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 document B document, we prove that the linearized Boltzmann operator document L document of this model is a Fredholm operator. For this, we write document L document as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator document K document is compact. The result is established after inspecting the kernel form of document K document and proving it to be document L² document integrable over its domain using elementary arguments. This implies that document K document is a Hilbert-Schmidt operator.