2022 Fiscal Year Research-status Report
Singular integral operators and special functions in scattering theory
| Project/Area Number |
21K03292
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| Research Institution | Nagoya University |
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Principal Investigator |
Richard Serge 名古屋大学, 教養教育院, 教授 (70725241)
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| Project Period (FY) |
2021-04-01 – 2025-03-31
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| Keywords | Singular integrals / Special functions / scattering theory / surface states / Index theorems |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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| Causes of Carryover |
The amount is very small and will be used appropriately in the new fiscal year.
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Research Products
(8 results)
