Geometric extensions of many-particle Hardy inequalities

TL;DR

This paper introduces geometric extensions of many-particle Hardy inequalities using ground state representations and Clifford algebra, with implications for quantum systems and particle interaction models.

Contribution

It provides a systematic derivation of geometric Hardy inequalities involving volumes of simplices, expanding their applicability to complex quantum and particle interaction models.

Findings

Derived new Hardy inequalities involving simplex volumes

Simplified geometric computations using Clifford algebra

Applicable to quantum systems like Calogero-Sutherland models

Abstract

Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of $\mathbb{R}^n$. This includes geometric extensions of the standard Hardy inequalities to involve volumes of simplices spanned by a subset of points. Clifford/multilinear algebra is employed to simplify geometric computations. These results and the techniques involved are relevant for classes of exactly solvable quantum systems such as the Calogero-Sutherland models and their higher-dimensional generalizations, as well as for membrane matrix models, and models of more complicated particle interactions of geometric character.