Wave transmission and its universal fluctuations in one-dimensional systems with L\'evy-like disorder: Schr\"odinger, Klein-Gordon and Dirac equations

Abstract

We investigate the propagation of waves in one-dimensional systems with L\'evy-type disorder. We perform a complete analysis of non-relativistic and relativistic wave transmission submitted to potential barriers whose width, separation or both follow L\'evy distributions characterized by an exponent $0 < \alpha <1$. For the first two cases, where one of the parameters is fixed, non-relativistic and relativistic waves present anomalous localization, $\langle T \rangle \propto L^{-\alpha}$. However, for the latter case, in which both parameters follow a L\'evy distribution, non-relativistic and relativistic waves present a transition between anomalous and standard localization as the incidence energy increases relative to the barrier height. Moreover, we obtain the localization diagram delimiting anomalous and standard localization regimes, in terms of incidence angle and energy. Finally, we verify that transmission fluctuations, characterized by its standard deviation, are universal, independent of barrier architecture, wave equation type, incidence energy and angle, further extending earlier studies on electronic localization.