  Analytical complexity of quantum wavefunction whose argument is extended into
the complex plane provides an important information about the potentiality of
manifesting complex quantum dynamics such as time-irreversibility, dissipation
and so on. We examine Pade approximation and some complementary methods to
investigate the complex-analytical properties of some complex quantum states
such as impurity states, Anderson-localized states and localized states of
Harper model. The impurity states can be characterized by simple poles of the
Pade approximation, and the localized states of Anderson model and Harper model
can be characterized by an accumulation of poles and zeros of the Pade
approximated function along a critical border, which implies a natural boundary
(NB). A complementary method based on shifting the expansion-center is used to
confirm the existence of the NB numerically, and it is strongly suggested that
the both Anderson-localized state and localized states of Harper model have NBs
in the complex extension. Moreover, we discuss an interesting relationship
between our research and the natural boundary problem of the potential function
whose close connection to the localization problem was discovered quite
recently by some mathematicians. In addition, we also examine the usefulness of
the Pade approximation for numerically predicting the existence of NB by means
of two typical examples, lacunary power series and random power series.
