# Interference of Non-Hermiticity with Hermiticity at Exceptional Points

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Non-Hermitian and Hermitian Operators in Interaction

#### 2.1. Motivation: The Access to EPs in Quantum Physics

#### 2.2. Paradox of Non-Locality of Complex Delta-Function Interactions

## 3. Finite-Dimensional Toy-Model Hamiltonians

#### 3.1. Partitioned Structure of the Relevant Hamiltonians

#### 3.2. $\phantom{\rule{3.33333pt}{0ex}}\mathcal{PT}$-Symmetry Requirement and Reparametrization

- [A] T elements of H lying on the main diagonal will form a real and equidistant sequence simulating the spectrum of the most common harmonic oscillator;
- [B] As long as a broad class of general matrices can be routinely tridiagonalized, we assume that all of our Hs are tridiagonal. Moreover, for the methodical reasons formulated in [49], their off-diagonal part was chosen to be antisymmetric;
- [C] After a finite-dimensional truncation, our benchmark N by N matrices $H={H}^{\left(N\right)}$ is required to be $\mathcal{PT}$-symmetric, $H\mathcal{PT}=\mathcal{PT}H$ [50,51]. Here, $\mathcal{P}$ is defined as the antidiagonal unit matrix (“parity” [49]), and the symbol $\mathcal{T}$ represents the antilinear Hermitian conjugation mimicking the time reversal [38].

## 4. Exceptional Points

#### 4.1. Illustrative $N=8$ Example

**Lemma**

**1**

#### 4.2. Hamiltonians in the EP Limit

**Lemma**

**2**

## 5. Hamiltonians in the Vicinity of EPs

#### 5.1. The Perturbed Schrödinger Equation

#### 5.2. Leading-Order Solution

**Lemma**

**3.**

**Corollary**

**1.**

#### 5.3. Fine-Tuned anomaly ${W}_{N,1}=0$ in an Illustrative $N=4$ Example

## 6. Step-by-Step Hermitizations in the Generic Case

#### 6.1. Elementary Generic $N=4$ Model

#### 6.2. Linear Unfoldings at Arbitrary $N=2J$

**Lemma**

**4.**

#### 6.3. Illustrative Model with $N=8$

## 7. Summary

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Quantum Mechanics Using Non-Hermitian Operators

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**Figure 1.**The numerically calculated unfoldings and smoothness of the real eigenvalues of our toy model ${H}^{\left(8\right)}\left(t\right)$.

${\mathit{z}}_{1}>0$ | ${\mathit{z}}_{1}\le 0$ | ${\mathit{z}}_{2}\le 0$ | … | ${\mathit{z}}_{\mathit{J}-1}\le 0$ | − | |
---|---|---|---|---|---|---|

− | ${\mathbf{z}}_{\mathbf{2}}>\mathbf{0}$ | ${\mathbf{z}}_{\mathbf{3}}>\mathbf{0}$ | … | ${\mathbf{z}}_{\mathbf{J}}>\mathbf{0}$ | ${\mathbf{z}}_{\mathbf{J}}=\mathbf{0}$ | |

M | − | 1 | 2 | … | $J-1$ | J |

K | J | $J-1$ | $J-2$ | … | 1 | − |

t | 0 | 7 | 12 | 15 | 16 |
---|---|---|---|---|---|

$M=M\left(t\right)$ | − | 1 | 2 | 3 | 4 |

EP(2K) | EP(8) | EP(6) | EP(4) | EP(2) | − |

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Znojil, M.
Interference of Non-Hermiticity with Hermiticity at Exceptional Points. *Mathematics* **2022**, *10*, 3721.
https://doi.org/10.3390/math10203721

**AMA Style**

Znojil M.
Interference of Non-Hermiticity with Hermiticity at Exceptional Points. *Mathematics*. 2022; 10(20):3721.
https://doi.org/10.3390/math10203721

**Chicago/Turabian Style**

Znojil, Miloslav.
2022. "Interference of Non-Hermiticity with Hermiticity at Exceptional Points" *Mathematics* 10, no. 20: 3721.
https://doi.org/10.3390/math10203721