  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 random Gaussian potential. We determine, in the
form of bounds, the precise form of this dependence and show that the
disorder-average ground-state energy decreases with an increase of the size $R$
of the sample as a stretched-exponential function, $\exp( - R^{z})$, 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.
