Skip to main content
SpringerLink
Log in

Computer Assisted Proofs of Stability of Matter

  • Chapter
Computer Aided Proofs in Analysis

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 28))

Abstract

We review some recent progress in the study of Schrodinger equations for arbitrarily many fermions interacting via Coulomb forces. The goal is to prove lower bounds for the infimum of the spectrum which are reasonably close to optimal. Some of the key estimates are established with the help of a computer.

Supported in part by N.S.F. grants

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Y. F. Chang: The ATOMCC toolbox. Byte 11,4, 215–226 (1986).

    Google Scholar 

  2. J. Conlon: The ground state energy for a classical gas. Comm. Math. Phys. 94, 439 XX (1984).

    Article  MathSciNet  Google Scholar 

  3. F. Dyson, A. Lenard: Stability of matter I. Jour. Math. Phys. 8, 423–434 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Dyson, A. Lenard: Stability of matter II. Jour. Math. Phys. 9, 698–711 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Daubechies, E. H. Lieb: One-electron relativistic molecules with Coulomb interaction. Comm. Math. Phys. 90, 497–510 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  6. J-P. Eckmann, H. Koch and P. Wittwer: A computer assisted proof of universality in area preserving maps. Mem. of the A.M.S. 289, (1984).

    Google Scholar 

  7. J-P. Eckmann, P. Wittwer: “Computer methods and Borel summability applied to Feigenbaum’s equation”, Springer Verlag, NY (1985).

    Book  MATH  Google Scholar 

  8. C. Fefferman, R. de la Llave: Relativistic stability of matter I. Rev. Mat. Iber. 2, 119–213 (1986).

    Google Scholar 

  9. C. Fefferman: The atomic and molecular nature of matter. Rev. Mat. Iber. 1, 1–44 (1985).

    MathSciNet  MATH  Google Scholar 

  10. C. FefFerman: The N-body problem in quantum mechanics. Comm. Pure and Appl. Math. 39, S67–S109 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  11. C. FefFerman: Graduate courses taught at Princeton 1986–1987,1987–1988.

    Google Scholar 

  12. I. Herbst: Spectral theory of the operator (p 2+m 2)1/2 - Ze 2/r. Comm. Math. Phys. 53, 285–294 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  13. E. W. Kaucher, W. Miranker: “Self-Validating numerics for function space problems”, Academic Press, Orlando (1984).

    MATH  Google Scholar 

  14. T. Kato: Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Am. Math. Soc. 70, 195–211 (1951).

    MATH  Google Scholar 

  15. J. Lebowitz, E. H. Lieb: The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Adv. in Math 9, 316–398 (1972).

    Article  MathSciNet  Google Scholar 

  16. R. de la Llave, D. Rana: Algorithms for the proof of existence of special orbits, preprint

    Google Scholar 

  17. E. H. Lieb, W. Thirring: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975).

    Article  Google Scholar 

  18. E. H. Lieb, W. Thirring: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. of Phys. 155, 494–512 (1984).

    Article  MathSciNet  Google Scholar 

  19. E. H. Lieb, H.-T. Yau: The stability and instability of relativistic matter. Comm. Math. Phys. 118, 177–213 (1984).

    Article  MathSciNet  Google Scholar 

  20. O.E. Lanford III: Computer assisted proofs in analysis. Physica 124A, 465–470 (1984).

    MathSciNet  Google Scholar 

  21. E. H. Lieb: Stability of matter. Rev. Mod. Phys. 48, 553–569 (1976).

    Article  MathSciNet  Google Scholar 

  22. E. H. Lieb: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–642 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  23. R. MacKay, I.C. Percival: Converse K.A.M.: theory and practice. Comm. Math. Phys. 98, 469–512 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  24. R.E. Moore: “Methods and applications of interval analysis”, S.I.A.M. Philadelphia (1979).

    MATH  Google Scholar 

  25. H. Poincare: “Les methodes nouvelles de la mechanique celeste”, Gauthier Villars, Paris (1891–1899).

    Google Scholar 

  26. W. Thirring: “A course in mathematical Physics IV, Quantum mechanics of large systems”, Springer Verlag (1981).

    Google Scholar 

  27. R. Weder: Spectral analysis of pseudodifferential operators. Jour. Funct. Anal. 20, 319–377 (1975).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Math., University of Texas, Austin, TX, 78712, USA

    R. de la Llave

Authors
  1. R. de la Llave
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departments of Mathematics and Computer Science, University of Cincinnati, Cincinnati, OH, 45221, USA

    Kenneth R. Meyer

  2. Department of Computer Science, University of Cincinnati, Cincinnati, OH, 45221, USA

    Dieter S. Schmidt

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

de la Llave, R. (1991). Computer Assisted Proofs of Stability of Matter. In: Meyer, K.R., Schmidt, D.S. (eds) Computer Aided Proofs in Analysis. The IMA Volumes in Mathematics and Its Applications, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9092-3_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-9092-3_11

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9094-7

  • Online ISBN: 978-1-4613-9092-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation