Self-Adjoint Extensions of Dirac Operator with Coulomb Potential

@article{Gallone2017SelfAdjointEO,
  title={Self-Adjoint Extensions of Dirac Operator with Coulomb Potential},
  author={Matteo Gallone},
  journal={arXiv: Mathematical Physics},
  year={2017},
  pages={169-185}
}
  • M. Gallone
  • Published 5 October 2017
  • Mathematics
  • arXiv: Mathematical Physics
In this note we give a concise review of the present state-of-art for the problem of self-adjoint realisations for the Dirac operator with a Coulomb-like singular scalar potential V (x) = ϕ(x)I4. We try to follow the historical and conceptual path that leads to the present understanding of the problem and to highlight the techniques employed and the main ideas. In the final part we outline a few major open questions that concern the topical problem of the multiplicity of self-adjoint… 
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References

SHOWING 1-10 OF 31 REFERENCES

On the contribution of the Coulomb singularity of arbitrary charge to the Dirac Hamiltonian

. We study the self-adjoint extensions of the Dirac operator α (cid:1) ( p − B )+ µ 0 β − W , where the electrical potential W contains a Coulomb singularity of arbitrary charge and the magnetic…

SELF-ADJOINT EXTENSIONS OF DIRAC OPERATORS WITH COULOMB TYPE SINGULARITY

In this work we construct self-adjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by…

Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms

Some general results about perturbations of not-semibounded self-adjoint operators by quadratic forms are obtained. These are applied to obtain the distinguished self-adjoint extension for Dirac…

Some essentially self-adjoint one-electron dirac operators

A class of smooth functions is introduced and it is shown that, the one-electron Dirac operator corresponding to an element with atomic number less than 102, is essentially self-adjoint on this class…

The overcritical Dirac–Coulomb operator

Considering relativistic hydrogenic systems with one electron moving around a fixed nuclear centre of charge Ze0/α, we analyse the corresponding Dirac–Coulomb operator; our focus here lies on the…

Characterization and uniqueness of distinguished self-adjoint extensions of Dirac operators

Distinguished self-adjoint extensions of Dirac operators are characterized by Nenciu and constructed by means of cut-off potentials by Wüst. In this paper it is shown that the existence and a more…

Self-adjointness of Dirac operators via Hardy-Dirac inequalities

Distinguished selfadjoint extensions of Dirac operators are constructed for a class of potentials including Coulombic ones up to the critical case, $-|x|^{-1}$. The method uses Hardy-Dirac…

Fundamental properties of Hamiltonian operators of Schrödinger type

Introduction. The fundamental quality required of operators representing physical quantities in quantum mechanics is that they be hypermaximal(l) or self-adjoint(2) in the strict sense employed in…

Spectral Theory of Ordinary Differential Operators

Formally self-adjoint differential expressions.- Appendix to section 1: The separation of the Dirac operator.- Fundamental properties and general assumptions.- Appendix to section 2: Proof of the…

Spectral Theory of Dirac's Radial Relativistic Wave Equation

The analytical methods developed by Weyl and Titchmarsh for the study of the Sturm‐Liouville equation are extended to the investigation of the spectral properties of the Dirac radial wave equation.…