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REVIEW 3 major objections 4 minor 48 references

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

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Pith's one-line read Highly excited Schrödinger-Poisson eigenstates obey scaling laws that collapse all their rotation curves onto one universal shape.

desk verdict Useful heuristic scaling laws for excited Schrödinger-Poisson states, but the universal collapse claim needs real residuals and error bars before it convinces. read the letter →

arxiv 2502.05030 v2 pith:JK55TFFS submitted 2025-02-07 math-ph math.MP

Gaia Marangon , Antonio Ponno , Lorenzo Zanelli This is my paper · ORCID
classification math-phmath.MP MSC 35Q4135Q5581Q05
keywords Schrödinger-PoissonChoquardequationexcitedstationarystateseigenvelocitiesrotationcurvesscalinglawsuniversalitynumericaleigenstates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Using numerical solutions of the spherically symmetric Schrödinger-Poisson system up to excitation index $n=80$, the paper establishes heuristic scaling laws for excited eigenstates. The effective support of the matter density grows quadratically with $n$, nodal spacings follow a regular pattern that converges after normalization, and the oscillation amplitude decays as a power law whose exponent approaches $-1$. From these eigenfunctions the paper defines eigenvelocities, the tangential speeds of test particles in circular orbits, and shows that their mid-range slope decays to zero as $n^{-2.86}$. The central result is that, after rescaling radius and velocity by the outermost eigenvelocity extremum ($\tilde r_{2n}(n)=133n^2+245n-185$ and $\tilde v_{2n}=0.27\tilde r_{2n}^{-1/2}$), all numerically computed eigenvelocity profiles collapse onto one universal curve. The result matters because it gives the Schrödinger-Poisson model an intrinsic, $n$-independent rotation-curve shape that can be compared with observed galaxy rotation curves.

What carries the argument

The load-bearing construct is the eigenvelocity $v_n(r)=\sqrt{(\int_0^r f_n^2(s)s^2\,ds)/r}$ and the two-point rescaling built from the outermost extremum of $v_n$. The paper's heuristic laws for the outermost radius, $\tilde r_{2n}(n)=133n^2+245n-185$, and the outermost velocity, $\tilde v_{2n}=0.27\tilde r_{2n}^{-1/2}$, define the dimensionless variables $R=r/\tilde r_{2n}$ and $V=v/\tilde v_{2n}$; the claim is that plotting $V$ against $R$ erases the dependence on $n$ and exposes the universal shape. The underlying eigenfunctions come from the Choquard stationary problem, the nonlinear eigenvalue equation obtained after eliminating the Poisson potential, solved numerically on a progressively extended domain.

What would settle it

Compute eigenstates for $n=81$ through $n=200$ with an independent high-accuracy solver and test whether equations (5)-(12) hold and whether the rescaled curves from (13) continue to collapse; if the points drift from the fitted curves or the curves spread apart, the universality claim is refuted.

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Extended reading notes

Core claim

The paper's central claim is that highly excited spherically symmetric stationary states of the Schrödinger-Poisson problem are governed by quantitative scaling laws in the excitation index $n$, and that their associated eigenvelocities are universal after a one-parameter rescaling. Specifically, the outermost radius of the eigenvelocity, $\tilde r_{2n}(n)=133n^2+245n-185$, and the velocity at that point, $\tilde v_{2n}(\tilde r_{2n})=0.27\,\tilde r_{2n}^{-1/2}$, define the rescaling $R=r/\tilde r_{2n}$, $V=v/\tilde v_{2n}$, under which the computed profiles for different $n$ collapse onto a common curve. The same analysis yields parabolic support growth, a regular nodal-distance pattern, power-law amplitude decay with exponent $a(n)=-1+0.24n^{-0.25}$, and mid-range velocity slopes $\sigma(n)=2.82\times10^{-5}n^{-2.86}$ that vanish in the large-$n$ limit. These are presented as heuristic laws inferred from numerical data, not as proven theorems.

Load-bearing premise

All fitted laws and the claimed universal collapse rest on the numerical solver accurately resolving eigenstates up to $n=80$; no convergence proof or error metrics are reported, and the large-$n$ statements extrapolate from that range.

