  The Constrained Adiabatic Trajectory Method (CATM) is reexamined as an
integrator for the Schr\"odinger equation. An initial discussion places the
CATM in the context of the different integrators used in the literature for
time-independent or explicitly time-dependent Hamiltonians. The emphasis is put
on adiabatic processes and within this adiabatic framework the interdependence
between the CATM, the wave operator, the Floquet and the (t,t') theories is
presented in detail. Two points are then more particularly analysed and
illustrated by a numerical calculation describing the $H_2^+$ ion submitted to
a laser pulse. The first point is the ability of the CATM to dilate the
Hamiltonian spectrum and thus to make the perturbative treatment of the
equations defining the wave function possible, possibly by using a Krylov
subspace approach as a complement. The second point is the ability of the CATM
to handle extremely complex time-dependencies, such as those which appear when
interaction representations are used to integrate the system.
