Monatshefte für Mathematik

, 155:43

Compactness properties for trace-class operators and applications to quantum mechanics

  • J. Dolbeault
  • P. Felmer
  • J. Mayorga-Zambrano
Article

Abstract.

Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics.

2000 Mathematics Subject Classification: 81Q10, 82B10; 26D15, 35J10, 47B34 
Key words: Compact self-adjoint operators, trace-class operators, mixed states, occupation numbers, Lieb-Thirring inequality, Gagliardo-Nirenberg inequality, logarithmic Sobolev inequality, optimal constants, orthonormal and sub-orthonormal systems, Schrödinger operator, asymptotic distribution of eigenvalues, free energy, embeddings, compactness results 

References

  1. Ben Abdallah, N, Dolbeault, J 2003Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness)Arch Ration Mech Anal168253298MATHMathSciNetCrossRefGoogle Scholar
  2. Benguria RD, Loss M (2004) Connection between the Lieb-Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane. In: Conca C et al (eds) Partial Differential Equations and Inverse Problems, pp 53–61. Providence, RI: Amer Math SocGoogle Scholar
  3. Brezis, H 1983Analyse fonctionnelle, Collection Mathématiques Appliquées pour la MaîtriseMassonParisGoogle Scholar
  4. Carrillo, JA, Jüngel, A, Markowich, PA, Toscani, G, Unterreiter, A 2001Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalitiesMonatsh Math133182MATHCrossRefMathSciNetGoogle Scholar
  5. Dolbeault, J, Felmer, P, Loss, M, Paturel, E 2006Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systemsJ Funct Anal238193220MATHCrossRefMathSciNetGoogle Scholar
  6. Dolbeault, J, Markowich, P, Oelz, D, Schmeiser, C 2007Non linear diffusions as limit of kinetic equations with relaxation collision kernelsArch Ration Mech Anal186133158CrossRefMathSciNetMATHGoogle Scholar
  7. Dolbeault, J, Markowich, PA, Unterreiter, A 2001On singular limits of mean-field equationsArch Ration Mech Anal158319351MATHCrossRefMathSciNetGoogle Scholar
  8. Eden, A, Foias, C 1991A simple proof of the generalized Lieb-Thirring inequalities in one-space dimensionJ Math Anal Appl162250254MATHCrossRefMathSciNetGoogle Scholar
  9. Ghidaglia, J-M, Marion, M, Temam, R 1988Generalization of the Sobolev-Lieb-Thirring inequalities and applications to the dimension of attractorsDiffer Integral Equ1121MATHMathSciNetGoogle Scholar
  10. Gilbarg, D, Trudinger, NS 1983Elliptic partial differential equations of second order2SpringerBerlin Heidelberg New YorkMATHGoogle Scholar
  11. Glaser, V, Martin, A 1983Comment on the paper: “Necessary conditions on potential functions for nonrelativistic bound states” by G. RosenLett Nuovo Cimento36519520CrossRefMathSciNetGoogle Scholar
  12. Guo, Y, Rein, G 1999Stable steady states in stellar dynamicsArch Ration Mech Anal147225243MATHCrossRefMathSciNetGoogle Scholar
  13. Guo, Y, Rein, G 1999Isotropic steady states in galactic dynamicsComm Math Phys219607629CrossRefMathSciNetGoogle Scholar
  14. Lieb, E, Thirring, W 1976Lieb, ESimon, BWightman, A eds. Essays Homan Valentine BargunannUniv PressPrinceton269303Google Scholar
  15. Lieb EH (1989) Kinetic energy bounds and their application to the stability of matter. Lect Notes Phys 345: Berlin: SpringerGoogle Scholar
  16. Lieb EH (1991) Bounds on Schrödinger operators and generalized Sobolev-type inequalities with applications in mathematics and physics. In: Ereritt WN (ed) Inequalities Lect Notes Pure Appl Math 129: 123–133. New York: DekkerGoogle Scholar
  17. Lions, P-L 1988

    Hartree-Fock and related equations

    Bréris, HLions, JL eds. Nonlinear Partial Differential Equations and Their ApplicationsLongmanHarlow304333
    Google Scholar
  18. Markowich, P, Rein, G, Wolansky, G 2002Existence and nonlinear stability of stationary states of the Schrödinger-Poisson systemJ Statist Phys10612211239MATHCrossRefMathSciNetGoogle Scholar
  19. Reed, M, Simon, B 1972Methods of Modern Mathematical Physics. IAcademic PressNew YorkGoogle Scholar
  20. Reed, M, Simon, B 1975Methods of Modern Mathematical Physics. IIAcademic PressNew YorkMATHGoogle Scholar
  21. Rein, G 2003Non-linear stability of gaseous starsArch Ration Mech Anal168115130MATHMathSciNetGoogle Scholar
  22. Rosen, G 1982Necessary conditions on potential functions for nonrelativistic bound statesPhys Rev Lett4918851887CrossRefMathSciNetGoogle Scholar
  23. Solovej, JP 2003The ionization conjecture in Hartree-Fock theoryAnn Math158509576MATHMathSciNetGoogle Scholar
  24. Thirring, W 1981A Course in Mathematical PhysicsSpringer Quantum Mechanics of Atoms and MoleculesNew YorkMATHGoogle Scholar
  25. Veling EJM (2002) Lower bounds for the infimum of the spectrum of the Schrödinger operator in \({\Bbb R}^N\) and the Sobolev inequalities. J Inequal Pure Appl Math 3: No 4, paper No 63, 22 pp (electronic)Google Scholar
  26. Weinstein, MI 1982/83Nonlinear Schrödinger equations and sharp interpolation estimatesComm Math Phys87567576CrossRefMathSciNetGoogle Scholar
  27. Weyl, H 1912Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)Math Ann71441479MATHCrossRefMathSciNetGoogle Scholar
  28. Wolansky, G, Ghil, M 1996An extension of Arnold’s second stability theorem for the Euler equationsPhys D94161167MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • J. Dolbeault
    • 1
  • P. Felmer
    • 2
  • J. Mayorga-Zambrano
    • 2
  1. 1.Ceremade Université Paris DauphineParisFrance
  2. 2.Universidad de ChileSantiagoChile

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