  Using exact formulae for the scattering data of the Benjamin-Ono equation
valid for general rational potentials recently obtained by Miller and Wetzel
(2015), we rigorously analyze the scattering data in the small-dispersion
limit. In particular, we deduce precise asymptotic formulae for the reflection
coefficient, the location of the eigenvalues and their density, and the
asymptotic dependence of the phase constant (associated with each eigenvalue)
on the eigenvalue itself. Our results give direct confirmation of conjectures
in the literature that have been partly justified by means of inverse
scattering, and they also provide new details not previously reported in the
literature.
