Self-Adjoint Extensions of Dirac Operator with Coulomb Potential

  • Matteo GalloneEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 18)


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 realisations of the model, and which are worth addressing in the future.


Dirac-Coulomb operator Distinguished self-adjoint extension Self-adjoint extensions von Neumann extension theory 



This work is partially supported by INdAM (GNFM) and by the 2014–2017 MIUR-FIR grant Cond-Math: Condensed Matter and Mathematical Physics code RBFR13WAET.


  1. 1.
    M. Arai, On essential self-adjointness of Dirac operators. RIMS Kokyuroku, Kyoto Univ. 242, 10–21 (1975)Google Scholar
  2. 2.
    M. Arai, On essential self-adjointness, distinguished self-adjoint extension and essential spectrum of Dirac operators with matrix valued potentials. Publ. RIMS, Kyoto Univ. 19, 33–57 (1983)Google Scholar
  3. 3.
    N. Arrizabalaga, Distinguished self-adjoint extensions of Dirac operators via Hardy-Dirac inequalities. J. Math. Phys. 52, 092301 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    N. Arrizabalaga, J. Duoandikoetxea, L. Vega, Self-Adjoint extensions of Dirac operators with Coulomb-like singularity. J. Math. Phys. 54, 041504 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P.R. Chernhoff, Schrödinger and Dirac operators with singular potentials and hyperbolic equations. Pac. J. Math. 72, 361–382 (1977)CrossRefGoogle Scholar
  6. 6.
    M. Esteban, L. Loss, Self-adjointness for Dirac operators via Hardy-Dirac inequalities. J. Math. Phys. 48(11), 112107 (2007)Google Scholar
  7. 7.
    W.D. Evans, On the unique self-adjoint extension of the Dirac operator and the existence of the Green matrix. Proc. Lond. Math. Soc. (3) 20, 537–557 (1970)Google Scholar
  8. 8.
    G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup. Pisa (3), 22, 425–513 (1968)Google Scholar
  9. 9.
    G. Grubb, Distributions and Operators, vol. 252 of Graduate Texts in Mathematics (Springer, New York, 2009)Google Scholar
  10. 10.
    K.E. Gustafson, P.A. Rejtö, Some essentially self-adjoint Dirac operators with spherically symmetric potentials. Isr. J. Math. 14, 63–75 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. Hogreve, The overcritical Dirac-Coulomb Operator. J. Phys. A Math. Theor. 46, 025301 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    T. Kato, Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Am. Math. Soc. 70, 195–211 (1951)zbMATHGoogle Scholar
  13. 13.
    T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966)CrossRefzbMATHGoogle Scholar
  14. 14.
    M. Klaus, R. Wüst, Characterization and uniqueness of distinguished self-adjoint extensions of Dirac operators. Commun. Math. Phys., 64, 171–176 (1979)CrossRefzbMATHGoogle Scholar
  15. 15.
    J.J. Landgren, P.A. Rejtö, On a theorem of Jörgens and Chernoff concerning essential self-adjointness of Dirac operators. J. Reine Angew. Math. 332, 1–14 (1981)zbMATHGoogle Scholar
  16. 16.
    A. Michelangeli, The Kreĭn-Višik-Birman self-adjoint extension theory revisited. SISSA preprint 59/2015/MAT (2015)
  17. 17.
    G. Nenciu, Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms. Commun. Math. Phys. 48, 235–247 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. 2 (Academic Press, New York, 1975)zbMATHGoogle Scholar
  19. 19.
    P.A. Rejtö, Some essentially self-adjoint one-electron Dirac operators. Isr. J. Math. 9, 144–171 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    F. Rellich, Eigenwerttheorie Partieller Differentialgleichungen II (Vorlesungsmanuskript, Göttingen, 1953)Google Scholar
  21. 21.
    B.W. Roos, W.C. Sangren, Spectral theory of Dirac’s radial relativistic wave equations. J. Math. Phys. 3, 702–723 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    U.W. Schmincke, Essential selfadjointness of Dirac operators with a strongly singular potential. Math. Z. 126, 71–81 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    U.W. Schmincke, Distinguished self-adjoint extensions of Dirac Operators. Math. Z. 129, 335–349 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    B. Thaller, The Dirac Equation (Springer, Berlin, 1992)CrossRefzbMATHGoogle Scholar
  25. 25.
    J. Weidmann Oszillationsmethoden für Systeme gewöhnlicher Differentialgleichungen. Math. Z. 119, 349–373 (1971)Google Scholar
  26. 26.
    J. Weidmann, Spectral Theory of Ordinary Differential Operators (Springer, Berlin, 1987)CrossRefzbMATHGoogle Scholar
  27. 27.
    R. Wüst, Distinguished self-adjoint extensions of Dirac operators constructed by means of cut-off potentials. Math. Z. 141, 93–98 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    R. Wüst, Dirac Operations with Strongly Singular Potentials–distinguished self-adjoint extensions constructed with a spectral gap theorem and cut-off potentials. Math. Z. 152, 259–271 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    J. Xia, On the contribution of the coulomb singularity of arbitrary charge to the Dirac Hamiltonian. Trans. Am. Math. Soc. 351, 1989–2023 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.SISSATriesteItaly

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