2014 Fiscal Year Research-status Report
Probing crystal defects with scattering theory and non-commutative topology
| Project/Area Number |
26707005
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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, その他 (70725241)
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | scattering theory / wave operators / index theorems / periodic systems |
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| Outline of Annual Research Achievements |
The goals for this first year of the project have been reached: With R. Tiedra, we studied a model of scattering in a half-space with periodic boundary conditions, and this had led to a first preprint and additional investigations are in progress. With K. Pankrashkin, we started the investigations on a discrete model with an almost periodic potential, which mimics some quasi-periodic structure. With T. Umeda and J. Uchiyama, we also finished an earlier project on Schroedinger operators with several embedded eigenvalues. A new project with J. Derezinski has also been initiated on a system with an infinite number of bound states. All those projects are intended to expand our expertise at the interface between scattering theory and non-commutative topology.
At the beginning of the year, we also organized a 2-day workshop entitled "Quantum walks, random walks and scattering theory on periodic systems", which facilitates the discussions between experts of these three fields. The supervision of the work of a PhD student on scattering theory on crystal lattices is going on successfully, and the work has made lots of progress. In November, I have been invited to deliver a 3-hour course at an international conference in Chile on my research program and on my results.
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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
In mathematics, one can never be sure if the progress will be made in the right directions or in a slightly different directions. During this year, we have obtained several results which are pointing towards our final goal. A lot of work has been done, and this is why more progress have been obtained compared to what was expected.
Another reason for this successful year is that the long stay of R. Tiedra in Nagoya, initially planned during the second year of the project, has already taken place at the end of the first year. This stay gave a nice boost to the global project.
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| Strategy for Future Research Activity |
The future work will go on as planned in the original project. We expect further investigations on purely periodic systems, on almost periodic systems and also on systems living in a half-space. The perturbations of such systems will be investigated as well.
Besides our research, a rather large amount of time is spent on the dissemination of the information and on the education of the younger generations to this interdisciplinary research project. For that purpose, a master course has been given at Nagoya University during the spring semester 2014, and another one is currently delivered during the spring semester 2015. This is part of our commitment for the dissemination of this research proposal and of its outcomes.
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| Causes of Carryover |
Since the stay of R. Tiedra in Nagoya started during the previous fiscal year (1st year of the project) and ended during the new fiscal year (second year of the project), only half of the flight ticket was paid during the previous fiscal, and the second half will be paid during this new fiscal year. The leftover money is a provision for that purpose.
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| Expenditure Plan for Carryover Budget |
The leftover money will be entirely used for paying the return ticket to R. Tiedra.
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Research Products
(5 results)
