Existence of a crystal with infinitely many distinct eigenvalues

Determine whether there exists a Z^d-periodic graph with a finite fundamental cell and summable symmetric nonnegative edge weights (a "crystal" in the sense of Assumption 2.6) whose associated Schrödinger operator H_Γ has infinitely many distinct eigenvalues (i.e., an infinite set of eigenvalues in its point spectrum).

Background

Throughout the paper, a "crystal" denotes a Z^d-periodic graph with a finite fundamental cell and summable symmetric edge weights, equipped with associated adjacency/Laplacian/Schrödinger operators via Floquet theory.

The authors demonstrate new point-spectrum phenomena in the non-locally finite setting, including existence of flat bands without compactly supported eigenvectors. They then ask whether one can go further and obtain infinitely many distinct eigenvalues in this periodic framework.