  We investigate the topological classification of the subgap bands induced in
a two-dimensional superconductor by a densely packed chain of magnetic moments
with ferromagnetic or spiral alignments. The wave functions for these bands are
composites of Yu-Shiba-Rusinov-type states and magnetic scattering states and
have a significant spatial extension away from the magnetic moments. We show
that this spatial structure prohibits a straightforward extraction of a
Hamiltonian useful for the topological classification. To address the latter
correctly we construct a family of spatially varying topological Hamiltonians
for the subgap bands adapted for the broken translational symmetry caused by
the chain. The spatial dependence in particular captures the transition to the
topologically trivial bulk phase when moving away from the chain by showing how
this, necessarily discontinuous, transition can be understood from an alignment
of zeros with poles of Green's functions. Through the latter the topological
Hamiltonians reflect a characteristic found otherwise primarily in strongly
interacting systems.
