The adiabatic approximation is a natural approach for the description of
phenomena induced by low frequency laser radiation because the ratio of the
laser frequency to the characteristic frequency of an atom or a molecule is a
small parameter. Since the main aim of this work is the study of ionization
phenomena, the version of the adiabatic approximation that can account for the
transition from a bound state to the continuum must be employed. Despite much
work in this topic, a universally accepted adiabatic approach of bound-free
transitions is lacking. Hence, based on Savichev's modified adiabatic
approximation [Sov. Phys. JETP 73, 803 (1991)], we first of all derive the most
convenient form of the adiabatic approximation for the problems at hand.
Connections of the obtained result with the quasiclassical approximation and
other previous investigations are discussed. Then, such an adiabatic
approximation is applied to single-electron ionization and non-sequential
double ionization of atoms in a strong low frequency laser field.
Single-electron molecular ionization is also studied. Based on the geometrical
approach to tunnelling by P. D. Hislop and I. M. Sigal [Memoir. AMS 78, No. 399
(1989)], we introduce the concept of a leading tunnelling trajectory. It is
then proven that leading tunnelling trajectories for single active electron
models of molecular tunnelling ionization (i.e., theories where a molecular
potential is modelled by a single-electron multi-centre potential) are linear
in the case of short range interactions and "almost" linear in the case of long
range interactions. The results are presented on both the formal and physically
intuitive levels. Physical implications of the proven statements are discussed.
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