Number Theory and Atomic Densities

  • Charles L. Fefferman
  • Luis A. Seco
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 109)


After the initial success to explain the hydrogen atom, one of the early challenges of quantum mechanics was to study larger atoms. The problems encountered in this process were numerous, and the quest for an understanding quickly became a search for simplified quantum atomic models that would explain different properties of the atom.


Atomic Density Semiclassical Approximation Analytic Number Theory Large Atom Schrodinger Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [Ba]
    Bach, V., “Accuracy of Mean Field Approximations for Atoms and Molecules”, Comm. Math. Phys. 155, no. 2, 295–310 (1993).MathSciNetMATHCrossRefGoogle Scholar
  2. [CFS1]
    Córdoba, A., Fefferman, C., Seco, L., “A Trigonometric Sum relevant to the Non-relativistic Theory of Atoms”, Proc. Nat. Acad. Sci. USA 91, 5776–5778, June 1994.MATHCrossRefGoogle Scholar
  3. [CFS2]
    Córdoba, A., Fefferman, C., Seco, L., “Weyl Sums and Atomic Energy Oscillations”, Revista Matemática Iberoamericana 11, no. 1, 167–228 (1995).Google Scholar
  4. [CFS3]
    Córdoba, A., Fefferman, C., Seco, L., “A Number-Theoretic Estimate for the Thomas-Fermi Density”, Comm. P. D. E. 21 (1996), 1087–1102.MATHCrossRefGoogle Scholar
  5. [E]
    Englert, B.G., “Semiclassical Theory of the Atom”, Springer Verlag Lecture Notes in Physics 301.Google Scholar
  6. [FS1]
    Fefferman, C. and Seco, L., “The Ground-State Energy of a Large Atom”, Bull. A.M.S., 23, no. 2, 525–530, 1990.MathSciNetMATHCrossRefGoogle Scholar
  7. [FS2]
    Fefferman, C. and Seco, L., “Eigenvalues and Eigenfunctions of Ordinary Differential Operators”, Adv. Math. 95, no. 2, 145–305 (1992).MathSciNetMATHCrossRefGoogle Scholar
  8. [FS3]
    Fefferman, C. and Seco, L., “The Eigenvalue Sum for a One-Dimensional Potential”, Advances in Math 108, no. 2, 263–335, Oct 1994.MathSciNetMATHCrossRefGoogle Scholar
  9. [FS4]
    Fefferman, C. and Seco, L., “The Density in a One-Dimensional Potential”, Advances in Math. 107, no. 2, 187–236, Sep 1994.MathSciNetMATHCrossRefGoogle Scholar
  10. [FS5]
    Fefferman, C. and Seco, L., “The Eigenvalue Sum for a Three-Dimensional Radial Potential”, To appear in Adv. Math.Google Scholar
  11. [FS6]
    Fefferman, C. and Seco, L., “The Density in a Three-Dimensional Radial Potential”, Advances in Math. 111, no. 1, 88–161, March 1995.MathSciNetMATHCrossRefGoogle Scholar
  12. [FS7]
    Fefferman, C. and Seco, L., “On the Dirac and Schwinger Corrections to the Ground-State Energy of an Atom”, Advances in Math. 107, no. 1, 1–185, Aug 1994.MathSciNetMATHCrossRefGoogle Scholar
  13. [FS8]
    Fefferman, C. and Seco, L., “Aperiodicity of the Hamiltonian Flow in the Thomas-Fermi Potential”, Revista Matemática Iberoamericana 9, no. 3, 409–551 (1993).MathSciNetMATHCrossRefGoogle Scholar
  14. [FS9]
    Fefferman, C., Seco, L., “Asymptotic Neutrality of Large Ions”, Comm. Math. Phys. 128, 109–130 (1990).MathSciNetMATHCrossRefGoogle Scholar
  15. [Fef]
    Fefferman, C., “Atoms and Analytic Number Theory”, A.M.S. Centennial Publication II, 27–36 (1992).MathSciNetGoogle Scholar
  16. [Fer]
    Fermi, E., “Un Metodo Statistico per la Determinazione di alcune Priorieta dell’Atome”, Rend. Accad. Naz. Lincei 6, 602–607 (1927).Google Scholar
  17. [GS]
    Graf, G.M. and Solovej, J.P., “A Correlation Estimate with Applications to Quantum Systems with Coulomb Interactions”, Reviews in Math. Phys. 6 No. 5a, 977–997 (1994).MathSciNetMATHCrossRefGoogle Scholar
  18. [GK]
    S.W. Graham and G. Kolesnik, “Van der Corput’s Method of Exponential Sums”, Cambridge University Press. London Math. Soc. Lecture Notes Series, 126.Google Scholar
  19. [HL]
    O.J. Heilmann and E.H. Lieb, “Electron Density near the Nucleus of a large Atom”, Phys. Rev A 52, 3628–3643 (1995).CrossRefGoogle Scholar
  20. [Hug]
    Hughes, W., “An Atomic Energy Lower Bound that Agrees with Scott’s Correction”, Advances in Mathematics 79, 213–270, 1990.MathSciNetMATHCrossRefGoogle Scholar
  21. [ILS]
    A. Iantchenko, E. Lieb and H. Siedentop, “Proof of a Conjecture about Atomic and Molecular Cores Related to Scott’s Correction”, J. Reine u. Ang. Math. (in press).Google Scholar
  22. [IS]
    Ivrii, V. and Sigal, I.M., “Asymptotics of the Ground State Energies of Large Coulomb Systems “, Annals of Math. 138, no. 2, 243–335 (1993).MathSciNetMATHCrossRefGoogle Scholar
  23. [LS]
    Lieb, E. and Simon, B., “Thomas-Fermi Theory of Atoms, Molecules and Solids”, Adv. Math. 23, 22–116 (1977).MathSciNetCrossRefGoogle Scholar
  24. [L1]
    Lieb, E., “Thomas-Fermi and Related Theories of Atoms and Molecules”, Reviews of Modern Physics 53, no. 4, 603–641 (1981).MathSciNetMATHCrossRefGoogle Scholar
  25. [L2]
    Lieb, E.H., “A Lower Bound for Coulomb Energies”, Phys. Lett. 70A, 444–446 (1979).MathSciNetGoogle Scholar
  26. [L3]
    Lieb, E.H., “Atomic and Molecular Negative Ions”, Phys. Rev. Lett. 52, 315.Google Scholar
  27. [L4]
    Lieb, E.H., “Bound on the Maximum Negative Ionization of Atoms and Molecules”, Phys. Rev. A29, 3018–3028.Google Scholar
  28. [LSST]
    Lieb, E.H, Sigal, I., Simon, B. & Thirring, W., “Approximate Neutrality of Large-Z Ions”, Communications in Mathematical Physics 116(4), 635–644 (1988).MathSciNetCrossRefGoogle Scholar
  29. [Ru]
    Ruskai, M.B., “Absence of Discrete Spectrum in Highly Negative Ions”, I & II Comm. Math. Phys. 82, 457–469 and 85, 325-327 (1982).MathSciNetCrossRefGoogle Scholar
  30. [Sch]
    Schwinger, J. (1981) “Thomas—Fermi Model: The Second Correction”, Physical Review A24 5, 2353–2361 (1981).MathSciNetCrossRefGoogle Scholar
  31. [Sco]
    Scott, J.M.C., “The Binding Energy of the Thomas—Fermi Atom”, Phil. Mag. 43, 859–867 (1952).Google Scholar
  32. [SW1]
    Siedentop, H., Weikard, R., “On the Leading Energy Correction for the Statistical Model of the Atom: Interacting Case”, Communications in Mathematical Physics 112, 471–490 (1987).MathSciNetMATHCrossRefGoogle Scholar
  33. [SW2]
    Siedentop, H., Weikard, R., “On the Leading Correction of the Thomas—Fermi Model: Lower Bound” and an appendix by A.M.K. Müller. Inv. Math., 97, 159–193 (1989).MathSciNetMATHCrossRefGoogle Scholar
  34. [SW3]
    Siedentop, H., Weikard, R. (1990)}, “A New Phase Space Localization Technique with Applications to the Sum of Negative Eigenvalues of Schrödinger Operators.” Ann. Scient. École Normale Supérieure 24, 215–225 (1991).MathSciNetGoogle Scholar
  35. [Si]
    Sigal, I.M., “Geometric Methods in the Quantum Many-Body Problem. Nonexistence of Very Negative Ions”, Comm. Math. Phys. 85, 309–324 (1982).MathSciNetMATHCrossRefGoogle Scholar
  36. [T]
    Thomas, L.H., “The Calculation of Atomic Fields”, Proc. Cambridge Philos. Soc. 23, 542–548 (1927).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Charles L. Fefferman
    • 1
  • Luis A. Seco
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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