Approximating Spectral Invariants of Harper Operators on Graphs

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada [Sun]. A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group, can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory. Introduction Both the Harper operator and the discrete magnetic Laplacian (DML) on the Cayley graph of Z2 have been extremely well studied in mathematical physics, as they arise as the Hamiltonian for the discrete model describing the quantum mechanics of free electrons in the presence of a magnetic field. In particular, the DML is the Hamiltonian of the discrete model for the integer quantum Hall effect, cf. [Bel]. These operators can be easily generalized to the Cayley graph of an arbitrary discrete group. This and a further generalization to general graphs with a free co-compact action of a discrete group with finite quotient, was defined by Sunada [Sun] and studied in the context of noncommutative Bloch theory and the quantum Hall effect in [CHMM], [CHM], [MM], [MM2]. In this paper, we study certain aspects of the spectral theory of the DML on graphs X with a free group action with finite fundamental domain by a discrete amenable group Γ. We will use the following characterization of amenable groups, due to Følner (see also [Ad].) A discrete group Γ is said to be amenable if there is a sequence of finite subsets