Editorial extensions

If this is right

  • In the large-$n$ limit the eigenfunction amplitudes become $f_n(r)\sim r^{-1}$, which plugged into the eigenvelocity definition gives an approximately flat mid-range curve; flat rotation-curve plateaux are thus a natural asymptotic feature of the model.
  • The mid-range slope decays as $\sigma(n)=2.82\times10^{-5}n^{-2.86}$, so the flattening is quantitative: higher excited states produce flatter plateaux at a predictable rate.
  • The universal rescaling (13) reduces every computed eigenvelocity profile to one common shape, giving an $n$-independent curve intrinsic to the Schrödinger-Poisson model.
  • The parabolic laws for support, outermost node, and outermost eigenvelocity radius connect the excited-state structure to Bohr/Kepler-type scaling, offering a concrete target for analytic derivations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the collapse is exact beyond the fitted range, the master curve could be tabulated once and used as a template for multimodal dark-matter halos built from many excited states, avoiding repeated numerical solution of the Choquard equation.
  • The model's norm-scaling invariance suggests the universal shape may survive renormalization of the total mass, so the same rescaled curve could apply across very different physical mass scales; the paper does not itself make this claim.
  • The fitted correction $0.24n^{-0.25}$ means plain $r^{-1}$ behavior is only approached slowly; at $n=80$ the exponent is still about $-0.92$, so finite-$n$ rotation curves should retain noticeable residual slope and oscillations even after rescaling.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper numerically computes spherically symmetric stationary states of the Schrödinger-Poisson equation up to excitation index n=80 and proposes heuristic scaling laws: the effective support rhat_n(n) ~ 131 n^2, the outer node z_n(n) and outer nodal distance d_{n-1}(n) with parabolic fits, the amplitude modulation |f_i| = b(n) rhat_i^{a(n)} with a(n) approaching -1, the mid-range eigenvelocity slope sigma(n) ~ n^{-2.86}, and the outer extremum scaling vtilde_{2n}(rtilde_{2n}) = 0.27 rtilde_{2n}^{-0.5}. The paper's central claim is that all eigenvelocity curves collapse onto a single universal shape after the rescaling R = r/rtilde_{2n}(n), V = v/vtilde_{2n}(n) defined in Eq. (13).

Significance. If the claimed universal rotation-curve shape holds, it would provide a compact empirical characterization of highly excited Schrödinger-Poisson eigenstates and could inform studies of multimodal dark-matter configurations. The paper is honest that the laws are heuristic, and it documents numerical validation through grid refinements. However, the universal collapse is not an independent prediction: the normalizing quantities in Eq. (13) are least-squares fits to the same data being collapsed, and no residual or scatter metric is reported. The absence of error bars on fitted coefficients, post hoc fit-region choices, and the lack of code or data make the central claims plausible but not yet quantitatively established. With targeted robustness and reproducibility analyses, the result would be a useful contribution; as it stands, the evidence for universality is largely visual.

major comments (3)
  1. [§3.5, Eq. (13) and Fig. 13] The claimed universal collapse is partly constructed: the rescaling uses rtilde_{2n}(n) and vtilde_{2n}(n) obtained from the same eigenvelocity curves through the fits (12a)-(12b), so systematic fit errors or post hoc choices in those fits are absorbed into the normalization. The collapse is supported only visually, with no quantitative residual or spread metric. Please report the RMS deviation of the rescaled curves from a common shape as a function of n, and provide an independent test, e.g., fit (12) on n ≤ 50 and examine whether the predicted collapse holds for n = 60,...,80.
  2. [§3.4, Eqs. (8)-(9)] The amplitude exponent law a(n) = -1 + 0.24 n^{-0.25} depends on post hoc fit choices: amplitudes are fitted only up to 0.95 r_min, the first and last extremum are excluded, and the onset n ≥ 20 is selected after inspecting the data. No error bars or goodness-of-fit measures are reported for a(n) and b(n), so the asymptotic statement a → -1 is an extrapolation from a fit region that was itself chosen from the data. Please provide uncertainties on the fitted coefficients and a sensitivity analysis with respect to the cutoff 0.95 r_min and the onset index.
  3. [§2.1] The numerical accuracy of the eigenstates is described only qualitatively as 'confirming reliability up to n = 80'. Because every heuristic law in Eqs. (5)-(12) is a fit to these numerical data, the paper should report concrete convergence diagnostics, including grid sizes, tolerances, eigenvalue errors, and node or extremum position errors as functions of n. Releasing the code or data, even as supplementary material, would substantially strengthen the reproducibility of the empirical claims.
minor comments (4)
  1. [§1.1] There is a typo in 'demostrated' near the discussion of Tod and Moroz; please correct it.
  2. [§3.5, Eq. (2)] The definition of v_n(r) omits the 4π factor from angular integration; the omission is stated, but it would be clearer to also note that this is an overall normalization convention that does not affect the scaling or universality claims.
  3. [§3.3, Eqs. (5) and (7a)] The parabolic fits for rhat_n(n) and z_n(n) are inconsistent for small n (e.g., at n=1 the fitted z_1 exceeds the fitted rhat_1, although the outermost node must lie inside the outermost extremum); please state the range of n for which each fit is intended to be valid.
  4. [Fig. 7] The caption refers to 'Red regions' to indicate excluded points, but the figure may not be colorblind-safe; please add hatching or a grayscale-readable marker.

