As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no eigenfunctions in the Hilbert space.
On the other hand, both bound and continuous eigenfunctions are required to have a complete set, to be able to expand an arbitrary wave function in terms of the eigenfunctions (Landau&Lifshitz, Quantum Mechanics, p.19). How are these results connected, how to explain the apparent contradiction?
Is a formulation that the Hilbert space is spanned by both bound and continuous hydrogen atom eigenfunctions correct?
Update: I just found Scattering states of Hydrogen atom in non-relativistic perturbation theory which is related (but only partially answers this question).