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Kinetic Energy Bounds and their Application to the Stability of Matter

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Abstract

he Sobolev inequality on Rn,n ≥ 3 is very important because it gives a lower bound for the kinetic energy f| ∇f|2 in terms of an L p norm of f.

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References

  1. M. Aizenman and E.H. Lieb, On semiclassical bounds for eigenvalues of Schrödinger operators, Phys. Lett. 66k, 427–429 (1978).

    MathSciNet  Google Scholar 

  2. J. Conlon, E.H. Lieb and H.-T. Yau, The N 7/5 law for bosons, Commun. Math. Phys. (submitted).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. I. Daubechies, Commun. Math. Phys. 90, 511–520 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  5. F.J. Dyson, Ground state energy of a finite systems of charged particles, J. Math. Phys. 8, 1538–1545 (1967).

    Article  MathSciNet  Google Scholar 

  6. E.H. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, A.M.S. Proc. Symp. in Pure Math. 36, 241–251 (1980). The results were announced in Bull. Ann. Math. Soc. 82, 751-753 (1976).

    Article  MathSciNet  Google Scholar 

  7. E.H. Lieb, An L p bound for the Riesz and Bessel potentials of orthonormal functions, J. Funct. Anal. 51, 159–165 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  8. E.H. Lieb, On characteristic exponents in turbulence, Commun. Math. Phys. 92, 473–480 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  9. E.H. Lieb and M. Loss, Stability of Coulomb systems with magnetic fields: II. The many-electron atom and the one-electron molecule, Commun. Math. Phys. 104, 271–282 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  10. E.H. Lieb and W.E. Thirring, Bounds for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett. 35, 687–689 (1975). Errata 35, 1116 (1975).

    Article  Google Scholar 

  11. E.H. Lieb and W.E. Thirring, “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities” in Studies in Mathematical Physics (E. Lieb, B. Simon, A. Wightman eds.) Princeton University Press, 1976, pp. 269–304.

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

    Article  MathSciNet  Google Scholar 

  13. E.H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys. 112, 147–174 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  14. G.V. Rosenbljum, Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR 202, 1012–1015 (1972). (MR 45 # 4216). The details are given in Izv. Vyss. Ucebn. Zaved. Matem. 164, 75-86 (1976). (English trans. Sov. Math. (Iz VUZ) 20, 63-71 (1976).)

    MathSciNet  Google Scholar 

  15. E.H. Lieb and H.T. Yau, The stability and instability of relativistic matter, Commun. Math. Phys. 118, 177–213 (1988). For a short summary see: Many body stability implies a bound on the fine structure constant, Phys. Rev. Lett. 61, 1695-1697 (1988).

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, P.O. Box 708, Princeton, NJ, 08544, USA

    Elliott H. Lieb

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  1. Elliott H. Lieb
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Editor information

Editors and Affiliations

  1. School of Mathematics, Georgia Tech, Atlanta, GA, 30332-0160, USA

    Michael Loss

  2. Department of Mathematics, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    Mary Beth Ruskai

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© 2002 Springer-Verlag Berlin Heidelberg

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Lieb, E.H. (2002). Kinetic Energy Bounds and their Application to the Stability of Matter. In: Loss, M., Ruskai, M.B. (eds) Inequalities. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55925-9_27

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  • DOI: https://doi.org/10.1007/978-3-642-55925-9_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62758-3

  • Online ISBN: 978-3-642-55925-9

  • eBook Packages: Springer Book Archive

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