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# 数学代写|数学物理代写Mathematical Physics代考|PHYS509 Notations

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## 数学代写|数学物理代写Mathematical Physics代考|Notations

Let us introduce first a few notations. We define the Hamiltonian flow
$$\left{\begin{array}{l} \dot{x}(t)=\xi(t) \quad ; \quad x(0)=x_0, \ \dot{\xi}(t)=-\lambda^{\prime}(x(t)) \quad ; \quad \xi(0)=\xi_0 . \end{array}\right.$$
Note that $x, \xi \in C^{\infty}\left(\mathbf{R} ; \mathbf{R}^d\right)$ and that the subquadraticity of $V(x)$ yields an exponential control in time on the norm of the trajectories. We define the classical action as
$$S(t)=\int_0^t\left(\frac{1}{2}|\xi(s)|^2-\lambda(x(s))\right) d s,$$
and we perform the rescaling:
$$\psi^{\varepsilon}(t, x) \sim \frac{1}{\varepsilon^{1 / 4}} u^{\varepsilon}\left(t, \frac{x-x(t)}{\sqrt{\varepsilon}}\right) e^{i(S(t)+\xi(t) \cdot(x-x(t))) / \varepsilon} \chi(t, x) .$$
In the scalar case, $V(x)=\lambda(x) \operatorname{Id}$ (and $\chi(x)=1)$, the equation for $\psi^{\varepsilon}$ writes
$$i \partial_t u^{\varepsilon}+\frac{1}{2} \partial_y^2 u^{\varepsilon}=\mathcal{V}^{\varepsilon} u^{\varepsilon}+\varepsilon^{\alpha-3 / 2}\left|u^{\varepsilon}\right|^2 u^{\varepsilon},$$
where $\mathcal{V}^{\varepsilon}(t, y)=\frac{1}{\varepsilon}\left(\lambda(x(t)+\sqrt{\varepsilon} y)-\lambda(x(t))-\sqrt{\varepsilon} \lambda^{\prime}(x(t)) y\right)$.

## 数学代写|数学物理代写Mathematical Physics代考|Main Results

Theorem 1 (Adiabatic Theorem, [5]). Let $a \in \mathcal{S}(\mathbf{R})$. Under (H1) and (H2), there exists $C>0$ such that $w^{\varepsilon}(t, x)=\psi^{\varepsilon}(t, x)-\varphi^{\varepsilon}(t, x) \chi^1(t, x)$ satisfies
$$\sup {|t| \leq C \log \log \frac{1}{\varepsilon}}\left(\left|w^{\varepsilon}(t)\right|{L^2}+\left|x w^{\varepsilon}(t)\right|_{L^2}+\left|\varepsilon \partial_x w^{\varepsilon}(t)\right|_{L^2}\right) \underset{\varepsilon \rightarrow 0}{\longrightarrow} 0$$
We prove this theorem by energy estimates on the function $w^{\varepsilon}+\varepsilon g^{\varepsilon}$ where $g^{\varepsilon}$ are correction terms belonging to the orthogonal to $\operatorname{Ker}(V(x)-\lambda(x)$ Id). These correction terms are here to compensate the component of the vector $r(t, x)=$ $\partial_t \chi+\xi(t) \partial_x \chi$ on $\operatorname{Ker}(V(x)-\lambda(x) \mathrm{Id})^{\perp}$. We point out that if $V(x)=\lambda(x) \mathrm{Id}$, one can prove that the asymptotics holds up to (an analogue of) Ehrenfest time by using Strichartz estimates.
It is also possible to prove a nonlinear superposition result. We suppose
$$\psi_0^{\varepsilon}(x)=\varphi_1^{\varepsilon}(0, x) \chi_1(x)+\varphi_2^{\varepsilon}(0, x) \chi_2(x),$$

with $\left(x_1, \xi_1\right) \neq\left(x_2, \xi_2\right)$ when $\chi_1(x)$ and $\chi_2(x)$ are in the same eigenspace. We naturally associate with $\left(x_j, \xi_j, \chi_j\right), j \in{1,2}$ an ansatz $\varphi_j^{\varepsilon}$ and we obtain the following result.

