  We study a simple system described by a 2x2 Hamiltonian and the evolution of
the quantum states under the influence of a perturbation. More precisely, when
the initial Hamiltonian is not degenerate,we check analytically the validity of
the adiabatic approximation and verify that, even if the evolution operator has
no limit for adiabatic switchings, the Gell-Mann and Low formula allows to
follow the evolution of eigenstates. In the degenerate case, for generic
initial eigenstates, the adiabatic approximation (obtained by two different
limiting procedures) is either useless or wrong, and the Gell-Mann and Low
formula does not hold. We show how to select initial states in order to avoid
such failures.
