Singular integral operators and special functions in scattering theory

散射理论中的奇异积分算子和特殊函数

• 批准号: 21K03292
• 负责人: Richard Serge
• 金额: $ 2.66万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2021
• 资助国家: 日本
• 起止时间: 2021-04-01 至 2025-03-31
• 项目状态: 未结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21K03292/
• 关键词: Singular integrals; Special functions; scattering theory; surface states; Index theorems; Scattering theory; Wave operators; Surface states; Decay estimate; spectral theory; singular operators; special functions

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.

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。

项目成果

Decay Estimates for Unitary Representations with Applications to Continuous- and Discrete-Time Models

酉表示的衰减估计及其在连续和离散时间模型中的应用

• DOI: 10.1007/s00023-022-01199-5
• 发表时间: 2022
• 期刊: Annales Henri Poincare
• 影响因子: 1.5
• 作者: S. Richard;R. Tiedra de Aldecoa
• 通讯作者: R. Tiedra de Aldecoa

Theorie de la diffusion et un theoreme d'indice sur la partie radiale de SL(2,R)

扩散理论和 SL(2,R) 径向方指数定理

• 发表时间: 2022
• 作者: Matsuoka Katsuo;Mizuta Yoshihiro;Shimomura Tetsu;松谷茂樹;Serge Richard
• 通讯作者: Serge Richard

Scattering theory and an index theory theorem on the radial part of SL(2,R)

SL(2,R)径向部分的散射理论和指数理论定理

• 发表时间: 2022
• 作者: Mizuta Yoshihiro;Shimomura Tetsu;松谷茂樹;Serge Richard
• 通讯作者: Serge Richard

Scattering theory and an index theorem on the radial part of SL(2,R)

SL(2,R)径向部分的散射理论和指数定理

• 发表时间: 2022
• 作者: Ohno Takao;Shimomura Tetsu;Serge Richard
• 通讯作者: Serge Richard

Spectral and scattering theory for topological crystals perturbed by infinitely many new edges

受无限多新边扰动的拓扑晶体的光谱和散射理论

• DOI: 10.1142/s0129055x22500106
• 发表时间: 2022
• 期刊: Reviews in Mathematical Physics
• 影响因子: 1.8
• 作者: S. Richard;N. Tsuzu
• 通讯作者: N. Tsuzu