Circularity Check

2 steps flagged · score 6.0 of 10

Universal-collapse claims are partly constructed by rescaling each curve with its own fitted endpoint values; otherwise the heuristic laws are honest fits.

  1. fitted input called prediction [Section 3.5, Eqs. (12)-(13) and Fig. 13]
    "Overall, the heuristic laws (12) suggest a natural scaling for the eigenvelocities, that depends solely on the excitation index n: R≡ r/˜r2n(n) ; V≡ v/˜v2n(n), (13) where ˜v2n(n) is obtained by combining Equations (12a) and (12b), ˜v2n(n)≡ ˜v2n(˜r2n(n)). Figure 13 reports eigenvelocities rescaled according to Equation (13). The plot shows how the numerically computed rotation curves, originally shown in Figure10, collapse onto a single average curve after rescaling, revealing an intrinsic universal behavior."

    The two normalizing constants in Eq. (13) are not independent scales: Eq. (12a) is a power-law fit to the outermost-extremum velocities and Eq. (12b) is a parabolic fit to the outermost-extremum radii of the very same curves that Fig. 13 then rescales. Consequently each rescaled curve has its outermost extremum placed at (R,V)=(1,1) by construction, so the endpoint coincidence in the 'collapse' is built into the plot.

  2. fitted input called prediction [Section 3.3, Eqs. (6)-(7) and Fig. 6]
    "A natural approach is to normalize the patterns using their outermost point (zn, dn−1), which comprises the outermost node zn and the outermost nodal distance dn−1≡ zn− zn−1: Zi+1≡ zi+1/zn(n) ; Di≡ di/dn−1(n) ; for i = 1,..., n− 1. ... As shown in Figure 6, the rescaled nodal distance patterns, while not perfectly coincident, demonstrate increasing convergence with larger values of n, approaching a universal curve in the large n limit."

    Eq. (6) normalizes each curve by its own outermost point, and Eq. (7) fits that outermost point's dependence on n from the same curves. Hence the outer endpoint of every rescaled pattern is (1,1) by definition, and the 'increasing convergence' is at least partly manufactured by the normalization. As with the velocity rescaling, no quantitative residual is provided to show that the interior (non-normalized) portion converges independently of this constructed common endpoint.

full rationale

The paper is mostly an honest numerical characterization: the eigenstates are computed with an external scheme ([20], [39]) and the heuristic laws (5), (7), (9), (11), (12) are explicitly labeled fits to the same n≤80 data set. I find no load-bearing self-citation: no uniqueness theorem or ansatz is imported from the authors' own prior work, and the cited numerical scheme is external. The circularity concern is concentrated in the two 'data collapse' claims. In Eq. (13), the rescaled radius and velocity are defined using the fitted coordinates of each eigenvelocity's outermost extremum (Eqs. (12a)-(12b)); because those coordinates were fitted to the very curves being collapsed, every rescaled curve has its endpoint placed at (1,1) by construction. The reported collapse is therefore not an independent check: the endpoint agreement is definitional, and the interior agreement is asserted visually with no residual metric. The nodal-distance normalization in Eqs. (6)-(7) has the same structure. These are partial circularities: the fitted endpoint alignment is built into the 'universal' curve, although the inner portion of the collapse retains some independent content. The asymptotic extrapolations (a→-1, σ→0) are separate extrapolation risks, not circularity.

