  We consider the one-dimensional Schr\"odinger equation with a random
potential and study the cumulant generating function of the logarithm of the
wave function $\psi(x)$, known in the literature as the "generalized Lyapunov
exponent"; this is tantamount to studying the statistics of the so-called
"finite size Lyapunov exponent". The problem reduces to that of finding the
leading eigenvalue of a certain non-random non-self-adjoint linear operator
defined on a somewhat unusual space of functions. We focus on the case of
Cauchy disorder, for which we derive a secular equation for the generalized
Lyapunov exponent. Analytical expressions for the first four cumulants of
$\ln|\psi(x)|$ for arbitrary energy and disorder are deduced. In the universal
(weak-disorder/high-energy) regime, we obtain simple asymptotic expressions for
the generalized Lyapunov exponent and for all the cumulants. The large
deviation function controlling the distribution of $\ln|\psi(x)|$ is also
obtained in several limits. As an application, we show that, for a disordered
region of size $L$, the distribution $\mathcal{W}_L$ of the conductance $g$
exhibits the power law behaviour $\mathcal{W}_L(g)\sim g^{-1/2}$ as $g\to0$.
