  We investigate the effect of strong interactions on the spectral properties
of quantum wires with strong Rashba spin-orbit interaction in a magnetic field,
using a combination of Matrix Product State and bosonization techniques.
Quantum wires with strong Rashba spin-orbit interaction and magnetic field
exhibit a partial gap in one-half of the conducting modes. Such systems have
attracted wide-spread experimental and theoretical attention due to their
unusual physical properties, among which are spin-dependent transport, or a
topological superconducting phase when under the proximity effect of an s-wave
superconductor. As a microscopic model for the quantum wire we study an
extended Hubbard model with spin-orbit interaction and Zeeman field. We obtain
spin resolved spectral densities from the real-time evolution of excitations,
and calculate the phase diagram. We find that interactions increase the pseudo
gap at $k = 0$ and thus also enhance the Majorana-supporting phase and
stabilize the helical spin order. Furthermore, we calculate the optical
conductivity and compare it with the low energy spiral Luttinger Liquid result,
obtained from field theoretical calculations. With interactions, the optical
conductivity is dominated by an excotic excitation of a bound
soliton-antisoliton pair known as a breather state. We visualize the
oscillating motion of the breather state, which could provide the route to
their experimental detection in e.g. cold atom experiments.
