Fuzzy Dark Matter Soliton Cores around Supermassive Black Holes

We explore the effect of a supermassive black hole (SMBH) on the density profile of a fuzzy dark matter (FDM) soliton core at the centre of a dark matter halo. We numerically solve the Schr\"odinger-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 dark matter, we obtain a constraint on the FDM particle mass, finding that the range $10^{-22.12}$ eV $\lesssim m \lesssim 10^{-22.06}$ eV should be forbidden (taking into account additional factors concerning the life-time of the soliton 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 FDM mass region, while self-consistent theoretical modeling of the soliton--black hole system can extend the upper-bound.