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On the energy of a large atom. (English) Zbl 0722.35072

We announce a proof of an asymptotic formula for the groundstate energy of a large atom. The early work of Thomas-Fermi, Hartree-Fock, Dirac, and Scott predicted that for an atomic number Z, the energy is \(E(Z)\approx - c_ 0Z^{7/3}+c_ 1Z^ 2-c_ 2Z^{5/3}\) for known \(c_ 0,c_ 1\), and \(c_ 2\) [see E. H. Lieb, Rev. Mod. Phys 53, 603-641 (1981)]. J. Schwinger [Phys. Rev. A 24, 2353-2361 (1981)] observed an additional effect and set down the modified formula \(E(Z)\approx -c_ 0Z^{7/3}+c_ 1Z^ 2-(10/9)c_ 2Z^{5/3}\). Our proof shows that Schwinger’s formula is correct.

MSC:

35Q40 PDEs in connection with quantum mechanics
81V45 Atomic physics
81Q15 Perturbation theories for operators and differential equations in quantum theory
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[1] Webster Hughes, An atomic energy lower bound that agrees with Scott’s corrections, Adv. Math. 79 (1990), no. 2, 213 – 270. · Zbl 0715.46046
[2] C. L. Fefferman and L. A. Seco, An upper bound for the number of electrons in a large ion, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 10, 3464 – 3465.
[3] Charles L. Fefferman and Luis A. Seco, Asymptotic neutrality of large ions, Comm. Math. Phys. 128 (1990), no. 1, 109 – 130. · Zbl 0695.60098
[4] Elliott H. Lieb, A lower bound for Coulomb energies, Phys. Lett. A 70 (1979), no. 5-6, 444 – 446.
[5] Elliott H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys. 53 (1981), no. 4, 603 – 641. , https://doi.org/10.1103/RevModPhys.53.603 Elliott H. Lieb, Erratum: ”Thomas-Fermi and related theories of atoms and molecules”, Rev. Modern Phys. 54 (1982), no. 1, 311. · Zbl 1049.81679
[6] Elliott H. Lieb and Barry Simon, The Thomas-Fermi theory of atoms, molecules and solids, Advances in Math. 23 (1977), no. 1, 22 – 116. · Zbl 0938.81568
[7] Julian Schwinger, Thomas-Fermi model: the second correction, Phys. Rev. A (3) 24 (1981), no. 5, 2353 – 2361.
[8] L. A. Seco, I. M. Sigal, and J. P. Solovej, Bound on the ionization energy of large atoms, Comm. Math. Phys. 131 (1990), no. 2, 307 – 315. · Zbl 0714.35059
[9] Heinz Siedentop and Rudi Weikard, On the leading energy correction for the statistical model of the atom: interacting case, Comm. Math. Phys. 112 (1987), no. 3, 471 – 490. · Zbl 0920.35120
[10] Heinz Siedentop and Rudi Weikard, On the leading correction of the Thomas-Fermi model: lower bound, Invent. Math. 97 (1989), no. 1, 159 – 193. With an appendix by A. M. Klaus Müller. · Zbl 0689.34011
[11] H. Siedentop and R. Weikard, A lower bound of Scott type by a new microlocalization technique (to appear). · Zbl 0802.35128
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