Let $g \in C_{b}(\mathbb{R})$ satisfying hypothesis A (i.e., $g$ is of class $C^{\infty}$ on $\mathbb{R}$, bounded, and all its derivative functions of all orders are bounded on $\mathbb{R}$). Deduce the set of functions $g$ satisfying hypothesis A and such that $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$.

We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Let $x _ { 0 }$ in $] 0 , + \infty [$. We set for $x > 0$
$$h ( x ) = \int _ { x _ { 0 } } ^ { x } | z ( u ) | \frac { \mathrm { d } u } { u ^ { 2 } }$$
a) Show that there exist real constants $\mu$ and $M$ such that $h$ satisfies the differential inequality for $x \geqslant x _ { 0 }$
$$h ^ { \prime } ( x ) - \frac { \mu } { x ^ { 2 } } h ( x ) \leqslant \frac { M } { x ^ { 2 } }$$
Specify the constants $\mu$ and $M$ in terms of $A , B$ and $\lambda$.
b) Deduce that $h$ is bounded on $\left[ x _ { 0 } , + \infty [ \right.$ and then that $z$ is bounded on the same interval.
Multiply by $e ^ { \mu / x }$ and integrate the inequality from the previous question.

We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Justify that
$$\int _ { x } ^ { + \infty } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } = O \left( \frac { 1 } { x } \right)$$
near $+ \infty$. Deduce the existence of constants $\alpha$ and $\beta$ such that near $+ \infty$,
$$z ( x ) = \alpha \cos ( x - \beta ) + O \left( \frac { 1 } { x } \right)$$

Let $n \in \mathbb { N }$. Show that there exists a pair of real numbers $( \alpha _ { n } , \beta _ { n } )$ such that for $x \rightarrow + \infty$,
$$\varphi _ { n } ( x ) = \frac { \alpha _ { n } } { \sqrt { x } } \cos \left( x - \beta _ { n } \right) + O \left( \frac { 1 } { x \sqrt { x } } \right)$$

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 $f^{\prime}(0) = 0$.
We set $F_{1} = \Re(f)$ and $f_{2} = \Im(f)$. Express $f_{1}^{\prime\prime}$ and $f_{2}^{\prime\prime}$ in terms of $f_{1}^{\prime}, f_{2}^{\prime}, f_{1}$ and $f_{2}$.

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 $f^{\prime}(0) = 0$.
Show that $$\forall x \in \mathbb{R}, \left|f^{\prime}(x)\right|^{2} + \frac{1}{4\alpha}\left(\alpha|f(x)|^{2} + 1\right)^{2} = \frac{1}{4\alpha}(\alpha + 1)^{2}$$

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 $f^{\prime}(0) = 0$.
Show that $\forall x \in \mathbb{R}, |f(x)| \leq 1$ and that $$\forall x \in \mathbb{R}, \left|f^{\prime}(x)\right| \leq \frac{1}{2\sqrt{\alpha}}(\alpha + 1)$$

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 $f^{\prime}(0) = 0$.
The purpose of this question is to prove that there exist $(\ell, M_{0}) \in \mathbb{R}_{+}^{2}$ such that $$\forall x > 0, \left||f(x)|^{2} - 1\right| \leq \frac{M_{0}}{x}$$
(a) Show that $$\forall x \in \mathbb{R}, \Im\left(f^{\prime}(x)\overline{f(x)}\right) + \frac{x}{4}|f(x)|^{2} - \frac{1}{4}\int_{0}^{x}|f(t)|^{2}\,dt = 0$$
(b) Show that $$\forall x > 0, \frac{d}{dx}\left(\frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt\right) = -\frac{4}{x^{2}}\Im\left(f^{\prime}(x)\overline{f(x)}\right)$$
(c) Deduce that there exists $\ell \in \mathbb{R}_{+}$ such that $$\lim_{x \rightarrow +\infty} \frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt = \ell$$
(d) Show that there exists $M \in \mathbb{R}_{+}$ such that $$\forall x > 0, \left|\frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt - \ell\right| \leq \frac{M}{x}$$
(e) Conclude.

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 $f^{\prime}(0) = 0$.
Show that $|f|$ is not periodic.

