A spectral study of the linearized Boltzmann operator in $ L^2 $-spaces with polynomial and Gaussian weights

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 \begin{document}$ L^2\left( {\mathbb R}^d, \exp(|v|^2/4)\right) $\end{document} by B. Nicolaenko [ 27 ] in the case of hard spheres, then generalized to hard and Maxwellian potentials by R. Ellis and M. Pinsky [ 13 ], and S. Ukai proved the existence of a spectral gap for large frequencies [ 33 ]. The aim of this paper is to extend to the spaces \begin{document}$ L^2\left( {\mathbb R}^d, (1+|v|)^{k}\right) $\end{document} 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 ].