  Structure of eigenstates in a periodic quasi-1D waveguide with a rough
surface is studied both analytically and numerically. We have found a large
number of "regular" eigenstates for any high energy. They result in a very slow
convergence to the classical limit in which the eigenstates are expected to be
completely ergodic. As a consequence, localization properties of eigenstates
originated from unperturbed transverse channels with low indexes, are strongly
localized (delocalized) in the momentum (coordinate) representation. These
eigenstates were found to have a quite unexpeted form that manifests a kind of
"repulsion" from the rough surface. Our results indicate that standard
statistical approaches for ballistic localization in such waveguides seem to be
unappropriate.
