Singular integral operators and special functions in scattering theory

主要研究内容如下：1）以H.井上关于散射理论和SL（2，R）径向部分的指数定理已经完成并提交了手稿。这是列文森定理在群表示中的第一个应用，涉及到许多特殊的函数。Miyoshi和Q.孙有完成，提交出版，一个已经出版，另一个接受。这些工作涉及数据同化技术，并因COVID-19大流行而受到限制。3）地表状态调查是本财年的主要课题，工作已接近完成。D. Parra和A. Rennie加入了这个项目的团队。手稿可能会在2023年春季提交。4）一个新的研究项目已经开始与A。Rennie关于在0能量下具有特殊奇异性的2D Schroedinger算子。初始问题涉及一个非常奇异的积分核，预计解将涉及一个特殊函数的乘积。5）在N. 2022年秋季在名古屋的Boussaid。初步结果是有希望的，但需要进一步调查。

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.