Levinson’s theorem for two-dimensional scattering systems: It was a surprise, it is now topological!
Angus Alexander
University of Wollongong, Wollongong, AustraliaThanh Duc Nguyen
Nagoya University, Nagoya, JapanAdam Rennie
University of Wollongong, Wollongong, AustraliaSerge Richard
Nagoya University, Nagoya, Japan
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Abstract
We prove a general Levinson’s theorem for Schrödinger operators in two dimensions with threshold obstructions at zero energy. Our results confirm and simplify earlier seminal results of Bollé, Gesztesy et al., while providing an explicit topological interpretation. We also derive explicit formulas for the wave operators, and so show that they are elements of a -algebra introduced by Cordes. As a consequence of our approach, we provide an evaluation of the spectral shift function at zero in the presence of -resonances.
Cite this article
Angus Alexander, Thanh Duc Nguyen, Adam Rennie, Serge Richard, Levinson’s theorem for two-dimensional scattering systems: It was a surprise, it is now topological!. J. Spectr. Theory 14 (2024), no. 3, pp. 991–1031
DOI 10.4171/JST/499