Assumptions & free parameters 9 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the existence of the discrete eigenstate family, the spherically symmetric ansatz, the chosen normalization, and the numerical accuracy of the solver up to n=80. The scaling laws themselves are fits, so the fitted coefficients are free parameters. No new physical entities are introduced.

free parameters (9)
  • Support parabolic fit coefficients (Eq 5) = 131, 53.53, 340
    Coefficients of rhat_n(n)=131n^2+53.53n+340, fitted to computed outermost extrema.
  • Outermost node parabolic fit coefficients (Eq 7a) = 130, -125, 795
    Coefficients of zn(n)=130n^2-125n+795, fitted to computed outermost node.
  • Outermost nodal distance parabolic fit coefficients (Eq 7b) = 1.80, 248, -565
    Coefficients of dn-1(n)=1.80n^2+248n-565, fitted to computed outermost nodal distance.
  • Amplitude exponent law coefficients (Eq 9) = -1, 0.24, -0.25
    a(n)=-1+0.24 n^{-0.25}, fitted to local extrema amplitudes for n>=20.
  • Velocity slope power-law coefficients (Eq 11) = 2.82e-5, -2.86
    sigma(n)=2.82e-5 n^{-2.86}, fitted to mid-range linear fits of eigenvelocities.
  • Last extremum velocity power-law coefficient (Eq 12a) = 0.27, -0.5
    v2n(r)=0.27 r^{-0.5}, fitted to outermost eigenvelocity extrema.
  • Last extremum radius parabolic fit coefficients (Eq 12b) = 133, 245, -185
    r2n(n)=133n^2+245n-185, fitted to outermost eigenvelocity extrema.
  • Amplitude fit upper cutoff = 0.95 rmin
    Amplitudes fitted only up to 0.95 rmin to exclude outer deviations; choice affects fitted exponent.
  • Onset index for amplitude exponent fit = n=20
    The power-law fit for a(n) starts at n=20, a post hoc choice based on apparent stabilization of the exponent.
assumptions (4)
  • standard math Existence and discreteness of spherically symmetric stationary states {ε_n, f_n} for the Choquard equation.
    Invoked in Section 2 via Lieb [17] and Lions [18]; the paper relies on these theorems without re-proving them.
  • domain assumption The restriction to spherically symmetric, radially real eigenfunctions without loss of generality.
    Section 2 states 'we focus on spherically symmetric solutions'; the numerical and scaling results are only for this symmetry class.
  • domain assumption The norm scaling invariance (3) and the choice of L2 norm equal to 1.
    The paper fixes normalization to define a unique eigenstate family; the scaling laws depend on this convention.
  • ad hoc to paper The adopted numerical scheme (Bernstein et al.) converges to the true eigenstates up to n=80.
    Section 2.1 asserts reliability based on grid refinement checks, but no convergence proof or accuracy metrics are provided.

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Pith. "Pith review of Scaling of highly excited Schr\"odinger-Poisson eigenstates and universality of their rotation curves." pith.science (2026). https://pith.science/paper/JK55TFFS

@misc{pith2026250205030,
  author       = {Pith},
  title        = {Pith review of: Scaling of highly excited Schr\"odinger-Poisson eigenstates and universality of their rotation curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JK55TFFS}},
  note         = {Machine review of arXiv:2502.05030}
}
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.

Figures

Figures reproduced from arXiv: 2502.05030 by the authors.

