Why is the photoelectric absorption coefficient finite at the threshold frequency? I mean the photoelectric effect of the hydrogen atom.
It is weird. By the Fermi golden rule, the transition or absorption rate is proportional to the density of the final states. At threshold, the electron has zero momentum and thus zero density of state. Therefore, the absorption coefficient should be zero at the threshold. 
Where does this argument go wrong?

 A: It is a very interesting question.
First point is that even with 1 meV of energy the electron will have plenty of states. I suppose the issue is what exactly is a zero energy electron. Hotop et al. have done very nice experiments in electron scattering where they control the electron energy to below 1 meV by laser photoionization. They can then observe the photoelectrons interacting with molecules - e.g. electron attachment to SF$_6$.
So here I should point out that 1 meV is very low energy and the scale of the nice graph that you show is not fine enough to be able to distinguish it. (and probably the experimental data quoted (or calculations) do not have sufficient detail (or resolution))
So what controls the availability of states for the free electron? It is effectively a case of a particle in a 3D box, so it depends on the dimensions of the container, but generally the containers are so huge compared to atomic scales that electrons can be treated as more or less classical particles - for example, photoelectron energies can be measured with time-of-flight spectrometers where the energy is found by measuring the speed of the electron - the speed is found by the time taken to move a fixed distance - and $E={1\over2}mv^2$ works well.
p.s. I have found a relevant paper not behind a paywall at NIST's site - it is about molecule - generally the photoelectric effect for atoms is called photoionization and the effect for molecules is very similar
