Lattice operators in crystals for Bravais and reciprocal vectors

The $\mathrm{kq}$ representation is used for defining lattice operators whose eigenvalues are all the discrete vectors of the direct and the reciprocal lattices in crystals. The eigenstates of the lattice operators form a complete and orthonormal set of localized functions in both the configuration and the momentum spaces. It is shown that these eigenstates can be chosen to be closely connected to either the free electron or the extremely tightly bound electron Wannier functions. The lattice operators turn out to be conjugate to the $k$ and $q$ coordinates.