我们考虑薛定谔方程\begin{equation}\label{eq_abstract} i\partial_t u(t)=-Δu(t)~~~~~\text{ on }Ω(t) \tag{$\ast$} \end{equation}，其中$Ω(t)\subset\mathbb{R}$是一个依赖于时间$t\in [0,T]$的移动区域。这项工作的目的是提供这样一个方程的解决方案的意义。我们利用有界参考域$Ω_0$和酉映射$h^\sharp(t): L^2(Ω(t),\mathbb{C})\longrightarrow L^2(Ω_0,\mathbb{C})$族的存在性。我们证明了$h^\sharp$共轭提供了一个新的形式\begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }Ω_0\tag{$\ast\ast$} \end{equation}的方程，其中$h_\sharp=(h^\sharp)^{-1}$。哈密顿量$H(t)$是$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$形式的磁拉普拉斯算子，其中$A$是取决于域$Ω(t)$的变形的显式磁势。公式\eqref{eq_abstract2}能够确保初始问题\eqref{eq_abstract}在$Ω(t)$上具有Dirichlet边界条件的弱解和强解的存在性。此外，还指出了正确的Neumann型边界条件不是齐次边界条件，而是磁边界条件$$\partial_νu(t)+i\langleν| A\rangle u(t)=0,$$，即使\eqref{eq_abstract}没有磁项.所有以前的结果也研究了存在的扩散系数以及磁和电势。最后，我们证明了一些相关的副产品作为绝热结果的域和时间依赖的版本的所谓的“莫泽的把戏”的缓慢变形。我们使用这个结果，以简化方程\eqref{eq_abstract2}，并保证适定性稍微不太经常变形的$Ω(t)$。
英文摘要：
We consider the Schr\''odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-Δu(t)~~~~~\text{ on }Ω(t) \tag{$\ast$} \end{equation}where $Ω(t)\subset\mathbb{R}$ is a moving domain depending on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $Ω_0$ and a specific family of unitary maps $h^\sharp(t): L^2(Ω(t),\mathbb{C})\longrightarrow L^2(Ω_0,\mathbb{C})$. We show that the conjugation by $h^\sharp$ provides a newequation of the form \begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }Ω_0\tag{$\ast\ast$} \end{equation} where $h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian operator of the form$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$where $A$ is an explicit magnetic potential depending on the deformation of the domain $Ω(t)$. The formulation \eqref{eq_abstract2} enables to ensure the existence of weak and strong solutions of the initial problem \eqref{eq_abstract} on $Ω(t)$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for \eqref{eq_abstract} are not the homogeneous but the magnetic ones$$\partial_νu(t)+i\langleν| A\rangle u(t)=0,$$even though \eqref{eq_abstract} has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and atime-dependent version of the so-called ``Moser's trick''. We use this outcome in order to simplify Equation \eqref{eq_abstract2} and to guarantee the well-posedness for slightly less regular deformations of $Ω(t)$.
On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers
CNRS IRMAR - UMR 6625（法国国家科学研究中心 IRMAR）；Univ. Lille, CNRS, Inria UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，Inria UMR 8524 - 保罗·庞加莱实验室）；Department of Mathematics, McMaster University（麦克马斯特大学数学系）
On unique continuation for solutions of the Schr{ö}dinger equation on trees
我们证明了如果齐次树上的含时薛定谔方程的解在两个不同的时刻快速衰减，则该解是平凡的.对于自由薛定谔算子，我们利用Laplacian谱理论和复分析，得到了导致在任意时刻急剧衰减的初始条件的一个特征.然后，我们利用Colin de Verdi和Turc的紧支势的Schr dinger算子的谱分解，推广了我们在紧支势存在下的结果.最后，我们使用实变量的方法首先介绍了Escauriaza，Kenig，Ponce和Vega建立一个一般的尖锐的结果的情况下，有界的潜力。
Global well-posedness and scattering for the defocusing, energy -critical, nonlinear Schr{ö}dinger equation in the exterior of a convex obstacle when $d = 4$
