Extend finite element diversity results to higher dimensions and remove augmentation

Extend the diversity results for finite element discretization of random Schrödinger operators beyond D = 1 and establish analogous bounds for the non-augmented (vanilla) sample set. Specifically, determine whether the matrix distribution induced by finite element discretization in D > 1 yields that the centralizer of {A^(1), …, A^(N)} is trivial with high probability, without augmenting the sample set by the finite element Laplacian.

Background

In the finite element setting, the paper proves diversity results only in one spatial dimension and only for the augmented sample set that includes the deterministic Laplacian matrix.

The authors explicitly state that generalizing to higher dimensions and removing the augmentation assumption are unresolved and are left as future work.

References

We leave it as an open problem for future work to extend the results of this subsection to $D > 1,$ and to lift the augmentation assumption.

— A Theory of Diversity for Random Matrices with Applications to In-Context Learning of Schrödinger Equations  (2601.12587 - Cole et al., 18 Jan 2026) in Subsection 3.2 (Schrödinger operators under finite element discretization)