Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

Abstract

We provide lower bounds for the sum of the negative eigenvalues of the operator $|\sigma\cdot p_A|^{2s} - C_s/|x|^{2s} + V$ in three dimensions, where $s\in (0, 1]$, covering the interesting physical cases $s = 1$ and $s = 1/2$. Here $\sigma$ is the vector of Pauli matrices, $p_A = p - A$, with $p = -i\nabla$ the three-dimensional momentum operator and $A$ a given magnetic vector potential, and $C_s$ is the critical Hardy constant, that is, the optimal constant in the Hardy inequality $|p|^{2s} \geq C_s/|x|^{2s}$. If spin is neglected, results of this type are known in the literature as Hardy-Lieb-Thirring inequalities, which bound the sum of negative eigenvalues from below by $-M_s\int V_{-}^{1 + 3/(2s)}$, for a positive constant $M_s$. The inclusion of magnetic fields in this case follows from the non-magnetic case by diamagnetism. The addition of spin, however, offers extra challenges that make the result more elusive. It is the purpose of this article to resolve this problem by providing simple bounds for the sum of the negative eigenvalues of the operator in question. In particular, for $1/2 \leq s \leq 1$ we are able to express the bound purely in terms of the magnetic field energy $\|B\|_2^2$ and integrals of powers of the negative part of $V$.