Schrodinger operator with matrix potential and Green's function

Clearly Schrodinger operators with matrix potentials appear very naturally in molecular dynamics/quantum chemistry, particularly when considering a crude adiabatic basis or diabatic basis for an ansatz solution to the time independent Schrodinger equation of a molecule (see for example Sections 12.1/12.2 of Tannor's "Intro to Quantum Mechanics, a Time Dependent Perspective").


Would knowledge of the existence of a Green's function/fundamental solution for such Schrodinger operators with a matrix potential be of any scientific or computational use?


More precisely, I'm interested in proving the existence of Green's functions/the fundamental solution to Schrodinger operators of the form $-\Delta \otimes I\_{n \times n} + V$ on an open set $\Omega \subseteq \mathbb{R}^d$. Here, $I\_{n \times n}$ is the $n \times n$ identity matrix, $V = V\_+ + V\_\epsilon$ where $V\_+$ is a Hermitian, positive semidefinite (a.e.) $n \times n$ matrix function and $V\_\epsilon$ is some "small perturbation" of $V$ (for example perhaps some Kato type class? I'm not quite sure what would work yet.)


