Eur. Phys. J. B (2014) 87: 208
https://doi.org/10.1140/epjb/e2014-50210-6

Regular Article

Analyticity of quantum states in one-dimensional tight-binding model

¹ Yamada Physics Research Laboratory, Aoyama 5-7-14-205, Niigata 950-2002, Japan
² Department of Physics, Ritsumeikan University Noji-higashi 1-1-1, Kusatsu 525, Japan

^a e-mail: hyamada@uranus.dti.ne.jp

Received: 27 March 2014
Received in final form: 25 June 2014
Published online: 16 September 2014

Abstract

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 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 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.