Let $\left(a_{k}\right)_{k \in \mathbb{N}}$ be a sequence of complex terms such that the series with general term $k^{2} a_{k}$ is absolutely convergent. For $(t, x) \in \mathbb{R}^{2}$ we then denote
$$\Phi_{0}(t, x) = \sum_{k=0}^{+\infty} a_{k} e^{-ik^{2}t + ikx}$$

(a) Show that $\Phi_{0}$ is well defined on $\mathbb{R}^{2}$.

(b) Show that for all $(t, x) \in \mathbb{R}^{2}$ we have $\Phi_{0}(., x) \in C^{1}(\mathbb{R}, \mathbb{C})$ and $\Phi_{0}(t, .) \in C^{2}(\mathbb{R})$. Calculate $\frac{\partial \Phi_{0}}{\partial t}(t, x)$ and $\frac{\partial^{2} \Phi_{0}}{\partial x^{2}}(t, x)$.

(c) Let $(c_{k})$ be a sequence of complex terms such that the series with general term $k^{2} c_{k}$ is absolutely convergent. For $x \in \mathbb{R}$, we set
$$f_{0}(x) = \sum_{k=0}^{+\infty} c_{k} e^{ikx}$$
Construct a function $\Psi_{0}$ defined on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ which satisfies
\begin{itemize}
  \item For all $(t, x) \in \mathbb{R}_{+}^{*}$, $\Psi_{0}(., x) \in C^{1}(\mathbb{R}_{+}^{*}, \mathbb{C})$ and $\Psi_{0}(t, .) \in C^{2}(\mathbb{R}, \mathbb{C})$ and $\Psi_{0}$ is a solution of equation $(F_{0})$, that is
$$\forall (t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}, \quad i\frac{\partial \Psi_{0}}{\partial t}(t, x) + \frac{\partial^{2} \Psi_{0}}{\partial x^{2}}(t, x) + \frac{1}{2t}\Psi_{0}(t, x) = 0$$
  \item For all $t > 0$, $\Psi_{0}(t, .)$ is periodic.
  \item For all $x \in \mathbb{R}$, $\Psi_{0}(1, x) = f_{0}(x)$.
\end{itemize}