A relativistic Hardy-type inequality with minimisers

TL;DR

This paper establishes a sharp, weighted Hardy-type inequality for the Dirac operator, demonstrating that the inequality is attained with explicit minimizers, advancing the understanding of spectral properties in relativistic quantum mechanics.

Contribution

It introduces a new sharp Hardy-type inequality for the Dirac operator and constructs explicit minimizers, extending prior spectral analysis work.

Findings

Proved a sharp weighted Hardy inequality for the Dirac operator.

Constructed explicit functions that attain equality in the inequality.

Extended spectral property analysis in relativistic quantum mechanics.

Abstract

In this paper, we prove a sharp, weighted Hardy-type inequality for the Dirac operator. A key feature of our result is that the inequality is not only sharp but also attained, and we construct explicit minimizers that satisfy the equality case. This extends previous work on the spectral properties of Dirac operators, especially in the context of relativistic quantum mechanics and Coulomb-like potentials.