  We study the localization of wave functions for one-dimensional Schr\"odinger
Hamiltonians with random potentials $V(x)$ with short range correlations and
large local fluctuations such that $\int\D{x} \smean{V(x)V(0)}=\infty$. A
random supersymmetric Hamiltonian is also considered. Depending on how large
the fluctuations of $V(x)$ are, we find either new energy dependences of the
localization length, $\ell_\mathrm{loc}\propto{}E/\ln{E}$,
$\ell_\mathrm{loc}\propto{}E^{\mu/2}$ with $0<\mu<2$ or
$\ell_\mathrm{loc}\propto\ln^{\mu-1}E$ for $\mu>1$, or superlocalization (decay
of the wave functions faster than a simple exponential).
