Singular integral operators and special functions in scattering theory
| Project/Area Number |
21K03292
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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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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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2024: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2023: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | Scattering theory / Singular integrals / Index theorems / Special functions / scattering theory / surface states / Wave operators / Surface states / Decay estimate / spectral theory / singular operators / special functions |
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| Outline of Research at the Start |
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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| Outline of Annual Research Achievements |
The research activities can be summarized as follows:
1) The manuscript on scattering theory and an index theorem on the radial part of SL(2,R), jointly written with H. Inoue, has been revised and accepted in a fairly good mathematical journal.
2) The investigations on the 2D Schroedinger operator with threshold singularities have been successfully completed. This work provides a definitive answer to some questions and doubtful results raised about 40 years ago. The key of this paper is precisely the resolution of a singular integral operator in terms of simpler special functions. The resulting paper is accepted in an excellent mathematical journal.
3) The investigations on surface states led to new results for families of discrete magnetic operators. A long manuscript has been submitted, and contains results of different nature on scattering theory, K-theory, and on integrable models. A surface of resonances is also exhibited, probably for the first time. This project has been done with collaborators in Australia, Chili, and Japan.
4) New investigations on the scattering theory and on index theorems for quantum walks have also been initiated, and the completion of this work is expected in Fall 2024.
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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
This research program has reached its maturity, and the relations between several topics have been established. These different topics and approaches complement each other and lead to a wide set of results. The participation of researchers of different origins and of students to this research project had also a positive impact.
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| Strategy for Future Research Activity |
The research activity 4) will continue, and a collaboration project with N. Boussaid (started in 2022 but temporarily paused in 2023) will resume. A new project involving singular integral operators, special functions, and also some number theory is now under discussion with J. Faupin. This project would correspond to unexpected new developments of this research proposal and could open new directions of research in the future. Finally, the project of writing a book with my long term collaborator R. Tiedra de Aldecoa has started. This project will take a long time, but the anticipated book should become a reference in spectral and scattering theory.
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Report
(3 results)
Research Products
(16 results)
