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### Abstract

This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution $\mu $. We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that $\mu $ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain using a quadratic form associated with the upper spinor, following earlier works [EL07, EL08] by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigenvalues have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac–Coulomb operators corresponding to $\mu $ being a finite sum of deltas.

### References

[ADV13] Self-adjoint extensions of Dirac operators with Coulomb type singularity, J. Math. Phys., Volume 54 (2013) no. 4, 041504 | Article | MR 3088216 | Zbl 1281.81031

[ASI + 10] Finite basis set approach to the two-centre Dirac problem in Cassini coordinates, J. Phys. B: At. Mol. Opt. Phys., Volume 43 (2010) no. 23, 235207 | Article

[BDE08] Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal., Volume 7 (2008) no. 3, pp. 533-562 | Article | MR 2379440 | Zbl 1162.35329

[BH03] Two-centre Dirac–Coulomb operators: regularity and bonding properties, Ann. Phys., Volume 306 (2003) no. 2, pp. 159-192 | Article | MR 1991620 | Zbl 1029.81024

[CPWS13] Evidence for Low-Temperature Melting of Mercury owing to Relativity, Angew. Chem. Int. Ed., Volume 52 (2013) no. 29, pp. 7583-7585 | Article

[DD88] The minimax technique in relativistic Hartree–Fock calculations, Pramana, Volume 30 (1988) no. 5, pp. 387-405 | Article

[DELV04] An analytical proof of Hardy-like inequalities related to the Dirac operator, J. Funct. Anal., Volume 216 (2004) no. 1, pp. 1-21 | Article | MR MR2091354 | Zbl 1060.35120

[DES00a] On the eigenvalues of operators with gaps. Application to Dirac operators, J. Funct. Anal., Volume 174 (2000) no. 1, pp. 208-226 | Article | MR MR1761368 | Zbl 0982.47006

[DES00b] Variational characterization for eigenvalues of Dirac operators, Calc. Var. Partial Differ. Equ., Volume 10 (2000) no. 4, pp. 321-347 | Article | MR 1767717 | Zbl 0968.49025

[DES03] A variational method for relativistic computations in atomic and molecular physics, Int. J. Quantum Chem., Volume 93 (2003) no. 3, pp. 149-155 | Article

[DES06] General results on the eigenvalues of operators with gaps, arising from both ends of the gaps. Application to Dirac operators, J. Eur. Math. Soc., Volume 8 (2006) no. 2, pp. 243-251 | Article | MR MR2239275 | Zbl 1157.47306

[EL07] Self-adjointness for Dirac operators via Hardy–Dirac inequalities, J. Math. Phys., Volume 48 (2007) no. 11, 112107 | MR MR2370231 | Zbl 1152.81423

[EL08] Self-adjointness via partial Hardy-like inequalities, Mathematical results in quantum mechanics. Proceedings of the QMath10 conference, Moieciu, Romania, 10–15 September 2007, World Scientific, 2008, pp. 41-47 | Article | MR 2466677 | Zbl 1156.81369

[ELS08] Variational methods in relativistic quantum mechanics, Bull. Am. Math. Soc., Volume 45 (2008) no. 4, pp. 535-593 | Article | MR 2434346 | Zbl 1288.49016

[ELS19] Domains for Dirac–Coulomb min-max levels, Rev. Mat. Iberoam., Volume 35 (2019) no. 3, pp. 877-924 | Article | MR 3960263 | Zbl 1450.81039

[ELS21] Dirac–Coulomb operators with general charge distribution. II. The lowest eigenvalue, Proc. Lond. Math. Soc., Volume 123 (2021) no. 4, pp. 345-383 | Article | MR 4332485

[ES99] Solutions of the Dirac–Fock equations for atoms and molecules, Commun. Math. Phys., Volume 203 (1999) no. 3, pp. 499-530 | Article | MR MR1700174 | Zbl 0938.35149

[GAD10] Relativistic effects on the linear optical properties of $Au$, $Pt$, $Pb$ and $W$, New J. Phys., Volume 12 (2010) no. 10, 103048 | Article

[GS99] A minimax principle for the eigenvalues in spectral gaps, J. Lond. Math. Soc., Volume 60 (1999) no. 2, pp. 490-500 | Article | MR 1724845 | Zbl 0952.47022

