Skip to main content
SpringerLink
Log in

Stability of Matter

  • Conference paper
  • First Online:
Spectral Analysis of Quantum Hamiltonians

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 224))

  • 800 Accesses

Abstract

These are extended notes based on a series of four lectures on the Stability of Matter, given by the author at the “Spectral Days” conference in Santiago de Chile in September, 2010.

Mathematics Subject Classification (2000). Primary 35P15; Secondary 35J05, 49R05.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. M.A. Aizenman, and E.H. Lieb, On semiclassical bounds for eigenvalues of Schrödinger operators, Phys. Letts. A 66 (1978), 427-429.

    Article  MathSciNet  Google Scholar 

  2. R.D. Benguria, Isoperimetric inequalities for eigenvalues of the laplacian, in Entropy and the Quantum II, (Robert Sims and Daniel Ueltschi, eds.), Contemporary Mathematics 552 (2011), 21-60.

    Article  MathSciNet  Google Scholar 

  3. R.D. Benguria, G.A. Bley, and M. Loss, An improved estimate on the indirect Coulomb Energy, International Journal of Quantum Chemistry, (published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/qua.23148) (2011).

  4. R.D. Benguria, P. Gallegos, and M. Tušek, A New Estimate on the Two-Dimensional Indirect Coulomb Energy, submitted.

    Google Scholar 

  5. R.D. Benguria and M. Loss, A simple proof of a theorem of Laptev and Weidl, Mathematical Research Letters 7 (2000), 195-203.

    MathSciNet  MATH  Google Scholar 

  6. R.D. Benguria and M. Loss, Connection between the Lieb-Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane, in Partial Differential Equations and Inverse Problems, C. Conca, R. Manásevich, G. Uhlmann, and M.S. Vogelius (eds.), Contemporary Mathematics 362, Amer. Math. Soc., Providence, R.I., pp. 53-61 (2004).

    Google Scholar 

  7. R.D. Benguria, M. Loss, and H. Siedentop, Stability of atoms and molecules in an ultrarelativistic Thomas-Fermi-Weizsacker model, J. Math. Phys. 49 (2008), article 012302.

    Google Scholar 

  8. N. Bohr, On the constitution of atoms and molecules, Philosophical Magazine and Journal of Science, 26 (1913), 1-25.

    Article  MATH  Google Scholar 

  9. N. Bohr, On the constitution of atoms and molecules. Part II. Systems Containing only a Single Nucleus, Philosophical Magazine and Journal of Science, 26 (1913), 476-501.

    Google Scholar 

  10. N. Bohr, On the constitution of atoms and molecules. Part III. Systems Containing Several Nuclei, Philosophical Magazine and Journal of Science, 26 (1913), 857-875.

    Google Scholar 

  11. N. Bohr, On the constitution of atoms and molecules, Papers of 1913 reprinted from the Philosophical Magazine with an Introduction by L. Rosenfeld; Munksgaard Ltd. Copenhagen, W.A. Benjamin Inc., NY, 1963.

    Google Scholar 

  12. V.S. Buslaev, and L.D. Faddeev, Formulas for traces for a singular Sturm-Liouville differential operator, Dokl. Akad. Nauk SSSR 132 (1960), 13-16 (Russian); English translation in Soviet Math. Dokl. 1 (1960), 451-454.

    MathSciNet  MATH  Google Scholar 

  13. J. Conlon, A new proof of the Cwikel-Lieb-Rosenbljum bound, Rocky Mountain J. Math., 15 (1985), 117-122.

    Article  MathSciNet  MATH  Google Scholar 

  14. M.M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. 2, 6 (1955), 121-127.

    Article  MathSciNet  Google Scholar 

  15. M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Annals of Math., 106 (1977), 93-100.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Darboux, Sur une proposition relative aux équations linéaires, C. R. Acad. Sci. (Paris), 94 (1882), 1456-1459.

    MATH  Google Scholar 

  17. L. de Broglie, Recherches sur la théorie des quanta, Thesis (Paris), 1924 (see also, Ann. Phys. (Paris) 3, (1925), 22).

    Google Scholar 

  18. P.A. Deift, Applications of a commutation formula, Duke Math. J. 45, (1978), 267310.

    Article  MathSciNet  Google Scholar 

  19. J. Dolbeault, A. Laptev, and M. Loss, Lieb-Thirring inequalities with improved constants, J. Eur. Math. Soc., 10 (2008), 1121-1126.

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Dyson and A. Lenard, Stability of Matter, I, J. Math. Phys. 8 (1967), 423-434.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Eden, and C. Foias, A simple proof of the generalized Lieb-Thirring inequalities in one-space dimension, J. Math. Anal. Appl. 162 (1991), 250-254.

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Engel and R.M. Dreizler, Field-theoretical Approach to a Relativistic Thomas- Fermi-Weizsäcker Model, Phys. Rev. A 35 (1987), 3607-3618.

    Article  Google Scholar 

  23. E. Engel and R.M. Dreizler, Solution of the relativistic Thomas-Fermi-Dirac- Weizsäcker Model for the Case of Neutral Atoms and Positive Ions, Phys. Rev. A 38 (1988), 3909-3917.

