A precise analytical description of the Earth matter effect on   oscillations of low energy neutrinos
  We present a formalism for the matter effects in the Earth on low energy neutrino fluxes which is both accurate and has all advantages of a full analytic treatment. The oscillation probabilities are calculated up to second order term in $\epsilon(x) \equiv 2V(x)E/\Delta m^2$ where $V(x)$ is the neutrino potential at position $x$. We show the absence of large undamped phases which makes the expansion in $\epsilon$ well behaved. An improved expansion is presented in terms of the variation of $V(x)$ around a suitable mean value which allows to treat energies up to those relevant for Supernova neutrinos. We discuss also the case of three-neutrino mixing. 
A. N. Ioannisian, N. A. Kazarian, A. Yu. Smirnov, D. Wyler