(This week's colloquium has been canceled.)

We study the spectrum of a linear advection-diffusion equation in a periodic domain, where the diffusion coefficient changes its sign. We prove that the spectrum of an associated linear operator consists of an infinite set of simple eigenvalues on the imaginary axis and the set of corresponding eigenfunctions is complete. However, we also show, assisted with numerical approximations, that the complete set of linearly independent eigenfunctions does not form a basis in a space of square integrable functions and that the Cauchy problem for the advection-diffusion equation is ill-posed.