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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M2">\begin{document}$ L^2\left( {\mathbb R}^d, \exp(|v|^2/4)\right) $\end{document}</tex-math></inline-formula> by B. Nicolaenko [<xref ref-type="bibr" rid="b27">27</xref>] in the case of hard spheres, then generalized to hard and Maxwellian potentials by R. Ellis and M. Pinsky [<xref ref-type="bibr" rid="b13">13</xref>], and S. Ukai proved the existence of a spectral gap for large frequencies [<xref ref-type="bibr" rid="b33">33</xref>]. The aim of this paper is to extend to the spaces <inline-formula><tex-math id="M3">\begin{document}$ L^2\left( {\mathbb R}^d, (1+|v|)^{k}\right) $\end{document}</tex-math></inline-formula> the spectral studies from [<xref ref-type="bibr" rid="b13">13</xref>,<xref ref-type="bibr" rid="b33">33</xref>]. 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 [<xref ref-type="bibr" rid="b22">22</xref>] as well as enlargement arguments from [<xref ref-type="bibr" rid="b25">25</xref>,<xref ref-type="bibr" rid="b19">19</xref>].</p>