# Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato’s Exceptional Points

## Abstract

**:**

## 1. Introduction

## 2. The Context of Quantum Mechanics

## 3. The Context of the Theory of Catastrophes

#### 3.1. A Remark on Terminology

#### 3.2. The Dyson’s Maps in Quantum Theory

#### 3.3. Special Models with Real Exceptional Points

## 4. The Context of Classical Optics

#### 4.1. Solvable Quantum Models

#### 4.2. Paraxial Approximation and Anomalous Diffusion Phenomena

## 5. The Context of Quantum Cosmology

#### 5.1. Problems with Quantization

#### 5.2. The Consistent Quantum Singularity at Big Bang

## 6. Toy Models

#### 6.1. Toy Model Matrices with $N=2$

#### 6.2. Toy-Model Matrices with $N=3$

**Lemma 1.**

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Proof.**

#### 6.3. Toy-Model Matrices with $N=4$

**Lemma 4.**

**Proof.**

## 7. Discussion

## 8. Summary

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The Alice-Bob distances $d\left(t\right)$: (1) measured by the methods of classical physics (large dots), (2) measured, in a Gedankenexperiment, by quantum-theory methods (including, perhaps, also the hypothetical “superluminal-velocity” Inflation period, small dots), (3) extrapolated down to $t=0$ (smooth curve, quantum Big-Bang-singularity hypothesis).

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Znojil, M.
Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato’s Exceptional Points. *Symmetry* **2016**, *8*, 52.
https://doi.org/10.3390/sym8060052

**AMA Style**

Znojil M.
Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato’s Exceptional Points. *Symmetry*. 2016; 8(6):52.
https://doi.org/10.3390/sym8060052

**Chicago/Turabian Style**

Znojil, Miloslav.
2016. "Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato’s Exceptional Points" *Symmetry* 8, no. 6: 52.
https://doi.org/10.3390/sym8060052