Dirac–Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas

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

doi:10.5802/ahl.106

Esteban, Maria J.¹; Lewin, Mathieu ¹; Séré, Éric ²

¹ CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75016 Paris (France)
² CEREMADE, Université Paris-Dauphine, PSL Research University, CNRS, Place de Lattre de Tassigny, 75016 Paris (France)

Annales Henri Lebesgue, Volume 4 (2021), pp. 1421-1456.

Abstract
Résumé

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 $μ$ . We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that $μ$ 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 $μ$ being a finite sum of deltas.

Dans ce premier article nous étudions les propriétés spectrales d’un opérateur de Dirac auquel on ajoute le potentiel de Coulomb généré par une distribution de charge $μ$ quelconque. Nous montrons l’existence d’une unique extension auto-adjointe distinguée, sous la seule condition que $μ$ ne possède aucun atome de poids supérieur ou égal à un. Ensuite, lorsque la mesure $μ$ est positive nous caractérisons le domaine à l’aide d’une forme quadratique pour le spineur haut, suivant une méthode introduite par Esteban et Loss dans [EL07, EL08]. Ceci nous permet de prouver des formules de min-max pour les valeurs propres situées dans le trou spectral. La formule reste valable même dans le cas où certaines des valeurs propres ont plongé dans le spectre continu inférieur. À la fin de l’article nous discutons du cas d’une molécule, ce qui correspond à prendre $μ$ égal à une somme finie de deltas.

Received: 2020-03-14
Revised: 2021-02-04
Accepted: 2021-04-01
Published online: 2021-12-03

DOI: 10.5802/ahl.106

Classification: 81Q05, 47A10, 49J35
Mots-clés : Dirac-Coulomb operators, self-adjointness, min-max formulas

Author's affiliations:

Esteban, Maria J. ¹; Lewin, Mathieu ¹; Séré, Éric ²

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     title = {Dirac{\textendash}Coulomb operators with general charge distribution {I.} {Distinguished} extension and min-max formulas},
     journal = {Annales Henri Lebesgue},
     pages = {1421--1456},
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Esteban, Maria J.; Lewin, Mathieu; Séré, Éric. Dirac–Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas. Annales Henri Lebesgue, Volume 4 (2021), pp. 1421-1456. doi : 10.5802/ahl.106. http://www.numdam.org/articles/10.5802/ahl.106/

References
Cited by

[ADV13] Arrizabalaga, Naiara; Duoandikoetxea, Javier; Vega, Luis Self-adjoint extensions of Dirac operators with Coulomb type singularity, J. Math. Phys., Volume 54 (2013) no. 4, 041504 | DOI | MR | Zbl

[ASI + 10] Artemyev, Anton N.; Surzhykov, Andrey; Indelicato, Paul; Plunien, Günter; Stöhlker, Thomas 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 | DOI

[BDE08] Bosi, Roberta; Dolbeault, Jean; Esteban, Maria J. 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 | DOI | MR | Zbl

[BH03] Briet, Philippe; Hogreve, H. Two-centre Dirac–Coulomb operators: regularity and bonding properties, Ann. Phys., Volume 306 (2003) no. 2, pp. 159-192 | DOI | MR | Zbl

[CPWS13] Calvo, Florent; Pahl, Elke; Wormit, Michael; Schwerdtfeger, Peter Evidence for Low-Temperature Melting of Mercury owing to Relativity, Angew. Chem. Int. Ed., Volume 52 (2013) no. 29, pp. 7583-7585 | DOI

[DD88] Datta, Shambhu N.; Devaiah, G. The minimax technique in relativistic Hartree–Fock calculations, Pramana, Volume 30 (1988) no. 5, pp. 387-405 | DOI

[DELV04] Dolbeault, Jean; Esteban, Maria J.; Loss, Michael; Vega, Luis An analytical proof of Hardy-like inequalities related to the Dirac operator, J. Funct. Anal., Volume 216 (2004) no. 1, pp. 1-21 | DOI | MR | Zbl

[DES00a] Dolbeault, Jean; Esteban, Maria J.; Séré, Éric On the eigenvalues of operators with gaps. Application to Dirac operators, J. Funct. Anal., Volume 174 (2000) no. 1, pp. 208-226 | DOI | MR | Zbl

