Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

Frank, Rupert; Lieb, Elliott; Seiringer, Robert

doi:10.1090/S0894-0347-07-00582-6

Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
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by Rupert L. Frank, Elliott H. Lieb and Robert Seiringer

J. Amer. Math. Soc. 21 (2008), 925-950

DOI: https://doi.org/10.1090/S0894-0347-07-00582-6

Published electronically: October 10, 2007

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Abstract:

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrödinger-like operators remain true, with possibly different constants, when the critical Hardy-weight $C |x|^{-2}$ is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schrödinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge $Z\alpha =2/\pi$, for $\alpha$ less than some critical value.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
William Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1897–1905. MR 1254832, DOI 10.1090/S0002-9939-1995-1254832-9

Pitt’s inequality with sharp error estimates

Haïm Brezis and Elliott H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), no. 1, 73–86. MR 790771, DOI 10.1016/0022-1236(85)90020-5
Haim Brezis and Juan Luis Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. MR 1605678
Ingrid Daubechies, An uncertainty principle for fermions with generalized kinetic energy, Comm. Math. Phys. 90 (1983), no. 4, 511–520. MR 719431, DOI 10.1007/BF01216182
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR 1103113
William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556, DOI 10.1007/978-3-642-65755-9
T. Ekholm and R. L. Frank, On Lieb-Thirring inequalities for Schrödinger operators with virtual level, Comm. Math. Phys. 264 (2006), no. 3, 725–740. MR 2217288, DOI 10.1007/s00220-006-1521-z

Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value

275

Ira W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys. 53 (1977), no. 3, 285–294. MR 436854
Elliott H. Lieb, The number of bound states of one-body Schroedinger operators and the Weyl problem, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 241–252. MR 573436
Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI 10.2307/2007032
Elliott H. Lieb, The stability of matter: from atoms to stars, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 1–49. MR 1014510, DOI 10.1090/S0273-0979-1990-15831-8
Elliott H. Lieb, The stability of matter and quantum electrodynamics, Fundamental physics—Heisenberg and beyond, Springer, Berlin, 2004, pp. 53–68. MR 2091506
Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014

Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities

Elliott H. Lieb and Horng-Tzer Yau, The stability and instability of relativistic matter, Comm. Math. Phys. 118 (1988), no. 2, 177–213. MR 956165, DOI 10.1007/BF01218577
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
G. Rozenblyum and M. Solomyak, The Cwikel-Lieb-Rozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups, Algebra i Analiz 9 (1997), no. 6, 214–236 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1195–1211. MR 1610184
Barry Simon, Maximal and minimal Schrödinger forms, J. Operator Theory 1 (1979), no. 1, 37–47. MR 526289
Barry Simon, Functional integration and quantum physics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. MR 2105995, DOI 10.1090/chel/351
D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal. 168 (1999), no. 1, 121–144. MR 1717839, DOI 10.1006/jfan.1999.3462