The KTY formalism and nonadiabatic contributions to the neutrino   oscillation probability
  It is shown how to obtain the analytical expression for the effective mixing angle in matter using the formalism which was developed by Kimura, Takamura and Yokomakura. If the baseline of the neutrino path is long enough so that averaging over rapid oscillations is a good approximation, then with the help of Landau's method, the nonadiabatic contribution to the oscillation probability can be expressed analytically by this formalism. We give two examples in which the present method becomes useful. 
Osamu Yasuda
hep-ph