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

<p style='text-indent:20px;'>The spectrum structure of the linearized Boltzmann operator has been a subject interest for over fifty years and inspected in space <inline-formula><tex-math id="M2">\begin{document}$ L^2\left( {\mathbb R}^d, \exp(|v|^2/4)\right) $\end{document}</tex-math></inline-formula> by B. Nicolaenko [<xref ref-type="bibr" rid="b27">27</xref>] case hard spheres, then generalized to Maxwellian potentials R. Ellis M. Pinsky rid="b13">13</xref>], S. Ukai proved existence spectral gap large frequencies rid="b33">33</xref>]. The aim this paper is extend spaces id="M3">\begin{document}$ (1+|v|)^{k}\right) studies from rid="b13">13</xref>,<xref More precisely, we look at Fourier transform variable inhomogeneous consider dual as fixed parameter. We perform precise study small (by seeing it perturbation homogeneous one) also semigroup point views. Our approach based on Kato's theory linear operators rid="b22">22</xref>] well enlargement arguments rid="b25">25</xref>,<xref rid="b19">19</xref>].</p>