发表机构
New York University; Instituto de Matemática Pura e Aplicada (IMPA); University of Oxford（纽约大学; 纯数学与应用数学研究所; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明一维 Schrödinger 演化中，L² 解由质量密度模全局相位唯一确定，并扩展至非线性模型，同时指出高维失效。
AI 中文摘要
考虑一维 Schrödinger 演化方程 $$ i\partial_tw+\partial_x^2w=Vw,\hspace{5mm}(t,x)\in [0,T]\times\mathbb{R},$$ 其中 $V$ 是实值时间依赖势，属于 $L^1_tL^\infty_x+L^2_{t,x}$。我们证明上述方程的一个 $L^2$ 解由其质量密度 $|w|^2$ 决定，模一个全局单位模相位因子。这回答了 Jaming 最近对一大类势提出的一个问题，并恢复了他早先对自由 Schrödinger 演化的结果。我们的证明简单且稳健。它结合了上述 Schrödinger 演化的基本局部平滑性质、Strichartz 估计以及 Ionescu 和 Kenig 的唯一延拓定理的条带版本，以证明交互函数 $$ F(t,x,y)=u(t,x)v(t,y)-u(t,y)v(t,x) $$ 在 $[0,T]\times\mathbb{R}^2$ 上恒为零，只要 $u$ 和 $v$ 是带有势 $V$ 且具有相同质量密度的 Schrödinger 方程的 $L^2$ 解。该方法立即扩展到几个重要的非线性模型，如三次 NLS 方程，该方程在 $L^2(\mathbb{R})$ 中全局适定，其中 $V=|u|^2$。有趣的是，我们还表明上述相位恢复性质在二维及更高维中失效，即使对于具有 Schwartz 初始数据的自由方程也是如此。
英文摘要
Consider the one-dimensional Schrödinger evolution $$ i\partial_tw+\partial_x^2w=Vw,\hspace{5mm}(t,x)\in [0,T]\times\mathbb{R},$$ where $V$ is a real-valued time-dependent potential belonging to $L^1_tL^\infty_x+L^2_{t,x}$. We prove that an $L^2$ solution to the above equation is determined by its mass density $|w|^2$ modulo a global unimodular phase factor. This answers a question recently posed by Jaming for a large class of potentials, and recovers his earlier result for the free Schrödinger evolution. Our proof is simple and robust. It combines the basic local smoothing properties of the above Schrödinger evolutions, Strichartz estimates, and a stripwise version of a unique continuation theorem of Ionescu and Kenig to show that the interaction function $$ F(t,x,y)=u(t,x)v(t,y)-u(t,y)v(t,x) $$ vanishes identically on $[0,T]\times\mathbb{R}^2$ whenever $u$ and $v$ are $L^2$ solutions to a Schröinder euqation with potential $V$ with the same mass density. The method immediately extends to several important nonlinear models such as the cubic NLS equation, which is globally well-posed in $L^2(\mathbb{R})$ with $V=|u|^2$. Interestingly, we also show that the above phase recovery property fails in dimensions two and higher, even for the free equation with Schwartz initial data.
Comments23 pages
