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What is the Wigner function closest to a given square integrable function?. (arXiv:1807.05635v1 [math-ph])
来源于:arXiv
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 查看全文>>