It is well known that the scattering states in quantum mechanics are not a part of the Hilbert space comprised of the bound states in a given quantum potential. It is often said that the (negative energy) eigenstates are not dense in the larger "rigged" Hilbert space that includes scattering states.
However, scattering theory is often described with the "complete set" of hydrogen eigenfunctions - the negative energy eigenstates. This is, e.g. the phase theory of scattering as discussed in Sakurai.
My question is: how is it possible to give a description of scattering with the negative energy eigenstates at all? Mathematically speaking, it seems that one is describing a process that should lie strictly in the extended (rigged) space with the limited negative energy Hilbert space. What is going on?