# Perturbation of One-Dimensional Time Independent Schrödinger Equation With a Symmetric Parabolic Potential Wall

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

Let ${G}_{1}$ be a group and let ${G}_{2}$ be a metric group with the metric $d(\xb7,\xb7)$. Given $\epsilon >0$, does there exist a $\delta >0$ such that if a function $h:{G}_{1}\to {G}_{2}$ satisfies the inequality $d\left(h\right(xy),h(x\left)h\right(y\left)\right)<\delta $ for all $x,y\in {G}_{1}$, then there exists a homomorphism $H:{G}_{1}\to {G}_{2}$ with $d\left(h\right(x),H(x\left)\right)<\epsilon $ for all $x\in {G}_{1}$?

## 2. Preliminaries

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

## 3. A Type of Hyers-Ulam Stability

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Proof.**

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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$\mathbf{Lemma}$ 2 | in (6) |
---|---|

$y\left(x\right)$ | $\varphi \left(x\right)$ |

$f\left(x\right)$ | $\alpha x-\beta $ |

$g\left(x\right)$ | 0 |

$\phi \left(x\right)$ | $\frac{2m}{{\hslash}^{2}}\epsilon $ |

$\mathbf{Lemma}$ 2 | in (11) |
---|---|

$y\left(x\right)$ | $\psi \left(x\right)$ |

$f\left(x\right)$ | $\beta -\alpha x$ |

$g\left(x\right)$ | ${\varphi}_{0}\left(x\right)$ |

$\phi \left(x\right)$ | $\frac{2m}{{\hslash}^{2}}\epsilon \left|{\int}_{0}^{x}exp\left(\beta (x-s)-\frac{\alpha}{2}\left({x}^{2}-{s}^{2}\right)\right)ds\right|$ |

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**MDPI and ACS Style**

Jung, S.-M.; Kim, B.
Perturbation of One-Dimensional Time Independent Schrödinger Equation With a Symmetric Parabolic Potential Wall. *Symmetry* **2020**, *12*, 1089.
https://doi.org/10.3390/sym12071089

**AMA Style**

Jung S-M, Kim B.
Perturbation of One-Dimensional Time Independent Schrödinger Equation With a Symmetric Parabolic Potential Wall. *Symmetry*. 2020; 12(7):1089.
https://doi.org/10.3390/sym12071089

**Chicago/Turabian Style**

Jung, Soon-Mo, and Byungbae Kim.
2020. "Perturbation of One-Dimensional Time Independent Schrödinger Equation With a Symmetric Parabolic Potential Wall" *Symmetry* 12, no. 7: 1089.
https://doi.org/10.3390/sym12071089