  Non-homogeneous chain environments (e.g. segmented ion traps) are
investigated through an exact diagonalization approach.
  Different spectral densities, including band-gaps, can be engineered to
separately assess memory effects. Environment non-Markovianity is quantified
with recently introduced measures of information flow-back and non-divisibility
of the system dynamical map. By sweeping the bath spectrum via tuning of the
system frequency we show strongest memory effect at band-gap edges and provide
an interpretation based on energy flow between system and environment. A system
weakly coupled to a stiff chain ensures a Markovian dynamics, while the size of
the environment as well as the local density of modes are not substantial
factors. We show an opposite effect when increasing the temperature inside or
outside the spectral band-gap. Further, non-Markovianity arises for larger
(negative and positive) powers of algebraic spectral densities, being the Ohmic
case not always the most Markovian one.
