2019 Fiscal Year Research-status Report
Index theorems in scattering theory: beyond a finite number of bound states
| Project/Area Number |
18K03328
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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) |
2018-04-01 – 2021-03-31
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| Keywords | Scattering theory / Wave operators / Index theorems / Integrable models |
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| Outline of Annual Research Achievements |
During this year, our research has been centered on three topics related to this project.
1) In collaboration with J. Derezinski, J. Faupin, and Q.N. Nguyen, we have further investigated the integrable model of a Schroedinger operator on a half-line with Coulomb and inverse square potentials. In this framework, we have considered the most general situation with all coupling constants complex valued. In the past, a simplified version of this model had led to several new index theorems in scattering theory, and we expect similar new results on this extended model.
2) With S.H. Nguyen and R. Tiedra de Aldecoa, we completed the analytic part of our investigations on a model presenting some surface states. The main feature of this project is that the notion of an infinite number of bound states is transformed into a continuum of bound states. This continuum leads to surface states.
3) In a review paper with T. Umeda, we have presented in a unified and simpler form some results about singular integral operators and the explicit formulas that we can deduce from them. For the last 10 years, these formulas have played a key role for the research of index theorems in scattering theory. It was necessary to gather these formulas in a single place, and to simplify the presentation.
The outcomes of 1) and 3) have already been accepted for publication, while a paper corresponding to 2) is submitted.
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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
Prior to this project, some preliminary investigations had been performed, and the possible extension of Levinson's theorem to a situation with an infinite number of bound states had been carefully considered. In that respect, the project was mature. The good conditions provided by the grant have also helped for further developing some collaborations. The subject is also flexible enough for providing several possible directions of investigation.
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| Strategy for Future Research Activity |
During this final year of the project, we shall finish the investigations on surface states, concentrating on the algebraic side of the problem (the analytic part being already solved). A new project on discrete graphs has also been initiated during the second year, we hope that it will be completed within the next 12 months. There exist very few investigations on a topological approach of Levinson's theorem in the discrete setting, which makes this project even more valuable. Research outcomes will also depend on the worldwide developments of the coronavirus outbreak.
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