  Recent experiments announced an observation of topological superconductivity
and Majorana quasiparticles in Shiba chains, consisting of an array of magnetic
atoms deposited on top of a superconductor. In this work we study helical Shiba
chains and generalize the microscopic theory of subgap energy bands to a regime
where the decoupled magnetic impurity energy and the hybridization of different
impurity states can be significant compared to the superconducting gap of the
host material. From exact solutions of the Bogoliubov-de Gennes equation we
extract expressions for the topological phase boundaries for arbitrary values
of the superconducting coherence length. The subgap spectral problem can be
formulated as a nonlinear matrix eigenvalue problem from which we obtain an
analytical solution for energy bands in the long coherence length limit.
Physical consequences and departures from the previously obtained results in
the deep-dilute impurity limit are discussed in detail.
