What is the Wigner function closest to a given square integrable function?

TL;DR

This paper investigates the problem of finding the Wigner function closest to a given square integrable function in phase space, providing exact solutions for radial functions and finite-dimensional approximations for general cases, with applications in quantum mechanics.

Contribution

It offers an exact solution for radial functions, develops finite-dimensional approximations for general functions, and analyzes the Wigner approximation's validity in dispersive media.

Findings

Exact solution for radial functions on phase space

Finite-dimensional approximation with error estimates

Wigner approximation does not produce a true Wigner function

Abstract

We consider an arbitrary square integrable function $F$ on the phase space and look for the Wigner function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a {\it bona fide} Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics.