  We study a quantum walker on a one-dimensional lattice with a single defect
site characterized by a phase. The spread and localization of discrete-time
quantum walks starting at the impurity site are affected by the appearance of
bound states and their reflection symmetry. We quantify the localization in
terms of an effective localization length averaged over all eigenstates and an
effective participation ratio after time evolution averaged over all initial
states. We observe that the reduced coin system dynamics undergoes oscillations
in the long-time limit, the frequencies of which are related to the unitary
sublattice operator and the bound state quasi-energy differences. The
oscillations give rise to non-Markovian evolution, which we quantify using the
trace distance and entanglement based measures of non-Markovianity. Indeed, we
reveal that the degree of the non-Markovian behavior is closely related to the
emergence of bound states due to the phase impurity. We also show that the
considered measures give qualitatively different results depending on the
number and symmetries of supported bound states. Finally, comparing
localization and non-Markovianity measures, we demonstrate that the degree of
non-Markovianity becomes maximum when the walker is most localized in position
space.
