  Whereas cold dark matter (CDM) simulations predict central dark matter cusps
with densities that diverge as $\rho$(r)$\sim$ 1/r observations often indicate
constant density cores with finite central densities $\rho_0$ and a flat
density distribution within a core radius r$_0$. This paper investigates
whether this core-cusp problem can be solved by fuzzy dark matter (FDM), a
hypothetical particle with a mass of order m$\approx$10$^{-22}$eV and a
corresponding de Broglie wavelength on astrophysical scales. We show that
galaxies with CDM halo virial masses M$_{vir} \leq 10^{11}$M$_{\odot}$ follow
two core scaling relations. In addition to the well known universal core column
density $\Sigma_0 \equiv \rho_0 \times$r$_0$ = 75 M$_{\odot}$pc$^{-2}$ core
radii increase with virial masses as r$_0 \sim$ M$_{vir}^{\gamma}$ with
$\gamma$ of order unity. Using the simulations by Schive et al. (2014) we
demonstrate that FDM can explain the r$_0$-M$_{vir}$ scaling relation if the
virial masses of the observed galaxy sample scale with formation redshift z as
M$_{vir}\sim$(1+z)$^{-0.4}$. The observed constant $\Sigma_0$ is however in
complete disagreement with FDM cores which are characterised by a steep
dependence $\Sigma_0 \sim$r$_0^{-3}$, independent of z. More high-resolution
simulations are now required to confirm the Schive et al. simulations and
explore especially the transition region between the soliton core and the
surrounding halo. If these results hold, FDM can be ruled out as the origin of
observed dark matter cores and other physical processes are required to account
for their formation.
