# Perturbation Theory Near Degenerate Exceptional Points

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Exceptional Points

#### 2.1. Bose–Hubbard Model and Exceptional Points of Geometric Multiplicity One, $L=1$

#### 2.2. Generic Non-Hermitian Degeneracies with Geometric Multiplicities Larger than One, $L>1$

## 3. Unitary Processes of Collapse at L = 2

#### 3.1. Quantum Physics behind “Degenerate Degeneracies” with $L=2$

#### 3.2. Unfoldings of Degeneracies under Random Perturbations at $L=2$

## 4. Perturbation Theory at L = 2

#### 4.1. The Recent Change of the Unitary-Evolution Paradigm

#### 4.2. Rearrangement of Schrödinger Equation

#### 4.3. Solutions

## 5. Schrödinger Equation in Leading-Order Approximation

#### 5.1. Generic Case: Perturbations without Vanishing Elements

#### 5.2. Hierarchy of Relevance and Reduced Approximations

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

#### 5.3. Unitary Case: Re-Scaled Perturbations

**Lemma**

**3.**

**Proof.**

## 6. Discussion

#### 6.1. Schrödinger Picture and Quasi-Hermitian Hamiltonians

#### 6.2. Non-Hermitian Degeneracies with $L>2$

#### 6.3. The Next-to-Leading-Order Approximation

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. An Exhaustive Classification of the Degeneracies of Exceptional Points

K | List |
---|---|

2 | 2 |

3 | 3 |

4 | 4 2 + 2 |

5 | 5 3 + 2 |

6 | 6 4 + 2 3 + 3 2 + 2 + 2 |

7 | 7 5 + 2 4 + 3 3 + 2 + 2 |

8 | 8 6 + 2 5 + 3 4 + 4 4 + 2 + 2 3 + 3 + 2 2 + 2 + 2 + 2 |

9 | 9 7 + 2 6 + 3 5 + 4 5 + 2 + 2 4 + 3 + 2 3 + 3 + 3 3 + 2 + 2 + 2 |

10 | 10 8 + 2 7 + 3 6 + 4 6 + 2 + 2 5 + 5 5 + 3 + 2 |

4 + 4 + 2 4 + 3 + 3 4 + 2 + 2 + 2 3 + 3 + 2 + 2 2 + 2 + 2 + 2 + 2 | |

⋮ | ... |

0, 0, 1, 1, 3, 3, 6, 7, 11, 13, 20, 23, 33, 40, 54, 65, 87, 104, 136, 164, |

209, 252, 319, 382, 477, 573, 707, 846, 1038, 1237, 1506, 1793, … . |

## Appendix B. Perturbation Theory Near Non-Degenerate Exceptional Points

#### Appendix B.1. The Choice of Basis at L=1

#### Appendix B.2. The Description of the Unfolding of the Degeneracy at L=1

#### Appendix B.3. Unitary-Evolution Process of Unfolding at L = 1

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Znojil, M.
Perturbation Theory Near Degenerate Exceptional Points. *Symmetry* **2020**, *12*, 1309.
https://doi.org/10.3390/sym12081309

**AMA Style**

Znojil M.
Perturbation Theory Near Degenerate Exceptional Points. *Symmetry*. 2020; 12(8):1309.
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**Chicago/Turabian Style**

Znojil, Miloslav.
2020. "Perturbation Theory Near Degenerate Exceptional Points" *Symmetry* 12, no. 8: 1309.
https://doi.org/10.3390/sym12081309