标题： 一个量子宇宙学的精确可解模型：类氢原子与尘埃及宇宙常数的类比 显示英文标题

标题： An exactly solvable model of quantum cosmology: the Hydrogen atom analogy with dust and Cosmological constant

摘要： 我们研究了在单模引力框架下，无压尘（通过布朗-库查尔形式主义建模）和一个动态宇宙常数$\Lambda$的空间平坦的弗里德曼-莱马特-罗伯逊-沃克（FLRW）宇宙的惠勒-德威特量子化，其中单模时间作为关系时钟。 值得注意的是，该系统的量子动力学表现出与非相对论氢原子的数学对应性——$\Lambda$对应于能量本征值，体积变量对应于径向坐标，尘埃能量密度参数对应于库仑势强度。这种类比产生了正$\Lambda$的连续谱，类似于散射态。 对于$\Lambda > 0$，我们证明了单模哈密顿量的自伴随性，保证了单模时间下的幺正演化。通过从归一化定态构造波包，我们展示了量子反弹，这解决了经典的宇宙大爆炸奇点。 动力学从远离反弹点的半经典行为过渡到由于反弹点附近的干涉而特征性地出现“回声”振荡的量子主导区域。我们通过宇宙可观测量的期望值和涨落来量化量子效应，发现晚期宇宙中存在持久的量子效应的证据。 因此，我们的结果表明，量子引力效应可能在反弹后的时间宇宙学中留下印记。 显示英文摘要

摘要： We study the Wheeler-DeWitt quantization of a spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe with pressureless dust (modeled via the Brown-Kucha\v{r} formalism) and a dynamical cosmological constant $\Lambda$ treated in the unimodular gravity framework, where unimodular time serves as a relational clock. Remarkably, the quantum dynamics of this system exhibit a mathematical correspondence to a non-relativistic hydrogen atom -- $\Lambda$ maps to energy eigenvalues, the volume variable to the radial coordinate, and the dust energy density parameter to the Coulomb potential strength. This analogy yields a continuous spectrum for positive $\Lambda$, analogous to scattering states. For $\Lambda > 0$, we prove the self-adjointness of the unimodular Hamiltonian, guaranteeing unitary evolution in unimodular time. By constructing wave packets from normalized stationary states, we demonstrate a quantum bounce that resolves the classical Big Bang singularity. The dynamics transition from semiclassical behavior far from the bounce to quantum-dominated regions featuring characteristic "ringing" oscillations due to interference near the bounce. We quantify quantum effects through expectation values and fluctuations of cosmological observables, finding evidence for persistent quantum effects in the late universe. Thus our results suggest that quantum gravitational effects may leave imprints on late-time cosmology, even beyond the bounce.