  Within the "pseudo-Dirac" scenario for massive neutrinos the existence of
sterile neutrinos which are almost degenerate in mass with the active ones is
hypothesized. The presence of these sterile neutrinos can affect the flavor
composition of cosmic neutrinos arriving at Earth after traveling large
distances from astrophysical objects. We examine the prospects of neutrino
telescopes such as IceCube to probe the very tiny mass squared differences
10^(-12) eV^2<\Delta m^2<10^(-19) eV^2, by analyzing the ratio of $\mu$-track
events to shower-like events. Considering various sources of uncertainties
which enter this analysis, we examine the capability of neutrino telescopes to
verify the validity of the pseudo-Dirac neutrino scenario and especially to
discriminate it from the conventional scenario with no sterile neutrino. We
also discuss the robustness of our results with respect to the uncertainties in
the initial flavor ratio of neutrinos at the source.
