  Chiral solitons with baryon number one are investigated in the Spectral Quark
Model. In this model the quark propagator is a superposition of complex-mass
propagators weighted with a suitable spectral function. This technique is a
method of regularizing the effective quark theory in a way preserving many
desired features crucial in analysis of solitons. We review the model in the
vacuum sector, stressing the feature of the absence of poles in the quark
propagator. We also investigate in detail the analytic structure of meson
two-point functions. We provide an appropriate prescription for constructing
valence states in the spectral approach. The valence state in the baryonic
soliton is identified with a saddle point of the Dirac eigenvalue treated as a
function of the spectral mass. Due to this feature the valence quarks never
become unbound nor dive into the negative spectrum, hence providing stable
solitons as absolute minima of the action. This is a manifestation of the
absence of poles in the quark propagator. Self-consistent mean field hedgehog
solutions are found numerically and some of their properties are determined and
compared to previous chiral soliton models. Our analysis constitutes an
involved example of a treatment of a relativistic complex-mass system.
