Uniform Lieb-Thirring inequality for the three dimensional Pauli operator with a strong non-homogeneous magnetic field

TL;DR

This paper establishes a new Lieb-Thirring inequality for the three-dimensional Pauli operator in strong, non-homogeneous magnetic fields, with optimal growth in field strength and applications to Dirac eigenfunctions.

Contribution

It introduces a novel Lieb-Thirring inequality that captures the optimal growth with magnetic field strength in non-homogeneous fields and provides bounds on Dirac eigenfunctions.

Findings

Optimal growth of the inequality with magnetic field strength

New localization scheme for elliptic operators

Bound on zero energy eigenfunctions of Dirac operator

Abstract

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. A new Lieb-Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.