# 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? – Willie Wong May 27 '20 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... – Joshua Isralowitz May 27 '20 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? – Willie Wong May 27 '20 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. – Joshua Isralowitz May 27 '20 at 20:03
• Please edit your post to include the above ^^ two comments. They provide important context. – Willie Wong May 27 '20 at 22:20