we denote by ${\cal A}$ the Banach space of all matrix valued functions obtained by the completion of functions of the form $$\begin{aligned}
\label{bFS}
{\bf F}(y,k)=\left[\begin{array}{ll}
G(y,k)&H(y,k)\\
H^\star(y,k)&G(y,-k)
\end{array}\right],\quad (y,k)\in{{\mathbb R}}\times {{\mathbb T}},\end{aligned}$$ with $C^\infty$ smooth entries satisfying $G$ is real valued and $H$ is even in $k$. The completion is taken in the norm given by the maximum of the ${\cal A}$ norms of the entries, see .

The Wigner distribution, corresponding to the wave function $\psi^{({\epsilon})}(t)$, is a $2\times 2$-matrix tensor ${\bf W}_{\epsilon}(t)$, whose entries are distributions, given by their respective Fourier transforms $$\begin{aligned}
\label{hbw}
&
\widehat {{\bf W}}_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}\left[\hat\Psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\otimes \left(\hat\Psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right]
\\
&
=\left[\begin{array}{ll}
\widehat W_{{\epsilon},+}(t,\eta,k)&\widehat Y_{{\epsilon},+}(t,\eta,k)\\
\widehat Y_{{\epsilon},-}(t,\eta,k)&\widehat W_{{\epsilon},-}(t,\eta,k)
\end{array}\right], \quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}},\end{aligned}$$ with $$\begin{aligned}
&
\widehat W_{{\epsilon},+}(t,\eta,k):=\widehat W_{{\epsilon}}(t,\eta,k)=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \left(\hat\psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y_{{\epsilon},+}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,-k+\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y_{{\epsilon},-}(t,\eta,k):=\widehat
  Y_{{\epsilon},+}^\star(t,-\eta,k),\quad 
\widehat W_{{\epsilon},-}(t,\eta,k):=\widehat W_{{\epsilon},+}(t,\eta,-k).\end{aligned}$$ Then ${\bf W}_{\epsilon}(t)$ belongs to $ {\cal A}'$ - the dual to ${\cal A}$ that is made of all distributions ${\bf W}$, whose Fourier transform in the first variable equals $$\begin{aligned}
\label{bW}
\widehat{\bf W}(\eta,k)=\left[\begin{array}{ll}
\widehat W_{+}(\eta,k)&\widehat Y_{+}(\eta,k)\\
\widehat Y_{-}(\eta,k)&\widehat W_{-}(\eta,k)
\end{array}\right],\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}},\end{aligned}$$ whose entries belong to ${\cal A}'$ and satisfy $$\begin{aligned}
&\widehat W_{+}^\star(\eta,k)=\widehat W_{+}(-\eta,k),\quad \widehat
  Y_{+}(\eta,k)=\widehat Y_{+}(\eta,-k),\\
