We perform a complete spectral analysis of the linear three-dimensional Boltzmann BGK operator resulting in an explicit transcendental equation for the eigenvalues. Using the theory of finite-rank perturbations, we confirm the existence of a critical wave number \(k_{\textrm{crit}}\) which limits the number of hydrodynamic modes in the frequency space. This implies that there are only finitely many isolated eigenvalues above the essential spectrum at each wave number, thus showing the existence of a finite-dimensional, well-separated linear hydrodynamic manifold as a combination of invariant eigenspaces. The obtained results can serve as a benchmark for validating approximate theories of hydrodynamic closures and moment methods and provides the basis for the spectral closure operator.

我们对线性三维玻尔兹曼 BGK 算子进行了完整的谱分析，从而得到特征值的显式超越方程。利用有限秩扰动理论，我们确认了临界波数 \（k_{\textrm{crit}}\） 的存在，它限制了频率空间中流体动力学模式的数量。这意味着在每个波数上，基本光谱以上只有有限数量的孤立特征值，从而表明存在一个有限维、分离良好的线性流体动力学流形作为不变特征空间的组合。所得结果可以作为验证流体动力学闭合和矩法近似理论的基准，并为谱闭合算子提供基础。