An adiabatic theorem for singularly perturbed hamiltonians
Résumé
The adiabatic approximation in quantum mechanics is considered in the case where the selfadjoint Hamiltonian H 0 (t), satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form εH 1 (t). Here ε→0 is the adiabaticity parameter and H 1 (t) is a selfadjoint operator defined on a smaller domain than the domain of H 0 (t). Thus the total Hamiltonian H 0 (t)+εH 1 (t) does not necessarily satisfy the gap assumption, ∀ε>0.
It is shown that an adiabatic theorem can be proved in this situation under reasonable 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.