Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

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 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 α = 2 / π Z\alpha =2/\pi , for α \alpha less than some critical value.

The best constant in a fractional Hardy inequality

“…We like to note that in some respects, the censored stable process is a better analogue of the killed Brownian motion than the killed stable process is (see [14,7,32], and [37,31]). We suggest the former as a possible setup for studying Dirichlet boundary value problems for non-local integro-differential operators and the corresponding stochastic processes ( [30], [38]) on subdomains of R d ( [3]), beyond the "convolutional" case of the whole of R d ( [21,4]). In this connection, we refer to [25,26,24] for Green-type formulas for the censored process.…”

A Simple Proof of Hardy-Lieb-Thirring Inequalities

“…This can be seen using Trotter's product formula. By an approximation argument the inequality holds also for W (x) = −C s,d |x| −2s ; see [FrLiSe1].…”

“…For d ≥ 3 and 1 < s < d/2 this is a new result, even for integer values of s when the operator is local. This result can not be attained with the method of [FrLiSe1], since positivity properties of the heat kernel break down for s > 1. (3) Though our new proof of (1.3) does not work in the presence of a magnetic field, we shall prove a new operator-theoretic result, which says that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a different constant).…”

“…Here, as usual, D = −i∇ and the operator |D − A| 2s := ((D − A) 2 ) s is defined by means of the spectral theorem. Using the magnetic version of (1.3) we could prove stability of relativistic matter in magnetic fields up to and including the critical value of the nuclear charge α Z = 2/π = C 1/2,3 ; see [FrLiSe1] and also [FrLiSe2].…”

“…While the method in [FrLiSe1] relied on rather involved relations between Sobolev inequalities and decay estimates on heat kernels, the present proof uses nothing more than (1.4) (with γ = 0 and with s replaced by some t < s) and the generalization of a powerful (though elementary to prove) new inequality by Solovej, Sørensen and Spitzer [SoSøSp]. (2) We will extend (1.3) to its optimal parameter range 0 < s < d/2.…”