Distinguished self-adjoint extensions of Dirac operators via Hardy-Dirac inequalities

Abstract

We prove some Hardy-Dirac inequalities with two different weights including measure valued and Coulombic ones. Those inequalities are used to construct distinguished self-adjoint extensions of Dirac operators for a class of diagonal potentials related to the weights in the above mentioned inequalities.2000 Mathematics Subject Classification. 81Q10, 35P05, 35Q40.

Self-adjoint extensions for the Dirac operator with Coulomb-type spherically symmetric potentials

“…In this context the case δ > 0 is sub-critical, while it is critical if δ = 0 for some k and supercritical if δ < 0 for some k. This formulation of criticality is different from the one in [14,3,2] but it appears to be suited to this problem, where a particular structure of V is assumed. In fact, in the particular case that λ = ν = 0 and V = µ |x| β for all µ ∈ R, the operator H is essentially self-adjoint on C ∞ c (R 3 ) 4 and self-adjoint on D(H 0 ) = H 1 (R 3 ) 4 , see Corollary 1.6.…”

Discrete spectra for critical Dirac-Coulomb Hamiltonians

“…[13,Section 2]), one sees that the operator h m j ,κ j is essentially self-adjoint in the Hilbert space H m j ,κ j if and only if 13) and it has deficiency indices (1, 1) otherwise. Thus, the operator h 1 2 ,1 ⊕ h 1 2 ,−1 ⊕ h − 1 2 ,1 ⊕ h − 1 2 ,−1 , and hence H itself, has deficiency indices (4,4), and therefore a 16-real-parameter family of self-adjoint extensions.…”

Boundary triples for the Dirac operator with Coulomb-type spherically symmetric perturbations

“…with b ∈ R and a < 1/2, see [20,Theorem V 5.10]. In [3,4,19] more general 4 × 4 matrixvalued measured functions V are considered, in the assumption that |x||V(x)| ≤ ν < 1, and a distinguished self-adjoint extension (in the sense of (1.5)) is constructed, exploiting the Kato-Nenciu inequality…”