  We study the radial Schr\"odinger equation for a particle of mass $m$ in the
field of a singular attractive $g^2/{r^4}$ potential with particular emphasis
on the bound states problem. Using the regularization method of Beane
\textit{et al.}, we solve analytically the corresponding ``renormalization
group flow" equation. We find in agreement with previous studies that its
solution exhibits a limit cycle behavior and has infinitely many branches. We
show that a continuous choice for the solution corresponds to a given fixed
number of bound states and to low energy phase shifts that vary continuously
with energy. We study in detail the connection between this regularization
method and a conventional method modifying the short range part of the
potential with an infinitely repulsive hard core. We show that both methods
yield bound states results in close agreement even though the regularization
method of Beane \textit{et al.} does not include explicitly any new scale in
the problem. We further illustrate the use of the regularization method in the
computation of electron bound states in the field of neutral polarizable
molecules without dipole moment. We find the binding energy of s-wave
polarization bound electrons in the field of C$_{60}$ molecules to be 17 meV
for a scattering length corresponding to a hard core radius of the size of the
molecule radius ($\sim 3.37$ \AA). This result can be further compared with
recent two-parameter fits using the Lennard-Jones potential yielding binding
energies ranging from 3 to 25 meV.
