Scaling of highly excited Schrödinger-Poisson eigenstates and universality of their rotation curves

This work provides a comprehensive numerical characterization of the excited spherically symmetric stationary states of the Schrödinger-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.