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Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems

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Abstract

We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.

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References

  1. Adams R.A.: Sobolev spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Aubin T.: Problèmes isopèrimétriques et espaces de Sobolev. J. Diff. Géom. 11, 573–598 (1976)

    MathSciNet  MATH  Google Scholar 

  3. Bellazzini J., Frank R.L., Visciglia N.: Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems. Math. Ann. 360, 653–673 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellazzini, J., Ozawa, T., Visciglia, N.: Ground states for semi-relativistic Schrödinger-Poisson-Slater energies, arXiv:1103.2649 (2011)

  5. Calogero F.: Ground state of a one-dimensional N-body system. J. Math. Phys. 10, 2197–2200 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  6. Cotsiolis A., Tavoularis N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, 225–236 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin, Heidelberg (1987)

  8. Daubechies I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90, 511–520 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Ekholm T., Enblom A.: Critical Hardy–Lieb–Thirring inequalities for fourth-order operators in low dimensions. Lett. Math. Phys. 94, 293–312 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Ekholm T., Frank R.L.: On Lieb–Thirring inequalities for Schrödinger operators with virtual level. Commun. Math. Phys. 264(3), 725–740 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  13. Frank, R.L., Geisinger, L.: Refined semiclassical asymptotics for fractional powers of the Laplace operator. J. Reine Angew. Math. doi:10.1515/crelle-2013-0120 (ahead of print)

  14. Frank R.L., Lieb E.H., Seiringer R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc. 21, 925–950 (2007)

    Article  MathSciNet  Google Scholar 

  15. Frank R. L.: A simple proof of Hardy–Lieb–Thirring inequalities. Commun. Math. Phys. 290, 789–800 (2009)

    Article  ADS  Google Scholar 

  16. Frank R.L., Seiringer R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Frank R.L., Seiringer R.: Lieb–Thirring inequality for a model of particles with point interactions. J. Math. Phys. 53, 095201 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  18. Girardeau M.: Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1, 516–523 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Grubb, G.: Regularity of spectral fractional Dirichlet and Neumann problems. arXiv:1412.3744

  20. Hainzl C., Seiringer R.: General decomposition of radial functions on \({{\mathbb{R}}^n}\) and applications to N-body quantum systems. Lett. Math. Phys. 61, 75–84 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Herbst I.W.: Spectral theory of the operator \({(p^2+m^2)^{1/2}-Ze^2/r}\). Commun. Math. Phys. 53, 285–294 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Hoffmann-Ostenhof M., Hoffmann-Ostenhof T.: Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A 16, 1782–1785 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  23. Hurri-Syrjänen R., Vähäkangas A.V.: On fractional Poincaré inequalities. J. Anal. Math. 120, 85–104 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Knuepfer H., Muratov C.B.: On an isoperimetric problem with a competing non-local term. I. The planar case. Comm. Pure Appl. Math. 66, 1129–1162 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Knuepfer H., Muratov C.B.: On an isoperimetric problem with a competing non-local term. II. The general case. Comm. Pure Appl. Math. 67, 1174–1194 (2014)

    Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Lieb E.H.: A lower bound for Coulomb energies. Phys. Lett. A 70, 444–446 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  28. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, 2nd edn, vol. 14. American Mathematical Society, Providence (2001)

  29. Lieb E.H., Oxford S.: Improved lower bound on the indirect Coulomb energy. Int. J. Quantum Chem. 19, 427–439 (1980)

    Article  Google Scholar 

  30. Lieb E.H., Seiringer R.: The stability of matter in quantum mechanics. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  31. Lieb E.H., Solovej J.P., Yngvason J.: Ground states of large quantum dots in magnetic fields. Phys. Rev. B 51, 10646–10665 (1995)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  33. Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics. pp. 269–303. Princeton University Press, Princeton (1976)

  34. Lieb E.H., Yau H.-T.: The stability and instability of relativistic matter. Commun. Math. Phys. 118(2), 177–213 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Lieb E.H., Yngvason J.: The ground state energy of a dilute two-dimensional Bose gas. J. Stat. Phys. 103, 509–526 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lundholm D., Portmann F., Solovej J.P.: Lieb–Thirring bounds for interacting Bose gases. Commun. Math. Phys. 335, 1019–1056 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Lundholm D., Solovej J.P.: Hardy and Lieb–Thirring inequalities for anyons. Commun. Math. Phys. 322, 883–908 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Lundholm D., Solovej J.P.: Local exclusion and Lieb–Thirring inequalities for intermediate and fractional statistics. Ann. Henri Poincaré 15, 1061–1107 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Muratov C.: Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions. Commun. Math. Phys. 299, 45–87 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Musina R., Nazarov A.I.: On fractional Laplacians. Comm. Part. Differ. Equ. 39, 1780–1790 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rumin A.: Balanced distribution-energy inequalities and related entropy bounds. Duke Math. J. 160, 567–597 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Servadei R., Valdinoci E.: On the spectrum of two different fractional operators. Proc. R. Soc. A 144, 831–855 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Solomyak M.: A remark on the Hardy inequalities. Integral Equ. Oper. Theory 19, 120–124 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  44. Solovej J.P., Sørensen T.Ø., Spitzer W.L.: Relativistic Scott correction for atoms and molecules. Comm. Pure Appl. Math. 63, 39–118 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton. (1971)

  46. Sutherland B.: Quantum many-body problem in one dimension: ground state. J. Math. Phys. 12, 246–250 (1971)

    Article  ADS  Google Scholar 

  47. Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  48. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)

  49. Yafaev D.: Sharp Constants in the Hardy–Rellich inequalities. J. Func. Anal. 168, 121–144 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. KTH Royal Institute of Technology, Stockholm, Sweden

    Douglas Lundholm

  2. Institute of Science and Technology Austria, Klosterneuburg, Austria

    Phan Thành Nam

  3. University of Copenhagen, Copenhagen, Denmark

    Fabian Portmann

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  1. Douglas Lundholm
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  2. Phan Thành Nam
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  3. Fabian Portmann
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Correspondence to Phan Thành Nam.

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Communicated by C. Le Bris

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Lundholm, D., Nam, P.T. & Portmann, F. Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems. Arch Rational Mech Anal 219, 1343–1382 (2016). https://doi.org/10.1007/s00205-015-0923-5

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  • DOI: https://doi.org/10.1007/s00205-015-0923-5

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