Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

Authors:
Rupert L. Frank, Elliott H. Lieb and Robert Seiringer

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

MSC (2000):
Primary 35P15; Secondary 81Q10

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

Published electronically:
October 10, 2007

MathSciNet review:
2425175

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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 , for less than some critical value.

**[AbSt]**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****[Be1]**William Beckner,*Pitt’s inequality and the uncertainty principle*, Proc. Amer. Math. Soc.**123**(1995), no. 6, 1897–1905. MR**1254832**, https://doi.org/10.1090/S0002-9939-1995-1254832-9**[Be2]**W. Beckner,*Pitt's inequality with sharp error estimates*, preprint arXiv:math/0701939.**[BL]**Haïm Brezis and Elliott H. Lieb,*Sobolev inequalities with remainder terms*, J. Funct. Anal.**62**(1985), no. 1, 73–86. MR**790771**, https://doi.org/10.1016/0022-1236(85)90020-5**[BVa]**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****[Dau]**Ingrid Daubechies,*An uncertainty principle for fermions with generalized kinetic energy*, Comm. Math. Phys.**90**(1983), no. 4, 511–520. MR**719431****[D]**E. B. Davies,*Heat kernels and spectral theory*, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR**1103113****[Do]**William F. Donoghue Jr.,*Monotone matrix functions and analytic continuation*, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 207. MR**0486556****[EkFr]**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**, https://doi.org/10.1007/s00220-006-1521-z**[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, 479-489.**[He]**Ira W. Herbst,*Spectral theory of the operator (𝑝²+𝑚²)^{1/2}-𝑍𝑒²/𝑟*, Comm. Math. Phys.**53**(1977), no. 3, 285–294. MR**0436854****[L1]**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****[L2]**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**, https://doi.org/10.2307/2007032**[L3]**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**, https://doi.org/10.1090/S0273-0979-1990-15831-8**[L4]**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,*The stability of matter and quantum electrodynamics*, Milan J. Math.**71**(2003), 199–217. MR**2120921**, https://doi.org/10.1007/s00032-003-0020-3**[LLo]**Elliott H. Lieb and Michael Loss,*Analysis*, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR**1817225****[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, 269-303. Princeton University Press, Princeton, NJ, 1976.**[LY]**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****[ReSi1]**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****[ReSi2]**Michael Reed and Barry Simon,*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419**

Michael Reed and Barry Simon,*Methods of modern mathematical physics. II. Fourier analysis, self-adjointness*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0493420**

Michael Reed and Barry Simon,*Methods of modern mathematical physics. IV. Analysis of operators*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**0493421****[RoSo]**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****[Si1]**Barry Simon,*Maximal and minimal Schrödinger forms*, J. Operator Theory**1**(1979), no. 1, 37–47. MR**526289****[Si2]**Barry Simon,*Functional integration and quantum physics*, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. MR**2105995****[Ya]**D. Yafaev,*Sharp constants in the Hardy-Rellich inequalities*, J. Funct. Anal.**168**(1999), no. 1, 121–144. MR**1717839**, https://doi.org/10.1006/jfan.1999.3462

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

Keywords:
Hardy inequality,
relativistic Schr\"odinger operator,
Lieb-Thirring 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 non-commercial purposes.