12 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
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
On the wave operators for the Friedrichs-Faddeev model
We provide new formulae for the wave operators in the context of the
Friedrichs-Faddeev model. Continuity with respect to the energy of the
scattering matrix and a few results on eigenfunctions corresponding to embedded
eigenvalues are also derived.Comment: 10 page
One-dimensional Dirac operators with zero-range interactions: Spectral, scattering, and topological results
17 pagesInternational audienceThe spectral and scattering theory for 1-dimensional Dirac operators with mass and with zero-range interactions are fully investigated. Explicit expressions for the wave operators and for the scattering operator are provided. These new formulae take place in a representation which links, in a suitable way, the energies and , and which emphasizes the role of . Finally, a topological version of Levinson's theorem is deduced, with the threshold effects at automatically taken into account
On some integral operators appearing in scattering theory, and their resolutions
We discuss a few integral operators and provide expressions for them in terms of smooth functions of some natural self-adjoint operators. These operators appear in the context of scattering theory, but are independent of any perturbation theory. The Hilbert transform, the Hankel transform, and the finite interval Hilbert transform are among the operators considered