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arXiv · 2502.05030

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

Gaia Marangon·Antonio Ponno·Lorenzo Zanelli
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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.

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Gaia Marangon, Antonio Ponno, Lorenzo Zanelli. 2025-02-07. Scaling of highly excited Schr\"odinger-Poisson eigenstates and universality of their rotation curves. https://doi.org/10.1016/j.physleta.2025.130761

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