Interval Arithmetic in Quantum Mechanics

  • Charles L. Fefferman
  • Luis A. Seco
Part of the Applied Optimization book series (APOP, volume 3)


Quantum mechanics is an area which, over the last ten years or so, has sparked a respectable amount of rigorous computer assisted work (see, for example [9, 13, 20, 39, 38], and their applications in [10, 11, 12, 3, 4, 5,14,15,16,17,18,19, 20]). The purpose of this review is to select a piece of that body of work, and to give a more or less detailed account, both of the quantum mechanical problem surrounding the computer work and of the computer assisted proof itself. We hope this will be enlightening since much of the other computer assisted work in quantum mechanics shares many of the main features presented below.


Nial Seco 


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  1. [1]
    V. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Math. No. 60, Springer-Verlag, N.Y., 1978.Google Scholar
  2. [2]
    V. Bach, “Accuracy of Mean Field Approximations for Atoms and Molecules”, Comm. Math. Phys., 1993, Vol. 155, No. 2, pp. 295–310.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Córdoba, C. Fefferman, and L. Seco, “A Trigonometric Sum relevant to the Non-relativistic Theory of Atoms”, Proc. Nat. Acad. Sci. USA, June 1994, Vol. 91, pp. 5776–5778.MATHCrossRefGoogle Scholar
  4. [4]
    A. Córdoba, C. Fefferman, and L. Seco, “Weyl Sums and Atomic Energy Oscillations”, Revista Matemática Iberoamericana, 1995, Vol. 11, No. 1, pp. 167–228.Google Scholar
  5. [5]
    A. Córdoba, C. Fefferman, and L. Seco, “A Number-Theoretic Estimate for the Thomas-Fermi Density”, 1995 (to appear).Google Scholar
  6. [6]
    P. Dirac, “Note on Exchange Phenomena in the Thomas-Fermi Atom”, Proc. Cambridge Philos. Soc., 1930, Vol. 26, pp. 376–385.MATHCrossRefGoogle Scholar
  7. [7]
    J. P. Eckmann, H. Koch, and P. Wittwer, A computer Assisted Proof of Universality in Area Preserving Maps, Memoirs A.M.S., Vol 289, American Mathematical Society, Providence, R.I., 1984.Google Scholar
  8. [8]
    J. P. Eckmann and P. Wittwer, Computer Methods and Borel Summability Applied to Feigenbaum’s equation, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, N.Y., 1985.Google Scholar
  9. [9]
    C. Falcolini, C. Fefferman, and R. Llave, In preparation.Google Scholar
  10. [10]
    C. Fefferman, “Atoms and Analytic Number Theory”, In: A.M.S. Centennial Publication, Vol. II, American Math. Society, Providence, RI, 1992, p. 27–36.Google Scholar
  11. [11]
    C. Fefferman, “The Atomic and Molecular Structure of Matter”, Revista Matemática Iberoamericana, 1985, Vol. 1, No. 1, pp. 1–44.MATHMathSciNetGoogle Scholar
  12. [12]
    C. Fefferman, “The N-Body Problem in Quantum Mechanics”, Comm. Pure and Appl Math., 1986, Vol. 39, No. S, pp. S67–S110.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    C. Fefferman and R. Llave, “Relativistic Stability of Matter, I”, Revista Matematica Iberoamericana, 1986, Vol. 2, No. 1&2, pp. 119–213.MathSciNetGoogle Scholar
  14. [14]
    C. Fefferman and L. Seco, “The Ground-State Energy of a Large Atom”, Bull. A.M.S., 1990, Vol. 23, No. 2, pp. 525–530.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    C. Fefferman and L. Seco, “Eigenvalues and Eigenfunctions of Ordinary Differential Operators”, Adv. Math., 1992, Vol. 95, No. 2, pp. 145–305.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    C. Fefferman and L. Seco, “The Eigenvalue Sum for a One-Dimensional Potential”, Advances in Math, Oct. 1994, Vol. 108, No. 2, pp. 263–335.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    C. Fefferman and L. Seco, “The Density in a One-Dimensional Potential”, Advances in Math., Sept. 1994, Vol. 107, No. 2, pp. 187–364.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    C. Fefferman and L. Seco, “The Eigenvalue Sum for a Three-Dimensional Radial Potential”, Advances in Math., 1995 (to appear).Google Scholar
  19. [19]
    C. Fefferman and L. Seco, “The Density in a Three-Dimensional Radial Potential”, Advances in Math., 1995 (to appear).Google Scholar
  20. [20]
    C. Fefferman and L. Seco, “On the Dirac and Schwinger Corrections to the Ground-State Energy of an Atom”, Advances in Math., Aug. 1994, Vol. 107, No. 1, pp. 1–185.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    C. Fefferman and L. Seco, “Aperiodicity of the Hamiltonian Flow in the Thomas-Fermi Potential”, Revista Matemática Iberoamericana, 1993, Vol. 9, No. 3, pp. 409–551.MATHMathSciNetGoogle Scholar
