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arXiv · 1807.05635

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

J. S. Ben-Benjamin·L. Cohen·N. C. Dias·P. Loughlin·J. N. Prata
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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.

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J. S. Ben-Benjamin, L. Cohen, N. C. Dias, P. Loughlin, J. N. Prata. 2018-07-15. What is the Wigner function closest to a given square integrable function?. https://doi.org/10.1137/18m116633x

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