Parametric resonance of neutrino oscillations and passage of solar and atmospheric neutrinos through the earth

We present an exact analytic solution to the neutrino evolution equation in matter with periodic step-function density profile and discuss in detail the parametric resonance of neutrino oscillations that can occur in such a system. Solar and atmospheric neutrinos traversing the earth pass through layers of alternating density and can therefore experience parametric resonance of their oscillations. Atmospheric neutrinos can undergo parametrically enhanced oscillations in the earth when their trajectories deviate from the vertical by about $26^\circ - 32^\circ$. Solar neutrinos traversing the earth can experience a strong parametric resonance of their oscillations in a wide range of zenith angles. If the small mixing angle MSW effect is the solution of the solar neutrino problem, the oscillations of solar neutrinos crossing the core of the earth {\em must} undergo strong parametric resonance; this phenomenon should facilitate significantly the observation of the day-night effect in oscillations of solar neutrinos. If observed, the enhanced day-night effect for core crossing neutrinos would therefore confirm both the MSW solution of the solar neutrino problem and the parametric resonance of neutrino oscillations.