  22. [22]
    E. Fermi, “Un Metodo Statistico per la Determinazione di alcune Priorieta dell’Atome”, Rend. Accad. Naz. Lincei, 1927, Vol. 6, pp. 602–607.Google Scholar
  23. [23]
    G. M. Graf and J. P. Solovej, “A Correlation Estimate with Applications to Quantum Systems with Coulomb Interactions”, Reviews in Math. Phys., 1994, Vol. 6, No. 5a, pp. 977–997.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    W. Hughes, “An Atomic Energy Lower Bound that Agrees with Scott’s Correction”, Advances in Mathematics, 1990, Vol. 79, pp. 213–270.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    B. Helffer, A. Knauf, H. Siedentop, and R. Weikard, “On the Absence of a First Order Correction for the Number of Bound States of a Schrodinger Operator with Coulomb Singularity”, Comm. P.D.E., 1992, Vol. 17, No. 3&4, pp. 615–639.MATHMathSciNetGoogle Scholar
  26. [26]
    E. Hille, “On the Thomas-Fermi Equation”, Proc. Nat. Acad. Sci. USA, 1969, Vol. 62, pp. 7–10.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    V. Ivrii and I. M. Sigal, “Asymptotics of the Ground State Energies of Large Coulomb Systems”, Annals of Math., 1993, Vol. 138, No. 2, pp. 243–335.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    E. W. Kaucher and W. L. Miranker, Self-validating Numerics for Function Space Problems, Academic Press, N.Y., 1984.MATHGoogle Scholar
  29. [29]
    O. Lanford, “A Computer-Assisted Proof of the Feigenbaum Conjecture”, Bull. AMS, 1986, Vol. 6, pp. 427–434.CrossRefMathSciNetGoogle Scholar
  30. [30]
    E. Lieb, “Thomas-Fermi and Related Theories of Atoms and Molecules”, Reviews of Modern Physics, 1981, Vol. 53, No. 4, pp. 603–641.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    E. Lieb and B. Simon, “Thomas-Fermi Theory of Atoms, Molecules and Solids”, Adv. Math., 1977, Vol. 23, pp. 22–116.CrossRefMathSciNetGoogle Scholar
  32. [32]
    R. Llave, “Computer Assisted Bounds in Stability of Matter”, Computer Aided Proofs in Analysis, IMA Series in Math, and Appl., Vol 28, Springer, Cincinatti, 1989.Google Scholar
  33. [33]
    R. Lohner, Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen, Dissertation, Universität Karlsruhe (TH), 1988.Google Scholar
  34. [34]
    R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, PA, 1979.MATHGoogle Scholar
  35. [35]
    D. Rana, Proof of Accurate Upper and Lower Bounds for Stability Domains in Denominator Problems, Thesis, Princeton University, 1987.Google Scholar
  36. [36]
    J. Schwinger, “Thomas-Fermi Model: The Second Correction”, Physical Review, 1981, Vol. A24, No. 5, pp. 2353–2361.MathSciNetGoogle Scholar
  37. [37]
    J. M. C. Scott, “The Binding Energy of the Thomas-Fermi Atom”, Phil Mag., 1952, Vol. 43, pp. 859–867.Google Scholar
  38. [38]
    L. Seco, “Lower Bounds for the Ground State Energy of Atoms”, Thesis, Princeton University, 1989.Google Scholar
  39. [39]
    L. Seco, “Computer Assisted Lower Bounds for Atomic Energies”, Computer Aided Proofs in Analysis, IMA Series in Math, and Appl., Vol 28, Springer, Cincinatti, 1989, pp. 241–251.Google Scholar
  40. [40]
    H. Siedentop and R. Weikard, “On the Leading Energy Correction for the Statistical Model of the Atom: Interacting Case”, Communications in Mathematical Physics, 1987, Vol. 112, pp. 471–490.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    H. Siedentop and R. Weikard, “On the Leading Correction of the Thomas-Fermi Model: Lower Bound”, and an appendix by A. M. K. Müller, Inv. Math., 1989, Vol. 97, pp. 159–193.MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    H. Siedentop and R. Weikard, “A New Phase Space Localization Technique with Applications to the Sum of Negative Eigenvalues of Schrödinger Operators”, Ann. Scient. Ecole Normale Superieure, 1991, Vol. 24, pp. 215–225.MATHMathSciNetGoogle Scholar
  43. [43]
    L. H. Thomas, “The Calculation of Atomic Fields”, Proc. Cambridge Phi los. Soc., 1927, Vol. 23, pp. 542–548.MATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

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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