  The goal of this presentation is to highlight various computational
techniques used to study dynamics of quantum many-body systems. We examine the
projection and variable phase methods being applied to multi-channel problems
of scattering and tunneling; here the virtual, energy-forbidden channels and
their treatment are of particular importance. The direct time-dependent
solutions using Trotter-Suzuki propagator expansion provide yet another
approach to exploring the complex dynamics of unstable systems. While
presenting computational tools, we briefly revisit the general theory of the
quantum decay of unstable states. The list of questions here includes those of
the internal dynamics in decaying systems, formation and evolution of the
radiating state, and low-energy background that dominates at remote times.
Mathematical formulations and numerical approaches to time-dependent problems
are discussed using the quasi-stationary methods involving effective
Non-Hermitian Hamiltonian formulation.
