研究課題/領域番号 |
21K03292
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研究種目 |
基盤研究(C)
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配分区分 | 基金 |
応募区分 | 一般 |
審査区分 |
小区分12010:基礎解析学関連
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研究機関 | 名古屋大学 |
研究代表者 |
Richard Serge 名古屋大学, 教養教育院, 教授 (70725241)
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研究期間 (年度) |
2021-04-01 – 2025-03-31
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研究課題ステータス |
交付 (2022年度)
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配分額 *注記 |
4,160千円 (直接経費: 3,200千円、間接経費: 960千円)
2024年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2023年度: 1,170千円 (直接経費: 900千円、間接経費: 270千円)
2022年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2021年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
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キーワード | Singular integrals / Special functions / scattering theory / surface states / Index theorems / Scattering theory / Wave operators / Surface states / Decay estimate / spectral theory / singular operators / special functions |
研究開始時の研究の概要 |
In this project, we want to study more systematically singular integral operators through their representations in terms of special functions, and derive new analytical estimates. The investigations are divided into 3 main tasks: the discovery part which corresponds to the systematic transcription of wave operators with special functions, the technical part in which refined estimates will be obtained through the study of regularity properties of C0-group in Banach spaces, the visionary part in which this program will be extended to representation theory, and ultimately to number theory.
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研究実績の概要 |
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.
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現在までの達成度 (区分) |
現在までの達成度 (区分)
1: 当初の計画以上に進展している
理由
With the end of the pandemic and the opening of the borders, it has been possible to resume the activities, invite colleagues from abroad, and organize business trips. All researchers enjoy having again in person interactions. This enthusiasm is visible on research activities and on the research progress.
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今後の研究の推進方策 |
The research activities 3), 4), and 5) will continue. Invitations or business trips will be organized according to these projects. In addition, there are discussions about the project of writing a book with my best collaborator R. Tiedra de Aldecoa. This project would be the culmination of 20 years of collaboration, and is a huge but very appealing project. Preliminary investigations are currently performed.
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