Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

• Published in: Journal of the American Mathematical Society
• Main Authors: Frank, Rupert L., Lieb, Elliott H., Seiringer, Robert
• Format: Article in Journal/Newspaper
• Language: unknown
• Published: American Mathematical Society 2007
• Subjects: Hardy inequality; relativistic Schr¨odinger operator; Lieb-Thirring inequalities; Sobolev inequalities; stability of matter; diamagnetic inequality.; Ari; Laplace; laptev
• Online Access: https://doi.org/10.1090/S0894-0347-07-00582-6

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. © 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.). Published - S0894-0347-07-00582-6.pdf Submitted - 0610593.pdf