  We introduce and develop a novel approach to extend the ordinary two-flavor
neutrino oscillation formalism in matter using a non-Hermitian PT symmetric
effective Hamiltonian. The condition of PT symmetry is weaker and less
mathematical than that of hermicity, but more physical, and such an extension
of the formalism can give rise to sub-leading effects in neutrino flavor
transitions similar to the effects by so-called non-standard neutrino
interactions. We derive the necessary conditions for the spectrum of the
effective Hamiltonian to be real as well as the mappings between the
fundamental and effective parameters. We find that the real spectrum of the
effective Hamiltonian will depend on all new fundamental parameters introduced
in the non-Hermitian PT symmetric extension of the usual neutrino oscillation
formalism and that either i) the spectrum is exact and the effective leptonic
mixing must always be maximal or ii) the spectrum is approximate and all new
fundamental parameters must be small.
