# Approximate Analytical Solutions of the Schrödinger Equation with Hulthén Potential in the Global Monopole Spacetime

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## Abstract

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## 1. Introduction

## 2. Nonrelativistic Quantum Mechanics in the Global Monopole Spacetime

## 3. Scattering Phase Shift

## 4. Analysis of Bound States

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Effective potential (Equation (14)) as a function of r for different values of $\alpha <1$. Four situations involving $\xi $ and l are considered: (

**a**) $\xi =0.1$ and $l=1$, (

**b**) $\xi =0.65$ and $l=1$, (

**c**) $\xi =0.21$ and $l=2$, and (

**d**) $\xi =0.65$ and $l=3$.

**Figure 2.**Effective potential (Equation (14)) as a function of r for different values of $\alpha >1$. Four situations involving $\xi $ and l are considered: (

**a**) $\xi =0.1$ and $l=1$, (

**b**) $\xi =0.65$ and $l=1$, (

**c**) $\xi =0.21$ and $l=2$, and (

**d**) $\xi =0.65$ and $l=3$.

**Figure 3.**The plots of ${\left|u\left(r\right)\right|}^{2}$ as a function of r for different values of n displayed for (

**a**) $\alpha =0.2$, $\xi =0.1$, (

**b**) $\alpha =0.8$, $\xi =0.1$, (

**c**) $\alpha =0.2$, $\xi =0.65$, and (

**d**) $\alpha =0.8$, $\xi =0.65$. We use the parameters $\hslash =1$, $M=1$, $e=1$, $k=1$, and $l=1$.

**Figure 4.**The plots of ${\left|u\left(r\right)\right|}^{2}$ as a function of r for different values of n displayed for (

**a**) $\alpha =1.2$, $\xi =0.1$, (

**b**) $\alpha =1.8$, $\xi =0.1$, (

**c**) $\alpha =1.2$, $\xi =0.65$, and (

**d**) $\alpha =1.8$, $\xi =0.65$. We use the parameters $\hslash =1$, $M=1$, $e=1$, $k=1$, and $l=1$.

**Figure 5.**Energy levels (Equation (77)) with $n=1$ as a function of l for $\xi =0.1$. In (

**a**), we display the energies corresponding to different values of $\alpha >1$ and, in (

**b**), values of $\alpha <1$. An inversion between the two profiles is observed at $l=2$.

**Figure 6.**Energy levels (Equation (77)) with $n=1$ as a function of $\alpha $ for (

**a**) $\xi =0.01$ and (

**b**) $\xi =0.5$.

**Figure 7.**Energy levels (Equation (77)) with $n=1$ as a function of $\xi $ for $\alpha =1.5$. In (

**b**), we show the range from (

**a**) to $\xi =0.5$.

**Table 1.**Energies in the limit $\xi \to 0$ for $n=1$ and different values of l (Equation (77)). We assume that $\hslash =1,M=1,Z=1,e=1,n=1$ and $\alpha =0.7$.

${\mathit{E}}_{\mathit{n},\mathit{l}}$ | Values |
---|---|

${E}_{1,1}$ | $-0.0472842$ |

${E}_{1,2}$ | $-0.0236029$ |

${E}_{1,3}$ | $-0.0141781$ |

${E}_{1,4}$ | $-0.0094620$ |

${E}_{1,5}$ | $-0.0067639$ |

${E}_{1,6}$ | $-0.0050758$ |

${E}_{1,7}$ | $-0.0039496$ |

${E}_{1,8}$ | $-0.0031608$ |

${E}_{1,9}$ | $-0.0025868$ |

${E}_{1,10}$ | $-0.0021561$ |

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**MDPI and ACS Style**

Alves, S.S.; Cunha, M.M.; Hassanabadi, H.; Silva, E.O.
Approximate Analytical Solutions of the Schrödinger Equation with Hulthén Potential in the Global Monopole Spacetime. *Universe* **2023**, *9*, 132.
https://doi.org/10.3390/universe9030132

**AMA Style**

Alves SS, Cunha MM, Hassanabadi H, Silva EO.
Approximate Analytical Solutions of the Schrödinger Equation with Hulthén Potential in the Global Monopole Spacetime. *Universe*. 2023; 9(3):132.
https://doi.org/10.3390/universe9030132

**Chicago/Turabian Style**

Alves, Saulo S., Márcio M. Cunha, Hassan Hassanabadi, and Edilberto O. Silva.
2023. "Approximate Analytical Solutions of the Schrödinger Equation with Hulthén Potential in the Global Monopole Spacetime" *Universe* 9, no. 3: 132.
https://doi.org/10.3390/universe9030132