  We compute the complete spectrum of the displacement Hessian operator, which
is obtained from the confined porous medium equation by linearization around
its stationary attractor, the Barenblatt profile. On a formal level, the
operator is conjugate to the Hessian of the entropy via similarity
transformation. We show that the displacement Hessian can be understood as a
self-adjoint operator and find that its spectrum is purely discrete. The
knowledge of the complete spectrum and the explicit information about the
corresponding eigenfunctions give new insights on the convergence and higher
order asymptotics of solutions to the porous medium equation towards its
attractor. More precisely, the inspection of the eigenfunctions allows to
identify symmetries in $\mathbbm{R}^N$ with flows whose rates of convergence
are faster than the uniform, translation-governed bound. The present work
complements the analogous study of Denzler & McCann for the fast-diffusion
equation.
