  The adiabatic approximation in quantum mechanics is considered in the case
where the self-adjoint hamiltonian $H_0(t)$, satisfying the usual spectral gap
assumption in this context, is perturbed by a term of the form $\epsilon
H_1(t)$. Here $\epsilon \to 0$ is the adiabaticity parameter and $H_1(t)$ is a
self-adjoint operator defined on a smaller domain than the domain of $H_0(t)$.
Thus the total hamiltonian $H_0(t)+\epsilon H_1(t)$ does not necessarily
satisfy the gap assumption, $\forall \epsilon >0$. It is shown that an
adiabatic theorem can be proven in this situation under reasonnable hypotheses.
The problem considered can also be viewed as the study of a time-dependent
system coupled to a time-dependent perturbation, in the limit of large coupling
constant.
