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Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei

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Abstract

We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kreĭn-Višik-Birman extension scheme, or also on Grubb’s universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.

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Notes

  1. In fact, with a slightly more elaborate argument we can better estimate the reminder in (2.13) as a \(O(r^{1-B})\) term; however, this is not needed in the analysis that follows.

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Acknowledgements

We are indebted to Naiara Arrizabalaga, Gianfausto Dell’Antonio, Marko Erceg, Diego Noja, and Giulio Ruzza for many instructive and inspirational discussions on the subject of this work.

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  1. International School for Advanced Studies – SISSA, via Bonomea 265, 34136, Trieste, Italy

    Matteo Gallone & Alessandro Michelangeli

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  1. Matteo Gallone
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  2. Alessandro Michelangeli
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Correspondence to Alessandro Michelangeli.

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All authors declare that they have no conflict of interest.

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This work was partially supported by the 2014-2017 MIUR-FIR grant “Cond-Math: Condensed Matter and Mathematical Physics” code RBFR13WAET.

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Gallone, M., Michelangeli, A. Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei. Anal.Math.Phys. 9, 585–616 (2019). https://doi.org/10.1007/s13324-018-0219-7

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