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 $C |x|^{-2}$ 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.

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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

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

Keywords:
Hardy inequality,
relativistic Schrödinger 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.