779 research outputs found
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
Cyclic cohomology for graded -algebras and its pairings with van Daele -theory
We consider cycles for graded -algebras (Real -algebras)
which are compatible with the -structure and the real structure. Their
characters are cyclic cocycles. We define a Connes type pairing between such
characters and elements of the van Daele -groups of the -algebra
and its real subalgebra. This pairing vanishes on elements of finite order. We
define a second type of pairing between characters and -group elements which
is derived from a unital inclusion of -algebras. It is potentially
non-trivial on elements of order two and torsion valued. Such torsion valued
pairings yield topological invariants for insulators. The two-dimensional
Kane-Mele and the three-dimensional Fu-Kane-Mele strong invariant are special
cases of torsion valued pairings. We compute the pairings for a simple class of
periodic models and establish structural results for two dimensional aperiodic
models with odd time reversal invariance.Comment: 57 page
The Local Structure of Tilings and their Integer Group of Coinvariants
The local structure of a tiling is described in terms of a multiplicative
structure on its pattern classes. The groupoid associated to the tiling is
derived from this structure and its integer group of coinvariants is defined.
This group furnishes part of the -group of the groupoid -algebra for
tilings which reduce to decorations of . The group itself as well as the
image of its state is computed for substitution tilings in case the
substitution is locally invertible and primitive. This yields in particular the
set of possible gap labels predicted by -theory for Schr\"odinger operators
describing the particle motion in such a tiling.Comment: 45 pages including 9 figures, LaTe
Rotation Numbers, Boundary Forces and Gap labelling
We review the Johnson-Moser rotation number and the -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 -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
Fractal spectral triples on Kellendonk's -algebra of a substitution tiling
We introduce a new class of noncommutative spectral triples on Kellendonk's
-algebra associated with a nonperiodic substitution tiling. These spectral
triples are constructed from fractal trees on tilings, which define a geodesic
distance between any two tiles in the tiling. Since fractals typically have
infinite Euclidean length, the geodesic distance is defined using
Perron-Frobenius theory, and is self-similar with scaling factor given by the
Perron-Frobenius eigenvalue. We show that each spectral triple is
-summable, and respects the hierarchy of the substitution system. To
elucidate our results, we construct a fractal tree on the Penrose tiling, and
explicitly show how it gives rise to a collection of spectral triples.Comment: Updated to agree with published versio
Boundary maps for -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
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