Fuzzy dark matter soliton cores around supermassive black holes

We explore the effect of a supermassive black hole (SMBII) on the density profile of a fuzzy dark matter (FDM) soliton core at the centre of a dark matter (DM) halo. We numerically solve the SchrOdinger Poisson equations, treating the black hole as a gravitational point mass, and demonstrate that this additional perturbing term has a 'squeezing' effect on the soliton density profile, decreasing the core radius, and increasing the central density. In the limit of large black hole mass, the solution approaches one akin to the hydrogen atom, with radius inversely proportional to the black hole mass. By applying our analysis to two specific galaxies (M87 and the Milky Way) and pairing it with known observational limits on the amount of centrally concentrated DM, we obtain a constraint on the FDM particle mass, finding that the range 10 22.12 eV en 10 22.06 eV should be forbidden (taking into account additional factors concerning the lifetime of the soli ton in the vicinity of a black hole). Improved observational mass measurements of the black hole and total enclosed masses will significantly extend the lower bound on the excluded 1DM mass region, while self-consistent theoretical modelling of the soliton black hole system can extend the upper bound.