Figure 1
Figure 1. Three-dimensional section of the matter distribution, as predicted by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Parabolic fit for the radial position ˆr [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Nodal distances {di} n−1 i=1 plotted against their corresponding right nodes {zi+1} n−1 i=1 , for several sample eigenfunctions. values of n. A natural approach is to normalize the patterns us￾ing their outermost point (zn, dn−1), which comprises the outer￾most node zn and the outermost nodal distance dn−1 ≡ zn − zn−1: Zi+1 ≡ zi+1 zn(n) ; Di ≡ di dn−1(n) ; for i = 1, . . . , n − 1 . (6) For this rescaling to be mean… view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: Variation with excitation index n of (a) the outermost node zn(n) and (b) the outermost nodal distance dn−1(n) of the eigenfunction. dependent on the excitation index n and with negative exponent a(n) < 0: | ˆfi | = b(n) ˆr a(n) i . (8) [PITH_FULL_IMAGE:figures/full_f…
Figure 6
Figure 6. Figure 6: Normalized nodal distances {Di} n−1 i=1 plotted against their normalized right nodes {Zi+1} n−1 i=1 , for several sample eigenfunctions. Normalization follows Equation (6). 5 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: Logarithmic plot showing the fitted amplitudes [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 9
Figure 9. Figure 9: Eigenvelocity v8(r) with detailed notation. Panel (a) shows the com￾plete eigenvelocity profile, with local extrema {(˜ri , v˜i)} 16 i=0 . Panel (b) provides a magnified view highlighting the notation. To conclude, we analyze the outermost local extremum (˜r2n, v˜2n) o…
Figure 10
Figure 10. Figure 10: Examples of eigenvelocity profiles vn(r) with mid-range linear fit. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]
Figure 11
Figure 11. Figure 11: Power law relationship between the slopes [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 12
Figure 12. Figure 12: Heuristic laws describing the outermost local extremum [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]
Figure 13
Figure 13. Figure 13: Universality in eigenvelocity profiles, after rescaling according to [PITH_FULL_IMAGE:figures/full_fig_p007_13.png]

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Works this paper leans on

48 extracted references · 26 canonical work pages

  1. [1]

    Paredes, D

    A. Paredes, D. N. Olivieri, H. Michinel, From optics to dark matter: A review on nonlinear Schr¨odinger–Poisson systems, Physica D: Nonlinear Phenomena 403 (2020) 132301. URL:https://www.sciencedirect. com/science/article/pii/S0167278919307079. doi: https: //doi.org/10.1016/j.physd.2019.132301

    work page arXiv 2020

  2. [2]

    Ru ffini, S

    R. Ru ffini, S. Bonazzola, Systems of self-gravitating particles in general relativity and the concept of an equation of state, Phys. Rev. 187 (1969) 1767–1783. URL: https://link.aps.org/doi/10.1103/PhysRev. 187.1767. doi:10.1103/PhysRev.187.1767

    doi:10.1103/physrev 1969

  3. [3]

    F. E. Schunck, E. W. Mielke, General relativistic boson stars, Classical and Quantum Gravity 20 (2003) R301. URL: https://dx.doi.org/10.1088/0264-9381/20/20/201. doi:10.1088/0264-9381/20/20/201

    doi:10.1088/0264-9381/20/20/201 2003

  4. [4]

    Matos, L

    T. Matos, L. A. Ure ˜na-L´opez, J.-W. Lee, Short review of the main achievements of the scalar field, fuzzy, ultralight, wave, BEC dark matter model, Frontiers in Astronomy and Space Sciences 11 (2024). URL: https://www.frontiersin.org/journals/ astronomy-and-space-sciences/articles/10.3389/fspas. 2024.1347518. doi:10.3389/fspas.2024.1347518

    work page arXiv 2024

  5. [5]

    L. Hui, J. P. Ostriker, S. Tremaine, E. Witten, Ultralight scalars as cos- mological dark matter, Phys. Rev. D 95 (2017) 043541. URL: https:// link.aps.org/doi/10.1103/PhysRevD.95.043541. doi:10.1103/ PhysRevD.95.043541

    doi:10.1103/physrevd.95.043541 2017

  6. [6]

    Di ´osi, Gravitation and quantum-mechanical localization of macro- objects, Physics Letters A 105 (1984) 199–202

    L. Di ´osi, Gravitation and quantum-mechanical localization of macro- objects, Physics Letters A 105 (1984) 199–202. URL: https://www. sciencedirect.com/science/article/pii/0375960184903979. doi:https://doi.org/10.1016/0375-9601(84)90397-9

    arXiv 1984

  7. [7]

    Penrose, P

    R. Penrose, P. Marcer, Quantum computation, entanglement and state reduction [and discussion], Philosophical Transactions: Mathematical, Physical and Engineering Sciences 356 (1998) 1927–1939. URL: http: //www.jstor.org/stable/55019

    work page 1998

  8. [8]

    Penrose, On the gravitization of quantum mechanics 1: Quantum state reduction, Foundations of Physics 44 (2014) 557–575

