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 on e-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 t hat the disorder-averaged groundstate energy decreases with an increas e of the size R of the sample as a stretched-exponential function exp( -R(z)) where the characteristic exponent z depends merely on the natur e of correlations in the random potential. In the particular case wher e 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.