This approach is based on the method of quasi boundary triples, originally introduced in [13] for the study of elliptic partial differential operators.…”

Dirac operators with Lorentz scalar shell interactions

“…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.…”