# Solution of Non-Autonomous Schrödinger Equation for Quantized de Sitter Klein-Gordon Oscillator Modes Undergoing Attraction-Repulsion Transition

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{*}-algebra generated by the Weyl operators [34]. Use of a countable basis of exact field solutions in a finite universe has resulted in the number of repulsive units remaining finite, although increasing with time. This means that although the repulsive modes have neither a natural vacuum representation nor a particle interpretation, Stone’s theorem guarantees that the Hilbert space representation is unique, modulo a unitary automorphism. This preferred physical representation enables meaningful calculations and predictions about the system. This result is significantly different from that of the approximate continuous Fourier expansion [18], since approximation of finite space by spatially infinite space, artificially leads to a problematic infinite number of unstable modes. The remaining infinitely many quantum oscillator modes still have a unique Fock-Cook representation, so overall there is still a well defined preferred physical representation.

## 2. Canonical Field Quantization and Unstable Modes

#### 2.1. Classical Klein-Gordon Field in an Accelerating Space-Time

#### 2.2. Reduction of Time-Frozen Hamiltonian by Bogoliubov Transformation

#### 2.3. Non-Autonomous Quantum Evolution: Solution with Finitely Many Extrema

#### 2.3.1. Transforming to an Autonomous Schrödinger Equation

#### 2.3.2. Harmonic Oscillator Initially in Ground State

#### 2.3.3. Superposition of Harmonic Oscillator Ground State and First Excited State

## 3. Discussion

#### 3.1. Cosmological Implications

#### 3.2. Existence of Preferred Physical Representation

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

FLRW | Friedmann Lemaitre Robertson Walker |

PDE | partial differential equation |

## Appendix A. Separation of Variables

## Appendix B. Choice of Boundary Conditions

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**Figure 1.**(

**a**) Time dependent dispersion relation (schematic) for minimally coupled Klein-Gordon equation. ${\omega}^{2}$ for the minimum discrete wave number k becomes negative at critical time ${t}_{c}$. (

**b**) Classical potential energy and quantum energy levels for time-dependent Hamiltonian. As time increases, the attractive force weakens and discrete energy gaps narrow between bound eigenstates. At times $t>{t}_{c}$, the force is repulsive and the Hamiltonian has continuous spectrum with no lower bound.

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**MDPI and ACS Style**

Broadbridge, P.; Deutscher, K.
Solution of Non-Autonomous Schrödinger Equation for Quantized de Sitter Klein-Gordon Oscillator Modes Undergoing Attraction-Repulsion Transition. *Symmetry* **2020**, *12*, 943.
https://doi.org/10.3390/sym12060943

**AMA Style**

Broadbridge P, Deutscher K.
Solution of Non-Autonomous Schrödinger Equation for Quantized de Sitter Klein-Gordon Oscillator Modes Undergoing Attraction-Repulsion Transition. *Symmetry*. 2020; 12(6):943.
https://doi.org/10.3390/sym12060943

**Chicago/Turabian Style**

Broadbridge, Philip, and Kathryn Deutscher.
2020. "Solution of Non-Autonomous Schrödinger Equation for Quantized de Sitter Klein-Gordon Oscillator Modes Undergoing Attraction-Repulsion Transition" *Symmetry* 12, no. 6: 943.
https://doi.org/10.3390/sym12060943