    R. Penrose, On the gravitization of quantum mechanics 1: Quantum state reduction, Foundations of Physics 44 (2014) 557–575. doi: 10.1007/ s10701-013-9770-0

    work page 2014

Show all 48 references
  1. [9]

    Bahrami, A

    M. Bahrami, A. Großardt, S. Donadi, A. Bassi, The Schr ¨odinger–Newton equation and its foundations, New Journal of Physics 16 (2014) 115007. URL: https://dx.doi.org/10.1088/1367-2630/16/11/115007. doi:10.1088/1367-2630/16/11/115007

    2014 doi

  2. [11]

    Bekenstein, R

    R. Bekenstein, R. Schley, M. Mutzafi, C. Rotschild, M. Segev, Opti- cal simulations of gravitational e ffects in the Newton–Schr ¨odinger sys- tem, Nature Physics 11 (2015) 872–878. URL: https://doi.org/10. 1038/nphys3451. doi:10.1038/nphys3451

    2015 doi

  3. [12]

    Lahaye, C

    T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, T. Pfau, The physics of dipolar bosonic quantum gases, Reports on Progress in Physics 72 (2009) 126401. URL: https://dx.doi.org/10.1088/0034-4885/72/12/ 126401. doi:10.1088/0034-4885/72/12/126401

    2009 doi

  4. [13]

    Giovanazzi, G

    S. Giovanazzi, G. Kurizki, I. E. Mazets, S. Stringari, Collective excitations of a ”gravitationally” self-bound bose gas, Europhysics Letters 56 (2001) 1. URL: https://dx.doi.org/10.1209/epl/ i2001-00478-8 . doi:10.1209/epl/i2001-00478-8

    2001 doi

  5. [14]

    J. T. Mendonc ¸a, Wave-kinetic approach to the schr¨odinger–newton equa- tion, New Journal of Physics 21 (2019) 023004. URL: https://dx. doi.org/10.1088/1367-2630/ab0045. doi: 10.1088/1367-2630/ ab0045

    2019 doi

  6. [15]

    J. J. Garc ´ıa-Ripoll, V . V . Konotop, B. Malomed, V . M. P ´erez- Garc´ıa, A quasi-local gross-pitaevskii equation for attractive bose- einstein condensates, Math. Comput. Simul. 62 (2003) 21–30. URL: https://doi.org/10.1016/S0378-4754(02)00190-8 . doi:10.1016/S0378-4754(02)00190-8

    2003 doi

  7. [16]

    Navarrete, A

    A. Navarrete, A. Paredes, J. R. Salgueiro, H. Michinel, Spa- tial solitons in thermo-optical media from the nonlinear schr ¨odinger- poisson equation and dark-matter analogs, Phys. Rev. A 95 (2017) 013844. URL: https://link.aps.org/doi/10.1103/PhysRevA. 95.013844. doi:10.1103/...

    2017 doi

  8. [17]

    E. H. Lieb, Existence and uniqueness of the minimizing so- lution of Choquard’s nonlinear equation, Studies in Applied Mathematics 57 (1977) 93–105. URL: https://onlinelibrary. wiley.com/doi/abs/10.1002/sapm197757293. doi:https://doi. org/10.1002/sapm197757293

    1977 doi

  9. [18]

    Lions, The Choquard equation and related questions, Nonlinear Analysis: Theory, Methods & Applications 4 (1980) 1063–1072

    P. Lions, The Choquard equation and related questions, Nonlinear Analysis: Theory, Methods & Applications 4 (1980) 1063–1072. URL: https://www.sciencedirect.com/science/article/ pii/0362546X80900164. doi: https://doi.org/10.1016/ 0362-546X(80)90016-4

    1980

  10. [19]

    Sin, Late-time phase transition and the galactic halo as a Bose liquid, Phys

    S.-J. Sin, Late-time phase transition and the galactic halo as a Bose liquid, Phys. Rev. D 50 (1994) 3650–3654. URL: https: //link.aps.org/doi/10.1103/PhysRevD.50.3650. doi:10.1103/ PhysRevD.50.3650

    1994 doi

  11. [20]

    Bernstein, E

    D. Bernstein, E. Gilardi, K. Jones, Eigenstates of the gravitational Schr¨odinger equation, Modern Physics Letters A 13 (1998) 2327–2336. URL: https://doi.org/10.1142/S0217732398002473. doi: 10. 1142/S0217732398002473

    1998 doi

  12. [22]

    Lelli, S

    F. Lelli, S. S. McGaugh, J. M. Schombert, SPARC: mass models for 175 disk galaxies with Spitzer photometry and ac- curate rotation curves, The Astronomical Journal 152 (2016)

    2016

  13. [23]

    H. Katz, F. Lelli, S. S. McGaugh, A. Di Cintio, C. B. Brook, J. M. Schombert, Testing feedback-modified dark matter haloes with galaxy rotation curves: estimation of halo parameters and consistency with λcdm scaling relations, Monthly Notices of the Royal Astronomical Society ...

