288 research outputs found
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
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 -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
Levinson's theorem and higher degree traces for Aharonov-Bohm operators
We study Levinson type theorems for the family of Aharonov-Bohm models from
different perspectives. The first one is purely analytical involving the
explicit calculation of the wave-operators and allowing to determine precisely
the various contributions to the left hand side of Levinson's theorem, namely
those due to the scattering operator, the terms at 0-energy and at infinite
energy. The second one is based on non-commutative topology revealing the
topological nature of Levinson's theorem. We then include the parameters of the
family into the topological description obtaining a new type of Levinson's
theorem, a higher degree Levinson's theorem. In this context, the Chern number
of a bundle defined by a family of projections on bound states is explicitly
computed and related to the result of a 3-trace applied on the scattering part
of the model.Comment: 33 page
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
Topological quantization of boundary forces and the integrated density of states
For quantum systems described by Schr\"odinger operators on the half-space
\RR^{d-1}\times\RR^{leq 0} the boundary force per unit area and unit energy
is topologically quantised provided the Fermi energy lies in a gap of the bulk
spectrum. Under this condition it is also equal to the integrated density of
states at the Fermi energy.Comment: 7 page
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