This dissertation presents a series of compact perturbative methods to calculate neutrino oscillations in matter with uniform density. In the method we implement multiple rotations to figure out zeroth order approximations. The rotations are able to resolve zeroth order degeneracy so the higher order corrections can converge for the complete matter potential versus baseline divided by neutrino energy plane. The rotations also diminish scales of perturbative terms of the Hamiltonian operator.

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... expansion parameter used in the method is not a constant. Its scale can be restricted to be smaller than ratio of the solar to the atmospheric $\Delta m^2$ globally. In vacuum the expansion parameter is strictly equal to zero, thus the zeroth order approximations return to the exact vacuum values. We use analytic derivations and numerical tests to prove the favorable properties. Moreover, the zeroth order eigenvalues of the Hamiltonian (differences of neutrino mass squares) can be inserted into a recently rediscovered identity which relates a Hermitian operator's eigenvalues to eigenvectors to give simple and symmetric expressions of the mixing angles and CP phase with extraordinary precision. Finally, the method can be extended from the standard three flavors neutrino scheme to a scheme with one more sterile neutrino, with its advantages inherited.