[HK83] On the double-well problem for Dirac operators, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 38 (1983) no. 2, pp. 153-166 | Numdam | MR 705337 | Zbl 0529.35062

[Kar85] Generalized Dirac-operators with several singularities, J. Oper. Theory, Volume 13 (1985) no. 1, pp. 171-188 | MR 768311 | Zbl 0569.47044

[Kat83] Holomorphic families of Dirac operators, Math. Z., Volume 183 (1983) no. 3, pp. 399-406 | Article | MR 706397 | Zbl 0515.47006

[Kla80] Dirac operators with several Coulomb singularities, Helv. Phys. Acta, Volume 53 (1980) no. 3, pp. 463-482 | MR 611770

[KW79] Characterization and uniqueness of distinguished selfadjoint extensions of Dirac operators, Commun. Math. Phys., Volume 64 (1978/79) no. 2, pp. 171-176 | Article | MR MR519923 | Zbl 0408.47022

[KW79] Spectral properties of Dirac operators with singular potentials, J. Math. Anal. Appl., Volume 72 (1979) no. 1, pp. 206-214 | Article | MR 552332 | Zbl 0423.47014

[McC13] Two centre problems in relativistic atomic physics (2013) (Ph. D. Thesis)

[MM15] On the minimax principle for Coulomb–Dirac operators, Math. Z., Volume 280 (2015) no. 3, pp. 733-747 | Article | MR 3369348 | Zbl 1320.49032

[Mül16] Minimax principles, Hardy–Dirac inequalities, and operator cores for two and three dimensional Coulomb–Dirac operators, Doc. Math., Volume 21 (2016), pp. 1151-1169 | MR 3578211 | Zbl 1350.49074

[Nen76] Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms, Commun. Math. Phys., Volume 48 (1976) no. 3, pp. 235-247 | Article | MR MR0421456 | Zbl 0349.47014

[Nen77] Distinguished self-adjoint extension for Dirac operator with potential dominated by multicenter Coulomb potentials, Helv. Phys. Acta, Volume 50 (1977) no. 1, pp. 1-3 | MR MR0462346

[RS75] Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness, Academic Press Inc., 1975 | MR MR0493421 | Zbl 0308.47002

[Sch72] Distinguished selfadjoint extensions of Dirac operators, Math. Z., Volume 129 (1972), pp. 335-349 | Article | MR MR0326448 | Zbl 0252.35062

[Sim05] Trace ideals and their applications, Mathematical Surveys and Monographs, 120, American Mathematical Society, 2005 | MR 2154153 | Zbl 1074.47001

[SST20] Friedrichs Extension and Min-Max Principle for Operators with a Gap, Ann. Henri Poincaré, Volume 21 (2020) no. 2, pp. 327-357 | Article | MR 4056270 | Zbl 1432.49065

[Tal86] Minimax principle for the Dirac equation, Phys. Rev. Lett., Volume 57 (1986), pp. 1091-1094 | Article | MR 854208

[Tha92] The Dirac equation, Texts and Monographs in Physics, Springer, 1992 | Article | MR MR1219537

[Tix98] Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall, Bull. Lond. Math. Soc., Volume 30 (1998) no. 3, pp. 283-290 | Article | MR 1608118 | Zbl 0939.35134

[Wüs73] A convergence theorem for selfadjoint operators applicable to Dirac operators with cutoff potentials, Math. Z., Volume 131 (1973), pp. 339-349 | Article | MR 0355655 | Zbl 0274.47008

[Wüs75] Distinguished self-adjoint extensions of Dirac operators constructed by means of cut-off potentials, Math. Z., Volume 141 (1975), pp. 93-98 | Article | MR 0365233 | Zbl 0311.47020

[Wüs77] Dirac operations with strongly singular potentials. Distinguished self-adjoint extensions constructed with a spectral gap theorem and cut-off potentials, Math. Z., Volume 152 (1977) no. 3, pp. 259-271 | MR 0437948 | Zbl 0361.35051

[ZEP11] Relativity and the mercury battery, Phys. Chem. Chem. Phys., Volume 13 (2011) no. 37, pp. 16510-16512 | Article