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

TL;DR

This paper numerically analyzes highly excited Schr"odinger-Poisson eigenstates, revealing universal scaling laws and characteristic behaviors of their eigenfunctions and eigenvelocities as excitation increases.

Contribution

It introduces novel heuristic laws describing how eigenstate features scale with excitation index in the Schr"odinger-Poisson problem.

Findings

Eigenfunction support scales parabolically with excitation index

Node spacing pattern varies regularly with excitation

Eigenvelocity oscillations follow a universal scaling law

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.