Neutrino Oscillations in Low Density Medium

We have solved the evolution equation for neutrinos in a low density medium, $V\ll \Delta m^2/2E$, where $V$ is the matter potential, developing the perturbation theory in $ϵ\equiv 2VE/\Delta m^2$. Simple and physically transparent formulas for the oscillation probabilities have been derived in the lowest order in $ϵ$ which are valid for an arbitrary density profile. The formulas can be applied for propagation of the solar and supernova neutrinos in matter of the Earth, substantially simplifying numerical calculations. Using these formulas we study sensitivity of the oscillations to structures of the density profile situated at different distances $d$ from a detector. For the mass-to-flavor transitions, e.g., ${\nu }_2\to {\nu }_e$, we have found the attenuation effect: a decrease of the sensitivity to remote structures, $d>l_{\nu }E/\Delta E$, where $l_{\nu }$ is the oscillation length and $\Delta E/E$ is the energy resolution of a detector.