Convergence of eigenvalues for a highly non-self-adjoint differential operator

In this paper we study a family of operators dependent on a small parameter $ϵ>0$, which arise in a problem in fluid mechanics. We show that the spectra of these operators converge to N as $ϵ\to 0$, even though, for fixed $ϵ>0$, the eigenvalue asymptotics are quadratic.