On the statistics of superlocalized states in self-affine disordered potentials

We investigate the statistics of eigenstates in a weak self-affine disordered potential in one dimension, whose Gaussian fluctuations grow with distance with a positive Hurst exponent $H$. Typical eigenstates are superlocalized on samples much larger than a well-defined crossover length, which diverges in the weak-disorder regime. We present a parallel analytical investigation of the statistics of these superlocalized states in the discrete and the continuum formalisms. For the discrete tight-binding model, the effective localization length decays logarithmically with the sample size, and the logarithm of the transmission is marginally self-averaging. For the continuum Schr\"odinger equation, the superlocalization phenomenon has more drastic effects. The effective localization length decays as a power of the sample length, and the logarithm of the transmission is fully non-self-averaging.