An Adiabatic Theorem for Singularly Perturbed Hamiltonians

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 $\epsilon H_1(t)$\mathrm{.}Here$\epsilon \to 0$istheadiabaticityparameterand$H_1(t)$ is a self-adjoint operator defined on a smaller domain than the domain of $H_0(t)$. Thus the total hamiltonian $H_0(t)+ϵH_1(t)$ does not necessarily satisfy the gap assumption, $\mathrm{\forall }ϵ>0$. 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.