Lets say $V : \mathbb{R}^n \rightarrow \mathbb{M}_d (\mathbb{R})$ is a $d \times d$ symmetric, positive semidefinite matrix function on $\mathbb{R}^n$ and consider the Schrodinger operator $- \Delta + V$ acting on $\mathbb{R}^d$ valued functions.
If the Fundamental solution (matrix valued) $\Gamma_V : \mathbb{R}^n \times \mathbb{R}^n \backslash \{(x, y) : x = y\} \rightarrow \mathbb{M}_d (\mathbb{R})$ exists (which I can prove under fairly mild growth conditions on $V$), then is there any expectation that it is positive semidefinite? (It is not hard to prove that it is symmetric.)
Has this question/answer appeared anywhere in the literature before?
