2021 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, 多元数理科学研究科(国際), G30特任教授 (70725241)
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| Project Period (FY) |
2021-04-01 – 2025-03-31
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| Keywords | Scattering theory / Wave operators / Special functions / Surface states / Decay estimate |
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| Outline of Annual Research Achievements |
The research activities have been adapted to the COVID-19 context, and have been centered on the following 4 topics:
1) With N. Tsuzu, we investigated the spectral and scattering theory of operators acting on topological crystals perturbed by infinitely many new edges. The setting of topological crystals corresponds to the most general framework for studying discrete periodic structures and their perturbations. The results have already been published.
2) With R. Tiedra de Aldecoa we have introduced a new technique to obtain polynomial decay estimates for the matrix coefficients of unitary operators. The result of these investigations have been submitted for publication.
3) With H. Inoue we have continued a project on the radial part of SL(2,R) and expect to get final results within a few months. This project corresponds to an application of Levinson's theorem in the context of groups representations.
4) With V. Austen, we have started a new project on the C*-algebraic framework for studying surface states. Special functions are also involved in this project. This project also corresponds to the second part of a project developed 3 years ago with R. Tiedra de Aldecoa and H.S. Nguyen and recently published.
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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
Because of the COVID-19 pandemic, the research program for the first year of this project was already adapted to this special situation. Thus, we have performed as much research as possible, but it has not been possible to invite any colleague to Nagoya or to visit any colleague for direct collaborations. Fortunately, research in mathematics does not necessitate big investments, which means that one can be more flexible and update the research directions according to the context. As a result, the outcomes for this first year are rather good, and better than anticipated
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| Strategy for Future Research Activity |
Since the first two projects mentioned above have been completed, we shall focus more on the projects 3 and 4. With T. Miyoshi and Q. Sun, we shall also continue our joint collaboration on a project involving the control of chaotic systems by data assimilation techniques, with an application to Lorenz 96 model. Additional investigations on singular integrals related to almost periodic systems are also planned.
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| Causes of Carryover |
Because of the COVID-19 pandemic, it has not been able to invite any colleague from abroad, or participate to any conferences on-site. During the new fiscal year, we shall come back to a more proactive attitude for research, by inviting colleagues or by visiting them for collaborations.
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Research Products
(4 results)
