# Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Effective Hamiltonians

#### 2.1. The Concept of Open Quantum Systems

- {A.1}
- about the pure-state nature of the initial conditions. This must be guaranteed by the preparation of the system at $t={t}_{ini}=0$;
- {A.2}
- about the unitarity of the evolution requiring, due to the Stone theorem [13], the self-adjointness of the Hamiltonian in ${\mathcal{H}}^{\left(T\right)}$, $H\left(t\right)={H}^{\u2020}\left(t\right)$;
- {A.3}

#### 2.2. The Feshbach’s Concept of Model Space

## 3. Antilinear Versus Nonlinear Interactions

#### 3.1. Quantum $\mathcal{PT}$-Symmetric Schrödinger Equations

#### 3.2. The Turn of Attention to Classical Optics

#### 3.3. The Turn of Attention to the Coupling of Channels

## 4. Nonlinear Schrödinger Equations on Complex-Plane Contours of the “Coordinate”

#### 4.1. Linear Equations on the Complex Contours

#### 4.2. Nonlinear Equations on the Complex Contours

- the latter two equations may be solved to determine the “correct” contour C,
- in a technically highly nontrivial manner, one can also guarantee that for one of the resulting contours, the “candidate for the anomalous quantum probability density” $\sigma \left(z\right)$ can be globally normalized to one, ${\int}_{C}\sigma \left(z\right)dz=1$.

## 5. Effective Nonlinearities

#### 5.1. Coupled Cluster Wave-Function Ansatz

#### 5.2. Broader Context in Physics

## 6. Roots in Linear Theory

#### 6.1. Pilot-Wave Approach

#### 6.2. Solvable and Partially-Solvable Models

## 7. Roots in Phenomenology

## 8. Constructions Strategies

## 9. Analytical Solutions

#### 9.1. Diagonal Case

#### 9.2. Off-Diagonal Case

## 10. Discussion

## 11. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Znojil, M.; Růžička, F.; Zloshchastiev, K.G.
Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear *PT*-Symmetry to the Nonlinear Coupling of Channels. *Symmetry* **2017**, *9*, 165.
https://doi.org/10.3390/sym9080165

**AMA Style**

Znojil M, Růžička F, Zloshchastiev KG.
Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear *PT*-Symmetry to the Nonlinear Coupling of Channels. *Symmetry*. 2017; 9(8):165.
https://doi.org/10.3390/sym9080165

**Chicago/Turabian Style**

Znojil, Miloslav, František Růžička, and Konstantin G. Zloshchastiev.
2017. "Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear *PT*-Symmetry to the Nonlinear Coupling of Channels" *Symmetry* 9, no. 8: 165.
https://doi.org/10.3390/sym9080165