# Vacuum Energy for a Scalar Field with Self-Interaction in (1 + 1) Dimensions

## Abstract

**:**

## 1. Introduction

## 2. Solutions of the NLS in Terms of Elliptic Functions

#### 2.1. The Repulsive Case

#### 2.2. The Attractive Case

## 3. The Vacuum Energy in a Finite Box

## 4. The Infinite Volume Case

## 5. The Limit of Strong Coupling

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**The vacuum energy ${E}_{0}^{\mathrm{ren}}$ of a massive scalar field in a box with repulsive self-interaction. All quantities are given in units of the mass ${m}_{e}$.Spectrum

**Figure 2.**The vacuum energy ${E}_{0}^{\mathrm{ren}}$ of a massive scalar field in a box with attractive self-interaction with ${m}_{e}=1$. The curves terminate where the energy becomes complex. All quantities are given in units of the mass ${m}_{e}$.Spectrum

**Figure 3.**The vacuum energy ${E}_{0}^{\mathrm{ren}}$ of a massless scalar field in a box with repulsive self-interaction and $\mu =1$. ${E}_{0}^{\mathrm{ren}}$ and $\alpha $ are given in units of $\mu $.Spectrum

**Figure 4.**The vacuum energy ${E}_{0}^{\mathrm{ren}}$ of a massless scalar field in a box with attractive self-interaction and $\mu =1$. The curves terminate when the energy becomes complex. ${E}_{0}^{\mathrm{ren}}$ and $\alpha $ are given in units of $\mu $.Spectrum

**Figure 5.**The vacuum energy ${E}_{0}^{\mathrm{ren}}$ of a massive scalar field with repulsive self-interaction on the whole axis. All quantities are given in units of the mass ${m}_{e}$.Spectrum

**Figure 6.**The vacuum energy ${E}_{0}^{\mathrm{ren}}$ of a massless scalar field with repulsive self-interaction on the whole axis. ${E}_{0}^{\mathrm{ren}}$ and $\alpha $ are given in units of $\mu $.Spectrum

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Bordag, M.
Vacuum Energy for a Scalar Field with Self-Interaction in (1 + 1) Dimensions. *Universe* **2021**, *7*, 55.
https://doi.org/10.3390/universe7030055

**AMA Style**

Bordag M.
Vacuum Energy for a Scalar Field with Self-Interaction in (1 + 1) Dimensions. *Universe*. 2021; 7(3):55.
https://doi.org/10.3390/universe7030055

**Chicago/Turabian Style**

Bordag, Michael.
2021. "Vacuum Energy for a Scalar Field with Self-Interaction in (1 + 1) Dimensions" *Universe* 7, no. 3: 55.
https://doi.org/10.3390/universe7030055