We prove that all the eigenvalues of a certain highly non-self-adjoint Sturm-Liouville differential operator are real. The results presented are motivated by and extend those recently found by various authors (Benilov et al. (2003) , Davies (2007)  and Weir (2008) ) on the stability of a model describing small oscillations of a thin layer of fluid inside a rotating cylinder. © 2010 Elsevier Inc.
- Non-Hermitian Hamiltonians
- Sturm-Liouville operators