An improved one-dimensional Hardy inequality

TL;DR

This paper presents a sharper one-dimensional Hardy inequality on the halfline with the best possible constant, enhancing classical results and applicable to Schrödinger operator theory.

Contribution

It introduces an improved Hardy inequality with a sharp constant, providing a new tool for analysis in quantum mechanics and related fields.

Findings

Established a Hardy inequality with the optimal constant.

Rederived existing doubly weighted Hardy inequalities using the new result.

Highlighted applications to Schrödinger operators.

Abstract

We prove a one-dimensional Hardy inequality on the halfline with sharp constant, which improves the classical form of this inequality. As a consequence of this new inequality we can rederive known doubly weighted Hardy inequalities. Our motivation comes from the theory of Schr\"odinger operators and we explain the use of Hardy inequalities in that context.