four-way break up, or the so called “compass-like” state emerges at $T_{rev}/8$. As shown by Ref. [@zurek], the latter produces sub-Planck scale structures in the phase space Wigner distribution. The behavior at fractional revival times can be derived in a straightforward way from Eq. (\[timeevolution\]) in the particular case when both $\rho$ and $\kappa$ are even. When the PT potential is symmetric ($\rho=\kappa$), the property of the Jacobi polynomial allows us to write $$\begin{aligned}
\label{symm}
&&\psi_{n}(x)=(-1)^{n}\psi_{n}(\pi/2\alpha-x)\nonumber\\
\quad &or&\;\;
\psi_{n}(\pi/4\alpha-x)=(-1)^{n}\psi_{n}(\pi/4\alpha+x).\end{aligned}$$ Using this fact at $T_{rev}/4$, a straightforward calculation yields $$\chi_{\beta}(x,T_{rev}/4)=\frac{1}{\sqrt{2}}\left[e^{-i\pi/4}\chi_{\beta}(x,0) + e^{i\pi/4}\chi_{\beta}(\frac{\pi}{2\alpha}-x,0)\right],$$ that is, the CS is split into two parts at time $T_{rev}/4$ forming the well known Schrödinger cat state. Each element of the superposition is proportional to the initial wave-packet, even though they are situated at the opposite ends of the potential well. In the asymmetric case instead one gets

$$\begin{aligned}
&&\chi_{\beta}(x,T_{rev}/4)=\chi_{\beta}^{e}(x,0)- i\chi_{\beta}^{o}(x,0);\;\;\quad  \eta=\rho+\kappa=\mathrm{multiple\;of}\;
4\nonumber\\
\mathrm{and}\;\;&&\chi_{\beta}(x,T_{rev}/4)=-i\chi_{\beta}^{e}(x,0)+
\chi_{\beta}^{o}(x,0);\;\;\quad\eta=\mathrm{multiple\;of}\; 2,\end{aligned}$$

where $\chi_{\beta}^{e,o}(x,0)=\sum_{n,\; even,odd} d_n \psi_n
(x)$. For the analysis at time $T_{rev}/8$, it is instead convenient to define a classical wave-packet [@averbukh; @robinett]: $$\chi_{\beta}^{cl}(x,t)=\sum_{n=0}^{\infty}d_{n}^{\beta}\psi_{n}(x)e^{-2\pi
i n t/T_{cl}}, \label{cl}$$ which evolves with the classical periodicity $T_{cl}=\pi/(\rho+\kappa)\alpha^{2}$. It can be easily seen that at $t=T_{rev}/2$, the CS will revive with an extra half 