Engineering spectral properties of non-interacting lattice Hamiltonians

We investigate the spectral properties of one-dimensional lattices with position-dependent hopping amplitudes and on-site potentials that are smooth bounded functions position. find an exact integral form for density states (DOS) in limit infinite number sites, which we derive using a mixed Bloch-Wannier basis consisting piecewise Wannier functions. Next, provide solution inverse problem constructing position-dependence lattice model yielding given DOS. confirm analytic results by comparing them to numerics obtained diagonalization various incarnations hoppings potentials. Finally, generalize DOS multi-orbital tight-binding models longer-range higher dimensions.