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 $\epsilon > 0$, which arise in a problem in fluid mechanics. We show that the spectra of these operators converge to N as $\epsilon \to 0$, even though, for fixed $\epsilon > 0$, the eigenvalue asymptotics are quadratic. 
E. B. Davies, John Weir