Almost quantum adiabatic dynamics and generalized time dependent wave operators

We consider quantum dynamics for which the strict adiabatic approximation fails but which do not escape too far from the adiabatic limit. To treat these systems we introduce a generalisation of the time dependent wave operator theory which is usually used to treat dynamics which do not escape too far from an initial subspace called the active space. Our generalisation is based on a time dependent adiabatic deformation of the active space. The geometric phases associated with the almost adiabatic representation are also derived. We use this formalism to study the adiabaticity of a dynamics surrounding an exceptional point of a non-hermitian hamiltonian. We show that the generalized time dependent wave operator can be used to correct easily the adiabatic approximation which is very unperfect in this situation.