On the Density of Eigenvalues on Periodic Graphs

Suppose that $\Gamma=(V,E)$ is a graph with vertices $V$, edges $E$, a free group action on the vertices $\mathbb{Z}^d \curvearrowright V$ with finitely many orbits, and a linear operator $D$ on the Hilbert space $l^2(V)$ such that $D$ commutes with the group action. Fix $\lambda \in \mathbb{R}$ in the pure-point spectrum of $D$ and consider the vector space of all eigenfunctions of finite support $K$. Then $K$ is a non-trivial finitely generated module over the ring of Laurent polynomials, and the density of $\lambda$ is given by an Euler-characteristic type formula by taking a finite free resolution of $K$. Furthermore, these claims generalize under suitable assumptions to the non-commutative setting of a finite generated amenable group acting on the vertices freely with finitely many orbits, and commuting with the operator $D$.