  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, thus proving the "strong conjecture" of Marzari and
Vanderbilt. If the system has some translation symmetries (e.g. usual
translations, screw transformations), they are "inherited" by the Wannier
basis.
