What is the Wigner function closest to a given square integrable function?
| Autor: | Ben-Benjamin, J. S., Cohen, L., Dias, N. C., Loughlin, P., Prata, J. N. |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | SIAM J. Math. Anal. 50-5 (2018), pp. 5161-5197 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1137/18M116633X |
| Popis: | 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. Comment: 50 pages, to appear in SIAM J. Math. Anal |
| Databáze: | arXiv |
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