2018 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 theorem |
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| Outline of Annual Research Achievements |
Investigations on a system with an infinite number of bound states have been performed, and the C*-algebraic framework has indeed led to new index theorems in scattering theory. Clearly, the new equality is not of the meaningless form "infinity = infinity" but provides a relation between the density of bound states and the density of winding numbers of the scattering operator. In that respect, it is the first time that Levinson's theorem reaches infinity. The outcomes of these investigations have been published in two joint papers with H. Inoue.
The first paper focus on self-adjoint operators and provides several new index theorems in scattering theory. From a different perspective, these results illustrate in one single example several famous and distinct index theorems. Namely, the corresponding wave operators belong to some C*-algebras of pseudo-differential operators with coefficients which either have limits at + and - infintity, or which are periodic or asymptotically periodic, or which are uniformly almost periodic. Then, the precise form of the index theorem depends precisely of the choice of these C*-algebras.
The second publication deals with operators which are not self-adjoint, and shows that one can give a meaningful interpretation of Levinson's theorem also in the presence of an infinite number of complex eigenvalues. Exceptional cases are also discussed, but these situations do not lead to new 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
Prior to this project, the possible extension of Levinson's theorem to a situation with an infinite number of bound states had been carefully considered and the necessary C*-algebraic framework chosen with attention. The model used for these investigations had also been developed by the PI in collaboration with J. Derezinski, and it means that the background material was well prepared. A key factor is also the C*-algebraic framework itself, which is flexible but simultaneously quite rigid: it provides the natural extension of the usual trace for dealing with an infinite dimensional projection.
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| Strategy for Future Research Activity |
Our investigations will now focus on a different extension of Levinson's theorem: the finite number of bound states is going to be replaced by surface states. This situation takes place for example when the background physical system is periodic. As a consequence, the projection on the bound states does not correspond to a finite or infinite sum of projections on isolated eigenfunctions but to a continuum of projections. The C*-algebraic framework has to be adapted to this situation, but once again it should lead to a new type of index theorem in scattering theory.
Additional preliminary investigations are also taking place for an application of these ideas in the context of Lie group theory. In this context, the notion of bound states will be replaced by finite dimensional representations.
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| Causes of Carryover |
The remaining money was not sufficient for an appropriate use during the past fiscal year, it will be used more efficiently for a business trip or an invitation in the new fiscal year.
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