    Article  Google Scholar 

  24. G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Flache und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften zu Munchen Jahrgang, pp. 169-172 (1923).

    Google Scholar 

  25. L.D. Faddeev, and V.E. Zakharov, The Korteweg de Vries equation is a completely integrable Hamiltonian system, Funktsional. Anal. i Prilozhen. 5 (1971), 18-27 (Russian); English translation in Functional Anal. Appl. 5 (1971), 280-287.

    Google Scholar 

  26. E. Fermi, Un metodo statistico per la determinazione di alcune prioretà dell átome, Rend. Acad. Naz. Lincei 6 (1927), 602-607.

    Google Scholar 

  27. F. Gesztesy, A Complete Spectral Characterization of the Double Commutator Method, J. Funct. Anal. 117, (1993), 401-446.

    Article  MathSciNet  MATH  Google Scholar 

  28. G.H. Hardy, Note on a Theorem of Hilbert, Math. Z. 6 (1920), 314-317.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Hochstadt, The functions of Mathematical Physics, Pure and Applied Mathematics 23, Wiley-Interscience, NY, 1971.

    Google Scholar 

  30. D. Hundertmark, E.H. Lieb, and L.E. Thomas, A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator, Adv. Theor. Math. Phys. 2 (1998), 719-731.

    MathSciNet  MATH  Google Scholar 

  31. C.G.J. Jacobi, Zur Theorie der Variationsrechnung und der Differentialgleichungen, J. Reine Angew. Math. 17 (1837), 68-82.

    Article  MATH  Google Scholar 

  32. J.H. Jeans, The Mathematical Theory of Electricity and Magnetism, Cambridge University Press, Cambridge, UK, 1915.

    Google Scholar 

  33. E. Krahn, Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), 97-100.

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9, 1-44 (1926). [English translation: Minimal properties of the sphere in three and more dimensions, Edgar Krahn 1894-1961: A Centenary Volume, Ü. Lumiste and J. Peetre, editors, IOS Press, Amsterdam, The Netherlands, pp. 139-174 (1994).]

    Google Scholar 

  35. J.B. Keller, Lower Bounds and Isoperimetric Inequalities for Eigenvalues of the Schrödinger Equation, J. Math. Phys. 2 (1961), 262-266.

    Article  MATH  Google Scholar 

  36. A. Laptev, and T. Weidl, Sharp Lieb-Thirring Inequalities in High Dimensions, Acta Mathematica 184 (2000) 87-111.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Lenard and F. Dyson, Stability of Matter, II, J. Math. Phys. 9 (1968), 698-711.

    Article  MathSciNet  MATH  Google Scholar 

  38. W. Lenz, Über die Anwendbarkeit der statistischen Methode auf Ionengitter, Z. Phys. 77 (1932), 713-721.

    Article  Google Scholar 

  39. E.H. Lieb, The stability of matter, Reviews in Modern Physics 48 (1976), 553-569.

    Article  MathSciNet  Google Scholar 

  40. E.H. Lieb, The Number of Bound States of One-Body Schrödinger Operators and the Weyl Problem, Proc. Am. Math. Soc. Symposia Pure Math, 36 (1980), 241-252.

    MathSciNet  Google Scholar 

  41. E.H. Lieb, Thomas-Fermi and Related Theories of Atoms and Molecules, Reviews in Modern Physics 53 (1981), 603-641.

    Article  MathSciNet  MATH  Google Scholar 

  42. E.H. Lieb, Kinetic energy bounds and their applications to the stability of matter, in Springer Lecture Notes in Physics 345 (1989), 371-382 (H. Holden and A. Jensen, eds.).

    Google Scholar 

  43. E.H. Lieb, The stability of matter: from atoms to stars, Bulletin Amer. Math. Soc. 22, 1-49 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  44. E.H. Lieb, Lieb-Thirring Inequalities, in Encyclopaedia of Mathematics, (M. Haze- winkel, ed.), Supplement vol. 2, pp. 311-313, Kluwer Academic Pub. (2000).

    Google Scholar 

  45. E.H. Lieb, The stability of matter: From Atoms to Stars, Selecta of Elliott H. Lieb, edited by W. Thirring, 4th. Edition, Springer-Verlag, Berlin, 2005.

    Google Scholar 

  46. E.H. Lieb and M. Loss, Analysis, Second Edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, 2001.

    Google Scholar 

  47. E.H. Lieb and H.-T. Yau, The Stability and Instability of Relativistic Matter, Commun. Math. Phys. 118 (1988), 177-213.

    Article  MathSciNet  MATH  Google Scholar 

  48. E.H. Lieb and H.-T. Yau, Many-Body Stability Implies a Bound on the Fine-Structure Constant, Phys. Rev. Lett. 61 (1988), 1695-1697.

    Article  MathSciNet  Google Scholar 

  49. E.H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, Cambridge, UK, 2009.

    Google Scholar 

  50. E.H. Lieb and B. Simon, Thomas-Fermi Theory Revisited, Phys. Rev. Lett. 31 (1973), 681.

    Article  Google Scholar 

  51. E.H. Lieb and W. Thirring, Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter, Phys. Rev. Lett. 35 (1975), 687-689.

