Rapidly decaying Wigner functions are Schwartz functions

A standard choice is to specialize to a Gaussian coherent state $χ(y)=exp(−x2/2)/(2π)n$ (especially when used as in Subsection II B as the reference wavefunction with respect to which Husimi function is defined). However, this specialization is not necessary and one could instead take χ to be, e.g., a smooth and compactly supported wavefunction.

Note that the multi-indices a, b, c, d in this section are n-dimensional rather than 2n-dimensional because the wavefunction ψ and the kernel $Kρ$ take arguments in position space rather than phase space.

For instance, the plateau wavefunction ψ(y) = {1 if 0 ≤ y ≤ 1, 0 otherwise} is compactly supported in position space but in momentum space decays to infinity only as a polynomial.

Consider the n = 1 quantum state $ρ=∑k=0∞|ψk〉〈ψk|$ with $ψk(y)=ψ0(y−zk)=(6/π2)k−2⁡exp[−(y−zk)2/2]/2π$ with z_k = k³. This is a mixture of Gaussians of equal variance, so the derivatives are all rapidly decreasing and $Fρ∈D(R2n)$⁠, but the mean tr[ρX] diverges, so $Wρ$ is not a Schwartz function.

Of course, the peculiar features of the Moyal product are not just a matter of practicalities: Because the Moyal bracket has phase space derivatives of arbitrarily high order, the dynamics of the Wigner function are non-local.