Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in K-theory

In this paper, we study a refined L2 version of the semiclassical approximation of projectively invariant elliptic operators with invariant Morse type potentials on covering spaces of compact manifolds. We work on the level of spectral projections (and not just their traces) and obtain an information about classes of these projections in K-theory in the semiclassical limit as the coupling constant goes to zero. An important corollary is a vanishing theorem for the higher traces in cyclic cohomology for the spectral projections. This result is then applied to the quantum Hall effect. We also give a new proof that there are arbitrarily many gaps in the spectrum of the operators under consideration in the semiclassical limit.Comment: 41 pages, latex2e, uses xypic package. Minor clarifications made, some references added. Final versio

Quantum Hall Effect on the Hyperbolic Plane in the presence of disorder

We study both the continuous model and the discrete model of the integer quantum Hall effect on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian. [CHMM] A. Carey, K. Hannabuss, V. Mathai and P. McCann, Quantum Hall Effect on the Hyperbolic Plane, Communications in Mathematical Physics, 190 vol. 3, (1998) 629-673.Comment: LaTeX2e, 17 page

Quantum Hall Effect and Noncommutative Geometry

We study magnetic Schrodinger operators with random or almost periodic electric potentials on the hyperbolic plane, motivated by the quantum Hall effect in which the hyperbolic geometry provides an effective Hamiltonian. In addition we add some refinements to earlier results. We derive an analogue of the Connes-Kubo formula for the Hall conductance via the quantum adiabatic theorem, identifying it as a geometric invariant associated to an algebra of observables that turns out to be a crossed product algebra. We modify the Fredholm modules defined in [CHMM] in order to prove the integrality of the Hall conductance in this case.Comment: 18 pages, paper rewritte

Semiclassical asymptotics and gaps in the spectra of magnetic Schroedinger operators

In this paper, we study an L2 version of the semiclassical approximation of magnetic Schroedinger operators with invariant Morse type potentials on covering spaces of compact manifolds. In particular, we are able to establish the existence of an arbitrary large number of gaps in the spectrum of these operators, in the semiclassical limit as the coupling constant goes to zero.Comment: 18 pages, Latex2e, more typos correcte

Cyclic cocycles on twisted convolution algebras

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper \'etale groupoids, Tu and Xu provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to a construction of Mathai and Stevenson. When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras. The results in this article were originally published in the author's Ph.D. thesis.Comment: 39 page

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

Twisted K-theory and K-theory of bundle gerbes

In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in non-trivial backgrounds are discussed.Comment: 29 pages, corrected typos, added references, included new section on twisted Chern character in non-torsion cas