  We investigate many body localization in the presence of a single particle
mobility edge. By considering an interacting deterministic model with an
incommensurate potential in one dimension we find that the single particle
mobility edge in the noninteracting system leads to a many body mobility edge
in the corresponding interacting system for certain parameter regimes. Using
exact diagonalization, we probe the mobility edge via energy resolved
entanglement entropy (EE) and study the energy resolved applicability (or
failure) of the eigenstate thermalization hypothesis (ETH). Our numerical
results indicate that the transition separating area and volume law scaling of
the EE does not coincide with the non-thermal to thermal transition.
Consequently, there exists an extended non-ergodic phase for an intermediate
energy window where the many body eigenstates violate the ETH while manifesting
volume law EE scaling. We also establish that the model possesses an infinite
temperature many body localization transition despite the existence of a single
particle mobility edge. We propose a practical scheme to test our predictions
in atomic optical lattice experiments which can directly probe the effects of
the mobility edge.
