Neutrino oscillations in low density medium

For the case of small matter effects: $V \ll \Delta m^2/2E$, where $V$ is the matter potential, we develop the perturbation theory using $\epsilon \equiv 2VE/\Delta m^2$ as the expansion parameter. We derive simple and physically transparent formulas for the oscillation probabilities in the lowest order in $\epsilon$ which are valid for an arbitrary density profile. They can be applied for the solar and supernova neutrinos propagating in the matter of the Earth. Using these formulas we study features of averaging of the oscillation effects over the neutrino energy. Sensitivity of these effects to remote (from a detector) $d > l_{\nu} E/\Delta E$ structures of the density profile is suppressed.