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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
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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 \vert x\vert^{-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.


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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.

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