For self-adjoint pseudodifferential operators of order 0, Colin de Verdiere and Saint-Raymond introduced natural dynamical conditions (motivated by the study of internal waves in fluids) guaranteeing absolute continuity of the spectrum. I will present an alternative approach to obtaining such results based on Melrose's radial propagation estimates from scattering theory (joint work with S Dyatlov). I will then explain how an adaptation of the Helffer–Sjoestrand theory of scattering resonances shows that in a complex neighbourhood of the continuous spectrum viscosity eigenvalues have limits as viscosity goes to 0. Here the viscosity eigenvalues are the eigen- values of the original operator to which an anti-self-adjoint elliptic 2nd order operator is added (joint work with J Galkowski).