[DES00b] Dolbeault, Jean; Esteban, Maria J.; Séré, Éric Variational characterization for eigenvalues of Dirac operators, Calc. Var. Partial Differ. Equ., Volume 10 (2000) no. 4, pp. 321-347 | DOI | MR | Zbl

[DES03] Dolbeault, Jean; Esteban, Maria J.; Séré, Éric A variational method for relativistic computations in atomic and molecular physics, Int. J. Quantum Chem., Volume 93 (2003) no. 3, pp. 149-155 | DOI

[DES06] Dolbeault, Jean; Esteban, Maria J.; Séré, Éric 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 | DOI | MR | Zbl

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

[EL08] Esteban, Maria J.; Loss, Michael 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 | DOI | MR | Zbl

[ELS08] Esteban, Maria J.; Lewin, Mathieu; Séré, Éric Variational methods in relativistic quantum mechanics, Bull. Am. Math. Soc., Volume 45 (2008) no. 4, pp. 535-593 | DOI | MR | Zbl

[ELS19] Esteban, Maria J.; Lewin, Mathieu; Séré, Éric Domains for Dirac–Coulomb min-max levels, Rev. Mat. Iberoam., Volume 35 (2019) no. 3, pp. 877-924 | DOI | MR | Zbl

[ELS21] Esteban, Maria J.; Lewin, Mathieu; Séré, Éric Dirac–Coulomb operators with general charge distribution. II. The lowest eigenvalue, Proc. Lond. Math. Soc., Volume 123 (2021) no. 4, pp. 345-383 | DOI | MR

[ES99] Esteban, Maria J.; Séré, Éric Solutions of the Dirac–Fock equations for atoms and molecules, Commun. Math. Phys., Volume 203 (1999) no. 3, pp. 499-530 | DOI | MR | Zbl

[GAD10] Glantschnig, Kathrin; Ambrosch-Draxl, Claudia Relativistic effects on the linear optical properties of $A u$ , $P t$ , $P b$ and $W$ , New J. Phys., Volume 12 (2010) no. 10, 103048 | DOI

[GS99] Griesemer, Marcel; Siedentop, Heinz A minimax principle for the eigenvalues in spectral gaps, J. Lond. Math. Soc., Volume 60 (1999) no. 2, pp. 490-500 | DOI | MR | Zbl

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

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

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

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

[KW79] Klaus, Martin; Wüst, Rainer Characterization and uniqueness of distinguished selfadjoint extensions of Dirac operators, Commun. Math. Phys., Volume 64 (1978/79) no. 2, pp. 171-176 | DOI | MR | Zbl

[KW79] Klaus, Martin; Wüst, Rainer Spectral properties of Dirac operators with singular potentials, J. Math. Anal. Appl., Volume 72 (1979) no. 1, pp. 206-214 | DOI | MR | Zbl

[McC13] McConnell, Sean R. Two centre problems in relativistic atomic physics, Ph. D. Thesis, University of Heidelberg, Germany (2013)

[MM15] Morozov, Sergey; Müller, David On the minimax principle for Coulomb–Dirac operators, Math. Z., Volume 280 (2015) no. 3, pp. 733-747 | DOI | MR | Zbl

[Mül16] Müller, David 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 | Zbl

[Nen76] Nenciu, Gheorghe 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 | DOI | MR | Zbl

[Nen77] Nenciu, Gheorghe 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

[RS75] Reed, Michael; Simon, Barry Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness, Academic Press Inc., 1975 | MR | Zbl

[Sch72] Schmincke, Upke-Walther Distinguished selfadjoint extensions of Dirac operators, Math. Z., Volume 129 (1972), pp. 335-349 | DOI | MR | Zbl

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

[SST20] Schimmer, Lukas; Solovej, Jan Philip; Tokus, Sabiha Friedrichs Extension and Min-Max Principle for Operators with a Gap, Ann. Henri Poincaré, Volume 21 (2020) no. 2, pp. 327-357 | DOI | MR | Zbl

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

[Tha92] Thaller, Bernd The Dirac equation, Texts and Monographs in Physics, Springer, 1992 | DOI | MR

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

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

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

[Wüs77] Wüst, Rainer 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 | Zbl

[ZEP11] Zaleski-Ejgierd, Patryk; Pyykkö, Pekka Relativity and the mercury battery, Phys. Chem. Chem. Phys., Volume 13 (2011) no. 37, pp. 16510-16512 | DOI

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