For a scalar field in an exponentially expanding universe, constituent modes of elementary excitation become unstable consecutively at shorter wavelength. After canonical quantization, a Bogoliubov transformation reduces the minimally coupled scalar field to independent 1D modes of two inequivalent types, leading eventually to a cosmological partitioning of energy. Due to accelerated expansion of the coupled space-time, each underlying mode transits from an attractive oscillator with discrete energy spectrum to a repulsive unit with continuous unbounded energy spectrum. The underlying non-autonomous Schrodinger equation is solved here as the wave function evolves through the attraction-repulsion transition and ceases to oscillate.

中文翻译：

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非自治薛定谔方程的求解量化 de Sitter Klein-Gordon 振荡器模式经历吸引-排斥转变

对于指数膨胀宇宙中的标量场，基本激发的组成模式在较短波长处连续变得不稳定。在规范量化之后，Bogoliubov 变换将最小耦合的标量场减少为两种不等价类型的独立一维模式，最终导致能量的宇宙学划分。由于耦合时空的加速膨胀，每个潜在模式从具有离散能谱的有吸引力的振荡器转变为具有连续无界能谱的排斥单元。随着波函数通过吸引-排斥转变演变并停止振荡，这里解决了潜在的非自治薛定谔方程。