Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators

It is proved that for general, not necessarily periodic, quasi one-dimensional systems the band position operator corresponding to an isolated part of the energy spectrum has discrete spectrum and its eigenfunctions have the same spatial localization as the corresponding spectral projection. As a consequence, an eigenbasis of the band position operator provides a basis of optimally localized (generalized) Wannier functions for quasi one-dimensional systems, and this proves the strong Marzari-Vanderbilt conjecture. If the system has some translation symmetries (e. g. usual translations, screw transformations), they are 'inherited' by the Wannier basis.