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

In this paper, a discrete form of the Kato inequality for discrete magnetic Laplacians on graphs is used to study asymptotic properties of the spectrum of discrete magnetic Schrodinger operators. We use the existence of a ground state with suitable properties for the ordinary combinatorial Laplacian and semigroup domination to relate the combinatorial Laplacian with the discrete magnetic Laplacian.Comment: 14 pages, latex2e, final version, to appear in "Contemporary Math.

Approximating L^2 invariants of amenable covering spaces: A heat kernel approach

In this paper, we prove that the L^2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L^2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the "principle of not feeling the boundary" (due to M. Kac), for heat kernels associated to boundary value problems.Comment: 14 pages, AMS-LaTeX, replaces an earlier version and contains a much strengthened version of one of the main results. 1991 Mathematics Subject Classification. 58G11, 58G18, 58G2

Approximating L^2 invariants of amenable covering spaces: A combinatorial approach

In this paper, we prove that the $L^2$ Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, proving a conjecture that we made in an earlier paper. We also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class.Comment: 14 pages, AMS-LaTeX, a minor revision of an earlier version containing new references to earlier work in the fiel

Arithmetic properties of eigenvalues of generalized Harper operators on graphs

Let \Qbar denote the field of complex algebraic numbers. A discrete group $G$ is said to have the $\sigma$-multiplier algebraic eigenvalue property, if for every matrix $A$ with entries in the twisted group ring over the complex algebraic numbers M_d(\Qbar(G,\sigma)), regarded as an operator on $l^2(G)^d$, the eigenvalues of $A$ are algebraic numbers, where $\sigma$ is an algebraic multiplier. Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier $\sigma$. In the special case when $\sigma$ is rational ($\sigma^n$=1 for some positive integer $n$) this property holds for a larger class of groups, containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.Comment: 28 pages, latex2e, paper revise