Adiabatic approximation, Gell-Mann and Low theorem, and degeneracies: A pedagogical example

We study a simple system described by a 2 by 2 Hamiltonian and the evolution of its 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 the evolution of eigenstates to be followed. 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.