Spectral Properties of Self-Adjoint Extensions of an Unbounded Hamiltonian as a Model for Dark Interactions

There is compelling evidence of massive dark matter around galactic structures. This observation poses difficulties for local detection of the matter, either due to extremely small mass or interaction that is very weak. As an analogy, one envisions the core of a galactic structure as the nucleus of an atom and the dark matter as being bound to the core like electrons. This model, if gravity were the only interaction, is known exactly as the Coulomb model for hydrogen-like atoms. The Hamiltonian can then be modified by a lowest order harmonic potential as an approximation for dark interactions. The resulting Hamiltonian is an unbounded linear operator-- one that is not even semi-bounded, meaning there is neither an upper nor a lower bound. This property is desirable for dark interactions at the galactic level because the upper bound, if it exists, is many times the excess of the average galactic radius. While, the non-existence of a lower bound implies room for as of yet undetected local particle structure. The resulting spectrum, if observable, will give a lowest order approximation to the dark interaction coupling.