  We study the properties of a quantum particle interacting with a one
dimensional structure of equidistant scattering centres. We derive an
analytical expression for the dispersion relation and for the Bloch functions
in the presence of both even and odd scattering waves within the
pseudopotential approximation. This generalises the well-known solid-state
physics text-book result known as the Kronig-Penney model. Our generalised
model can be used to describe systems such as degenerate Fermi gases
interacting with ions or with another neutral atomic species confined in an
optical lattice, thus enabling the investigation of polaron or Kondo physics
within a simple formalism. We focus our attention on the specific atom-ion
system and compare our findings with quantum defect theory. Excellent agreement
is obtained within the regime of validity of the pseudopotential approximation.
This enables us to derive a Bose-Hubbard Hamiltonian for a degenerate quantum
Bose gas in a linear chain of ions.
