论文标题
移动域中的schrödinger方程
Schrödinger equation in moving domains
论文作者
论文摘要
我们考虑schr \''odinger方程\ begin {equation} \ label {eq_abstract} i \ partial_t u（t）= - Δu（t）~~~~~~~~~~~~~~~~~~~~ text {on}ω（on}ω（on}ω（t）一个移动域，具体取决于[0，t] $中的时间$ t \。这项工作的目的是为该方程的解决方案提供含义。 We use the existence of a bounded reference domain $Ω_0$ and a specific family of unitary maps $h^\sharp(t): L^2(Ω(t),\mathbb{C})\longrightarrow L^2(Ω_0,\mathbb{C})$.我们表明，$ h^\ sharp $的共轭提供了形式\ begin {equination} \ label {eq_abstract2} i \ partial_t v = h^\ sharp（t） \ end {equation}其中$ h_ \ sharp =（h^\ sharp）^{ - 1} $。 Hamiltonian $ h（t）$是表格$$ h（t）= - （div+ia）\ circ（grad+ia） - | a |^2 $$的磁性拉普拉斯运算符，其中$ a $是明确的磁性电位，具体取决于域$ω（t）$的变形。公式\ eqref {eq_abstract2}使能够确保在$ω（t）上存在初始问题\ eqref {eq_abstract}的弱和强解决方案，并赋予了Dirichlet边界条件。此外，它还表明\ eqref {eq_abstract}的正确的neumann类型边界条件不是均匀的，而是磁性的$ \partial_νu（t）+i \langleν| \ rangle u（t）= 0，$$，即使\ eqref {eq_abstract}没有磁性术语。在存在扩散系数以及磁性和电势的存在下，所有先前的结果也进行了研究。最后，我们证明了一些相关的副产品，作为域慢变形的绝热结果，以及所谓的``Moser's Tract''的当场依赖性版本。我们使用此结果来简化方程式\ eqref {eq_abstract2}，并保证适合良好的$ω（t）$的定期变形。
We consider the Schr\''odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-Δu(t)~~~~~\text{ on }Ω(t) \tag{$\ast$} \end{equation}where $Ω(t)\subset\mathbb{R}$ is a moving domain depending on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $Ω_0$ and a specific family of unitary maps $h^\sharp(t): L^2(Ω(t),\mathbb{C})\longrightarrow L^2(Ω_0,\mathbb{C})$. We show that the conjugation by $h^\sharp$ provides a newequation of the form \begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }Ω_0\tag{$\ast\ast$} \end{equation} where $h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian operator of the form$$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 $Ω(t)$.
