Abstract
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
ϵ
H
1
(t)
. Here
ϵ→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)+ϵ
H
1
(t)
does not necessarily satisfy the gap assumption,
∀ϵ>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.