  Based on our recently proposed plane wave framework, we theoretically study
the localized-extended transition in the one dimensional incommensurate systems
with cosine type of potentials, which are in close connection to many recent
experiments in the ultracold atom and photonic crystal. We formulate a
propagator based scattering picture for the transition at the ground state and
single particle mobility edge, in which the deeper connection between the
incommensurate potentials, eigenstate compositions and transition mechanism is
revealed. We further show that there exists a upper limit of localization
length for all localized eigenstates, leading to an fundamental difference to
the Anderson localization. Numerical calculations are presented alongside the
analysis to justify our statements. The theoretical analysis and numerical
methods can also be generalized to systems in higher dimensions, with different
potentials or beyond the single particle regime, which would benefit the future
studies in the related fields.
