# The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (i)
- The related Birman–Schwinger operator$${B}_{E}={V}^{1/2}{({H}_{0}-E)}^{-1}{V}^{1/2},\phantom{\rule{1.em}{0ex}}E\in \rho \left({H}_{0}\right),$$
- (ii)

**Remark**

**1.**

## 3. Calculation of the Ground State Energy

## 4. Final Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Some Mathematical Results

#### Appendix A.1. First Theorem

**Theorem**

**A1.**

**Proof.**

#### Appendix A.2. Second Theorem

**Theorem**

**A2.**

**Proof.**

#### Appendix A.3. Third Theorem

**Theorem**

**A3.**

**Proof.**

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**Figure 1.**Plot of the ground state energy ${E}_{0}\left(\lambda \right)$ as a function of $\lambda $, given in (33).

**Figure 2.**Plot of the energy dependence of the norms of the following four operators: ${P}_{{\u03f5}_{0}}$ in (34) (blue curve), ${M}_{{\u03f5}_{0}}$ in (35) (red curve), ${K}_{{\u03f5}_{0},1}$ in (38), (cyan curve) and ${K}_{{\u03f5}_{0},2}$ in (39) (magenta curve). In the inset, we can see more details of the two lowest curves.

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**MDPI and ACS Style**

Fassari, S.; Gadella, M.; Nieto, L.M.; Rinaldi, F.
The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity. *Symmetry* **2021**, *13*, 1561.
https://doi.org/10.3390/sym13091561

**AMA Style**

Fassari S, Gadella M, Nieto LM, Rinaldi F.
The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity. *Symmetry*. 2021; 13(9):1561.
https://doi.org/10.3390/sym13091561

**Chicago/Turabian Style**

Fassari, Silvestro, Manuel Gadella, Luis Miguel Nieto, and Fabio Rinaldi.
2021. "The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity" *Symmetry* 13, no. 9: 1561.
https://doi.org/10.3390/sym13091561