The invariant measure and Lyapunov exponent associated to the one-dimensional Schrdinger operator with a random potential (or, in other words, to the damped linear oscillator with random restoring force) are studied for small real noise (diffusions). Analytic expression are given via perturbation expansion. As a by-product, the well-known positivity of the Lyapunov exponent (in the undamped case) is reproved