The spectrum structure of the linearized Boltzmann operator has been a subject of interest for over fifty years and has been inspected in the space $ L^2\left( {\mathbb R}^d, \exp(|v|^2/4)\right) $ by B. Nicolaenko [27] in the case of hard spheres, then generalized to hard and Maxwellian potentials by R. Ellis and M. Pinsky [13], and S. Ukai proved the existence of a spectral gap for large frequencies [33]. The aim of this paper is to extend to the spaces $ L^2\left( {\mathbb R}^d, (1+|v|)^{k}\right) $ the spectral studies from [13,33]. More precisely, we look at the Fourier transform in the space variable of the inhomogeneous operator and consider the dual Fourier variable as a fixed parameter. We then perform a precise study of this operator for small frequencies (by seeing it as a perturbation of the homogeneous one) and also for large frequencies from spectral and semigroup point of views. Our approach is based on Kato's perturbation theory for linear operators [22] as well as enlargement arguments from [25,19].

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具有多项式和高斯权重的 \begin{document}$ L^2 $\end{document}-空间中线性化玻尔兹曼算子的谱研究

线性化玻尔兹曼算子的谱结构一直是一个有趣的主题五十多年，并在空间 $ L^2\left( {\mathbb R}^d, \exp(|v|^2/4 )\right) $ by B. Nicolaenko [27] 在硬球的情况下，然后由 R. Ellis 和 M. Pinsky 推广到硬势和麦克斯韦势 [13]，S. Ukai 证明了大频率的频谱间隙的存在 [33]。本文的目的是扩展到空间 $ L^2\left( {\mathbb R}^d, (1+|v|)^{k}\right) $ [13,33]。更准确地说，我们查看非齐次算子空间变量中的傅立叶变换，并将对偶傅立叶变量视为一个固定参数。然后，我们对小频率（通过将其视为齐次的扰动）以及从光谱和半群的角度对大频率进行了对该算子的精确研究。我们的方法基于 Kato 的线性算子微扰理论 [22] 以及来自 [25,19]。