  Recent developments in the search for topological superconductivity have
brought lattices of magnetic adatoms on a superconductor into intense focus. In
this work we will study ferromagnetic chains of adatoms on superconducting
surfaces with Rashba spin-orbit coupling. Generalising the deep-impurity
approach employed extensively in previous works to arbitrary subgap energies,
we formulate the theory of the subgap spectrum as a nonlinear matrix eigenvalue
problem. We obtain an essentially analytical description of the subgap
spectrum, allowing an efficient study of the topological properties. Employing
a flat-band Hamiltonian sharing the topological properties of the chain, we
evaluate the $\mathbb{Z}$-valued winding number and discover five distinct
topological phases. Our results also confirm that the topological band
formation does not require the decoupled Shiba energies to be fine-tuned to the
gap centre. We also study the properties of Majorana bound states in the
system.
