Domains for Dirac-Coulomb min-max levels. (English) Zbl 1450.81039

Esteban, Maria J.; Lewin, Mathieu; Séré, Éric

Rev. Mat. Iberoam. 35, No. 3, 877-924 (2019).

The authors start with showing self-adjointness of Dirac operators with Coulomb potentials both in a classic way going back to G. Nenciu [Commun. Math. Phys. 48, 235–247 (1976; Zbl 0349.47014)] and by a more recent variational method of M. J. Esteban and M. Loss [J. Math. Phys. 48, No. 11, 112107, 8 p. (2007; Zbl 1152.81423)].
Their main result is the variational characterization of the eigenvalues in the gap. While there were several previous partial results, a solution up to the critical constant was only offered by S. Morozov and D. Müller [Math. Z. 280, No. 3–4, 733–747 (2015; Zbl 1320.49032)] and D. Müller [Doc. Math. 21, 1151–1169 (2016; Zbl 1350.49074)]. The important new contribution of the authors concerns the allowed test functions in the variational principle. They show that it suffices to use functions which are easy to characterize, like smooth functions of compact support, as opposed to the two previously mentioned references of Morozov and Müller who require a characterization of the domain.

Reviewer: Heinz Siedentop (München)

MSC:

81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
81R20	Covariant wave equations in quantum theory, relativistic quantum mechanics
35P05	General topics in linear spectral theory for PDEs
35A15	Variational methods applied to PDEs
81V45	Atomic physics
47B25	Linear symmetric and selfadjoint operators (unbounded)

Keywords:

Dirac operator; Coulomb potential; selfadjointness; minimax

Citations:

Zbl 0349.47014; Zbl 1152.81423; Zbl 1320.49032; Zbl 1350.49074

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