Theorem 2 (Nonlinear Superposition, [5]). Set $E_j=\frac{\xi_j^2}{2}+\tilde{\lambda}j\left(x_j\right)$ and suppose $$\Gamma=\inf {x \in \mathbf{R}}\left|\tilde{\lambda}1(x)-\tilde{\lambda}_2(x)-\left(E_1-E_2\right)\right|>0 .$$ Then, there exists $C>0$ such that the function $w^{\varepsilon}(t)=\psi^{\varepsilon}(t)-\varphi_1^{\varepsilon} \chi^1(t, x)-$ $\varphi_2^{\varepsilon} \chi^2(t, x)$ satisfies $$\sup {t \leq C \log \log \frac{1}{\varepsilon}}\left(\left|w^{\varepsilon}(t)\right|_{L^2}+\left|x w^{\varepsilon}(t)\right|_{L^2}+\left|\varepsilon \partial_x w^{\varepsilon}(t)\right|_{L^2}\right) \underset{\varepsilon \rightarrow 0}{\longrightarrow} 0$$
The proof of this theorem relies on energy estimates and a careful analysis of the nonlinear interactions between $\varphi_1^{\varepsilon}$ and $\varphi_2^{\varepsilon}$. One can prove that interaction terms of the form $\left|\varphi_1^{\varepsilon}\right|^2\left|\varphi_2^{\varepsilon}\right|$ are small provided the gap between the trajectory, $\left|x_1(t)-x_2(t)\right|$ is large enough. Then, the constant $\Gamma$ allows to control the lengths of the time intervals where this gap is small.

## 数学代写|数学物理代写数学物理代考|符号

.注释

$$\left{\begin{array}{l} \dot{x}(t)=\xi(t) \quad ; \quad x(0)=x_0, \ \dot{\xi}(t)=-\lambda^{\prime}(x(t)) \quad ; \quad \xi(0)=\xi_0 . \end{array}\right.$$

$$S(t)=\int_0^t\left(\frac{1}{2}|\xi(s)|^2-\lambda(x(s))\right) d s,$$

$$\psi^{\varepsilon}(t, x) \sim \frac{1}{\varepsilon^{1 / 4}} u^{\varepsilon}\left(t, \frac{x-x(t)}{\sqrt{\varepsilon}}\right) e^{i(S(t)+\xi(t) \cdot(x-x(t))) / \varepsilon} \chi(t, x) .$$

$$i \partial_t u^{\varepsilon}+\frac{1}{2} \partial_y^2 u^{\varepsilon}=\mathcal{V}^{\varepsilon} u^{\varepsilon}+\varepsilon^{\alpha-3 / 2}\left|u^{\varepsilon}\right|^2 u^{\varepsilon},$$
，其中$\mathcal{V}^{\varepsilon}(t, y)=\frac{1}{\varepsilon}\left(\lambda(x(t)+\sqrt{\varepsilon} y)-\lambda(x(t))-\sqrt{\varepsilon} \lambda^{\prime}(x(t)) y\right)$ .

## 数学代写|数学物理代写数学物理代考|主要结果

$$\sup {|t| \leq C \log \log \frac{1}{\varepsilon}}\left(\left|w^{\varepsilon}(t)\right|{L^2}+\left|x w^{\varepsilon}(t)\right|_{L^2}+\left|\varepsilon \partial_x w^{\varepsilon}(t)\right|_{L^2}\right) \underset{\varepsilon \rightarrow 0}{\longrightarrow} 0$$

。这个定理的证明依赖于能量估计和对$\varphi_1^{\varepsilon}$和$\varphi_2^{\varepsilon}$之间非线性相互作用的仔细分析。我们可以证明形式$\left|\varphi_1^{\varepsilon}\right|^2\left|\varphi_2^{\varepsilon}\right|$的相互作用项很小，只要轨迹之间的差距足够大，$\left|x_1(t)-x_2(t)\right|$就足够大。然后，常量$\Gamma$允许控制时间间隔的长度，当这个间隔很小的时候

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