During the last few decades, laser technology has gone through a series of revolutionary improvements. With light pulses in the sub-femtosecond regime now experimentally available, an explicit time-dependent Hamilton operator is indispensable for a proper theoretical description of the interactions of atoms with such pulses. However, these theoretical studies are computationally very demanding, which motivates the search for new numerical methods and algorithms to approach time-dependent problems. This thesis contributes to this research field, with the main focus on the use of complex-scaled Hamilton operators. Thus, the formalism of complex scaling is studied in the context of its application to explicitly time-dependent atomic systems.

Both non-relativistic and relativistic dynamics are investigated. The numerical advantages and the possibilities to extract physical quantities from complex-scaled wave functions are discussed. Of special interest is the ability to treat resonance states. These are multiply excited electronic states with sufficient energy to decay through Auger transitions to the surrounding continuum. With complex scaling, the Hamilton operator is non-Hermitian and such resonance states can be obtained as eigenstates.

To analyze the non-bound part of the wave function requires essentially the construction of a second wave function; the left state vector. This additional wave function is, however, not easily constructed numerically in the complex scaling-method. To circumvent some of the numerical problems, we have proposed three different methods. These methods are based on Floquet theory, a propagation on a complex time-grid and time-dependent perturbation theory. By carefully investigating the numerical properties of the left state vector, we have thus studied the non-bound part of the system.