HardyLiebThirring inequalities for fractional Schrödinger operators
Authors:
Rupert L. Frank, Elliott H. Lieb and Robert Seiringer
Journal:
J. Amer. Math. Soc. 21 (2008), 925950
MSC (2000):
Primary 35P15; Secondary 81Q10
Published electronically:
October 10, 2007
MathSciNet review:
2425175
Fulltext PDF Free Access
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Abstract: We show that the LiebThirring inequalities on moments of negative eigenvalues of Schrödingerlike operators remain true, with possibly different constants, when the critical Hardyweight 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 Hardyweight 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 , for less than some critical value.
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 [AbSt]
 M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. Dover Publications, New York, 1992. MR 1225604 (94b:00012)
 [Be1]
 W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 18971905. MR 1254832 (95g:42021)
 [Be2]
 W. Beckner, Pitt's inequality with sharp error estimates, preprint arXiv:math/0701939.
 [BL]
 H. Brezis, E. H. Lieb, Sobolev inequalities with remainder terms. J. Funct. Anal. 62 (1985), no. 1, 7386. MR 0790771 (86i:46033)
 [BVa]
 H. Brezis, J.L. Vázquez, Blowup solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Comp. Madrid 10 (1997), 443469. MR 1605678 (99a:35081)
 [Dau]
 I. Daubechies, An uncertainty principle fermions with a generalized kinetic energy. Comm. Math. Phys. 90 (1983), 511520. MR 0719431 (85j:81008)
 [D]
 E. B. Davies, Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, 1990. MR 1103113 (92a:35035)
 [Do]
 W.F. Donoghue, Monotone matrix functions and analytic continuation. Springer, New YorkHeidelberg, 1974. MR 0486556 (58:6279)
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 T. Ekholm, R. L. Frank, On LiebThirring inequalities for Schrödinger operators with virtual level. Comm. Math. Phys. 264 (2006), no. 3, 725740. MR 2217288 (2006m:81101)
 [FLS]
 R.L. Frank, E.H. Lieb, R. Seiringer, Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value, Commun. Math. Phys. 275 (2007) no. 2, 479489.
 [He]
 I. W. Herbst, Spectral theory of the operator . Comm. Math. Phys. 53 (1977), no. 3, 285294. MR 0436854 (55:9790)
 [L1]
 E. H. Lieb, The number of bound states of onebody Schroedinger operators and the Weyl problem. Proc. Sympos. Pure Math., XXXVI, 241252. Amer. Math. Soc., Providence, R.I., 1980. MR 0573436 (82i:35134)
 [L2]
 E. H. Lieb, Sharp Constants in the HardyLittlewoodSobolev and Related Inequalities, Annals of Math. 118 (1983), 349374. MR 0717827 (86i:42010)
 [L3]
 E. H. Lieb, The stability of matter: from atoms to stars. Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 149. MR 1014510 (91f:81002)
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 [LLo]
 E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001. MR 1817225 (2001i:00001)
 [LTh]
 E. H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics, 269303. Princeton University Press, Princeton, NJ, 1976.
 [LY]
 E. H. Lieb, H.T. Yau, The stability and instability of relativistic matter. Comm. Math. Phys. 118 (1988), no. 2, 177213. MR 0956165 (90c:81251)
 [ReSi1]
 M. Reed, B. Simon, Methods of modern mathematical physics. I. Functional Analysis (Revised and enlarged edition). Academic Press, New YorkLondon, 1980. MR 0751959 (85e:46002)
 [ReSi2]
 M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New YorkLondon, 1978. MR 0493421 (58:12429c)
 [RoSo]
 G. Rozenblyum, M. Solomyak, The CwikelLiebRozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups. St. Petersburg Math. J. 9 (1998), no. 6, 11951211. MR 1610184 (99c:47059)
 [Si1]
 B. Simon, Maximal and minimal Schrödinger forms. J. Operator Theory 1 (1979), no. 1, 3747. MR 0526289 (81m:35104)
 [Si2]
 B. Simon, Functional integration and quantum physics. Second edition. AMS Chelsea Publishing, Providence, RI, 2005. MR 2105995 (2005f:81003)
 [Ya]
 D. Yafaev, Sharp constants in the HardyRellich inequalities. J. Funct. Anal. 168 (1999), no. 1, 121144. MR 1717839 (2001e:26027)
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Additional Information
Rupert L. Frank
Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
Address at time of publication:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email:
rupert@math.kth.se, rlfrank@math.princeton.edu
Elliott H. Lieb
Affiliation:
Departments of Mathematics and Physics, Princeton University, P. O. Box 708, Princeton, New Jersey 08544
Email:
lieb@princeton.edu
Robert Seiringer
Affiliation:
Department of Physics, Princeton University, P. O. Box 708, Princeton, New Jersey 08544
Email:
rseiring@princeton.edu
DOI:
http://dx.doi.org/10.1090/S0894034707005826
PII:
S 08940347(07)005826
Keywords:
Hardy inequality,
relativistic Schr\"odinger operator,
LiebThirring inequalities,
Sobolev inequalities,
stability of matter,
diamagnetic inequality
Received by editor(s):
October 18, 2006
Published electronically:
October 10, 2007
Article copyright:
© Copyright 2007
by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.
