time $t$ of Eq. (\[timeevolution\]) is given by $$W(x,p,t)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}\chi_{\beta}^{*}(x-z,t)\chi_{\beta}(x+z,t)
e^{-2ipz/\hbar}dz.$$ The PT potential is an infinite array of identical impenetrable wells of width $\pi/2 \alpha$ and therefore the wave-function can be taken equal to zero outside the interval $[0,\pi/2\alpha]$. By imposing this fact, one finds that the following limits of integration can be taken in the above integral: in the left half of the well $(0\leq x \leq\pi/4 \alpha)$, one has $-x\leq z \leq+x$, while in the right half $(\pi/4 \alpha\leq
x \leq\pi/2 \alpha)$, one has $-(\pi/2 \alpha-x)\leq z \leq(\pi/2 \alpha-x)$.

Let us now analyze the Wigner function of the time-evolved CS wave-packet at different fractional revival times.

Symmetric case
--------------

First we choose the symmetric case with parameter values $\rho=\kappa=50$ and $\alpha=2\;a.u.$ The CS parameter is taken as $\beta=0.6$, which corresponds to an initial wave-packet with an energy distribution peaked around the level $\bar{n}=12$. The Wigner functions of the CS wave-packet at different fractional revival times in phase space are shown in Fig. \[wigSymAll\]. At time $t=T_{rev}/4$ (see Fig. \[wigSymAll\](a)) the state is a cat-like state and its Wigner function can be explicitly written in the following two equivalent ways

$$\begin{aligned}
W(x,p,T_{rev}/4)&=&\frac{1}{2}\left[W(x,p,0)+W(\frac{\pi}{2\alpha}-x,p,0)\right]-2
Im(W_{oe}(x,p))\label{Woe}\\
&= &\frac{1}{2}\left[W(x,p,0)+W(\frac{\pi}{2\alpha}-x,p,0)\right]+\sum_{m,n}d_m d_n (-1)^m Im(W_{mn}(x,p))\nonumber .\end{aligned}$$

