发表机构
University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了薛定谔方程在Hardy临界权重下的紧支撑实值标量势端点例子，解决了开放问题，并记录了有理尾部实现，结果由Eureka系统发现并验证。
AI 中文摘要
在一维空间$[0,1]$上的薛定谔方程中，对于临界Hardy高斯权重，已知的加权$L^2$类中的非零标量例子具有复值势。Cassano和Fanelli指出，实值标量端点例子的存在性是开放的，并且他们仅在引入磁势后才构造出具有实电场和磁场势的例子。我们构造了一个非零光滑解$i\partial_t u+\partial_x^2 u=Vu$在$\mathbb{R}\times[0,1]$上，使得$e^{x^2/4}u(\cdot,0),e^{x^2/4}u(\cdot,1)\in L^2(\mathbb{R})$，且该势是有界、光滑、实值、纯标量的，并在所有时间上支撑于一个固定的紧空间区间内。这个紧支撑端点例子是主要结果。我们还记录了该构造所依据的显式有理尾部实现。本文的主要结果由多智能体系统Eureka获得，并随后由作者验证。
英文摘要
At the critical Hardy Gaussian weight for the Schrödinger equation in one space dimension on $[0,1]$, the known nonzero scalar example in the weighted $L^2$ class carries a complex-valued potential. Cassano and Fanelli observed that the existence of a real-valued scalar endpoint example was open, and produced examples with real electric and magnetic potentials only after introducing a magnetic potential. We construct a nonzero smooth solution of $i\partial_t u+\partial_x^2 u=Vu$ on $\mathbb{R}\times[0,1]$ such that $e^{x^2/4}u(\cdot,0),e^{x^2/4}u(\cdot,1)\in L^2(\mathbb{R})$ and the potential is bounded, smooth, real-valued, and purely scalar and is supported in one fixed compact spatial interval for all times. This compact-support endpoint example is the main result. We also record the explicit rational-tail realization underlying the construction. The main results of this paper were obtained by the multi-agent system Eureka and have subsequently been verified by the authors.
Comments10 pages
