On the Oscillation Length Resonance in the Transitions of Solar and   Atmospheric Neutrinos Crossing the Earth Core
  Assuming two-neutrino mixing takes place in vacuum, we study in detail the conditions under which the $\nu_2 \to \nu_{e}$ and $\nu_{\mu} \to \nu_{e}$ ($\nu_e \to \nu_{\mu(\tau)}$) transitions in the Earth are strongly enhanced by the neutrino oscillation length resonance when the neutrinos cross the Earth core. We show, in particular, that the neutrino oscillation length resonance is operative also in the $\bar{\nu}_{\mu} \to \bar{\nu}_{s}$ transitions at small mixing angles. The properties of the $\nu_2 \to \nu_{e}$ and $\nu_{\mu} \to \nu_{e}$ ($\nu_e \to \nu_{\mu(\tau)}$) transition probabilities in the corresponding resonance regions are examined. Some implications of our results for the transitions of solar and atmospheric neutrinos traversing the Earth core, relevant for the interpretation of the results of the solar and atmospheric neutrino experiments, are also discussed. 
M. Chizhov, M. Maris and S. T. Petcov