Approximating Spectral Invariants of Harper Operators on Graphs

Abstract

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. © 2002 Elsevier Science (USA)

A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

“…The same restriction applies to most of the papers mentioned in the introduction, for instance [MSY03,KLS03,Ves05,LV09]. In particular it is not clear how the results of these papers can be extended to the case when the difference operator on ℓ 2 (G) has a matrix kernel with very fast off-diagonal decay, albeit is not of finite range.…”

Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence

“…In [35,36] one is given a family s γ ,γ ∈ of maps s γ : V −→ {z ∈ C | |z| = 1}. In the other cases one just sets s γ ≡ 1,γ ∈ .…”

“…Let us now briefly discuss these earlier partial results recovered in our framework. Results about pointwise convergence were obtained in, e.g., [15,35,36,39]. Note that in the literature special attention was devoted to convergence at the bottom of the spectrum [7,8,10], since it corresponds to approximation of Betti-numbers.…”

“…We hope that this clarification will be helpful for the study of finer properties of the IDS, which may be indeed model-dependent. Among those are: the quantisation of jumps of the IDS [9,43], their location [6,36,45], and the low energy behaviour of the IDS. Note that this low energy behaviour has been a recent focus of attention in particular for percolation models (see, e.g., [1,20,38]) while surveys of results for other models can be, e.g., found in [19,34].…”

Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians

“…, v m } is a fundamental set of vertices of K and A is a bounded operator on C 0 (2) (K) that is in commutant of the projective (G, σ)-action on C 0 (2) (K) (cf. [12] for a description of this action), for instance the DML, then the von Neumann trace of A is given by…”