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 $f^{\prime}(0) = 0$.
For $(\ell, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$, we set: $$f_{\alpha}(x) = f(x)\exp\left(i\frac{x^{2}}{4}\right), \quad \Psi_{\alpha}(t, x) = \frac{1}{\sqrt{t}} f_{\alpha}\left(\frac{x}{\sqrt{t}}\right)$$
(a) Does there exist $t > 0$ such that $\Psi_{\alpha}(t, .)$ is periodic?
(b) Express $f_{\alpha}^{\prime}, f_{\alpha}^{\prime\prime}$ and $|f_{\alpha}|$ in terms of $f, f^{\prime}, f^{\prime\prime}$ and $|f|$.
(c) Justify that for all $(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$, we have $\Psi_{\alpha}(., x) \in C^{1}(\mathbb{R}^{+*}, \mathbb{C})$ and $\Psi_{\alpha}(t, .) \in C^{2}(\mathbb{R}, \mathbb{C})$, then prove that $\Psi_{\alpha}$ satisfies equation $(F_{\alpha})$: $$\left(F_{\alpha}\right) \quad i\frac{\partial \Psi_{\alpha}}{\partial t}(t, x) + \frac{\partial^{2} \Psi_{\alpha}}{\partial x^{2}}(t, x) + \frac{1}{2}\Psi_{\alpha}(t, x)\left(\alpha\left|\Psi_{\alpha}(t, x)\right|^{2} + \frac{1}{t}\right) = 0$$

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left((I_{3} + \mathcal{M})G(x)\right) \wedge G^{\prime}(x)$$ and that moreover $$\|G^{\prime}(0)\| = 1, \quad \left((I_{3} + \mathcal{M})G(0)\right) \cdot G^{\prime}(0) = 0$$ We set $\forall x \in \mathbb{R}, T(x) = G^{\prime}(x)$.
Show that for all $x \in \mathbb{R}$, $\|T(x)\| = 1$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left((I_{3} + \mathcal{M})G(x)\right) \wedge G^{\prime}(x)$$ and that moreover $$\|G^{\prime}(0)\| = 1, \quad \left((I_{3} + \mathcal{M})G(0)\right) \cdot G^{\prime}(0) = 0$$ We set $\forall x \in \mathbb{R}, T(x) = G^{\prime}(x)$.
Show that for all $x \in \mathbb{R}$, $\left(I_{3} + \mathcal{M}\right)G(x) - x G^{\prime}(x) = 2 G^{\prime}(x) \wedge G^{\prime\prime}(x)$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left((I_{3} + \mathcal{M})G(x)\right) \wedge G^{\prime}(x)$$ and that moreover $$\|G^{\prime}(0)\| = 1, \quad \left((I_{3} + \mathcal{M})G(0)\right) \cdot G^{\prime}(0) = 0$$ We set $\forall x \in \mathbb{R}, T(x) = G^{\prime}(x)$. $F(t)$ denotes the matrix exponential defined in question 9.
For $(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$, we define $\tilde{G}(t, x) = \sqrt{t}\, F\!\left(\frac{\ln(t)}{2}\right) G\!\left(\frac{x}{\sqrt{t}}\right)$. Show that for all $(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$ we have $\tilde{G}(., x) \in C^{1}(\mathbb{R}_{+}^{*}, \mathbb{R}^{3})$ and $\tilde{G}(t, .) \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$, then establish that $$\forall (t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}, \frac{\partial \tilde{G}}{\partial t}(x, t) = \frac{\partial \tilde{G}}{\partial x}(x, t) \wedge \frac{\partial^{2} \tilde{G}}{\partial x^{2}}(x, t)$$

We assume that $m = 0$, that is $$\mathcal{M} = \left(\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left(G(x)\right) \wedge G^{\prime}(x)$$ and that moreover there exists $\lambda > 0$ such that $$G(0) = (0, 0, 2\lambda), \quad G^{\prime}(0) = (1, 0, 0)$$ We set $\forall x \in \mathbb{R}, T(x) = G^{\prime}(x)$.
Show that for all $x \in \mathbb{R}$, we have $|G_{1}(x)| \leq |x|$, where we denote by $G_{1} \in C^{2}(\mathbb{R}, \mathbb{R})$ the first coordinate of $G = (G_{1}, G_{2}, G_{3})$.

We assume that $m = 0$, that is $$\mathcal{M} = \left(\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{\infty}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left(G(x)\right) \wedge G^{\prime}(x)$$ and that moreover there exists $\lambda > 0$ such that $$G(0) = (0, 0, 2\lambda), \quad G^{\prime}(0) = (1, 0, 0)$$ We set $\forall x \in \mathbb{R}, T(x) = G^{\prime}(x)$.
Show that for all $x \in \mathbb{R}$, we have $\|T^{\prime}(x)\| = \lambda$.

