Rotation Numbers, Boundary Forces and Gap labelling

We review the Johnson-Moser rotation number and the $K_0$-theoretical gap labelling of Bellissard for one-dimensional Schr\"odinger operators. We compare them with two further gap-labels, one being related to the motion of Dirichlet eigenvalues, the other being a $K_1$-theoretical gap label. We argue that the latter provides a natural generalisation of the Johnson-Moser rotation number to higher dimensions.Comment: 10 pages, version accepted for publicatio

Levinson's theorem for Schroedinger operators with point interaction: a topological approach

In this note Levinson theorems for Schroedinger operators in R^n with one point interaction at 0 are derived using the concept of winding numbers. These results are based on new expressions for the associated wave operators.Comment: 7 page

Boundary maps for C∗C^*-crossed products with R with an application to the quantum Hall effect

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with R is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t.Connes' pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schroedinger operators.Comment: to appear in Commun. Math. Phy

Pattern equivariant functions and cohomology

The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier accessible. It is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity.Comment: 8 pages including 2 figure

The topological meaning of Levinson's theorem, half-bound states included

We propose to interpret Levinson's theorem as an index theorem. This exhibits its topological nature. It furthermore leads to a more coherent explanation of the corrections due to resonances at thresholds.Comment: 4 page

Operators, Algebras and their Invariants for Aperiodic Tilings

We review the construction of operators and algebras from tilings of Euclidean space. This is mainly motivated by physical questions, in particular after topological properties of materials. We explain how the physical notion of locality of interaction is related to the mathematical notion of pattern equivariance for tilings and how this leads naturally to the definition of tiling algebras. We give a brief introduction to the K-theory of tiling algebras and explain how the algebraic topology of K-theory gives rise to a correspondence between the topological invariants of the bulk and its boundary of a material. 1.1 Tilings and the topology of their hulls In condensed matter theory tilings are used to describe the spatial arrangement of the constitutents which make up a material, for instance a quasicrys-tal. They describe the spatial structure of the material. Associated to a tiling are various topological spaces and topological dy-namical systems. Their topology is peculiar. It takes into account the topology of the space in which the tiling lies and, at the same time, its pattern structure , that is, the way how finite patterns repeat over the tiling. Continuity in the tiling topology is related to locality in physics