论文标题
奇点的传播和弗雷德姆分析的时间依赖性schrödinger方程
Propagation of singularities and Fredholm analysis for the time-dependent Schrödinger equation
论文作者
论文摘要
我们研究了与时间相关的schrödinger运算符$ p = d_t +δ_g + v $作用于在$ \ mathbb {r}^{n + 1} $上定义的函数，其中，使用coortinates $ z \ in \ mathbb {r} \ \ partial_t $，$δ_g$是相对于时间依赖的非捕获指标的家庭的积极的laplacian $ g_ {ij}（z，z，t）dz^i dz^i dz^j $ on $ \ mathbb {r mathbb {r}^n $，与euclidean compact in spacement in spacement in s Spacemple和z在时空中紧凑。在本文中，我们通过在操作员之间的一对希尔伯特空间进行了不可逆转的方式，介绍了一种研究$ p $的新方法。 使用这种可逆性，对于时间依赖的Schrödinger方程解决了“最终状态问题”，也就是说，找到一个全局解决方案$ u（z，t）$ $ pu = 0 $ of $ t \ pu = 0 $已将渐近剂规定为$ t \ to \ to \ infty $。这些渐近学是$$ u（z，t）\ sim t^{ - n/2} e^{ $ \ MATHCAL {W}^K（\ MATHBB {r}^n）$;在这里，$ k $是同时测量无穷大的平滑度和衰减的规律性参数。当然，我们同样可以与$ t \ to - \ infty $一样，同样有理由开处方渐近学；这导致传入数据$ f _- $。我们将“ Poisson运算符” $ \ Mathcal {P} _ \ PM：F_ \ PM \ \ to u $ $ $，然后精确地在$ \ Mathcal {w}^k（\ Mathbb {r}^n）$ space上表征了这些操作员的范围。最后，我们证明了散射矩阵，将$ f _- $映射到$ f _+$，保留了这些空间。
We study the time-dependent Schrödinger operator $P = D_t + Δ_g + V$ acting on functions defined on $\mathbb{R}^{n+1}$, where, using coordinates $z \in \mathbb{R}^n$ and $t \in \mathbb{R}$, $D_t$ denotes $-i \partial_t$, $Δ_g$ is the positive Laplacian with respect to a time dependent family of non-trapping metrics $g_{ij}(z, t) dz^i dz^j$ on $\mathbb{R}^n$ which is equal to the Euclidean metric outside of a compact set in spacetime, and $V = V(z, t)$ is a potential function which is also compactly supported in spacetime. In this paper we introduce a new approach to studying $P$, by finding pairs of Hilbert spaces between which the operator acts invertibly. Using this invertibility it is straightforward to solve the `final state problem' for the time-dependent Schrödinger equation, that is, find a global solution $u(z, t)$ of $Pu = 0$ having prescribed asymptotics as $t \to \infty$. These asymptotics are of the form $$ u(z, t) \sim t^{-n/2} e^{i|z|^2/4t} f_+\big( \frac{z}{2t} \big), \quad t \to +\infty $$ where $f_+$, the `final state' or outgoing data, is an arbitrary element of a suitable function space $\mathcal{W}^k(\mathbb{R}^n)$; here $k$ is a regularity parameter simultaneously measuring smoothness and decay at infinity. We can of course equally well prescribe asymptotics as $t \to -\infty$; this leads to incoming data $f_-$. We consider the `Poisson operators' $\mathcal{P}_\pm : f_\pm \to u$ and precisely characterize the range of these operators on $\mathcal{W}^k(\mathbb{R}^n)$ spaces. Finally we show that the scattering matrix, mapping $f_-$ to $f_+$, preserves these spaces.
