Projection operator approach to unfolding supercell band structures

While the methodology of band structure unfolding has appeared in several publications, the original derivations of the unfolding formulas can be considerably simplified by using the $k$-projection method. In this work, more transparent derivations of unfolded spectral weights are given by using the projection operator approach. A range of illustrative examples are also presented which include finite and random one-dimensional chains, and Kekul\'e-textured honeycomb lattice.