An indefinite convection-diffusion operator

Abstract: The author of this paper gives a mathematically rigorous analysis which confirms the surprising results in a recent paper of Benilov, O'Brien and Sazonov [J. Fluid Mech. 497 (2003) 201–224] about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics. He also shows that the set of eigenvectors does not form a basis for the operator.