UC Santa Cruz

This thesis is a study of the uniqueness of the higher stationary states of the Schr\"odinger--Newton system under the assumption of spherically symmetric solutions. We begin with a theory of dark matter put foward by Bray \cite{hubray} involving the Einstein--Klein--Gordon system of equations, and then pose the Schr\"odinger--Newton system as the low--field nonrelatavistic limit of the Einstein--Klein--Gordon system. From here, by imposing spherical symmetry, we show that the potential term in the Schr\"odinger--Newton system can be seen as a nonlinear perturbation from the Coulomb potential $\frac{1}{r}$ on the half--line $[0, \infty)$. After proving uniqueness of bound states for the Hydrogen atom on the half--line, we then proceed by defining weighted Banach spaces for which the Schr\"odinger operator representing the Hydrogen atom on the half--line is Fredholm of index 0. In the last chapter, we detail an iteration scheme involving the implicit function theorem to show a correspondence between bound state solutions of the Hydrogen atom on the half--line and bound state solutions of the full Schr\"odinger--Newton system to prove the uniqueness result.