Probing the LSND scale and four neutrino scenarios with a neutrino telescope

We show in this paper that the observation of the angular distribution of upward-going muons and cascade events induced by atmospheric neutrinos at the TeV energy scale, which can be performed by a kilometer-scale neutrino telescope, such as the IceCube detector, can be used to probe a large neutrino mass splitting, $| \Delta m^2 | \sim (0.5-2.0)$ eV$^2$, implied by the LSND experiment and discriminate among four neutrino mass schemes. This is due to the fact that such a large mass scale can promote non negligible muon neutrino to electron/tau neutrino (and/or anti-muon neutrino to anti-electron/anti-tau neutrino) conversions at these energies by the MSW effect as well as vacuum oscillation, unlike what is expected if all the neutrino mass splittings are small.