  This review explores the dynamics and the low-energy excitation spectra of
Bose-Einstein condensates (BECs) of interacting bosons in external potential
traps putting particular emphasis on the emerging many-body effects beyond
mean-field descriptions. To do so, methods have to be used that, in principle,
can provide numerically exact results for both the dynamics and the excitation
spectra in a systematic manner. Numerically exact results for the dynamics are
presented employing the well-established multicongurational time-dependent
Hartree for bosons (MCTDHB) method. The respective excitation spectra are
calculated utilizing the more recently introduced linear-response theory atop
it (LR-MCTDHB). The latter theory gives rise to an, in general, non-hermitian
eigenvalue problem. The theory and its newly developed implementation are
described in detail and benchmarked towards the exactly-solvable
harmonic-interaction model. Several applications to BECs in one- and
two-dimensional potential traps are discussed. With respect to dynamics, it is
shown that both the out-of-equilibrium tunneling dynamics and the dynamics of
trapped vortices are of many-body nature. Furthermore, many-body effects in the
excitation spectra are presented for BECs in different trap geometries. It is
demonstrated that even for essentially-condensed systems, the spectrum of the
lowest-in-energy excitations computed at the many-body level can differ
substantially from the standard mean-field description. In general, it is shown
that bosons carrying angular momentum are more sensitive to many-body effects
than bosons without. These effects are present in both the dynamics and the
excitation spectrum.
