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We study the sample-size dependence of the ground-state energy in aone-dimensional localization problem, based on a supersymmetric quantummechanical Hamiltonian with random Gaussian potential. We determine, in theform of bounds, the precise form of this dependence and show that thedisorder-average ground-state energy decreases with an increase of the size $R$of the sample as a stretched-exponential function, $\exp( - R^{z})$, where thecharacteristic exponent $z$ depends merely on the nature of correlations in therandom potential. In the particular case where the potential is distributed asa 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.Comment: 30 pages and 4 figures (not included). The figures are available upon  reques

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