Obtain FEM diversity bounds that improve with grid size M (analogous to FD)

Derive a diversity bound for the finite element discretization in one dimension such that the failure probability decreases with the grid size M, analogous to the improvement observed in Theorem FD for finite difference discretization. Demonstrate that the non-improving bound currently obtained for FEM can be strengthened to exhibit the same blessing-of-dimensionality behavior.

Background

Theorem FD shows a surprising improvement of the diversity bound with increasing grid size M for finite difference discretization, while the stated FEM bound does not currently improve as M increases.

The authors conjecture that this lack of improvement is due to analysis artifacts, and that an FEM bound analogous to FD should be provable.

References

While the stated bound does not improve as $M \rightarrow \infty$ (as was observed in Theorem \ref{thm: FD}), we conjecture that this is an artifact of our analysis, and that a similar bound to the one in Theorem \ref{thm: FD} can be proven for the finite element discretization.

— 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)