Localization problem and mapping of one-dimensional wave equations in random and quasiperiodic media

The one-dimensional Schrödinger equation with multiple scattering potentials is transformed to a discrete (tight-binding) form exactly. For a random configuration of potentials in which all the states are localized, it is shown (not argued) that the resistance ρ behaves as ρ∼exp(γl) at a large distance l, where γ is the Lyapunov exponent (inverse of the localization length) of corresponding transfer matrices. In a case where two (or more) types of scatterers are arranged in a quasiperiodic manner (for example, the Fibonacci series), it is shown that wave functions are always critical, namely they are either self-similar or chaotic, and are intermediate between localized and extended states.