Scaling of highly excited Schr\"odinger-Poisson eigenstates and universality of their rotation curves

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.