Describing Oscillations of High Energy Neutrinos in Matter Precisely

We present a formalism for precise description of oscillation phenomena in matter at high energies or high densities, $V>\Delta m^2/2E$, where $V$ is the matter-induced potential of neutrinos. The accuracy of the approximation is determined by the quantity ${sin}^{2}2{\theta }_m\Delta V/2\pi V$, where ${\theta }_m$ is the mixing angle in matter and $\Delta V$ is a typical change of the potential over the oscillation length ($l\sim 2\pi /V$). We derive simple and physically transparent formulas for the oscillation probabilities, which are valid for arbitrary matter density profiles. They can be applied to oscillations of high-energy ($E>10\mathrm{GeV}$) accelerator, atmospheric, and cosmic neutrinos in the matter of the Earth, substantially simplifying numerical calculations and providing an insight into the physics of neutrino oscillations in matter. The effect of parametric enhancement of the oscillations of high-energy neutrinos is considered.