Atmospheric neutrinos with three flavor mixing

We analyze the atmospheric neutrino data in the context of three flavor neutrino oscillations taking account of the matter effects in the earth. With the hierarchy among the vacuum mass eigenvalues $\mu_3^2 \gg \mu_2^2 \geq \mu_1^2$, the solution of the atmospheric neutrino problem depends on $\delta_{31}=\mu_3^2 - \mu_1^2$ and the $13$ and $23$ mixing angles $\phi$ and $\psi$. Whereas the sub-GeV atmospheric neutrino data imposes only a lower limit on $\delta_{31} > 10^{-3} eV^2$, the zenith angle dependent suppression observed in the multi-GeV data limits $\delta_{31}$ from above also. The allowed regions of the parameter space are strongly constrained by the multi-GeV data. Combined with our earlier solution to the solar neutrino problem which depends on $\delta_{21}= \mu_2^2-\mu_1^2$ and the $12$ and $13$ mixing angles $\omega$ and $\phi$, we have obtained the ranges of values of the five neutrino parameters which solve both the solar and the atmospheric neutrino problems simultaneously.