  We find explicitly the Green kernel of a random Schr\"odinger operator with
Brownian white noise. To do this, we first handle the random operator by
defining it weakly using the inner product of a Hilbert space. Then, using
classic Sturm-Liouville theory, we can build the Green kernel with linearly
independent solutions of a homogeneous problem. As a corollary we have that the
random operator has a discrete spectra.
