# Schrodinger operator with matrix potential

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.

• Is the Schrodinger operator only for motivational purposes? It looks to me as though your question is just about geometry, and I don't see why you tagged elliptic-pde, and I don't see why you are looking for papers in the Schrodinger operator literature. Should the metric have something to do with the Schrodinger operator in some way? May 27, 2020 at 19:28
• Really it's just for motivational purposes (and is also my general reason for asking, as that's what I'm researching right now). Also I find it hard to imagine this situation has't appeared in the theory of Schrodinger operators. I tagged it elliptic-pde because you can ask the same question for more general elliptic operators with a matrix potential and again it's hard to imagine such an issue has never came up in that context (at least to a non-expert like myself. I mostly do classical harmonic analysis). Though, maybe this also has appeared naturally in other contexts too... May 27, 2020 at 19:49
• But what is this metric supposed to tell you about the original Schrodinger operator? Why does it make sense to start from a matrix potential and ask about this metric? May 27, 2020 at 19:56
• To answer your other question, I'm essentially assuming apriori that a "useful" matrix valued Agmon function can be defined (analogous to how it is in Zhongwei Shen's 1999 "On fundamental solutions of generalized Schrodinger operators"), but this is naturally n times n valued, and thus doesn't naturally define an Agmon metric on $\mathbb{R}^d$... Here, it depends on the potential. What exactly this "Agmon function" is, is still work in progress. May 27, 2020 at 20:03
• Please edit your post to include the above ^^ two comments. They provide important context. May 27, 2020 at 22:20