Singular integral operators and special functions in scattering theory

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The research activities can be summarized as follows:1) The investigations with H. Inoue on scattering theory and an index theorem on the radial part of SL(2,R) have been completed and a manuscript submitted. It is the first application of Levinson's theorem in group representations, and involves numerous special functions.2) The two works with T. Miyoshi and Q. Sun have been completed, submitted for publication, and one has been published, the other one accepted. These works involve data assimilation techniques and have been developed because of the restrictions due to the COVID-19 pandemic.3) The investigations on surface states have been the main topic for this FY and the work is nearly completed. D. Parra and A. Rennie have joined the team for this project. A manuscript will probably be submitted in Spring 2023.4) A new research project has started with A. Rennie about 2D Schroedinger operators with exceptional singularities at 0 energy. The initial problem involves a very singular integral kernel, and it is expected that the solution will involve a product of special functions.5) A new project on Mourre theory and some propagation estimates has started during the stay of N. Boussaid in Nagoya in Fall 2022. Preliminary results are promising, but further investigations are necessary.