An indefinite convection-diffusion operator with real spectrum

has only real eigenvalues, which were shown to exist by Davies in [3]. In the same paper, he showed that the spectrum of −iH is equal to the set of its eigenvalues, so this implies that the spectrum is real. This was conjectured in [1], and Chugunova and Pelinovsky proved in [2] that all but finitely many eigenvalues are real, and gave numerical evidence that all eigenvalues are real. Our approach is to show that it is sufficient to consider H acting on the Hardy space H(−π, π) and analytically continue any solution of (3) to the unit disc, where the corresponding ODE (8) is now self-adjoint on [0, 1]