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We study the linear Vlasov equation with a given electric field $E \in\mathcal{S}$, where $\mathcal{S}$ is the space of Schwartz functions. Theassociated damped partial differential equation has a unique tempered solution,which fixes the needed Cauchy datum. This tempered solution then converges tothe causal solution of the linear Vlasov equation when the damping parametergoes to zero. This result allows us to define the plasma conductivity operator$\sigma$, which gives the current density $j = \sigma (E)$ induced by theelectric field $E$. We prove that $\sigma$ is continuous from $\mathcal{S}$ toits dual $\mathcal{S}^\prime$. We can treat rigorously the case of uniformnon-magnetized non-relativistic plasma (linear Landau damping) and the case ofuniform magnetized relativistic plasma (cyclotron damping). In both cases, wedemonstrate that the main part of the conductivity operator is apseudo-differential operator and we give its expression rigorously. Thismatches the formal results widely used in the theoretical physics community.

Uniqueness of the Cauchy datum for the tempered-in-time response and conductivity operator of a plasma