  An asymptotic approach for a Schroedinger type equation with a non
selfadjoint slowly varying Hamiltonian of a special type is developed. The
Hamiltonian is assumed to be the result of a small perturbation of an operator
with a twofold degeneracy (turning) point, which can be diagonalized at this
point. The non-adiabatic transformation of modes is studied in the case where
two small parameters are dependent: the parameter characterizing an order of
the perturbation is a square root of the adiabatic parameter. The perturbation
of the Hamiltonian produces a close pair of simple degeneracy points. Two
regimes of mode transformation for the Schroedinder type equation are
identified: avoided crossing of eigenvalues, corresponding to complex
degeneracy points, and an explicit unavoidable crossing (with real degeneracy
points).
  Both cases are treated by a method of matched asymptotic expansions in the
context of a unifying approach. An asymptotic expansion of the solution near a
crossing point containing the parabolic cylinder functions is constructed, and
the transition matrix connecting the coefficients of adiabatic modes to the
left and to the right of the degeneracy point is derived.
  Results are illustrated by an example: fermion scattering governed by the
Dirac equation.
