This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some matrix potential $V : \mathbb{R}^d \rightarrow \mathbb{M}^{n \times n}$ that is symmetric and positive definite a.e. on $\mathbb{R}^d$.

Suppose we have some matrix function $M : \mathbb{R}^d \rightarrow \mathbb{M}^{n \times n}$ that is symmetric and positive definite a.e. on $\mathbb{R}^d$ (intuitively, some sort of "Agmon function" on $\mathbb{R}^d$ related to our matrix potential). If we are lucky and have $n = d$ then $M$ canonically induces a metric on $\mathbb{R}^d$. Namely, just define \begin{equation} d(x, y) = \inf_\gamma \int_0^1 \langle M(\gamma(t)) \gamma '(t), \gamma'(t) \rangle_{ \mathbb{R}^d} ^\frac12 \, dt \tag{1}\label{Metric}\end{equation} where the infimum is over all absolutely continuous curves $\gamma : [0, 1] \rightarrow \mathbb{R}^d$ where $\gamma(0) = x$ and $\gamma(1) = y$.

If $n \neq d$ then there seems no natural way to do this while "preserving" the matrix structure of $M$. That is, I don't want to look at something like $$d(x, y) = \inf_\gamma \int_0^1 \|M(\gamma(t))\| |\gamma '(t)|_{\mathbb{R}^d} \, dt. $$

Surely $M$ can induce a bundle metric on some rank $n$ vector bundle, but it seems impossible to do this canonically in a way that reduces to \eqref{Metric} when $n = d$.

Are there any papers in the Schrödinger operator literature that touch on this? I can't imagine an issue like this hasn't come up somewhere before.

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