The book [14] and, in particular, its chapter 10 may be consulted to
settle the scene: The (unknown) Hamiltonian which describes an open
- scattering-admitting - abstract quantum system is replaced by the
``equivalent'' Feshbach's effective-Hamiltonian $H$ (i.e., a
projection on a subspace, assumed known and, naturally, manifestly
non-Hermitian and admitting complex eigenvalues). In the second step
one admits that $H=H(\vec{\lambda})$ varies with its parameters and,
``slowly'', with time, $\vec{\lambda}=\vec{\lambda}(t)$. It is
popular to make calculations using instantaneous, ``adiabatic''
eigenbases. In the paper, the authors consider the specific scenario
(called ``state-flip'') in which the parameters $\vec{\lambda}(t)$
encircle a branch point of the hypersurface of the eigenstates (so
that a ket-vector in consideration does not return to its initial
value). In this setting, the main result of the paper is that the
interplay between the non-Hermiticity and non-adiabatic factors may
mar the observability of the flip effect. A useful recommended
complementary reading is paper [19] which may help the readers to
understand the idea better since it describes the ``closed loop of
$\vec{\lambda}(t)$'' flipping paradox in a model which is solved,
exactly, in terms of Bessel functions.


MR2844994 Uzdin, Raam; Mailybaev, Alexei; Moiseyev, Nimrod On the
observability and asymmetry of adiabatic state flips generated by
exceptional points. J. Phys. A 44 (2011), no. 43, 435302, 8 pp.
81S22