Sample-size dependence of the ground-state energy in a one-dimensional localization problem

We study the sample-size dependence of the ground-state energy in a one-dimensional localization problem, based on a supersymmetric quantum mechanical Hamiltonian with a random Gaussian potential. We determine, in the form of bounds, the precise form of this dependence and show that the disorder-averaged ground-state energy decreases with an increase of the size R of the sample as a stretched-exponential function exp(-Rᶻ) where the characteristic exponent z depends merely on the nature of correlations in the random potential. In the particular case where the potential is distributed as a Gaussian white noise we prove that z=1/3. We also predict the value of z in the general case of Gaussian random potentials with correlations. {} 1996 The American Physical Society.