We assume that $m = 0$, that is $$\mathcal{M} = \left(\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{\infty}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left(G(x)\right) \wedge G^{\prime}(x)$$ and that moreover there exists $\lambda > 0$ such that $$G(0) = (0, 0, 2\lambda), \quad G^{\prime}(0) = (1, 0, 0)$$ We set $\forall x \in \mathbb{R}, T(x) = G^{\prime}(x)$. For $x \in \mathbb{R}$, we introduce the vectors $$n(x) = \frac{T^{\prime}(x)}{\lambda}, \quad b(x) = T(x) \wedge n(x)$$ so that $(T(x), n(x), b(x))$ forms a direct orthonormal basis.
(a) Using question 15, show that for all $x \in \mathbb{R}$, we have $2b^{\prime}(x) = -x\, n(x)$.
(b) Deduce that $n^{\prime}(x) = -\lambda T(x) + \frac{x}{2} b(x)$.
(c) Show that $G$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime\prime}(x) + \left(\lambda^{2} + \frac{x^{2}}{4}\right) G^{\prime}(x) - \frac{x}{4} G(x) = 0$$

Let $\lambda \in \mathbb{C}$ such that $\operatorname{Re}(\lambda) > 0$. Let $u$ be a function with complex values of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$.
Suppose that the function $v = u' + \lambda u$ is bounded on $\mathbb{R}^{+}$. Show that $u$ is bounded on $\mathbb{R}^{+}$.
One may consider the differential equation $y' + \lambda y = v$.

Let $T \in \mathcal{M}_{n}(\mathbb{C})$ be an upper triangular matrix with complex entries. Suppose that the diagonal entries of $T$ are complex numbers with strictly positive real part. Let $u_{1}, \ldots, u_{n}$ be functions with complex values, defined and of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$ and let, for all $t \in \mathbb{R}^{+}$, $$U(t) = \begin{pmatrix} u_{1}(t) \\ \vdots \\ u_{n}(t) \end{pmatrix}$$
Suppose that, for all $t \in \mathbb{R}^{+}$, $U'(t) + TU(t) = 0$.
Show that the functions $u_{j}$, where $1 \leqslant j \leqslant n$, are bounded on $\mathbb{R}^{+}$.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Recall that for any matrix $M \in \mathcal{M}_{n}(\mathbb{C})$, $\exp(M) = \sum_{k=0}^{\infty} \frac{M^{k}}{k!}$.
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix with complex eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ and let $\alpha$ be a real number such that $0 < \alpha < \min_{1 \leqslant j \leqslant n} \operatorname{Re}(\lambda_{j})$.
Show that the function $t \mapsto \mathrm{e}^{\alpha t}\exp(-tA)$ is bounded on $\mathbb{R}^{+}$.
One may apply question III.B.2 to an upper triangular matrix $T$ similar to $A - \alpha I_{n}$.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Which are the solutions of (II.1) that extend continuously to 0?

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$ a square matrix with complex coefficients, and we denote by $u$ the endomorphism of $\mathbf{C}^n$ canonically associated with this matrix. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$. For all real $t$ and for $(i,j) \in \llbracket 1,n \rrbracket^2$, we denote by $v_{i,j}(t)$ the coefficient with indices $(i,j)$ of the matrix $e^{tA}$.
$\mathbf{19}$ ▷ Show that, if $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$, then $\alpha < 0$.

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$. For all real $t$ and for $(i,j) \in \llbracket 1,n \rrbracket^2$, we denote by $v_{i,j}(t)$ the coefficient with indices $(i,j)$ of the matrix $e^{tA}$.
$\mathbf{22}$ ▷ Deduce from question 21) that there exists a natural integer $p$ such that, for all $(i,j) \in \llbracket 1,n \rrbracket^2$, we have $$v_{i,j}(t) = O\left(t^p e^{\alpha t}\right) \text{ as } t \rightarrow +\infty.$$

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$.
$\mathbf{23}$ ▷ Study the converse of question 19): that is, show that if $\alpha < 0$ then $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$.

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We introduce the following polynomials: $$\begin{aligned} & P_s(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) < 0}} (X - \lambda)^{m_\lambda}, \\ & P_i(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) > 0}} (X - \lambda)^{m_\lambda}, \\ & P_n(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) = 0}} (X - \lambda)^{m_\lambda}, \end{aligned}$$ and the subspaces $E_s = \operatorname{Ker}(P_s(A))$, $E_i = \operatorname{Ker}(P_i(A))$ and $E_n = \operatorname{Ker}(P_n(A))$ of $E = \mathbf{C}^n$. We have $E = E_s \oplus E_i \oplus E_n$.
$\mathbf{26}$ ▷ Show that $$E_n = \left\{ X \in E \mid \exists C \in \mathbf{R}_+^* \quad \exists p \in \mathbf{N} \quad \forall t \in \mathbf{R} \quad \left\| e^{tA} X \right\|_E \leq C(1 + |t|)^p \right\}.$$