We obtain asymptotics for eigenvalues of an operator which acts on bounded measurable functions vanishing outside some bounded domain in ${\rm \IR}\sp{r}$. The operator we consider is associated with a pure jump Markov process in the sense that the infinitesimal generator of the process acts on functions in the same way that the operator does. The asymptotics of the eigenvalues bear a resemblance to those found by A. D. Wentzell and M. I. Freidlin for a second order elliptic differential operator with a small parameter in the higher derivatives In a recent manuscript Professor Wentzell obtained asymptotics for the first eigenvalue of a process, possibly having jumps, assumed to have certain large deviations properties. We show that the pure jump Markov process we are treating has the large deviations behavior assumed in Professor Wentzell's paper, and thus obtain asymptotics for the first eigenvalue of our operator. We obtain asymptotics for additional eigenvalues of our operator by exploiting the large deviations properties of the associated Markov process