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Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

Frank, Rupert L. and Lieb, Elliott H. and Seiringer, Robert (2007) Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. Journal of the American Mathematical Society, 21 (4). pp. 925-950. ISSN 0894-0347. doi:10.1090/S0894-0347-07-00582-6. https://resolver.caltech.edu/CaltechAUTHORS:20170526-134159102

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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 Zɑ = 2/π, for ɑ less than some critical value.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/S0894-0347-07-00582-6DOIArticle
http://www.ams.org/journals/jams/2008-21-04/S0894-0347-07-00582-6/PublisherArticle
https://arxiv.org/abs/math/0610593v2arXivDiscussion Paper
ORCID:
AuthorORCID
Frank, Rupert L.0000-0001-7973-4688
Lieb, Elliott H.0000-0001-5843-3587
Additional Information:© 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. Received by the editors October 18, 2006. We thank Heinz Siedentop for suggesting that we study inequalities of this type, and we thank him, Ari Laptev and Jan Philip Solovej for helpful discussions. We also thank Renming Song for valuable comments on a previous version of this manuscript. This work was partially supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT) (R.F.), by U.S. National Science Foundation grants PHY 01 39984 (E.L.) and PHY 03 53181 (R.S.), and by an A.P. Sloan Fellowship (R.S.).
Funders:
Funding AgencyGrant Number
Swedish Foundation for International Cooperation in Research and Higher Education (STINT)UNSPECIFIED
NSFPHY-0139984
NSFPHY-0353181
Alfred P. Sloan FoundationUNSPECIFIED
Subject Keywords:Hardy inequality, relativistic Schr¨odinger operator, Lieb-Thirring inequalities, Sobolev inequalities, stability of matter, diamagnetic inequality.
Issue or Number:4
Classification Code:2000 MSC: Primary 35P15; Secondary 81Q10
DOI:10.1090/S0894-0347-07-00582-6
Record Number:CaltechAUTHORS:20170526-134159102
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170526-134159102
Official Citation:Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators Rupert L. Frank, Elliott H. Lieb and Robert Seiringer. J. Amer. Math. Soc. 21 (2008), 925-950
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77817
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:30 May 2017 17:08
Last Modified:15 Nov 2021 17:34

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