## 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

## 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

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## Bibliographic 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
- MR Author ID: 728268
- ORCID: 0000-0001-7973-4688
- 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
- Received by editor(s): October 18, 2006
- Published electronically: October 10, 2007
- © Copyright 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
- 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
- MathSciNet review: 2425175