  Using the Wentzel-Kramers-Brillouin (WKB) approximation we study the
formation and propagation of quantum bound states in the vicinity of a Galileon
black hole. We show that for given ranges of black hole horizon radii and of
the derivative coupling which appears in the metric of the Galileon black hole,
a Regge-Wheeler potential containing a local well is formed. Varying the
strength of the derivative coupling we investigate the behaviour of the bound
states trapped in the potential well or penetrating the horizon of the Galileon
black hole.
