Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

Frank, Rupert^1, 2 ; Lieb, Elliott³ ; Seiringer, Robert⁴

¹ Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
² Department of Mathematics, Princeton University, Princeton, New Jersey 08544
³ Departments of Mathematics and Physics, Princeton University, P. O. Box 708, Princeton, New Jersey 08544
⁴ Department of Physics, Princeton University, P. O. Box 708, Princeton, New Jersey 08544

Journal of the American Mathematical Society, Tome 21 (2008) no. 4, pp. 925-950

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-07-00582-6

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