Óscar Eduardo Rodríguez Villalba — Università di Parma #
Schrödinger-Poisson Equation with Contact Interactions #
We investigate the role of contact interactions in the dynamics of fuzzy dark matter (FDM), modeled through the Schrödinger–Poisson equation. While the $\Lambda$CDM paradigm successfully explains structure formation on large scales, its predictions at small scales remain in tension with observations. FDM provides an alternative framework in which local self-interactions may further influence the formation and evolution of structures. Using numerical simulations, we examine the impact of these interactions in three key scenarios: the properties of the lowest-energy stationary state, the relaxation of localized initial states, and the gravitational collapse of nonlocalized states. Our results show that the standard deviation of the stationary states follows a power law as a function of the contact interaction strength. Within the (1+1)-dimensional model, we confirm that the relaxed state--assessed via the quantum virial theorem--does not converge to the lowest-energy stationary solution, even in the presence of local self-interactions. Furthermore, by employing a quasi-probability distribution, we identify the shell-crossing event—a key stage in gravitational collapse—and characterize how it is affected by contact interactions. Taken together, these findings indicate that local self-interactions play a significant role in shaping the nonlinear dynamics of FDM.