Geodesic
Asymptotics of the ground state energy in the relativistic settings
Algebra i analiz, Tome 30 (2018) no. 3, pp. 76-92.

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The paper is aimed at deriving sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, and, in particular, calculating the relativistic Scott correction term and also Dirac, Schwinger, and relativistic correction terms. Also it is proved that the Thomas–Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.
Keywords: relativistic Schrödinger operator, heavy atoms and molecules, Thomas–Fermi theory, Scott correction term, microlocal analysis, sharp spectral asymptotics. 
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V. Ivrii. Asymptotics of the ground state energy in the relativistic settings. Algebra i analiz, Tome 30 (2018) no. 3, pp. 76-92. http://geodesic.mathdoc.fr/item/AA_2018_30_3_a4/

[1] Bach V., “Error bound for the Hartree–Fock energy of atoms and molecules”, Comm. Math. Phys., 147:3 (1992), 527–548 | DOI | MR | Zbl

[2] Daubechies I., “An uncertainty principle for fermions with generalized kinetic energy”, Comm. Math. Phys., 90:4 (1983), 511–520 | DOI | MR | Zbl

[3] Erdös L., Fournais S., Solovej J. P., “Scott correction for large atoms and molecules in a self-generated magnetic field”, Comm. Math. Phys., 312:3 (2012), 847–882 | DOI | MR | Zbl

[4] Erdös L., Fournais S., Solovej J. P., “Relativistic Scott correction in self-generated magnetic fields”, J. Math. Phys., 53 (2012), 095202 | DOI | MR | Zbl

[5] Frank R. L., Lieb E. H., Seiringer R., “Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators”, J. Amer. Math. Soc., 21:4 (2008), 925–950 | DOI | MR | Zbl

[6] Frank R. L., Siedentop H., Warzel S., “The ground state energy of heavy atoms: relativistic lowering of the leading energy correction”, Comm. Math. Phys., 278:2 (2008), 549–566 | DOI | MR | Zbl

[7] Graf G. M., Solovej J. P., “A correlation estimate with applications to quantum systems with Coulomb interactions”, Rev. Math. Phys., 6:5a (1994), 977–997 | DOI | MR | Zbl

[8] Herbst I. W., “Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$”, Comm. Math. Phys., 53:3 (1977), 285–294 | DOI | MR | Zbl

[9] Ivrii V., Microlocal analysis, sharp spectral asymptotics and applications, http://www.math.toronto.edu/ivrii/monsterbook.pdf

[10] Ivrii V., Asymptotics of the ground state energy in the relativistic settings and with self-generated magnetic field, arXiv: 1708.07737

[11] Lieb E. H., Thirring W. E., “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities”, Essays Honor Valentine Bargmann, Stud. Math. Phys., Princeton Univ. Press, Princeton, NJ, 1976, 269–303

[12] Lieb E. H., Yau H. T., “The stability and instability of relativistic matter”, Comm. Math. Phys., 118:2 (1988), 177–213 | DOI | MR | Zbl

[13] Solovej J. P., Sørensen T. Ø., Spitzer W. L., “The relativistic Scott correction for atoms and molecules”, Comm. Pure Appl. Math., 63:1 (2010), 9–118 | DOI | MR