    2016 doi

  14. [24]

    Harko, E

    T. Harko, E. J. Madarassy, Bose-Einstein Condensate dark matter models in the presence of baryonic matter and ran- dom confining potentials, The European Physical Journal C 82 (2022). URL: http://arxiv.org/abs/2205.00297http: //dx.doi.org/10.1140/epjc/s10052-022-10344-7 . doi:...

    2022 arXiv

  15. [25]

    F. S. Guzm ´an, F. D. Lora-Clavijo, Rotation curves of ultralight BEC dark matter halos with rotation, General Relativity and Gravitation 47 (2015) 21. URL: https://link.springer.com/article/10.1007/ s10714-015-1865-9 . doi:10.1007/s10714-015-1865-9

    2015 doi

  16. [26]

    Binney, S

    J. Binney, S. Tremaine, Galactic Dynamics: Second Edition, Princeton Series in Astrophysics, Princeton University Press, 2011. URL: https: //books.google.it/books?id=6mF4CKxlbLsC

    2011

  17. [27]

    S. U. Ji, S. J. Sin, Late-time phase transition and the galactic halo as a Bose liquid. II. The e ffect of visible matter, Physical Review D 50 (1994). URL: https://journals.aps.org/prd/abstract/10. 1103/PhysRevD.50.3655. doi:10.1103/PhysRevD.50.3655

    1994 doi

  18. [28]

    Lee, I.-g

    J.-w. Lee, I.-g. Koh, Galactic halos as boson stars, Phys. Rev. D 53 (1996) 2236–2239. URL: https://link.aps.org/doi/10.1103/ PhysRevD.53.2236. doi:10.1103/PhysRevD.53.2236

    1996 doi

  19. [29]

    Burstein, V

    D. Burstein, V . C. Rubin, The distribution of mass in spiral galaxies., Astrophysical Journal 297 (1985) 423–435. doi:10.1086/163541

    1985 doi

  20. [30]

    V . C. Rubin, J. Ford, W. K., N. Thonnard, Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc)., Astrophysical Journal 238 (1980) 471–487. doi:10.1086/158003

    1980 doi

  21. [31]

    V . C. Rubin, D. Burstein, W. K. Ford, Jr., N. Thonnard, Rotation velocities of 16 SA galaxies and a comparison of Sa, Sb, and SC rotation properties, Astrophys. J. 289 (1985) 81. doi:10.1086/162866

    1985 doi

  22. [32]

    F. S. Guzm ´an, L. A. Ure ˜na L ´opez, Evolution of the Schr ¨odinger- Newton system for a self-gravitating scalar field, Phys. Rev. D 69 (2004) 124033. URL: https://link.aps.org/doi/10.1103/ PhysRevD.69.124033. doi:10.1103/PhysRevD.69.124033

    2004 doi

  23. [33]

    F. S. Guzm ´an, L. A. Ure ˜na L ´opez, Gravitational Cooling of Self- gravitating Bose Condensates, The Astrophysical Journal 645 (2006)

    2006

  24. [34]

    Schive, T

    H.-Y . Schive, T. Chiueh, T. Broadhurst, Cosmic structure as the quantum interference of a coherent dark wave, Nature Physics 10 (2014). URL: https://doi.org/10.1038/nphys2996. doi:10.1038/nphys2996

    2014 doi

  25. [35]

    ´Alvarez-Rios, T

    I. ´Alvarez-Rios, T. Bernal, P.-H. Chavanis, F. S. Guzm ´an, Galac- tic rotation curves of low surface brightness galaxies using core- halo fuzzy dark matter configurations, Phys. Rev. D 110 (2024) 063502. URL: https://link.aps.org/doi/10.1103/PhysRevD. 110.063502. doi:10.1103...

