  We analyze the spectrum of massive scalar bound states in the background of
extremal Kerr black holes, focusing on modes in the superradiant regime, which
grow exponentially in time and quickly deplete the black hole's mass and spin.
Previous analytical estimates for the growth rate of this instability were
limited to the $\mu M\ll1$ and $\mu M\gg1$ regimes, where $\mu$ and $M$ denote
the scalar field and black hole masses, respectively. In this work, we discuss
an analytical method to compute the superradiant spectrum for generic values of
these parameters, namely in the phenomenologically interesting regime $\mu
M\sim 1$. To do this, we solve the radial mode equation in two overlapping
regions and match the solutions in their common domain of validity. We show
that matching the functional forms of these functions involves approximations
that are not valid for the whole range of scalar masses, exhibiting unphysical
poles that produce a large enhancement of the growth rate. Alternatively, we
match the functions at a single point and show that, despite the uncertainty in
the choice of the match point, this method eliminates the spurious poles and
agrees with previous numerical computations of the spectrum using a
continued-fraction method.