    Article  Google Scholar 

  52. E.H. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in Studies in Mathematical Physics (E.H. Lieb, B. Simon, and A. Wightman, eds.), Princeton University Press, 1976, pp. 269-303.

    Google Scholar 

  53. H. Linde, A lower bound for the ground state energy of a Schrödinger operator on a loop, Proc. Amer. Math. Soc. 134 (2006), 3629-3635.

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Loss, Stability of Matter, Lecture Notes, on the WEB site: http://www.mathematik.uni-muenchen.de / ~lerdos/WS 08/QM/lossstabmath.pdf, 2005.

  55. H. Nagaoka, Kinetics of a System of Particles illustrating the Line and the Band, Spectrum and the Phenomena of Radioactivity, Philosophical Magazine 7 (1904), 445-455.

    MATH  Google Scholar 

  56. I. Newton, Philosophiae Naturalis Principia Mathematica, London, 1687.

    Google Scholar 

  57. H. Poincaré, Sur les rapports de l’analyse pure et de la physique mathématique, Acta Mathematica 21(1897), 331-342.

    Article  MathSciNet  MATH  Google Scholar 

  58. G.V. Rosenbljum, Distribution of the discrete spectrum of singular differential operator, Dokl. Aka. Nauk SSSR, 202 (1972), 1012-1015. The details are given in Distribution of the discrete spectrum of singular differential operators, Izv. Vyss. Ucebn. Zaved. Matematika 164 (1976), 75-86. (English trans. Sov. Math. (Iz. VUZ) 20 (1976), 63-71.)

    Google Scholar 

  59. E. Rutherford, The Scattering of Alpha and Beta Particles by Matter and the Structure of the Atom, Philosophical Magazine 21 (1911), 669-688; (a brief account of this paper was communicated to the Manchester Literature and Philosophical Society in February of 1911).

    Google Scholar 

  60. E. Rutherford, The Structure of the Atom,, Philosophical Magazine 27 (1914), 488498.

    Google Scholar 

  61. R. Seiringer, Inequalities for Schrödinger Operators and Applications to the Stability of Matter Problem, in Entropy and the Quantum, Arizona School of Analysis with Applications (Tucson, AZ, 2009), (R. Sims, D. Ueltschi, eds.), Contemporary Mathematics 529 (2010), 53-72.

    Article  MathSciNet  Google Scholar 

  62. U.-W. Schmincke, On Schrödinger’s factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 67-84.

    Article  MathSciNet  MATH  Google Scholar 

  63. E. Schrödinger, Quantization as an Eigenvalue Problem, Annalen der Physik 79 (1926), 361-376.

    Article  MATH  Google Scholar 

  64. C.L. Siegel and J.K. Moser, Lectures in Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  65. S.L. Sobolev, On a theorem of functional analysis, Mat. Sb. 4 (1938), 471-479. (The English translation appears in: AMS Trans. Series (2) 34 (1963), 39-68.)

    Google Scholar 

  66. G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3, 697-718 (1976).

    Google Scholar 

  67. E. Teller, On the Stability of Molecules in Thomas-Fermi Theory, Reviews of Modern Physics 34 (1962), 627-631.

    Article  MATH  Google Scholar 

  68. L.H. Thomas, The calculation of atomic fields, Proc. Cambridge Phil. Soc. 23 (1927), 542-548.

    Article  MATH  Google Scholar 

  69. J.J. Thomson, On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure, Philosophical Magazine 7 (1904), 237-265.

    Article  MathSciNet  MATH  Google Scholar 

  70. Y. Tomishima and K. Yonei, Solution of the Thomas-Fermi-Dirac Equation with a Modified Weizsäcker Correction, J. Phys. Soc. Japan. 21 (1966), 142-153.

    Article  Google Scholar 

  71. G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques, Journal für die Reine und Angewandte Mathematik 133 (1907), 97-178.

    Google Scholar 

  72. G. Voronoi, Recherches sur les paralléloedres Primitives, J. Reine Angew. Math. 134 (1908), 198-287.

    MATH  Google Scholar 

  73. T. Weidl, On the Lieb-Thirring constants L γ, 1 for γ ≥ 1/2, Commun. Math. Phys. 178 (1996), 135-146.

    Article  MathSciNet  MATH  Google Scholar 

  74. C.F. von Weizsäcker, Zur Theorie der Kernmassen, Z. Physik 96 (1935), 431-458.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

    Rafael D. Benguria

Authors
  1. Rafael D. Benguria
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafael D. Benguria .

Editor information

Editors and Affiliations

  1. , Departamento de Física, P. Universidad Católica de Chile, Santiago, 22, Chile

    Rafael Benguria

  2. , Departamento de Matemáticas, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Eduardo Friedman

  3. , Departamento de Matemática, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Marius Mantoiu

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel

About this paper

Cite this paper

Benguria, R.D. (2012). Stability of Matter. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_3

Download citation

Publish with us

Policies and ethics

Search

Navigation