# Schrödinger–Newton Model with a Background

## Abstract

**:**

## 1. Introduction

## 2. Extended SN Equation

## 3. Gravitational Collapse

## 4. Diffusive Radiation

## 5. Modified Jeans Instability

## 6. Jeans-Bubble Regime

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Gravitational collapse, determined by the negative values of ${\omega}^{2}={\mathsf{\Omega}}_{J}^{2}$, for $({I}_{0}=0)$. The black curve represents the normalized quantity ${\omega}^{2}/{\omega}_{J}^{2}$ as a function of the normalized wavenumber ${k}^{2}{v}_{th}^{2}/{\omega}_{J}^{2}$, for quantum matter in a Yukawa potential, with $\Lambda =0.01$, and ${(\hslash /2M)}^{2}=0.3{\omega}_{J}^{2}$. For comparison, the red curve represents the same function in the classical Newtonian limit, with $\Lambda =0$ and $\hslash =0$.

**Figure 2.**Maximum growth rates of the Jeans-bubble instability $\mathsf{\Gamma}$, for ${\mathsf{\Omega}}_{J}^{2}=0$, as a function of the coupling constant $\beta \u03f5$. These quantities are normalized to ${\gamma}_{D}$. The black curve corresponds to gravitational collapse and the red curve to matter void regions.

**Figure 3.**Maximum growth rates of the Jeans-bubble instability $\mathsf{\Gamma}$, as a function of ${\mathsf{\Omega}}_{J}^{2}$, for fixed values of the coupling constants: $\beta \u03f5=-{\gamma}_{D}^{3}$ in black, and $\beta \u03f5=-{\gamma}_{D}^{3}/0.27$ in red.

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Mendonça, J.T.
Schrödinger–Newton Model with a Background. *Symmetry* **2021**, *13*, 1007.
https://doi.org/10.3390/sym13061007

**AMA Style**

Mendonça JT.
Schrödinger–Newton Model with a Background. *Symmetry*. 2021; 13(6):1007.
https://doi.org/10.3390/sym13061007

**Chicago/Turabian Style**

Mendonça, José Tito.
2021. "Schrödinger–Newton Model with a Background" *Symmetry* 13, no. 6: 1007.
https://doi.org/10.3390/sym13061007