arXiv 2201.031402022-01math.APmath.SP

含时薛定谔方程的奇性传播与Fredholm分析

Propagation of singularities and Fredholm analysis for the time-dependent Schrödinger equation

Jesse Gell-Redman, Sean Gomes, Andrew Hassell

备注：: 63 pages, 3 figures

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AI总结：

针对含时薛定谔算子，本文通过寻找其可逆作用的希尔伯特空间对的新方法，求解了终态问题，刻画了泊松算子值域，并证明散射矩阵保持相应函数空间不变。

摘要：

我们研究作用于$\u211d^{n+1}$上函数的含时薛定谔算子$P = D_t + Δ_g + V$，其中采用坐标$z \u2208 \u211d^n$与$t \u2208 \u211d$，$D_t$表示$-i \u2202_t$，$Δ_g$是关于$\u211d^n$上一族含时非捕获度量$g_{ij}(z, t) dz^i dz^j$的正拉普拉斯算子，该度量在时空中的紧集外等于欧几里得度量，而$V = V(z, t)$是同样在时空中具有紧支集的势函数。本文提出一种研究$P$的新方法，即寻找算子在其间可逆作用的希尔伯特空间对。\n利用这种可逆性可以直接求解含时薛定谔方程的“终态问题”，即找到$Pu = 0$的全局解$u(z, t)$，使其在$t \to \nfty$时具有指定的渐近行为。这些渐近行为的形式为$$ u(z, t) \u223c t^{-n/2} e^{i|z|^2/4t} f_+\big( \frac{z}{2t} \big), \u3000 t \to +\nfty $$其中$f_+$即“终态”或出射数据，是合适的函数空间$\u2131^k(\u211d^n)$中的任意元素；此处$k$是同时衡量光滑性与无穷远衰减性的正则性参数。我们当然也可以同样指定$t \to -\nfty$时的渐近行为，这对应入射数据$f_-$。我们研究“泊松算子”$\u213b_\u00b1 : f_\u00b1 \to u$，并精确刻画了这些算子在$\u2131^k(\u211d^n)$空间上的值域。最后我们证明，将$f_-$映射为$f_+$的散射矩阵保持这些空间不变。

英文摘要：

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.

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