A Magnetic Contribution to the Hardy Inequality

TL;DR

This paper investigates how an external magnetic field in three dimensions can enhance the classical Hardy inequality by adding a non-negative potential term, depending on the magnetic field's properties.

Contribution

It demonstrates that a non-vanishing radial magnetic field improves the Hardy inequality with an additional potential term.

Findings

Magnetic fields can strengthen Hardy inequalities in 3D.

The improvement depends on the radial component of the magnetic field.

A non-zero radial magnetic field yields a positive potential contribution.

Abstract

We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.