Title: Scaling of highly excited Schrödinger-Poisson eigenstates and universality of their rotation curves Show Chinese title

Title: 高度激发的薛定谔-泊松本征态的标度及其旋转曲线的普遍性

Abstract: This work provides a comprehensive numerical characterization of the excited spherically symmetric stationary states of the Schr\"odinger-Poisson problem. Through numerical computation of highly excited eigenstates, novel heuristic laws are proposed, which describe how their fundamental features scale with the excitation index $n$. Key characteristics of the eigenfunctions include: the effective support, which exhibits a parabolic dependence on the excitation index; the distances between adjacent nodes, whose pattern varies regularly with $n$; and the oscillation amplitude, which follows a power law with an exponent approaching $-1$ for large $n$. Based on the eigenfunctions, eigenvelocities are conveniently defined. They exhibit a mid-range oscillatory region with an average linear trend, whose slope approaches zero in the large $n$ limit; and they are characterized by heuristic scaling relationships with the excitation index $n$, revealing an intrinsic universal behavior. Show Chinese abstract

Abstract: 这项工作提供了对薛定谔-泊松问题激发的球对称定态的全面数值表征。 通过计算高度激发的本征态，提出了新的启发式定律，这些定律描述了它们的基本特征如何随着激发指数$n$进行变化。 本征函数的关键特性包括：有效支撑范围，其与激发指数呈抛物线关系；相邻节点之间的距离，其模式随着$n$的变化而有规律地变化；以及振荡幅度，其遵循幂律，指数在$n$较大时接近$-1$。 基于本征函数，可以方便地定义本征速度。 它们表现出一个中等范围的振荡区域，具有平均线性趋势，其斜率在$n$较大时趋于零；并且它们与激发指数$n$具有启发式的标度关系，揭示了一种内在的普遍行为。