Schrödinger-Poisson and beyond: dynamical models for dark matter

This Ph.D. thesis investigates the Schrödinger-Poisson system as a model for dark matter dynamics, conceived as an aggregate of self-gravitating particles. The dark matter density distribution is represented by the squared modulus of a matter field, while the self-interactions are described by a gravitational field. The two components evolve in a coupled manner: the matter field according to a Schrödinger equation, and the gravitational potential according to Poisson dynamics. The model is applied to dark matter halos in galaxies, describing both their dynamics and equilibrium configurations. It also provides predictions for rotation curves, which represent the most significant observational data in this context. Specifically, this thesis explores the connection between the Schrödinger-Poisson system and established physical theories, showing how it can equivalently emerge from different foundational assumptions. A heuristic approach to the problem is presented, illustrating how general physical hypotheses about the nature of dark matter naturally lead to the identification of the mathematical structure of the problem, suggesting characteristic parameters and rescalings needed to express the system in dimensionless form. Through numerical calculations, the structure of spherically symmetric stationary states of the system is systematically analyzed, and a set of heuristic laws is identified that describe how their fundamental features depend on the excitation index. The same analysis is extended to the rotation curves predicted from the stationary states. The relevance of these heuristic laws for model applications is demonstrated through the construction of fits to experimental rotation curves. In particular, it is shown how the presence of these laws facilitates the fitting process by significantly reducing the required computational effort. A possible relativistic extension of the model, the Klein-Gordon–Wave system, is then addressed. The typical regimes of this system are described, and it is shown how in the low-energy limit and weak field limit it reduces to the previously described Schrödinger-Poisson system. The connection between the two models is formalized through perturbative techniques typical of Hamiltonian systems, showing how the Schrödinger-Poisson system emerges from the first-order truncation of the Hamiltonian normal form of the Klein-Gordon–Wave problem. Finally, the thesis concludes with some observations on the stability of the Schrödinger-Poisson and Klein-Gordon–Wave systems. The challenges arising from nonlinearity are highlighted, and the linearized version of the perturbed Klein-Gordon–Wave system is examined both analytically and numerically, showing that its stationary states exhibit spectral instability.