Pseudospectra of the Convection-Diffusion Operator

The spectrum of the simplest 1D convection-diffusion operator is a discrete subset of the negative real axis, but the pseudospectra are regions in the complex plane bounded approximately by parabolas. Put another way, the norm of the resolvent is exponentially large as a function of the Péclet number throughout a certain approximately parabolic region. These observations have a simple physical basis, and suggest that conventional spectral analysis for convection-diffusion operators may be of limited value in some applications.