Localization of Multi-Dimensional Wigner Distributions

TL;DR

This paper extends a known Gaussian maximization property of the Wigner distribution from discs to higher-dimensional balls, broadening the understanding of phase space localization in quantum mechanics.

Contribution

It provides a new generalization of Flandrin's result, showing that Gaussians uniquely maximize the Wigner distribution integral over higher-dimensional balls.

Findings

Gaussians uniquely maximize the Wigner distribution over higher-dimensional balls.

The generalization from discs to balls is established.

The result broadens the understanding of phase space localization.

Abstract

A well known result of P. Flandrin states that a Gaussian uniquely maximizes the integral of the Wigner distribution over every centered disc in the phase plane. While there is no difficulty in generalizing this result to higher-dimensional poly-discs, the generalization to balls is less obvious. In this note we provide such a generalization.