Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$. (a) Suppose in this question that $\ell = 1$. Show that there exists $M_{1} \in \mathbb{R}_{+}$ such that $$\forall x > 0, \left||f(x)|^{2} - 1\right| \leq \frac{M_{1}}{x^{3/2}}$$ (b) Deduce that $\ell < 1$.
Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and
$$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$
In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
(a) Suppose in this question that $\ell = 1$. Show that there exists $M_{1} \in \mathbb{R}_{+}$ such that
$$\forall x > 0, \left||f(x)|^{2} - 1\right| \leq \frac{M_{1}}{x^{3/2}}$$
(b) Deduce that $\ell < 1$.