Singular integral operators and special functions in scattering theory

1)完成了H·井上关于SL(2，R)径向部分的散射理论和指数定理的研究，并提交了一份手稿。这是Levinson定理在群表示中的第一次应用，涉及到许多特殊函数。2)与T·Miyoshi和Q·Sun的两部作品已经完成并提交出版，一部已经出版，另一部被接受。这些工作涉及到数据同化技术，并由于新冠肺炎大爆炸的限制而发展起来。3)表面态的研究一直是本财年的主要议题，工作已接近完成。D·帕拉和A·伦尼加入了这个项目的团队。一份手稿可能会在2023.4年春季提交)一个新的研究项目已经与A.Rennie一起开始了，研究在0能量具有特殊奇点的2D薛定谔算符。初始问题涉及一个非常奇异的积分核，预计解将涉及特殊函数的乘积。5)2022年秋季N.Boussay在名古屋逗留期间，一个关于Mourre理论和一些传播估计的新项目已经启动。初步结果是有希望的，但还需要进一步的调查。

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.