  The ground state of a free-fermionic chain with inhomogeneous hoppings at
half-filling can be mapped into the Dirac vacuum on a static curved space-time,
which presents exactly homogeneous occupations due to particle-hole symmetry.
Yet, far from half-filling we observe density modulations and depletion
effects. The system can be described by a 1D Schr\"odinger equation on a
different static space-time, with an effective potential which accounts for the
depleted regions. We provide a semiclassical expression for the single-particle
modes and the density profiles associated to different hopping patterns and
filling fractions. Moreover, we show that the depletion effects can be
compensated for all filling fractions by adding a chemical potential
proportional to the hoppings. Interestingly, we can obtain exactly the same
density profiles on a homogeneous chain if we introduce a chemical potential
which is inverse to the hopping intensities, even though the ground state is
different from the original one.
