Communications in Mathematical Physics

, Volume 365, Issue 2, pp 651–683 | Cite as

Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

  • Gonzalo A. Bley
  • Søren FournaisEmail author


We provide lower bounds for the sum of the negative eigenvalues of the operator \({|\sigma\cdot p_A|^{2s} - C_s/|x|^{2s} + V}\) in three dimensions, where \({s\in (0, 1]}\), covering the interesting physical cases s =  1 and s =  1/2. Here \({\sigma}\) is the vector of Pauli matrices, \({p_A = p - A}\), with \({p = -i\nabla}\) the three-dimensional momentum operator and A a given magnetic vector potential, and Cs is the critical Hardy constant, that is, the optimal constant in the Hardy inequality \({|p|^{2s} \geq C_s/|x|^{2s}}\). If spin is neglected, results of this type are known in the literature as Hardy–Lieb–Thirring inequalities, which bound the sum of negative eigenvalues from below by \({-M_s\int V_{-}^{1 + 3/(2s)}}\), for a positive constant Ms. The inclusion of magnetic fields in this case follows from the non-magnetic case by diamagnetism. The addition of spin, however, offers extra challenges that make the result more elusive. It is the purpose of this article to resolve this problem by providing simple bounds for the sum of the negative eigenvalues of the operator in question. In particular, for \({1/2 \leq s \leq 1}\) we are able to express the bound purely in terms of the magnetic field energy \({\|B\|_2^2}\) and integrals of powers of the negative part of V.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors were partially supported by a Sapere Aude grant from the Independent Research Fund Denmark, Grant number DFF–4181-00221. They would also like to thank the anonymous referee for precise and useful comments that helped make the article better, in particular for pointing out the article by Hansen and Pedersen cited by Eq. (1.28), of which the authors were unaware.


