Compactness property of the linearized Boltzmann operator for a diatomic single gas model

In the following work, we consider the Boltzmann equation that models a diatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section \begin{document}$ \mathcal{B} $\end{document}, we prove that the linearized Boltzmann operator \begin{document}$ \mathcal{L} $\end{document} of this model is a Fredholm operator. For this, we write \begin{document}$ \mathcal{L} $\end{document} as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator \begin{document}$ \mathcal{K} $\end{document} is compact. The result is established after inspecting the kernel form of \begin{document}$ \mathcal{K} $\end{document} and proving it to be \begin{document}$ L^2 $\end{document} integrable over its domain using elementary arguments.This implies that \begin{document}$ \mathcal{K} $\end{document} is a Hilbert-Schmidt operator.