Green kernel for a random Schr\"odinger operator

TL;DR

This paper explicitly derives the Green kernel for a random Schrödinger operator with Brownian white noise, revealing its spectral properties using classical Sturm-Liouville theory.

Contribution

It introduces a method to explicitly construct the Green kernel for a stochastic Schrödinger operator with Brownian noise, linking it to spectral analysis.

Findings

Green kernel explicitly derived

Operator has discrete spectra

Method connects stochastic operators with classical theory

Abstract

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.