  1. 1.
    Benguria R., Loss M.: A simple proof of a theorem by Laptev and Weidl. Math. Res. Lett. 7, 195 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Birman, M.S., Koplienko, L.S., Solomyak, M.Z.: Estimates for the spectrum of the difference between fractional powers of two self-adjoint operators, Izv. Vysš. Učebn. Zaved. Matematika, no. 3 (154), 3–10 (1975). (Russian) Translation to English in: Soviet Mathematics, 19 (3), 1–6 (1975)Google Scholar
  3. 3.
    Bugliaro L., Fefferman C., Fröhlich J., Graf G.M., Stubbe J.: A Lieb–Thirring bound for a magnetic Pauli Hamiltonian. Commun. Math. Phys. 187, 567 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cwikel M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, S., Frank, R.L., Weth, T.: Remainder terms in the fractional sobolev inequality. Indiana Univ. Math. J. 62, 1381 (2013)Google Scholar
  6. 6.
    Daubechies I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90, 511 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dolbeault J., Laptev A., Loss M.: Lieb–Thirring inequalities with improved constants. J. Eur. Math. Soc. 10, 1121 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ekholm T., Frank R.L.: On Lieb–Thirring inequalities for Schrödinger operators with virtual level. Commun. Math. Phys. 264, 725 (2006)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Erdös L.: Magnetic Lieb–Thirring inequalities. Commun. Math. Phys. 170, 629 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Erdös L., Fournais S., Solovej J.P.: Relativistic Scott correction in self-generated magnetic fields. J. Math. Phys. 53, 095202 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Erdös L., Solovej J.P.: Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields, I: nonasymptotic Lieb–Thirring-type estimate. Duke Math. J 96, 127 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Erdös L., Solovej J.P.: Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields, II: leading order asymptotic estimates. Commun. Math. Phys. 188, 599 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Erdös L., Solovej J.P.: Uniform Lieb–Thirring inequality for the three-dimensional Pauli operator with a strong non-homogeneous magnetic field. Ann. Henri Poincaré 5, 671 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Frank R.L.: A simple proof of Hardy–Lieb–Thirring inequalities. Commun. Math. Phys. 290, 789 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Frank, R.L.: Eigenvalue bounds for the fractional laplacian: a review (2017). arXiv:1603.09736
  16. 16.
    Frank R.L., Lieb E.H., Seiringer R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21, 925 (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    Frank R.L., Lieb E.H., Seiringer R.: Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value. Commun. Math. Phys. 275, 479 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fröhlich J., Lieb E.H., Loss M.: Stability of Coulomb systems with magnetic fields I. The one-electron atom. Commun. Math. Phys. 104, 251 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hansen F., Pedersen G.: Jensen’s operator inequality. Bull. Lond. Math. Soc. 35, 553 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Herbst I.W.: Spectral theory of the operator \({\left(p^2 + m^2\right)^{1/2} - Ze^2/r}\). Commun. Math. Phys. 53, 285–294 (1977)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Hundertmark D., Lieb E.H., Thomas L.E.: A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. Adv. Theor. Math. Phys. 2, 719 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hundertmark D., Laptev A., Weidl T.: New bounds on the Lieb–Thirring constants. Invent. Math. 140, 693 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Laptev, A., Weidl, T.: Sharp Lieb–Thirring inequalities in high dimensions. Acta Math. 184, 87 (2000)Google Scholar
  24. 24.
    Lenzmann E., Lewin M.: Minimizers for the Hartree–Fock–Bogoliubov theory of neutron stars and white dwarfs. Duke Math J. 152, 257 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lieb, E.H.: Lieb–Thirring Inequalities, Kluwer Encyclopedia of Mathematics, Supplement Vol. II, 311 (2000). arXiv:math-ph/0003039
  26. 26.
    Lieb E.H.: Bounds on the eigenvalues of the Laplace and Schroedinger operators. Bull. Am. Math. Soc. 82, 751 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lieb, E.H.: The number of bound states of one-body Schrödinger operators and the Weyl Problem, Geometry of the Laplace Operator. In: Proceedings of Symposia in Pure Mathematics, vol. 36, p. 250. American Mathematical Society (1980)Google Scholar
  28. 28.
    Lieb E.H., Aizenman M.: On semi-classical bounds for eigenvalues of Schrödinger operators. Phys. Lett. A 66, 427 (1978)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Lieb E.H., Loss M.: Stability of Coulomb systems with magnetic fields II. The many-electron atom and the one-electron molecule. Commun. Math. Phys. 104, 271 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lieb E.H., Loss M., Solovej J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lieb E.H., Siedentop H., Solovej J.P.: Stability and instability of relativistic electrons in classical electromagnetic fields. J. Stat. Phys. 89, 37 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lieb, E.H., Siedentop, H., Solovej, J.P.: Stability of relativistic matter with magnetic fields. Phys. Rev. Lett. 79, 1785 (1997)Google Scholar
  33. 33.
    Lieb E.H., Solovej J.P., Yngvason J.: Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band region. Commun. Pure Appl. Math. 47, 513 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lieb E.H., Solovej J.P., Yngvason J.: Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Commun. Math. Phys. 161, 77 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lieb E.H., Solovej J.P., Yngvason J.: Ground states of large quantum dots in magnetic fields. Phys. Rev. B 51, 10646 (1995)ADSCrossRefGoogle Scholar
  36. 36.
    Lieb, E.H., Thirring, W.E.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687. Erratum: Phys. Rev. Lett. 35, 1116 (1975)Google Scholar
  37. 37.
    Lieb, E.H., Thirring, W.E.: Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. Princeton University Press (1976)Google Scholar
  38. 38.
    Lieb E.H., Yau H-.T.: The stability and instability of relativistic matter. Commun. Math. Phys. 118, 177 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Loss M., Yau H-.T.: Stability of Coulomb systems with magnetic fields. III. Zero energy bound states of the Pauli operator. Commun. Math. Phys. 104, 283 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Rosenbljum, G.V.: The distribution of the discrete spectrum for singular differential operators. Dokl. Akad. Nauk SSSR 202, 1012 (1972). See also Sov. Math. Dokl. 13, 245 (1972) (English), Izv. Vyss. Ucebn. Zaved. Matem. 164, 75 (1976), and Sov. Math. (Iz VUZ) 20, 63 (1976) (English)Google Scholar
  41. 41.
    Schwinger J.: On the bound states of a given potential. Proc. Natl. Acad. Sci. 47, 122 (1961)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Sobolev A.V.: Lieb–Thirring inequalities for the Pauli operator in three dimensions. IMA Vol. Math. Appl. 95, 155–188 (1997)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Sobolev A.V.: On the Lieb–Thirring estimates for the Pauli operator. Duke Math. J. 82, 607 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Solovej J.P., Østergaard Sørensen T., Spitzer W.L.: Relativistic Scott correction for atoms and molecules. Commun. Pure Appl. Math. 63, 39 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Weidl, T.: Remarks on virtual bound states for semi-bounded operators. Commun. Part. Differ. Equ. 24(1& 2), 25 (1999)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark

Personalised recommendations