| 研究実績の概要 |
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.
|
| 今後の研究の推進方策 |
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.
|
