Green kernel for a random Schrödinger operator

We find explicitly the Green kernel of a random Schrödinger 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.