Arithmetic properties of eigenvalues of generalized Harper operators on graphs (Q852386)

Let \(G\) be a discrete group and \(\sigma\) be an algebraic multiplier for \(G\). Denote by \(L^{2} (G)^{d}\) the Hilbert space of all square-integrable \(\mathbb{C}^{d}\)-valued functions on \(G\); \(l^{2} (G) = \{h : G \to \mathbb{C} : \sum_{g \in G} | h (g)|^{2} < \infty\}\). The paper is concerned with algebraic eigenvalue properties for self-adjoint matrix operators that are defined on \(l^{2} (G)^{d}\) by \[ (A f) (x) = \sum_{g \in x S} A (x^{-1} g) \sigma (x,x^{-1} g) f (g),\quad f \in l^{2} (G)^{d}, \] where \(A (g)\) is a \(d \times d\) complex matrix for each \(g\) and \(S\) is a finite subset of \(G\) which is symmetric, i.e., \(S = S^{-1}\). The self-adjointness condition is equivalent to demanding that the weights \(A (g)\) satisfy \(A (g)^{*} = A (g^{-1}) \sigma (g,g^{-1}).\) These operators include as a special case the Harper operator and the discrete magnetic Laplacian on the Cayley graph of \(G\), where \(S\) is the symmetric generating set and \(A (g)\) is identically \(1\) for \(g\) in \(S\). Further, let \(\overline{Q} (G,\sigma)\) be the twisted group algebra over the complex algebraic numbers \(\overline{Q}\) with multiplier \(\sigma\). Elements of \(\overline{Q} (G,\sigma)\) are finite sums \(\sum a_{g} g, \;a_{g} \in \overline{Q}\), with the (twisted) multiplication given by \[ \Bigl(\sum a_{g} g \Bigl) \cdot \Bigl(\sum b_{g} g \Bigl) = \sum_{gh = k} a_{g} b_{h} (g,h) k. \] The group \(G\) is said to have the \(\sigma\)-multiplier algebraic eigenvalue property if, for every matrix \(A \in M_{d} (\overline{Q} (G,\sigma))\) regarded as an operator on \(L^{2} (G)^{d}\), the eigenvalues of \(A\) are algebraic numbers, where \(\sigma \in Z^{2} (G,U (\overline{Q}))\) is an algebraic multiplier and \(U (\overline{Q})\)denotes the unitary elements of the field of algebraic numbers \(\overline{Q}\). As main results, the authors prove that any \(A \in M_{d} (\overline{Q}(G,\sigma))\) has only eigenvalues that are algebraic numbers, whenever \(\sigma\) is a rational (\(\sigma^{n} = I\) for some positive integer \(n\)) multiplier on \(G\) and \(G \in K\). The class \(K\) is a smallest of groups containing free groups and amenable groups, which is closed under taking extensions with amenable quotient and under taking directed unions. Moreover, the authors establish an equality between the von Neumann spectral density function of \(A \in M_{d} (C (G,\sigma))\) for an arbitrary multiplier \(\sigma\), the integrated density of states of \(A\) with respect to a generalized Folner exhaustion of \(G\), whenever \(G\) is a finitely generated amenable group or a surface group.