  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.