    2024 doi

  26. [36]

    L. A. Urena-Lopez, A. Bernal, Bosonic gas as a Galactic Dark Matter Halo, Phys. Rev. D 82 (2010) 123535. doi: 10.1103/PhysRevD.82. 123535. arXiv:1008.1231

    2010 arXiv

  27. [37]

    F. S. Guzm ´an, L. A. Ure ˜na L ´opez, Gravitational atoms: General frame- work for the construction of multistate axially symmetric solutions of the Schr ¨odinger-Poisson system, Phys. Rev. D 101 (2020) 081302. doi:10.1103/PhysRevD.101.081302. arXiv:1912.10585

    2020 arXiv

  28. [38]

    I. M. Moroz, R. Penrose, P. Tod, Spherically-symmetric solutions of the Schr ¨odinger-Newton equations, Classical and Quantum Gravity 15 (1998) 2733. URL: https://dx.doi.org/10.1088/0264-9381/15/ 9/019. doi:10.1088/0264-9381/15/9/019

    1998 doi

  29. [39]

    Harrison, I

    R. Harrison, I. Moroz, K. P. Tod, A numerical study of the Schr¨odinger-Newton equations, Nonlinearity 16 (2002) 101. URL: https://dx.doi.org/10.1088/0951-7715/16/1/307. doi:10.1088/0951-7715/16/1/307

    2002 doi

  30. [40]

    C. G. B ¨ohmer, T. Harko, Can dark matter be a Bose–Einstein conden- sate?, Journal of Cosmology and Astroparticle Physics 2007 (2007)

    2007

  31. [41]

    P. Tod, I. M. Moroz, An analytical approach to the Schr¨odinger-Newton equations, Nonlinearity 12 (1999) 201. URL: https://dx.doi.org/10.1088/0951-7715/12/2/002. doi:10.1088/0951-7715/12/2/002

    1999 doi

  32. [42]

    Tod, The ground state energy of the Schr ¨odinger-Newton equation, Physics Letters A 280 (2001) 173–176

    K. Tod, The ground state energy of the Schr ¨odinger-Newton equation, Physics Letters A 280 (2001) 173–176. URL: https://www.sciencedirect.com/science/article/ pii/S0375960101000597. doi: https://doi.org/10.1016/ S0375-9601(01)00059-7

    2001

  33. [43]

    URL: https://dx.doi.org/10.1088/1475-7516/2007/06/

    2007 doi

  34. [44]

    doi:10.1088/1475-7516/2007/06/025

    2007 doi

  35. [45]

    Gri ffiths, Introduction to Quantum Mechanics, Cambridge Uni- versity Press, 2017

    D. Gri ffiths, Introduction to Quantum Mechanics, Cambridge Uni- versity Press, 2017. URL: https://books.google.it/books?id= 0h-nDAAAQBAJ

    2017

  36. [46]

    Y . Sofue, Rotation curve decomposition for size–mass relations of bulge, disk, and dark halo components in spiral galaxies, Publications of the Astronomical Society of Japan 68 (2015) 2. URL: https://doi.org/ 10.1093/pasj/psv103. doi:10.1093/pasj/psv103. 9

    2015 doi

  37. [47]

    Greiner, G

    D. Greiner, G. Wunner, Quantum defect analysis of the eigen- value spectrum of the Newton-Schr ¨odinger equation, Phys. Rev. A 74 (2006) 052106. URL: https://link.aps.org/doi/10.1103/ PhysRevA.74.052106. doi:10.1103/PhysRevA.74.052106

    2006 doi

  38. [48]

    M. K. Kiessling, On the asymptotic decay of the Schr ¨odinger- Newton ground state, Physics Letters A 395 (2021) 127209. URL: https://www.sciencedirect.com/science/article/ pii/S0375960121000736. doi: https://doi.org/10.1016/j. physleta.2021.127209

    2021

  39. [157]

    doi:10.3847/0004-6256/152/6/157

    URL: https://dx.doi.org/10.3847/0004-6256/152/6/157. doi:10.3847/0004-6256/152/6/157

    doi

  40. [814]

    doi: 10.1086/ 504508

    URL: https://dx.doi.org/10.1086/504508. doi: 10.1086/ 504508

    doi

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